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---|---|---|---|---|---|---|
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F'β : Filtration I M
h : Stable F
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
hF : Submodule.FG (Filtration.submodule F)
F' : Filtration I M
hf : F' β€ F
this : IsNoetherian β₯(reesAlgebra I) β₯(Filtration.submodule F)
β’ Submodule.FG (Filtration.submodule F') | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
| rw [isNoetherian_submodule] at this | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
| Mathlib.RingTheory.Filtration.405_0.wQ6WBws0g3n9213 | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable | Mathlib_RingTheory_Filtration |
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F'β : Filtration I M
h : Stable F
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
hF : Submodule.FG (Filtration.submodule F)
F' : Filtration I M
hf : F' β€ F
this : β s β€ Filtration.submodule F, Submodule.FG s
β’ Submodule.FG (Filtration.submodule F') | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
| exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf) | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
| Mathlib.RingTheory.Filtration.405_0.wQ6WBws0g3n9213 | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable | Mathlib_RingTheory_Filtration |
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
x : M
β’ x β β¨
i, I ^ i β’ β€ β β r, βr β’ x = x | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
| let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M) | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
| Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213 | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x | Mathlib_RingTheory_Filtration |
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
x : M
N : Submodule R M := β¨
i, I ^ i β’ β€
β’ x β β¨
i, I ^ i β’ β€ β β r, βr β’ x = x | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
| have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
| Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213 | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x | Mathlib_RingTheory_Filtration |
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
x : M
N : Submodule R M := β¨
i, I ^ i β’ β€
k : β
β’ I ^ k β’ β€ = Filtration.N (stableFiltration I β€) k | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by | simp | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by | Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213 | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x | Mathlib_RingTheory_Filtration |
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
x : M
N : Submodule R M := β¨
i, I ^ i β’ β€
hN : β (k : β), Filtration.N (stableFiltration I β€ β trivialFiltration I N) k = N
β’ x β β¨
i, I ^ i β’ β€ β β r, βr β’ x = x | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
| constructor | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
| Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213 | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x | Mathlib_RingTheory_Filtration |
case mp
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
x : M
N : Submodule R M := β¨
i, I ^ i β’ β€
hN : β (k : β), Filtration.N (stableFiltration I β€ β trivialFiltration I N) k = N
β’ x β β¨
i, I ^ i β’ β€ β β r, βr β’ x = x | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· | obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm) | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· | Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213 | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x | Mathlib_RingTheory_Filtration |
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
x : M
N : Submodule R M := β¨
i, I ^ i β’ β€
hN : β (k : β), Filtration.N (stableFiltration I β€ β trivialFiltration I N) k = N
β’ N β€ I β’ N | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
| obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N) | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
| Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213 | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x | Mathlib_RingTheory_Filtration |
case intro
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
x : M
N : Submodule R M := β¨
i, I ^ i β’ β€
hN : β (k : β), Filtration.N (stableFiltration I β€ β trivialFiltration I N) k = N
k : β
hk :
β n β₯ k,
I β’ Filtration.N (stableFiltration I β€ β trivialFiltration I N) n =
Filtration.N (stableFiltration I β€ β trivialFiltration I N) (n + 1)
β’ N β€ I β’ N | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
| have := hk k (le_refl _) | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
| Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213 | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x | Mathlib_RingTheory_Filtration |
case intro
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
x : M
N : Submodule R M := β¨
i, I ^ i β’ β€
hN : β (k : β), Filtration.N (stableFiltration I β€ β trivialFiltration I N) k = N
k : β
hk :
β n β₯ k,
I β’ Filtration.N (stableFiltration I β€ β trivialFiltration I N) n =
Filtration.N (stableFiltration I β€ β trivialFiltration I N) (n + 1)
this :
I β’ Filtration.N (stableFiltration I β€ β trivialFiltration I N) k =
Filtration.N (stableFiltration I β€ β trivialFiltration I N) (k + 1)
β’ N β€ I β’ N | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
| rw [hN, hN] at this | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
| Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213 | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x | Mathlib_RingTheory_Filtration |
case intro
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
x : M
N : Submodule R M := β¨
i, I ^ i β’ β€
hN : β (k : β), Filtration.N (stableFiltration I β€ β trivialFiltration I N) k = N
k : β
hk :
β n β₯ k,
I β’ Filtration.N (stableFiltration I β€ β trivialFiltration I N) n =
Filtration.N (stableFiltration I β€ β trivialFiltration I N) (n + 1)
this : I β’ N = N
β’ N β€ I β’ N | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
| exact le_of_eq this.symm | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
| Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213 | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x | Mathlib_RingTheory_Filtration |
case mp.intro.intro
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
x : M
N : Submodule R M := β¨
i, I ^ i β’ β€
hN : β (k : β), Filtration.N (stableFiltration I β€ β trivialFiltration I N) k = N
r : R
hrβ : r β I
hrβ : β n β N, r β’ n = n
β’ x β β¨
i, I ^ i β’ β€ β β r, βr β’ x = x | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
| intro H | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
| Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213 | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x | Mathlib_RingTheory_Filtration |
case mp.intro.intro
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
x : M
N : Submodule R M := β¨
i, I ^ i β’ β€
hN : β (k : β), Filtration.N (stableFiltration I β€ β trivialFiltration I N) k = N
r : R
hrβ : r β I
hrβ : β n β N, r β’ n = n
H : x β β¨
i, I ^ i β’ β€
β’ β r, βr β’ x = x | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
| exact β¨β¨r, hrββ©, hrβ _ Hβ© | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
| Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213 | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x | Mathlib_RingTheory_Filtration |
case mpr
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
x : M
N : Submodule R M := β¨
i, I ^ i β’ β€
hN : β (k : β), Filtration.N (stableFiltration I β€ β trivialFiltration I N) k = N
β’ (β r, βr β’ x = x) β x β β¨
i, I ^ i β’ β€ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· | rintro β¨r, eqβ© | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· | Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213 | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x | Mathlib_RingTheory_Filtration |
case mpr.intro
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
x : M
N : Submodule R M := β¨
i, I ^ i β’ β€
hN : β (k : β), Filtration.N (stableFiltration I β€ β trivialFiltration I N) k = N
r : β₯I
eq : βr β’ x = x
β’ x β β¨
i, I ^ i β’ β€ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
| rw [Submodule.mem_iInf] | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
| Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213 | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x | Mathlib_RingTheory_Filtration |
case mpr.intro
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
x : M
N : Submodule R M := β¨
i, I ^ i β’ β€
hN : β (k : β), Filtration.N (stableFiltration I β€ β trivialFiltration I N) k = N
r : β₯I
eq : βr β’ x = x
β’ β (i : β), x β I ^ i β’ β€ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
| intro i | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
| Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213 | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x | Mathlib_RingTheory_Filtration |
case mpr.intro
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
x : M
N : Submodule R M := β¨
i, I ^ i β’ β€
hN : β (k : β), Filtration.N (stableFiltration I β€ β trivialFiltration I N) k = N
r : β₯I
eq : βr β’ x = x
i : β
β’ x β I ^ i β’ β€ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
| induction' i with i hi | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
| Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213 | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x | Mathlib_RingTheory_Filtration |
case mpr.intro.zero
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
x : M
N : Submodule R M := β¨
i, I ^ i β’ β€
hN : β (k : β), Filtration.N (stableFiltration I β€ β trivialFiltration I N) k = N
r : β₯I
eq : βr β’ x = x
β’ x β I ^ Nat.zero β’ β€ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· | simp | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· | Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213 | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x | Mathlib_RingTheory_Filtration |
case mpr.intro.succ
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
x : M
N : Submodule R M := β¨
i, I ^ i β’ β€
hN : β (k : β), Filtration.N (stableFiltration I β€ β trivialFiltration I N) k = N
r : β₯I
eq : βr β’ x = x
i : β
hi : x β I ^ i β’ β€
β’ x β I ^ Nat.succ i β’ β€ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· | rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq] | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· | Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213 | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x | Mathlib_RingTheory_Filtration |
case mpr.intro.succ
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : Module.Finite R M
x : M
N : Submodule R M := β¨
i, I ^ i β’ β€
hN : β (k : β), Filtration.N (stableFiltration I β€ β trivialFiltration I N) k = N
r : β₯I
eq : βr β’ x = x
i : β
hi : x β I ^ i β’ β€
β’ βr β’ x β I β’ I ^ i β’ β€ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq]
| exact Submodule.smul_mem_smul r.prop hi | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq]
| Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213 | theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x | Mathlib_RingTheory_Filtration |
R M : Type u
instββ΅ : CommRing R
instββ΄ : AddCommGroup M
instβΒ³ : Module R M
I : Ideal R
F F' : Filtration I M
instβΒ² : IsNoetherianRing R
instβΒΉ : LocalRing R
instβ : Module.Finite R M
h : I β β€
β’ β¨
i, I ^ i β’ β€ = β₯ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq]
exact Submodule.smul_mem_smul r.prop hi
#align ideal.mem_infi_smul_pow_eq_bot_iff Ideal.mem_iInf_smul_pow_eq_bot_iff
theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
| rw [eq_bot_iff] | theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
| Mathlib.RingTheory.Filtration.457_0.wQ6WBws0g3n9213 | theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ | Mathlib_RingTheory_Filtration |
R M : Type u
instββ΅ : CommRing R
instββ΄ : AddCommGroup M
instβΒ³ : Module R M
I : Ideal R
F F' : Filtration I M
instβΒ² : IsNoetherianRing R
instβΒΉ : LocalRing R
instβ : Module.Finite R M
h : I β β€
β’ β¨
i, I ^ i β’ β€ β€ β₯ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq]
exact Submodule.smul_mem_smul r.prop hi
#align ideal.mem_infi_smul_pow_eq_bot_iff Ideal.mem_iInf_smul_pow_eq_bot_iff
theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
| intro x hx | theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
| Mathlib.RingTheory.Filtration.457_0.wQ6WBws0g3n9213 | theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ | Mathlib_RingTheory_Filtration |
R M : Type u
instββ΅ : CommRing R
instββ΄ : AddCommGroup M
instβΒ³ : Module R M
I : Ideal R
F F' : Filtration I M
instβΒ² : IsNoetherianRing R
instβΒΉ : LocalRing R
instβ : Module.Finite R M
h : I β β€
x : M
hx : x β β¨
i, I ^ i β’ β€
β’ x β β₯ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq]
exact Submodule.smul_mem_smul r.prop hi
#align ideal.mem_infi_smul_pow_eq_bot_iff Ideal.mem_iInf_smul_pow_eq_bot_iff
theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
intro x hx
| obtain β¨r, hrβ© := (I.mem_iInf_smul_pow_eq_bot_iff x).mp hx | theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
intro x hx
| Mathlib.RingTheory.Filtration.457_0.wQ6WBws0g3n9213 | theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ | Mathlib_RingTheory_Filtration |
case intro
R M : Type u
instββ΅ : CommRing R
instββ΄ : AddCommGroup M
instβΒ³ : Module R M
I : Ideal R
F F' : Filtration I M
instβΒ² : IsNoetherianRing R
instβΒΉ : LocalRing R
instβ : Module.Finite R M
h : I β β€
x : M
hx : x β β¨
i, I ^ i β’ β€
r : β₯I
hr : βr β’ x = x
β’ x β β₯ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq]
exact Submodule.smul_mem_smul r.prop hi
#align ideal.mem_infi_smul_pow_eq_bot_iff Ideal.mem_iInf_smul_pow_eq_bot_iff
theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
intro x hx
obtain β¨r, hrβ© := (I.mem_iInf_smul_pow_eq_bot_iff x).mp hx
| have := LocalRing.isUnit_one_sub_self_of_mem_nonunits _ (LocalRing.le_maximalIdeal h r.prop) | theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
intro x hx
obtain β¨r, hrβ© := (I.mem_iInf_smul_pow_eq_bot_iff x).mp hx
| Mathlib.RingTheory.Filtration.457_0.wQ6WBws0g3n9213 | theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ | Mathlib_RingTheory_Filtration |
case intro
R M : Type u
instββ΅ : CommRing R
instββ΄ : AddCommGroup M
instβΒ³ : Module R M
I : Ideal R
F F' : Filtration I M
instβΒ² : IsNoetherianRing R
instβΒΉ : LocalRing R
instβ : Module.Finite R M
h : I β β€
x : M
hx : x β β¨
i, I ^ i β’ β€
r : β₯I
hr : βr β’ x = x
this : IsUnit (1 - βr)
β’ x β β₯ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq]
exact Submodule.smul_mem_smul r.prop hi
#align ideal.mem_infi_smul_pow_eq_bot_iff Ideal.mem_iInf_smul_pow_eq_bot_iff
theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
intro x hx
obtain β¨r, hrβ© := (I.mem_iInf_smul_pow_eq_bot_iff x).mp hx
have := LocalRing.isUnit_one_sub_self_of_mem_nonunits _ (LocalRing.le_maximalIdeal h r.prop)
| apply this.smul_left_cancel.mp | theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
intro x hx
obtain β¨r, hrβ© := (I.mem_iInf_smul_pow_eq_bot_iff x).mp hx
have := LocalRing.isUnit_one_sub_self_of_mem_nonunits _ (LocalRing.le_maximalIdeal h r.prop)
| Mathlib.RingTheory.Filtration.457_0.wQ6WBws0g3n9213 | theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ | Mathlib_RingTheory_Filtration |
case intro
R M : Type u
instββ΅ : CommRing R
instββ΄ : AddCommGroup M
instβΒ³ : Module R M
I : Ideal R
F F' : Filtration I M
instβΒ² : IsNoetherianRing R
instβΒΉ : LocalRing R
instβ : Module.Finite R M
h : I β β€
x : M
hx : x β β¨
i, I ^ i β’ β€
r : β₯I
hr : βr β’ x = x
this : IsUnit (1 - βr)
β’ (1 - βr) β’ x = (1 - βr) β’ 0 | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq]
exact Submodule.smul_mem_smul r.prop hi
#align ideal.mem_infi_smul_pow_eq_bot_iff Ideal.mem_iInf_smul_pow_eq_bot_iff
theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
intro x hx
obtain β¨r, hrβ© := (I.mem_iInf_smul_pow_eq_bot_iff x).mp hx
have := LocalRing.isUnit_one_sub_self_of_mem_nonunits _ (LocalRing.le_maximalIdeal h r.prop)
apply this.smul_left_cancel.mp
| simp [sub_smul, hr] | theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
intro x hx
obtain β¨r, hrβ© := (I.mem_iInf_smul_pow_eq_bot_iff x).mp hx
have := LocalRing.isUnit_one_sub_self_of_mem_nonunits _ (LocalRing.le_maximalIdeal h r.prop)
apply this.smul_left_cancel.mp
| Mathlib.RingTheory.Filtration.457_0.wQ6WBws0g3n9213 | theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ | Mathlib_RingTheory_Filtration |
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : LocalRing R
h : I β β€
β’ β¨
i, I ^ i = β₯ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq]
exact Submodule.smul_mem_smul r.prop hi
#align ideal.mem_infi_smul_pow_eq_bot_iff Ideal.mem_iInf_smul_pow_eq_bot_iff
theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
intro x hx
obtain β¨r, hrβ© := (I.mem_iInf_smul_pow_eq_bot_iff x).mp hx
have := LocalRing.isUnit_one_sub_self_of_mem_nonunits _ (LocalRing.le_maximalIdeal h r.prop)
apply this.smul_left_cancel.mp
simp [sub_smul, hr]
#align ideal.infi_pow_smul_eq_bot_of_local_ring Ideal.iInf_pow_smul_eq_bot_of_localRing
/-- **Krull's intersection theorem** for noetherian local rings. -/
theorem Ideal.iInf_pow_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
| convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h | /-- **Krull's intersection theorem** for noetherian local rings. -/
theorem Ideal.iInf_pow_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
| Mathlib.RingTheory.Filtration.467_0.wQ6WBws0g3n9213 | /-- **Krull's intersection theorem** for noetherian local rings. -/
theorem Ideal.iInf_pow_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ | Mathlib_RingTheory_Filtration |
case h.e'_2.h.e'_4.h
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : LocalRing R
h : I β β€
xβ : β
β’ I ^ xβ = I ^ xβ β’ β€ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq]
exact Submodule.smul_mem_smul r.prop hi
#align ideal.mem_infi_smul_pow_eq_bot_iff Ideal.mem_iInf_smul_pow_eq_bot_iff
theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
intro x hx
obtain β¨r, hrβ© := (I.mem_iInf_smul_pow_eq_bot_iff x).mp hx
have := LocalRing.isUnit_one_sub_self_of_mem_nonunits _ (LocalRing.le_maximalIdeal h r.prop)
apply this.smul_left_cancel.mp
simp [sub_smul, hr]
#align ideal.infi_pow_smul_eq_bot_of_local_ring Ideal.iInf_pow_smul_eq_bot_of_localRing
/-- **Krull's intersection theorem** for noetherian local rings. -/
theorem Ideal.iInf_pow_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h
| ext i | /-- **Krull's intersection theorem** for noetherian local rings. -/
theorem Ideal.iInf_pow_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h
| Mathlib.RingTheory.Filtration.467_0.wQ6WBws0g3n9213 | /-- **Krull's intersection theorem** for noetherian local rings. -/
theorem Ideal.iInf_pow_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ | Mathlib_RingTheory_Filtration |
case h.e'_2.h.e'_4.h.h
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : LocalRing R
h : I β β€
xβ : β
i : R
β’ i β I ^ xβ β i β I ^ xβ β’ β€ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq]
exact Submodule.smul_mem_smul r.prop hi
#align ideal.mem_infi_smul_pow_eq_bot_iff Ideal.mem_iInf_smul_pow_eq_bot_iff
theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
intro x hx
obtain β¨r, hrβ© := (I.mem_iInf_smul_pow_eq_bot_iff x).mp hx
have := LocalRing.isUnit_one_sub_self_of_mem_nonunits _ (LocalRing.le_maximalIdeal h r.prop)
apply this.smul_left_cancel.mp
simp [sub_smul, hr]
#align ideal.infi_pow_smul_eq_bot_of_local_ring Ideal.iInf_pow_smul_eq_bot_of_localRing
/-- **Krull's intersection theorem** for noetherian local rings. -/
theorem Ideal.iInf_pow_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h
ext i
| rw [smul_eq_mul, β Ideal.one_eq_top, mul_one] | /-- **Krull's intersection theorem** for noetherian local rings. -/
theorem Ideal.iInf_pow_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h
ext i
| Mathlib.RingTheory.Filtration.467_0.wQ6WBws0g3n9213 | /-- **Krull's intersection theorem** for noetherian local rings. -/
theorem Ideal.iInf_pow_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ | Mathlib_RingTheory_Filtration |
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : IsDomain R
h : I β β€
β’ β¨
i, I ^ i = β₯ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq]
exact Submodule.smul_mem_smul r.prop hi
#align ideal.mem_infi_smul_pow_eq_bot_iff Ideal.mem_iInf_smul_pow_eq_bot_iff
theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
intro x hx
obtain β¨r, hrβ© := (I.mem_iInf_smul_pow_eq_bot_iff x).mp hx
have := LocalRing.isUnit_one_sub_self_of_mem_nonunits _ (LocalRing.le_maximalIdeal h r.prop)
apply this.smul_left_cancel.mp
simp [sub_smul, hr]
#align ideal.infi_pow_smul_eq_bot_of_local_ring Ideal.iInf_pow_smul_eq_bot_of_localRing
/-- **Krull's intersection theorem** for noetherian local rings. -/
theorem Ideal.iInf_pow_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h
ext i
rw [smul_eq_mul, β Ideal.one_eq_top, mul_one]
#align ideal.infi_pow_eq_bot_of_local_ring Ideal.iInf_pow_eq_bot_of_localRing
/-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
| rw [eq_bot_iff] | /-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
| Mathlib.RingTheory.Filtration.475_0.wQ6WBws0g3n9213 | /-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ | Mathlib_RingTheory_Filtration |
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : IsDomain R
h : I β β€
β’ β¨
i, I ^ i β€ β₯ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq]
exact Submodule.smul_mem_smul r.prop hi
#align ideal.mem_infi_smul_pow_eq_bot_iff Ideal.mem_iInf_smul_pow_eq_bot_iff
theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
intro x hx
obtain β¨r, hrβ© := (I.mem_iInf_smul_pow_eq_bot_iff x).mp hx
have := LocalRing.isUnit_one_sub_self_of_mem_nonunits _ (LocalRing.le_maximalIdeal h r.prop)
apply this.smul_left_cancel.mp
simp [sub_smul, hr]
#align ideal.infi_pow_smul_eq_bot_of_local_ring Ideal.iInf_pow_smul_eq_bot_of_localRing
/-- **Krull's intersection theorem** for noetherian local rings. -/
theorem Ideal.iInf_pow_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h
ext i
rw [smul_eq_mul, β Ideal.one_eq_top, mul_one]
#align ideal.infi_pow_eq_bot_of_local_ring Ideal.iInf_pow_eq_bot_of_localRing
/-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
rw [eq_bot_iff]
| intro x hx | /-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
rw [eq_bot_iff]
| Mathlib.RingTheory.Filtration.475_0.wQ6WBws0g3n9213 | /-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ | Mathlib_RingTheory_Filtration |
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : IsDomain R
h : I β β€
x : R
hx : x β β¨
i, I ^ i
β’ x β β₯ | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq]
exact Submodule.smul_mem_smul r.prop hi
#align ideal.mem_infi_smul_pow_eq_bot_iff Ideal.mem_iInf_smul_pow_eq_bot_iff
theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
intro x hx
obtain β¨r, hrβ© := (I.mem_iInf_smul_pow_eq_bot_iff x).mp hx
have := LocalRing.isUnit_one_sub_self_of_mem_nonunits _ (LocalRing.le_maximalIdeal h r.prop)
apply this.smul_left_cancel.mp
simp [sub_smul, hr]
#align ideal.infi_pow_smul_eq_bot_of_local_ring Ideal.iInf_pow_smul_eq_bot_of_localRing
/-- **Krull's intersection theorem** for noetherian local rings. -/
theorem Ideal.iInf_pow_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h
ext i
rw [smul_eq_mul, β Ideal.one_eq_top, mul_one]
#align ideal.infi_pow_eq_bot_of_local_ring Ideal.iInf_pow_eq_bot_of_localRing
/-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
rw [eq_bot_iff]
intro x hx
| by_contra hx' | /-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
rw [eq_bot_iff]
intro x hx
| Mathlib.RingTheory.Filtration.475_0.wQ6WBws0g3n9213 | /-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ | Mathlib_RingTheory_Filtration |
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : IsDomain R
h : I β β€
x : R
hx : x β β¨
i, I ^ i
hx' : x β β₯
β’ False | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq]
exact Submodule.smul_mem_smul r.prop hi
#align ideal.mem_infi_smul_pow_eq_bot_iff Ideal.mem_iInf_smul_pow_eq_bot_iff
theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
intro x hx
obtain β¨r, hrβ© := (I.mem_iInf_smul_pow_eq_bot_iff x).mp hx
have := LocalRing.isUnit_one_sub_self_of_mem_nonunits _ (LocalRing.le_maximalIdeal h r.prop)
apply this.smul_left_cancel.mp
simp [sub_smul, hr]
#align ideal.infi_pow_smul_eq_bot_of_local_ring Ideal.iInf_pow_smul_eq_bot_of_localRing
/-- **Krull's intersection theorem** for noetherian local rings. -/
theorem Ideal.iInf_pow_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h
ext i
rw [smul_eq_mul, β Ideal.one_eq_top, mul_one]
#align ideal.infi_pow_eq_bot_of_local_ring Ideal.iInf_pow_eq_bot_of_localRing
/-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
rw [eq_bot_iff]
intro x hx
by_contra hx'
| have := Ideal.mem_iInf_smul_pow_eq_bot_iff I x | /-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
rw [eq_bot_iff]
intro x hx
by_contra hx'
| Mathlib.RingTheory.Filtration.475_0.wQ6WBws0g3n9213 | /-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ | Mathlib_RingTheory_Filtration |
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : IsDomain R
h : I β β€
x : R
hx : x β β¨
i, I ^ i
hx' : x β β₯
this : x β β¨
i, I ^ i β’ β€ β β r, βr β’ x = x
β’ False | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq]
exact Submodule.smul_mem_smul r.prop hi
#align ideal.mem_infi_smul_pow_eq_bot_iff Ideal.mem_iInf_smul_pow_eq_bot_iff
theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
intro x hx
obtain β¨r, hrβ© := (I.mem_iInf_smul_pow_eq_bot_iff x).mp hx
have := LocalRing.isUnit_one_sub_self_of_mem_nonunits _ (LocalRing.le_maximalIdeal h r.prop)
apply this.smul_left_cancel.mp
simp [sub_smul, hr]
#align ideal.infi_pow_smul_eq_bot_of_local_ring Ideal.iInf_pow_smul_eq_bot_of_localRing
/-- **Krull's intersection theorem** for noetherian local rings. -/
theorem Ideal.iInf_pow_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h
ext i
rw [smul_eq_mul, β Ideal.one_eq_top, mul_one]
#align ideal.infi_pow_eq_bot_of_local_ring Ideal.iInf_pow_eq_bot_of_localRing
/-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
rw [eq_bot_iff]
intro x hx
by_contra hx'
have := Ideal.mem_iInf_smul_pow_eq_bot_iff I x
| simp_rw [smul_eq_mul, β Ideal.one_eq_top, mul_one] at this | /-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
rw [eq_bot_iff]
intro x hx
by_contra hx'
have := Ideal.mem_iInf_smul_pow_eq_bot_iff I x
| Mathlib.RingTheory.Filtration.475_0.wQ6WBws0g3n9213 | /-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ | Mathlib_RingTheory_Filtration |
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : IsDomain R
h : I β β€
x : R
hx : x β β¨
i, I ^ i
hx' : x β β₯
this : x β β¨
i, I ^ i β β r, βr * x = x
β’ False | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq]
exact Submodule.smul_mem_smul r.prop hi
#align ideal.mem_infi_smul_pow_eq_bot_iff Ideal.mem_iInf_smul_pow_eq_bot_iff
theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
intro x hx
obtain β¨r, hrβ© := (I.mem_iInf_smul_pow_eq_bot_iff x).mp hx
have := LocalRing.isUnit_one_sub_self_of_mem_nonunits _ (LocalRing.le_maximalIdeal h r.prop)
apply this.smul_left_cancel.mp
simp [sub_smul, hr]
#align ideal.infi_pow_smul_eq_bot_of_local_ring Ideal.iInf_pow_smul_eq_bot_of_localRing
/-- **Krull's intersection theorem** for noetherian local rings. -/
theorem Ideal.iInf_pow_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h
ext i
rw [smul_eq_mul, β Ideal.one_eq_top, mul_one]
#align ideal.infi_pow_eq_bot_of_local_ring Ideal.iInf_pow_eq_bot_of_localRing
/-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
rw [eq_bot_iff]
intro x hx
by_contra hx'
have := Ideal.mem_iInf_smul_pow_eq_bot_iff I x
simp_rw [smul_eq_mul, β Ideal.one_eq_top, mul_one] at this
| obtain β¨r, hrβ© := this.mp hx | /-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
rw [eq_bot_iff]
intro x hx
by_contra hx'
have := Ideal.mem_iInf_smul_pow_eq_bot_iff I x
simp_rw [smul_eq_mul, β Ideal.one_eq_top, mul_one] at this
| Mathlib.RingTheory.Filtration.475_0.wQ6WBws0g3n9213 | /-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ | Mathlib_RingTheory_Filtration |
case intro
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : IsDomain R
h : I β β€
x : R
hx : x β β¨
i, I ^ i
hx' : x β β₯
this : x β β¨
i, I ^ i β β r, βr * x = x
r : β₯I
hr : βr * x = x
β’ False | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq]
exact Submodule.smul_mem_smul r.prop hi
#align ideal.mem_infi_smul_pow_eq_bot_iff Ideal.mem_iInf_smul_pow_eq_bot_iff
theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
intro x hx
obtain β¨r, hrβ© := (I.mem_iInf_smul_pow_eq_bot_iff x).mp hx
have := LocalRing.isUnit_one_sub_self_of_mem_nonunits _ (LocalRing.le_maximalIdeal h r.prop)
apply this.smul_left_cancel.mp
simp [sub_smul, hr]
#align ideal.infi_pow_smul_eq_bot_of_local_ring Ideal.iInf_pow_smul_eq_bot_of_localRing
/-- **Krull's intersection theorem** for noetherian local rings. -/
theorem Ideal.iInf_pow_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h
ext i
rw [smul_eq_mul, β Ideal.one_eq_top, mul_one]
#align ideal.infi_pow_eq_bot_of_local_ring Ideal.iInf_pow_eq_bot_of_localRing
/-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
rw [eq_bot_iff]
intro x hx
by_contra hx'
have := Ideal.mem_iInf_smul_pow_eq_bot_iff I x
simp_rw [smul_eq_mul, β Ideal.one_eq_top, mul_one] at this
obtain β¨r, hrβ© := this.mp hx
| have := mul_right_cancelβ hx' (hr.trans (one_mul x).symm) | /-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
rw [eq_bot_iff]
intro x hx
by_contra hx'
have := Ideal.mem_iInf_smul_pow_eq_bot_iff I x
simp_rw [smul_eq_mul, β Ideal.one_eq_top, mul_one] at this
obtain β¨r, hrβ© := this.mp hx
| Mathlib.RingTheory.Filtration.475_0.wQ6WBws0g3n9213 | /-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ | Mathlib_RingTheory_Filtration |
case intro
R M : Type u
instββ΄ : CommRing R
instβΒ³ : AddCommGroup M
instβΒ² : Module R M
I : Ideal R
F F' : Filtration I M
instβΒΉ : IsNoetherianRing R
instβ : IsDomain R
h : I β β€
x : R
hx : x β β¨
i, I ^ i
hx' : x β β₯
thisβ : x β β¨
i, I ^ i β β r, βr * x = x
r : β₯I
hr : βr * x = x
this : βr = 1
β’ False | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Polynomial.Module
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# `I`-filtrations of modules
This file contains the definitions and basic results around (stable) `I`-filtrations of modules.
## Main results
- `Ideal.Filtration`: An `I`-filtration on the module `M` is a sequence of decreasing submodules
`N i` such that `I β’ N β€ I (i + 1)`. Note that we do not require the filtration to start from `β€`.
- `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I β’ (N i) = N (i + 1)` for large
enough `i`.
- `Ideal.Filtration.submodule`: The associated module `β¨ Nα΅’` of a filtration, implemented as a
submodule of `M[X]`.
- `Ideal.Filtration.submodule_fg_iff_stable`: If `F.N i` are all finitely generated, then
`F.Stable` iff `F.submodule.FG`.
- `Ideal.Filtration.Stable.of_le`: In a finite module over a noetherian ring,
if `F' β€ F`, then `F.Stable β F'.Stable`.
- `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**.
given `N β€ M`, there exists a `k` such that `IβΏM β N = IβΏβ»α΅(Iα΅M β N)` for all `n β₯ k`.
- `Ideal.iInf_pow_eq_bot_of_localRing`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian local rings.
- `Ideal.iInf_pow_eq_bot_of_isDomain`:
**Krull's intersection theorem** (`β¨
i, I ^ i = β₯`) for noetherian domains.
-/
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial BigOperators
/-- An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I β’ (N i) β€ N (i + 1)`. Note that we do not require the filtration to start from `β€`. -/
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : β β Submodule R M
mono : β i, N (i + 1) β€ N i
smul_le : β i, I β’ N i β€ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by
induction' i with _ ih
Β· simp
Β· rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, β add_assoc]
exact (Submodule.smul_mono_right ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : β) : I ^ (i + k) β’ F.N j β€ I ^ k β’ F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact Submodule.smul_mono_right (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
/-- The trivial `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_of_eq rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
/-- The `sup` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Sup (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(le_of_eq (Submodule.smul_sup _ _ _)).trans <| sup_le_sup (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : SupSet (I.Filtration M) :=
β¨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
/-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/
instance : Inf (I.Filtration M) :=
β¨fun F F' =>
β¨F.N β F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)β©β©
/-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/
instance : InfSet (I.Filtration M) :=
β¨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ β¨β¨_, F, hF, rflβ©, rflβ©
exact β¨_, β¨β¨_, F, hF, rflβ©, rflβ©, F.mono iβ©
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine' Submodule.smul_iInf_le.trans _
apply iInf_mono _
rintro β¨_, F, hF, rflβ©
exact F.smul_le i }β©
instance : Top (I.Filtration M) :=
β¨I.trivialFiltration β€β©
instance : Bot (I.Filtration M) :=
β¨I.trivialFiltration β₯β©
@[simp]
theorem sup_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F β F').N = F.N β F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (β€ : I.Filtration M).N = β€ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (β₯ : I.Filtration M).N = β₯ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iSup f).N = β¨ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ΞΉ : Sort*} (f : ΞΉ β I.Filtration M) : (iInf f).N = β¨
i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
β¨β₯β©
/-- An `I` filtration is stable if `I β’ F.N n = F.N (n+1)` for large enough `n`. -/
def Stable : Prop :=
β nβ, β n β₯ nβ, I β’ F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
/-- The trivial stable `I`-filtration of `N`. -/
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i β’ N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : β nβ, β k, F.N (nβ + k) = I ^ k β’ F.N nβ := by
obtain β¨nβ, hnβ© := h
use nβ
intro k
induction' k with _ ih
Β· simp
Β· rw [Nat.succ_eq_add_one, β add_assoc, β hn, ih, add_comm, pow_add, mul_smul, pow_one]
linarith
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
obtain β¨nβ, hnββ© := h.exists_pow_smul_eq
use nβ
intro n hn
convert hnβ (n - nβ)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable β β nβ, β n β₯ nβ, F.N n = I ^ (n - nβ) β’ F.N nβ := by
refine' β¨Stable.exists_pow_smul_eq_of_ge, fun h => β¨h.choose, fun n hn => _β©β©
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, β pow_succ,
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 β€ F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n := by
obtain β¨nβ, hFβ© := h
use nβ
intro n
induction' n with n hn
Β· refine' (F.antitone _).trans e; simp
Β· rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, β hF]
exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
β nβ, β n, F.N (n + nβ) β€ F'.N n β§ F'.N (n + nβ) β€ F.N n := by
obtain β¨nβ, hββ© := h.exists_forall_le (le_of_eq e)
obtain β¨nβ, hββ© := h'.exists_forall_le (le_of_eq e.symm)
use max nβ nβ
intro n
refine' β¨(F.antitone _).trans (hβ n), (F'.antitone _).trans (hβ n)β© <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | β i, f i β F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro β¨j, kβ© e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f β F.submodule β β i, f i β F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F β F').submodule = F.submodule β F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
/-- `Ideal.Filtration.submodule` as an `InfHom`. -/
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (β i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
Β· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ β¨m, hm, rflβ© j
rw [single_apply]
split_ifs with h
Β· rwa [β h]
Β· exact (F.N j).zero_mem
Β· intro f hf
rw [β f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (β i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [β Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (nβ : β) :
F.submodule = Submodule.span _ (β i β€ nβ, single R i '' (F.N i : Set M)) β
β n β₯ nβ, I β’ F.N n = F.N (n + 1) := by
rw [β submodule_span_single, β LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; Β· exact Submodule.span_mono (Set.iUnionβ_subset_iUnion _ _)
constructor
Β· intro H n hn
refine' (F.smul_le n).antisymm _
intro x hx
obtain β¨l, hlβ© := (Finsupp.mem_span_iff_total _ _ _).mp (H _ β¨x, hx, rflβ©)
replace hl := congr_arg (fun f : β ββ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [β hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro β¨_, _, β¨n', rflβ©, _, β¨hn', rflβ©, m, hm, rflβ© -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' β€ n + 1 by linarith)]
have e : n' β€ n := by linarith
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, β add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
Β· let F' := Submodule.span (reesAlgebra I) (β i β€ nβ, single R i '' (F.N i : Set M))
intro hF i
have : β i β€ nβ, single R i '' (F.N i : Set M) β F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnionβ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
Β· exact this _ (zero_le _)
by_cases hj' : j.succ β€ nβ
Β· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [Nat.succ_eq_add_one, β hF _ hj']
rintro _ β¨m, hm, rflβ©
refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)
Β· rw [add_comm, β monomial_smul_single]
exact F'.smul_mem
β¨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])β© (hj <| Set.mem_image_of_mem _ hm')
Β· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
/-- If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. -/
theorem submodule_fg_iff_stable (hF' : β i, (F.N i).FG) : F.submodule.FG β F.Stable := by
classical
delta Ideal.Filtration.Stable
simp_rw [β F.submodule_eq_span_le_iff_stable_ge]
constructor
Β· rintro H
refine H.stablizes_of_iSup_eq
β¨fun nβ => Submodule.span _ (β (i : β) (_ : i β€ nβ), single R i '' β(F.N i)), ?_β© ?_
Β· intro n m e
rw [Submodule.span_le, Set.iUnionβ_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnionβ _ _ _ (fun i => fun _ => β((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
Β· dsimp
rw [β Submodule.span_iUnion, β submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
Β· rintro β¨-, i, -, eβ©; exact β¨i, eβ©
Β· rintro β¨i, eβ©; exact β¨i, i, le_refl i, eβ©
Β· rintro β¨n, hnβ©
rw [hn]
simp_rw [Submodule.span_iUnionβ, β Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro β¨i, hiβ©
obtain β¨s, hsβ© := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, β Submodule.span_span_of_tower R, β Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, β this]
exact β¨_, rflβ©
#align ideal.filtration.submodule_fg_iff_stable Ideal.Filtration.submodule_fg_iff_stable
variable {F}
theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' β€ F) : F'.Stable := by
rw [β submodule_fg_iff_stable] at hF β’
any_goals intro i; exact IsNoetherian.noetherian _
have := isNoetherian_of_fg_of_noetherian _ hF
rw [isNoetherian_submodule] at this
exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)
#align ideal.filtration.stable.of_le Ideal.Filtration.Stable.of_le
theorem Stable.inter_right [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F β F').Stable :=
hF.of_le inf_le_left
#align ideal.filtration.stable.inter_right Ideal.Filtration.Stable.inter_right
theorem Stable.inter_left [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable) :
(F' β F).Stable :=
hF.of_le inf_le_right
#align ideal.filtration.stable.inter_left Ideal.Filtration.Stable.inter_left
end Ideal.Filtration
variable (I)
/-- **Artin-Rees lemma** -/
theorem Ideal.exists_pow_inf_eq_pow_smul [IsNoetherianRing R] [Module.Finite R M]
(N : Submodule R M) : β k : β, β n β₯ k, I ^ n β’ β€ β N = I ^ (n - k) β’ (I ^ k β’ β€ β N) :=
((I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)).exists_pow_smul_eq_of_ge
#align ideal.exists_pow_inf_eq_pow_smul Ideal.exists_pow_inf_eq_pow_smul
theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) :
x β (β¨
i : β, I ^ i β’ β€ : Submodule R M) β β r : I, (r : R) β’ x = x := by
let N := (β¨
i : β, I ^ i β’ β€ : Submodule R M)
have hN : β k, (I.stableFiltration β€ β I.trivialFiltration N).N k = N :=
fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)
constructor
Β· obtain β¨r, hrβ, hrββ© :=
Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by
obtain β¨k, hkβ© := (I.stableFiltration_stable β€).inter_right (I.trivialFiltration N)
have := hk k (le_refl _)
rw [hN, hN] at this
exact le_of_eq this.symm)
intro H
exact β¨β¨r, hrββ©, hrβ _ Hβ©
Β· rintro β¨r, eqβ©
rw [Submodule.mem_iInf]
intro i
induction' i with i hi
Β· simp
Β· rw [Nat.succ_eq_one_add, pow_add, β smul_smul, pow_one, β eq]
exact Submodule.smul_mem_smul r.prop hi
#align ideal.mem_infi_smul_pow_eq_bot_iff Ideal.mem_iInf_smul_pow_eq_bot_iff
theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R]
[Module.Finite R M] (h : I β β€) : (β¨
i : β, I ^ i β’ β€ : Submodule R M) = β₯ := by
rw [eq_bot_iff]
intro x hx
obtain β¨r, hrβ© := (I.mem_iInf_smul_pow_eq_bot_iff x).mp hx
have := LocalRing.isUnit_one_sub_self_of_mem_nonunits _ (LocalRing.le_maximalIdeal h r.prop)
apply this.smul_left_cancel.mp
simp [sub_smul, hr]
#align ideal.infi_pow_smul_eq_bot_of_local_ring Ideal.iInf_pow_smul_eq_bot_of_localRing
/-- **Krull's intersection theorem** for noetherian local rings. -/
theorem Ideal.iInf_pow_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h
ext i
rw [smul_eq_mul, β Ideal.one_eq_top, mul_one]
#align ideal.infi_pow_eq_bot_of_local_ring Ideal.iInf_pow_eq_bot_of_localRing
/-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
rw [eq_bot_iff]
intro x hx
by_contra hx'
have := Ideal.mem_iInf_smul_pow_eq_bot_iff I x
simp_rw [smul_eq_mul, β Ideal.one_eq_top, mul_one] at this
obtain β¨r, hrβ© := this.mp hx
have := mul_right_cancelβ hx' (hr.trans (one_mul x).symm)
| exact I.eq_top_iff_one.not.mp h (this βΈ r.prop) | /-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ := by
rw [eq_bot_iff]
intro x hx
by_contra hx'
have := Ideal.mem_iInf_smul_pow_eq_bot_iff I x
simp_rw [smul_eq_mul, β Ideal.one_eq_top, mul_one] at this
obtain β¨r, hrβ© := this.mp hx
have := mul_right_cancelβ hx' (hr.trans (one_mul x).symm)
| Mathlib.RingTheory.Filtration.475_0.wQ6WBws0g3n9213 | /-- **Krull's intersection theorem** for noetherian domains. -/
theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I β β€) :
β¨
i : β, I ^ i = β₯ | Mathlib_RingTheory_Filtration |
x : β
β’ 0 β€ x β x β AddSubmonoid.closure (Set.range fun s => star s * s) | /-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Order
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Rat.Order
import Mathlib.GroupTheory.Submonoid.Membership
#align_import data.rat.star from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
/-! # Star order structure on β
Here we put the trivial `star` operation on `β` for convenience and show that it is a
`StarOrderedRing`. In particular, this means that every element of `β` is a sum of squares.
-/
namespace Rat
instance : StarRing β where
star := id
star_involutive _ := rfl
star_mul _ _ := mul_comm _ _
star_add _ _ := rfl
instance : TrivialStar β where star_trivial _ := rfl
instance : StarOrderedRing β :=
StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by
| refine'
β¨fun hx => _, fun hx =>
AddSubmonoid.closure_induction hx (by rintro - β¨s, rflβ©; exact mul_self_nonneg s) le_rfl
fun _ _ => add_nonnegβ© | instance : StarOrderedRing β :=
StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by
| Mathlib.Data.Rat.Star.30_0.tRpObn3UXJ6ySXp | instance : StarOrderedRing β | Mathlib_Data_Rat_Star |
x : β
hx : x β AddSubmonoid.closure (Set.range fun s => star s * s)
β’ β x β Set.range fun s => star s * s, 0 β€ x | /-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Order
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Rat.Order
import Mathlib.GroupTheory.Submonoid.Membership
#align_import data.rat.star from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
/-! # Star order structure on β
Here we put the trivial `star` operation on `β` for convenience and show that it is a
`StarOrderedRing`. In particular, this means that every element of `β` is a sum of squares.
-/
namespace Rat
instance : StarRing β where
star := id
star_involutive _ := rfl
star_mul _ _ := mul_comm _ _
star_add _ _ := rfl
instance : TrivialStar β where star_trivial _ := rfl
instance : StarOrderedRing β :=
StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by
refine'
β¨fun hx => _, fun hx =>
AddSubmonoid.closure_induction hx (by | rintro - β¨s, rflβ© | instance : StarOrderedRing β :=
StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by
refine'
β¨fun hx => _, fun hx =>
AddSubmonoid.closure_induction hx (by | Mathlib.Data.Rat.Star.30_0.tRpObn3UXJ6ySXp | instance : StarOrderedRing β | Mathlib_Data_Rat_Star |
case intro
x : β
hx : x β AddSubmonoid.closure (Set.range fun s => star s * s)
s : β
β’ 0 β€ (fun s => star s * s) s | /-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Order
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Rat.Order
import Mathlib.GroupTheory.Submonoid.Membership
#align_import data.rat.star from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
/-! # Star order structure on β
Here we put the trivial `star` operation on `β` for convenience and show that it is a
`StarOrderedRing`. In particular, this means that every element of `β` is a sum of squares.
-/
namespace Rat
instance : StarRing β where
star := id
star_involutive _ := rfl
star_mul _ _ := mul_comm _ _
star_add _ _ := rfl
instance : TrivialStar β where star_trivial _ := rfl
instance : StarOrderedRing β :=
StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by
refine'
β¨fun hx => _, fun hx =>
AddSubmonoid.closure_induction hx (by rintro - β¨s, rflβ©; | exact mul_self_nonneg s | instance : StarOrderedRing β :=
StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by
refine'
β¨fun hx => _, fun hx =>
AddSubmonoid.closure_induction hx (by rintro - β¨s, rflβ©; | Mathlib.Data.Rat.Star.30_0.tRpObn3UXJ6ySXp | instance : StarOrderedRing β | Mathlib_Data_Rat_Star |
x : β
hx : 0 β€ x
β’ x β AddSubmonoid.closure (Set.range fun s => star s * s) | /-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Order
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Rat.Order
import Mathlib.GroupTheory.Submonoid.Membership
#align_import data.rat.star from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
/-! # Star order structure on β
Here we put the trivial `star` operation on `β` for convenience and show that it is a
`StarOrderedRing`. In particular, this means that every element of `β` is a sum of squares.
-/
namespace Rat
instance : StarRing β where
star := id
star_involutive _ := rfl
star_mul _ _ := mul_comm _ _
star_add _ _ := rfl
instance : TrivialStar β where star_trivial _ := rfl
instance : StarOrderedRing β :=
StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by
refine'
β¨fun hx => _, fun hx =>
AddSubmonoid.closure_induction hx (by rintro - β¨s, rflβ©; exact mul_self_nonneg s) le_rfl
fun _ _ => add_nonnegβ©
/- If `x = p / q`, then, since `0 β€ x`, we have `p q : β`, and `p / q` is the sum of `p * q`
copies of `(1 / q) ^ 2`, and so `x` lies in the `AddSubmonoid` generated by square elements.
Note: it's possible to rephrase this argument as `x = (p * q) β’ (1 / q) ^ 2`, but this would
be somewhat challenging without increasing import requirements. -/
-- Porting note: rewrote proof to avoid removed constructor rat.mk_pnat
| suffices
(Finset.range (x.num.natAbs * x.den)).sum
(Function.const β ((1 : β) / x.den * ((1 : β) / x.den))) =
x
by exact this βΈ sum_mem fun n _ => AddSubmonoid.subset_closure β¨_, rflβ© | instance : StarOrderedRing β :=
StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by
refine'
β¨fun hx => _, fun hx =>
AddSubmonoid.closure_induction hx (by rintro - β¨s, rflβ©; exact mul_self_nonneg s) le_rfl
fun _ _ => add_nonnegβ©
/- If `x = p / q`, then, since `0 β€ x`, we have `p q : β`, and `p / q` is the sum of `p * q`
copies of `(1 / q) ^ 2`, and so `x` lies in the `AddSubmonoid` generated by square elements.
Note: it's possible to rephrase this argument as `x = (p * q) β’ (1 / q) ^ 2`, but this would
be somewhat challenging without increasing import requirements. -/
-- Porting note: rewrote proof to avoid removed constructor rat.mk_pnat
| Mathlib.Data.Rat.Star.30_0.tRpObn3UXJ6ySXp | instance : StarOrderedRing β | Mathlib_Data_Rat_Star |
x : β
hx : 0 β€ x
this : Finset.sum (Finset.range (Int.natAbs x.num * x.den)) (Function.const β (1 / βx.den * (1 / βx.den))) = x
β’ x β AddSubmonoid.closure (Set.range fun s => star s * s) | /-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Order
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Rat.Order
import Mathlib.GroupTheory.Submonoid.Membership
#align_import data.rat.star from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
/-! # Star order structure on β
Here we put the trivial `star` operation on `β` for convenience and show that it is a
`StarOrderedRing`. In particular, this means that every element of `β` is a sum of squares.
-/
namespace Rat
instance : StarRing β where
star := id
star_involutive _ := rfl
star_mul _ _ := mul_comm _ _
star_add _ _ := rfl
instance : TrivialStar β where star_trivial _ := rfl
instance : StarOrderedRing β :=
StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by
refine'
β¨fun hx => _, fun hx =>
AddSubmonoid.closure_induction hx (by rintro - β¨s, rflβ©; exact mul_self_nonneg s) le_rfl
fun _ _ => add_nonnegβ©
/- If `x = p / q`, then, since `0 β€ x`, we have `p q : β`, and `p / q` is the sum of `p * q`
copies of `(1 / q) ^ 2`, and so `x` lies in the `AddSubmonoid` generated by square elements.
Note: it's possible to rephrase this argument as `x = (p * q) β’ (1 / q) ^ 2`, but this would
be somewhat challenging without increasing import requirements. -/
-- Porting note: rewrote proof to avoid removed constructor rat.mk_pnat
suffices
(Finset.range (x.num.natAbs * x.den)).sum
(Function.const β ((1 : β) / x.den * ((1 : β) / x.den))) =
x
by | exact this βΈ sum_mem fun n _ => AddSubmonoid.subset_closure β¨_, rflβ© | instance : StarOrderedRing β :=
StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by
refine'
β¨fun hx => _, fun hx =>
AddSubmonoid.closure_induction hx (by rintro - β¨s, rflβ©; exact mul_self_nonneg s) le_rfl
fun _ _ => add_nonnegβ©
/- If `x = p / q`, then, since `0 β€ x`, we have `p q : β`, and `p / q` is the sum of `p * q`
copies of `(1 / q) ^ 2`, and so `x` lies in the `AddSubmonoid` generated by square elements.
Note: it's possible to rephrase this argument as `x = (p * q) β’ (1 / q) ^ 2`, but this would
be somewhat challenging without increasing import requirements. -/
-- Porting note: rewrote proof to avoid removed constructor rat.mk_pnat
suffices
(Finset.range (x.num.natAbs * x.den)).sum
(Function.const β ((1 : β) / x.den * ((1 : β) / x.den))) =
x
by | Mathlib.Data.Rat.Star.30_0.tRpObn3UXJ6ySXp | instance : StarOrderedRing β | Mathlib_Data_Rat_Star |
x : β
hx : 0 β€ x
β’ Finset.sum (Finset.range (Int.natAbs x.num * x.den)) (Function.const β (1 / βx.den * (1 / βx.den))) = x | /-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Order
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Rat.Order
import Mathlib.GroupTheory.Submonoid.Membership
#align_import data.rat.star from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
/-! # Star order structure on β
Here we put the trivial `star` operation on `β` for convenience and show that it is a
`StarOrderedRing`. In particular, this means that every element of `β` is a sum of squares.
-/
namespace Rat
instance : StarRing β where
star := id
star_involutive _ := rfl
star_mul _ _ := mul_comm _ _
star_add _ _ := rfl
instance : TrivialStar β where star_trivial _ := rfl
instance : StarOrderedRing β :=
StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by
refine'
β¨fun hx => _, fun hx =>
AddSubmonoid.closure_induction hx (by rintro - β¨s, rflβ©; exact mul_self_nonneg s) le_rfl
fun _ _ => add_nonnegβ©
/- If `x = p / q`, then, since `0 β€ x`, we have `p q : β`, and `p / q` is the sum of `p * q`
copies of `(1 / q) ^ 2`, and so `x` lies in the `AddSubmonoid` generated by square elements.
Note: it's possible to rephrase this argument as `x = (p * q) β’ (1 / q) ^ 2`, but this would
be somewhat challenging without increasing import requirements. -/
-- Porting note: rewrote proof to avoid removed constructor rat.mk_pnat
suffices
(Finset.range (x.num.natAbs * x.den)).sum
(Function.const β ((1 : β) / x.den * ((1 : β) / x.den))) =
x
by exact this βΈ sum_mem fun n _ => AddSubmonoid.subset_closure β¨_, rflβ©
| simp only [Function.const_apply, Finset.sum_const, Finset.card_range, nsmul_eq_mul] | instance : StarOrderedRing β :=
StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by
refine'
β¨fun hx => _, fun hx =>
AddSubmonoid.closure_induction hx (by rintro - β¨s, rflβ©; exact mul_self_nonneg s) le_rfl
fun _ _ => add_nonnegβ©
/- If `x = p / q`, then, since `0 β€ x`, we have `p q : β`, and `p / q` is the sum of `p * q`
copies of `(1 / q) ^ 2`, and so `x` lies in the `AddSubmonoid` generated by square elements.
Note: it's possible to rephrase this argument as `x = (p * q) β’ (1 / q) ^ 2`, but this would
be somewhat challenging without increasing import requirements. -/
-- Porting note: rewrote proof to avoid removed constructor rat.mk_pnat
suffices
(Finset.range (x.num.natAbs * x.den)).sum
(Function.const β ((1 : β) / x.den * ((1 : β) / x.den))) =
x
by exact this βΈ sum_mem fun n _ => AddSubmonoid.subset_closure β¨_, rflβ©
| Mathlib.Data.Rat.Star.30_0.tRpObn3UXJ6ySXp | instance : StarOrderedRing β | Mathlib_Data_Rat_Star |
x : β
hx : 0 β€ x
β’ β(Int.natAbs x.num * x.den) * (1 / βx.den * (1 / βx.den)) = x | /-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Order
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Rat.Order
import Mathlib.GroupTheory.Submonoid.Membership
#align_import data.rat.star from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
/-! # Star order structure on β
Here we put the trivial `star` operation on `β` for convenience and show that it is a
`StarOrderedRing`. In particular, this means that every element of `β` is a sum of squares.
-/
namespace Rat
instance : StarRing β where
star := id
star_involutive _ := rfl
star_mul _ _ := mul_comm _ _
star_add _ _ := rfl
instance : TrivialStar β where star_trivial _ := rfl
instance : StarOrderedRing β :=
StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by
refine'
β¨fun hx => _, fun hx =>
AddSubmonoid.closure_induction hx (by rintro - β¨s, rflβ©; exact mul_self_nonneg s) le_rfl
fun _ _ => add_nonnegβ©
/- If `x = p / q`, then, since `0 β€ x`, we have `p q : β`, and `p / q` is the sum of `p * q`
copies of `(1 / q) ^ 2`, and so `x` lies in the `AddSubmonoid` generated by square elements.
Note: it's possible to rephrase this argument as `x = (p * q) β’ (1 / q) ^ 2`, but this would
be somewhat challenging without increasing import requirements. -/
-- Porting note: rewrote proof to avoid removed constructor rat.mk_pnat
suffices
(Finset.range (x.num.natAbs * x.den)).sum
(Function.const β ((1 : β) / x.den * ((1 : β) / x.den))) =
x
by exact this βΈ sum_mem fun n _ => AddSubmonoid.subset_closure β¨_, rflβ©
simp only [Function.const_apply, Finset.sum_const, Finset.card_range, nsmul_eq_mul]
| rw [β Int.cast_ofNat, Int.ofNat_mul, Int.coe_natAbs,
abs_of_nonneg (num_nonneg_iff_zero_le.mpr hx), Int.cast_mul, Int.cast_ofNat] | instance : StarOrderedRing β :=
StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by
refine'
β¨fun hx => _, fun hx =>
AddSubmonoid.closure_induction hx (by rintro - β¨s, rflβ©; exact mul_self_nonneg s) le_rfl
fun _ _ => add_nonnegβ©
/- If `x = p / q`, then, since `0 β€ x`, we have `p q : β`, and `p / q` is the sum of `p * q`
copies of `(1 / q) ^ 2`, and so `x` lies in the `AddSubmonoid` generated by square elements.
Note: it's possible to rephrase this argument as `x = (p * q) β’ (1 / q) ^ 2`, but this would
be somewhat challenging without increasing import requirements. -/
-- Porting note: rewrote proof to avoid removed constructor rat.mk_pnat
suffices
(Finset.range (x.num.natAbs * x.den)).sum
(Function.const β ((1 : β) / x.den * ((1 : β) / x.den))) =
x
by exact this βΈ sum_mem fun n _ => AddSubmonoid.subset_closure β¨_, rflβ©
simp only [Function.const_apply, Finset.sum_const, Finset.card_range, nsmul_eq_mul]
| Mathlib.Data.Rat.Star.30_0.tRpObn3UXJ6ySXp | instance : StarOrderedRing β | Mathlib_Data_Rat_Star |
x : β
hx : 0 β€ x
β’ βx.num * βx.den * (1 / βx.den * (1 / βx.den)) = x | /-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Order
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Rat.Order
import Mathlib.GroupTheory.Submonoid.Membership
#align_import data.rat.star from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
/-! # Star order structure on β
Here we put the trivial `star` operation on `β` for convenience and show that it is a
`StarOrderedRing`. In particular, this means that every element of `β` is a sum of squares.
-/
namespace Rat
instance : StarRing β where
star := id
star_involutive _ := rfl
star_mul _ _ := mul_comm _ _
star_add _ _ := rfl
instance : TrivialStar β where star_trivial _ := rfl
instance : StarOrderedRing β :=
StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by
refine'
β¨fun hx => _, fun hx =>
AddSubmonoid.closure_induction hx (by rintro - β¨s, rflβ©; exact mul_self_nonneg s) le_rfl
fun _ _ => add_nonnegβ©
/- If `x = p / q`, then, since `0 β€ x`, we have `p q : β`, and `p / q` is the sum of `p * q`
copies of `(1 / q) ^ 2`, and so `x` lies in the `AddSubmonoid` generated by square elements.
Note: it's possible to rephrase this argument as `x = (p * q) β’ (1 / q) ^ 2`, but this would
be somewhat challenging without increasing import requirements. -/
-- Porting note: rewrote proof to avoid removed constructor rat.mk_pnat
suffices
(Finset.range (x.num.natAbs * x.den)).sum
(Function.const β ((1 : β) / x.den * ((1 : β) / x.den))) =
x
by exact this βΈ sum_mem fun n _ => AddSubmonoid.subset_closure β¨_, rflβ©
simp only [Function.const_apply, Finset.sum_const, Finset.card_range, nsmul_eq_mul]
rw [β Int.cast_ofNat, Int.ofNat_mul, Int.coe_natAbs,
abs_of_nonneg (num_nonneg_iff_zero_le.mpr hx), Int.cast_mul, Int.cast_ofNat]
| rw [β mul_assoc, mul_assoc (x.num : β), mul_one_div_cancel (Nat.cast_ne_zero.mpr x.pos.ne'),
mul_one, mul_one_div, Rat.num_div_den] | instance : StarOrderedRing β :=
StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by
refine'
β¨fun hx => _, fun hx =>
AddSubmonoid.closure_induction hx (by rintro - β¨s, rflβ©; exact mul_self_nonneg s) le_rfl
fun _ _ => add_nonnegβ©
/- If `x = p / q`, then, since `0 β€ x`, we have `p q : β`, and `p / q` is the sum of `p * q`
copies of `(1 / q) ^ 2`, and so `x` lies in the `AddSubmonoid` generated by square elements.
Note: it's possible to rephrase this argument as `x = (p * q) β’ (1 / q) ^ 2`, but this would
be somewhat challenging without increasing import requirements. -/
-- Porting note: rewrote proof to avoid removed constructor rat.mk_pnat
suffices
(Finset.range (x.num.natAbs * x.den)).sum
(Function.const β ((1 : β) / x.den * ((1 : β) / x.den))) =
x
by exact this βΈ sum_mem fun n _ => AddSubmonoid.subset_closure β¨_, rflβ©
simp only [Function.const_apply, Finset.sum_const, Finset.card_range, nsmul_eq_mul]
rw [β Int.cast_ofNat, Int.ofNat_mul, Int.coe_natAbs,
abs_of_nonneg (num_nonneg_iff_zero_le.mpr hx), Int.cast_mul, Int.cast_ofNat]
| Mathlib.Data.Rat.Star.30_0.tRpObn3UXJ6ySXp | instance : StarOrderedRing β | Mathlib_Data_Rat_Star |
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
β’ fourierCoeff (Periodic.lift (_ : Periodic (β(β' (n : β€), ContinuousMap.comp f (ContinuousMap.addRight (n β’ 1)))) 1))
m =
π βf βm | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
| let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β© | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
β’ fourierCoeff (Periodic.lift (_ : Periodic (β(β' (n : β€), ContinuousMap.comp f (ContinuousMap.addRight (n β’ 1)))) 1))
m =
π βf βm | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
| have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
β’ β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
| have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x)) | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
this : β (x : β), βe xβ = 1
β’ β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
| intro K g | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
this : β (x : β), βe xβ = 1
K : Compacts β
g : C(β, β)
β’ βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
| simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
β’ fourierCoeff (Periodic.lift (_ : Periodic (β(β' (n : β€), ContinuousMap.comp f (ContinuousMap.addRight (n β’ 1)))) 1))
m =
π βf βm | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
| have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
β’ β (n : β€), ContinuousMap.comp e (ContinuousMap.addRight βn) = e | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
| intro n | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
n : β€
β’ ContinuousMap.comp e (ContinuousMap.addRight βn) = e | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; | ext1 x | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; | Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
case h
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
n : β€
x : β
β’ (ContinuousMap.comp e (ContinuousMap.addRight βn)) x = e x | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
| have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m)) | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
case h
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
n : β€
x : β
this : Periodic (βe) 1
β’ (ContinuousMap.comp e (ContinuousMap.addRight βn)) x = e x | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
| simpa only [mul_one] using this.int_mul n x | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd : β (n : β€), ContinuousMap.comp e (ContinuousMap.addRight βn) = e
β’ fourierCoeff (Periodic.lift (_ : Periodic (β(β' (n : β€), ContinuousMap.comp f (ContinuousMap.addRight (n β’ 1)))) 1))
m =
π βf βm | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
| calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd : β (n : β€), ContinuousMap.comp e (ContinuousMap.addRight βn) = e
β’ fourierCoeff (Periodic.lift (_ : Periodic (β(β' (n : β€), ContinuousMap.comp f (ContinuousMap.addRight (n β’ 1)))) 1))
m =
β« (x : β) in 0 ..1, e x * (β' (n : β€), ContinuousMap.comp f (ContinuousMap.addRight βn)) x | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
| simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul] | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd : β (n : β€), ContinuousMap.comp e (ContinuousMap.addRight βn) = e
β’ β« (x : β) in 0 ..1, e x * (β' (n : β€), ContinuousMap.comp f (ContinuousMap.addRight βn)) x =
β« (x : β) in 0 ..1, β' (n : β€), (e * ContinuousMap.comp f (ContinuousMap.addRight βn)) x | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
| simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd : β (n : β€), ContinuousMap.comp e (ContinuousMap.addRight βn) = e
β’ β« (x : β) in 0 ..1, β' (n : β€), (e * ContinuousMap.comp f (ContinuousMap.addRight βn)) x =
β' (n : β€), β« (x : β) in 0 ..1, (e * ContinuousMap.comp f (ContinuousMap.addRight βn)) x | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
| refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd : β (n : β€), ContinuousMap.comp e (ContinuousMap.addRight βn) = e
β’ Summable fun i =>
βContinuousMap.restrict (β{ carrier := uIcc 0 1, isCompact' := (_ : IsCompact (uIcc 0 1)) })
(e * ContinuousMap.comp f (ContinuousMap.addRight βi))β | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
| convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
case h.e'_5
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd : β (n : β€), ContinuousMap.comp e (ContinuousMap.addRight βn) = e
β’ (fun i =>
βContinuousMap.restrict (β{ carrier := uIcc 0 1, isCompact' := (_ : IsCompact (uIcc 0 1)) })
(e * ContinuousMap.comp f (ContinuousMap.addRight βi))β) =
fun n =>
βContinuousMap.restrict (β{ carrier := uIcc 0 1, isCompact' := (_ : IsCompact (uIcc 0 1)) })
(ContinuousMap.comp f (ContinuousMap.addRight βn))β | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
| exact funext fun n => neK _ _ | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd : β (n : β€), ContinuousMap.comp e (ContinuousMap.addRight βn) = e
β’ β' (n : β€), β« (x : β) in 0 ..1, (e * ContinuousMap.comp f (ContinuousMap.addRight βn)) x =
β' (n : β€), β« (x : β) in 0 ..1, (ContinuousMap.comp (e * f) (ContinuousMap.addRight βn)) x | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
| simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’ | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd :
β (n : β€),
ContinuousMap.comp (ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk))
(ContinuousMap.addRight βn) =
ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
β’ β' (n : β€),
β« (x : β) in 0 ..1,
(ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) *
ContinuousMap.comp f (ContinuousMap.addRight βn))
x =
β' (n : β€),
β« (x : β) in 0 ..1,
(ContinuousMap.comp (ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk))
(ContinuousMap.addRight βn) *
ContinuousMap.comp f (ContinuousMap.addRight βn))
x | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
| simp_rw [eadd] | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd : β (n : β€), ContinuousMap.comp e (ContinuousMap.addRight βn) = e
β’ β' (n : β€), β« (x : β) in 0 ..1, (ContinuousMap.comp (e * f) (ContinuousMap.addRight βn)) x = β« (x : β), e x * f x | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
| suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd : β (n : β€), ContinuousMap.comp e (ContinuousMap.addRight βn) = e
β’ Integrable β(e * f) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
| apply integrable_of_summable_norm_Icc | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
case hf
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd : β (n : β€), ContinuousMap.comp e (ContinuousMap.addRight βn) = e
β’ Summable fun n => βContinuousMap.restrict (Icc 0 1) (ContinuousMap.comp (e * f) (ContinuousMap.addRight βn))β | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
| convert hf β¨Icc 0 1, isCompact_Iccβ© using 1 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
case h.e'_5
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd : β (n : β€), ContinuousMap.comp e (ContinuousMap.addRight βn) = e
β’ (fun n => βContinuousMap.restrict (Icc 0 1) (ContinuousMap.comp (e * f) (ContinuousMap.addRight βn))β) = fun n =>
βContinuousMap.restrict (β{ carrier := Icc 0 1, isCompact' := (_ : IsCompact (Icc 0 1)) })
(ContinuousMap.comp f (ContinuousMap.addRight βn))β | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
| simp_rw [mul_comp] at eadd β’ | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
case h.e'_5
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd :
β (n : β€),
ContinuousMap.comp (ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk))
(ContinuousMap.addRight βn) =
ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
β’ (fun n =>
βContinuousMap.restrict (Icc 0 1)
(ContinuousMap.comp (ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk))
(ContinuousMap.addRight βn) *
ContinuousMap.comp f (ContinuousMap.addRight βn))β) =
fun n =>
βContinuousMap.restrict (β{ carrier := Icc 0 1, isCompact' := (_ : IsCompact (Icc 0 1)) })
(ContinuousMap.comp f (ContinuousMap.addRight βn))β | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
| simp_rw [eadd] | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
case h.e'_5
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd :
β (n : β€),
ContinuousMap.comp (ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk))
(ContinuousMap.addRight βn) =
ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
β’ (fun n =>
βContinuousMap.restrict (Icc 0 1)
(ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) *
ContinuousMap.comp f (ContinuousMap.addRight βn))β) =
fun n =>
βContinuousMap.restrict (β{ carrier := Icc 0 1, isCompact' := (_ : IsCompact (Icc 0 1)) })
(ContinuousMap.comp f (ContinuousMap.addRight βn))β | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
| exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _ | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd : β (n : β€), ContinuousMap.comp e (ContinuousMap.addRight βn) = e
β’ β« (x : β), e x * f x = π βf βm | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
| rw [fourierIntegral_eq_integral_exp_smul] | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd : β (n : β€), ContinuousMap.comp e (ContinuousMap.addRight βn) = e
β’ β« (x : β), e x * f x = β« (v : β), cexp (β(-2 * Ο * v * βm) * I) β’ f v | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
| congr 1 with x : 1 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
case e_f.h
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd : β (n : β€), ContinuousMap.comp e (ContinuousMap.addRight βn) = e
x : β
β’ e x * f x = cexp (β(-2 * Ο * x * βm) * I) β’ f x | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
| rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply] | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
case e_f.h
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd : β (n : β€), ContinuousMap.comp e (ContinuousMap.addRight βn) = e
x : β
β’ cexp (2 * βΟ * I * β(-m) * βx / β1) * f x = cexp (β(-2 * Ο * x * βm) * I) * f x | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
| congr 2 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
case e_f.h.e_a.e_z
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd : β (n : β€), ContinuousMap.comp e (ContinuousMap.addRight βn) = e
x : β
β’ 2 * βΟ * I * β(-m) * βx / β1 = β(-2 * Ο * x * βm) * I | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
| push_cast | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
case e_f.h.e_a.e_z
f : C(β, β)
hf :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
m : β€
e : C(β, β) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)
neK : β (K : Compacts β) (g : C(β, β)), βContinuousMap.restrict (βK) (e * g)β = βContinuousMap.restrict (βK) gβ
eadd : β (n : β€), ContinuousMap.comp e (ContinuousMap.addRight βn) = e
x : β
β’ 2 * βΟ * I * -βm * βx / 1 = -2 * βΟ * βx * βm * I | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
| ring | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
| Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34 | /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
h_norm :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
h_sum : Summable fun n => π βf βn
β’ β' (n : β€), f βn = β' (n : β€), π βf βn | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
| let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β© | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
| Mathlib.Analysis.Fourier.PoissonSummation.108_0.1MbUAOzT9Ye0D34 | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
h_norm :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
h_sum : Summable fun n => π βf βn
F : C(UnitAddCircle, β) :=
ContinuousMap.mk
(Periodic.lift (_ : Periodic (β(β' (n : β€), ContinuousMap.comp f (ContinuousMap.addRight (n β’ 1)))) 1))
β’ β' (n : β€), f βn = β' (n : β€), π βf βn | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
| have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _ | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
| Mathlib.Analysis.Fourier.PoissonSummation.108_0.1MbUAOzT9Ye0D34 | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
h_norm :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
h_sum : Summable fun n => π βf βn
F : C(UnitAddCircle, β) :=
ContinuousMap.mk
(Periodic.lift (_ : Periodic (β(β' (n : β€), ContinuousMap.comp f (ContinuousMap.addRight (n β’ 1)))) 1))
β’ Summable (fourierCoeff βF) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
| convert h_sum | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
| Mathlib.Analysis.Fourier.PoissonSummation.108_0.1MbUAOzT9Ye0D34 | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) | Mathlib_Analysis_Fourier_PoissonSummation |
case h.e'_5.h
f : C(β, β)
h_norm :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
h_sum : Summable fun n => π βf βn
F : C(UnitAddCircle, β) :=
ContinuousMap.mk
(Periodic.lift (_ : Periodic (β(β' (n : β€), ContinuousMap.comp f (ContinuousMap.addRight (n β’ 1)))) 1))
xβ : β€
β’ fourierCoeff (βF) xβ = π βf βxβ | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
| exact Real.fourierCoeff_tsum_comp_add h_norm _ | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
| Mathlib.Analysis.Fourier.PoissonSummation.108_0.1MbUAOzT9Ye0D34 | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) | Mathlib_Analysis_Fourier_PoissonSummation |
f : C(β, β)
h_norm :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
h_sum : Summable fun n => π βf βn
F : C(UnitAddCircle, β) :=
ContinuousMap.mk
(Periodic.lift (_ : Periodic (β(β' (n : β€), ContinuousMap.comp f (ContinuousMap.addRight (n β’ 1)))) 1))
this : Summable (fourierCoeff βF)
β’ β' (n : β€), f βn = β' (n : β€), π βf βn | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
| convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1 | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
| Mathlib.Analysis.Fourier.PoissonSummation.108_0.1MbUAOzT9Ye0D34 | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) | Mathlib_Analysis_Fourier_PoissonSummation |
case h.e'_2
f : C(β, β)
h_norm :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
h_sum : Summable fun n => π βf βn
F : C(UnitAddCircle, β) :=
ContinuousMap.mk
(Periodic.lift (_ : Periodic (β(β' (n : β€), ContinuousMap.comp f (ContinuousMap.addRight (n β’ 1)))) 1))
this : Summable (fourierCoeff βF)
β’ β' (n : β€), f βn = F 0 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· | have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· | Mathlib.Analysis.Fourier.PoissonSummation.108_0.1MbUAOzT9Ye0D34 | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) | Mathlib_Analysis_Fourier_PoissonSummation |
case h.e'_2
f : C(β, β)
h_norm :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
h_sum : Summable fun n => π βf βn
F : C(UnitAddCircle, β) :=
ContinuousMap.mk
(Periodic.lift (_ : Periodic (β(β' (n : β€), ContinuousMap.comp f (ContinuousMap.addRight (n β’ 1)))) 1))
thisβ : Summable (fourierCoeff βF)
this :
β' (b : β€), (ContinuousMap.comp f (ContinuousMap.addRight βb)) 0 =
(β' (b : β€), ContinuousMap.comp f (ContinuousMap.addRight βb)) 0
β’ β' (n : β€), f βn = F 0 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
| simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
| Mathlib.Analysis.Fourier.PoissonSummation.108_0.1MbUAOzT9Ye0D34 | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) | Mathlib_Analysis_Fourier_PoissonSummation |
case h.e'_3
f : C(β, β)
h_norm :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
h_sum : Summable fun n => π βf βn
F : C(UnitAddCircle, β) :=
ContinuousMap.mk
(Periodic.lift (_ : Periodic (β(β' (n : β€), ContinuousMap.comp f (ContinuousMap.addRight (n β’ 1)))) 1))
this : Summable (fourierCoeff βF)
β’ β' (n : β€), π βf βn = β' (b : β€), fourierCoeff (βF) b β’ (fourier b) 0 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· | congr 1 with n : 1 | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· | Mathlib.Analysis.Fourier.PoissonSummation.108_0.1MbUAOzT9Ye0D34 | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) | Mathlib_Analysis_Fourier_PoissonSummation |
case h.e'_3.e_f.h
f : C(β, β)
h_norm :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
h_sum : Summable fun n => π βf βn
F : C(UnitAddCircle, β) :=
ContinuousMap.mk
(Periodic.lift (_ : Periodic (β(β' (n : β€), ContinuousMap.comp f (ContinuousMap.addRight (n β’ 1)))) 1))
this : Summable (fourierCoeff βF)
n : β€
β’ π βf βn = fourierCoeff (βF) n β’ (fourier n) 0 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· congr 1 with n : 1
| rw [β Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one] | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· congr 1 with n : 1
| Mathlib.Analysis.Fourier.PoissonSummation.108_0.1MbUAOzT9Ye0D34 | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) | Mathlib_Analysis_Fourier_PoissonSummation |
case h.e'_3.e_f.h
f : C(β, β)
h_norm :
β (K : Compacts β), Summable fun n => βContinuousMap.restrict (βK) (ContinuousMap.comp f (ContinuousMap.addRight βn))β
h_sum : Summable fun n => π βf βn
F : C(UnitAddCircle, β) :=
ContinuousMap.mk
(Periodic.lift (_ : Periodic (β(β' (n : β€), ContinuousMap.comp f (ContinuousMap.addRight (n β’ 1)))) 1))
this : Summable (fourierCoeff βF)
n : β€
β’ fourierCoeff (Periodic.lift (_ : Periodic (β(β' (n : β€), ContinuousMap.comp f (ContinuousMap.addRight (n β’ 1)))) 1))
n =
fourierCoeff (βF) n | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· congr 1 with n : 1
rw [β Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]
| rfl | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· congr 1 with n : 1
rw [β Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]
| Mathlib.Analysis.Fourier.PoissonSummation.108_0.1MbUAOzT9Ye0D34 | /-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
instβ : NormedAddCommGroup E
f : C(β, E)
b : β
hb : 0 < b
hf : βf =O[atTop] fun x => |x| ^ (-b)
R S : β
β’ (fun x => βContinuousMap.restrict (Icc (x + R) (x + S)) fβ) =O[atTop] fun x => |x| ^ (-b) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· congr 1 with n : 1
rw [β Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]
rfl
#align real.tsum_eq_tsum_fourier_integral Real.tsum_eq_tsum_fourierIntegral
section RpowDecay
variable {E : Type*} [NormedAddCommGroup E]
/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
| have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· positivity
Β· linarith
have hy' : 0 < y := hxR.trans_le hy
have : y ^ (-b) β€ (x + R) ^ (-b) := by
rw [rpow_neg, rpow_neg, inv_le_inv]
Β· gcongr
all_goals positivity
refine' this.trans _
rw [β mul_rpow, rpow_neg, rpow_neg]
Β· gcongr
linarith
all_goals positivity | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
| Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
instβ : NormedAddCommGroup E
f : C(β, E)
b : β
hb : 0 < b
hf : βf =O[atTop] fun x => |x| ^ (-b)
R S : β
β’ β (x : β), max 0 (-2 * R) < x β β (y : β), x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· congr 1 with n : 1
rw [β Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]
rfl
#align real.tsum_eq_tsum_fourier_integral Real.tsum_eq_tsum_fourierIntegral
section RpowDecay
variable {E : Type*} [NormedAddCommGroup E]
/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
| intro x hx y hy | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
| Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
instβ : NormedAddCommGroup E
f : C(β, E)
b : β
hb : 0 < b
hf : βf =O[atTop] fun x => |x| ^ (-b)
R S x : β
hx : max 0 (-2 * R) < x
y : β
hy : x + R β€ y
β’ y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· congr 1 with n : 1
rw [β Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]
rfl
#align real.tsum_eq_tsum_fourier_integral Real.tsum_eq_tsum_fourierIntegral
section RpowDecay
variable {E : Type*} [NormedAddCommGroup E]
/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
| rw [max_lt_iff] at hx | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
| Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
instβ : NormedAddCommGroup E
f : C(β, E)
b : β
hb : 0 < b
hf : βf =O[atTop] fun x => |x| ^ (-b)
R S x : β
hx : 0 < x β§ -2 * R < x
y : β
hy : x + R β€ y
β’ y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· congr 1 with n : 1
rw [β Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]
rfl
#align real.tsum_eq_tsum_fourier_integral Real.tsum_eq_tsum_fourierIntegral
section RpowDecay
variable {E : Type*} [NormedAddCommGroup E]
/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
| obtain β¨hx1, hx2β© := hx | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
| Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case intro
E : Type u_1
instβ : NormedAddCommGroup E
f : C(β, E)
b : β
hb : 0 < b
hf : βf =O[atTop] fun x => |x| ^ (-b)
R S x y : β
hy : x + R β€ y
hx1 : 0 < x
hx2 : -2 * R < x
β’ y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· congr 1 with n : 1
rw [β Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]
rfl
#align real.tsum_eq_tsum_fourier_integral Real.tsum_eq_tsum_fourierIntegral
section RpowDecay
variable {E : Type*} [NormedAddCommGroup E]
/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
| have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· positivity
Β· linarith | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
| Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
instβ : NormedAddCommGroup E
f : C(β, E)
b : β
hb : 0 < b
hf : βf =O[atTop] fun x => |x| ^ (-b)
R S x y : β
hy : x + R β€ y
hx1 : 0 < x
hx2 : -2 * R < x
β’ 0 < x + R | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· congr 1 with n : 1
rw [β Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]
rfl
#align real.tsum_eq_tsum_fourier_integral Real.tsum_eq_tsum_fourierIntegral
section RpowDecay
variable {E : Type*} [NormedAddCommGroup E]
/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
| rcases le_or_lt 0 R with (h | _) | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
| Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case inl
E : Type u_1
instβ : NormedAddCommGroup E
f : C(β, E)
b : β
hb : 0 < b
hf : βf =O[atTop] fun x => |x| ^ (-b)
R S x y : β
hy : x + R β€ y
hx1 : 0 < x
hx2 : -2 * R < x
h : 0 β€ R
β’ 0 < x + R | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· congr 1 with n : 1
rw [β Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]
rfl
#align real.tsum_eq_tsum_fourier_integral Real.tsum_eq_tsum_fourierIntegral
section RpowDecay
variable {E : Type*} [NormedAddCommGroup E]
/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· | positivity | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case inr
E : Type u_1
instβ : NormedAddCommGroup E
f : C(β, E)
b : β
hb : 0 < b
hf : βf =O[atTop] fun x => |x| ^ (-b)
R S x y : β
hy : x + R β€ y
hx1 : 0 < x
hx2 : -2 * R < x
hβ : R < 0
β’ 0 < x + R | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· congr 1 with n : 1
rw [β Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]
rfl
#align real.tsum_eq_tsum_fourier_integral Real.tsum_eq_tsum_fourierIntegral
section RpowDecay
variable {E : Type*} [NormedAddCommGroup E]
/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· positivity
Β· | linarith | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· positivity
Β· | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case intro
E : Type u_1
instβ : NormedAddCommGroup E
f : C(β, E)
b : β
hb : 0 < b
hf : βf =O[atTop] fun x => |x| ^ (-b)
R S x y : β
hy : x + R β€ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
β’ y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· congr 1 with n : 1
rw [β Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]
rfl
#align real.tsum_eq_tsum_fourier_integral Real.tsum_eq_tsum_fourierIntegral
section RpowDecay
variable {E : Type*} [NormedAddCommGroup E]
/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· positivity
Β· linarith
| have hy' : 0 < y := hxR.trans_le hy | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· positivity
Β· linarith
| Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case intro
E : Type u_1
instβ : NormedAddCommGroup E
f : C(β, E)
b : β
hb : 0 < b
hf : βf =O[atTop] fun x => |x| ^ (-b)
R S x y : β
hy : x + R β€ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
β’ y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· congr 1 with n : 1
rw [β Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]
rfl
#align real.tsum_eq_tsum_fourier_integral Real.tsum_eq_tsum_fourierIntegral
section RpowDecay
variable {E : Type*} [NormedAddCommGroup E]
/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· positivity
Β· linarith
have hy' : 0 < y := hxR.trans_le hy
| have : y ^ (-b) β€ (x + R) ^ (-b) := by
rw [rpow_neg, rpow_neg, inv_le_inv]
Β· gcongr
all_goals positivity | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· positivity
Β· linarith
have hy' : 0 < y := hxR.trans_le hy
| Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
instβ : NormedAddCommGroup E
f : C(β, E)
b : β
hb : 0 < b
hf : βf =O[atTop] fun x => |x| ^ (-b)
R S x y : β
hy : x + R β€ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
β’ y ^ (-b) β€ (x + R) ^ (-b) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· congr 1 with n : 1
rw [β Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]
rfl
#align real.tsum_eq_tsum_fourier_integral Real.tsum_eq_tsum_fourierIntegral
section RpowDecay
variable {E : Type*} [NormedAddCommGroup E]
/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· positivity
Β· linarith
have hy' : 0 < y := hxR.trans_le hy
have : y ^ (-b) β€ (x + R) ^ (-b) := by
| rw [rpow_neg, rpow_neg, inv_le_inv] | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· positivity
Β· linarith
have hy' : 0 < y := hxR.trans_le hy
have : y ^ (-b) β€ (x + R) ^ (-b) := by
| Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
instβ : NormedAddCommGroup E
f : C(β, E)
b : β
hb : 0 < b
hf : βf =O[atTop] fun x => |x| ^ (-b)
R S x y : β
hy : x + R β€ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
β’ (x + R) ^ b β€ y ^ b | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· congr 1 with n : 1
rw [β Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]
rfl
#align real.tsum_eq_tsum_fourier_integral Real.tsum_eq_tsum_fourierIntegral
section RpowDecay
variable {E : Type*} [NormedAddCommGroup E]
/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· positivity
Β· linarith
have hy' : 0 < y := hxR.trans_le hy
have : y ^ (-b) β€ (x + R) ^ (-b) := by
rw [rpow_neg, rpow_neg, inv_le_inv]
Β· | gcongr | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· positivity
Β· linarith
have hy' : 0 < y := hxR.trans_le hy
have : y ^ (-b) β€ (x + R) ^ (-b) := by
rw [rpow_neg, rpow_neg, inv_le_inv]
Β· | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case ha
E : Type u_1
instβ : NormedAddCommGroup E
f : C(β, E)
b : β
hb : 0 < b
hf : βf =O[atTop] fun x => |x| ^ (-b)
R S x y : β
hy : x + R β€ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
β’ 0 < y ^ b
case hb
E : Type u_1
instβ : NormedAddCommGroup E
f : C(β, E)
b : β
hb : 0 < b
hf : βf =O[atTop] fun x => |x| ^ (-b)
R S x y : β
hy : x + R β€ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
β’ 0 < (x + R) ^ b
case hx
E : Type u_1
instβ : NormedAddCommGroup E
f : C(β, E)
b : β
hb : 0 < b
hf : βf =O[atTop] fun x => |x| ^ (-b)
R S x y : β
hy : x + R β€ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
β’ 0 β€ x + R
case hx
E : Type u_1
instβ : NormedAddCommGroup E
f : C(β, E)
b : β
hb : 0 < b
hf : βf =O[atTop] fun x => |x| ^ (-b)
R S x y : β
hy : x + R β€ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
β’ 0 β€ y | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· congr 1 with n : 1
rw [β Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]
rfl
#align real.tsum_eq_tsum_fourier_integral Real.tsum_eq_tsum_fourierIntegral
section RpowDecay
variable {E : Type*} [NormedAddCommGroup E]
/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· positivity
Β· linarith
have hy' : 0 < y := hxR.trans_le hy
have : y ^ (-b) β€ (x + R) ^ (-b) := by
rw [rpow_neg, rpow_neg, inv_le_inv]
Β· gcongr
| all_goals positivity | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· positivity
Β· linarith
have hy' : 0 < y := hxR.trans_le hy
have : y ^ (-b) β€ (x + R) ^ (-b) := by
rw [rpow_neg, rpow_neg, inv_le_inv]
Β· gcongr
| Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case ha
E : Type u_1
instβ : NormedAddCommGroup E
f : C(β, E)
b : β
hb : 0 < b
hf : βf =O[atTop] fun x => |x| ^ (-b)
R S x y : β
hy : x + R β€ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
β’ 0 < y ^ b | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· congr 1 with n : 1
rw [β Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]
rfl
#align real.tsum_eq_tsum_fourier_integral Real.tsum_eq_tsum_fourierIntegral
section RpowDecay
variable {E : Type*} [NormedAddCommGroup E]
/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· positivity
Β· linarith
have hy' : 0 < y := hxR.trans_le hy
have : y ^ (-b) β€ (x + R) ^ (-b) := by
rw [rpow_neg, rpow_neg, inv_le_inv]
Β· gcongr
all_goals | positivity | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· positivity
Β· linarith
have hy' : 0 < y := hxR.trans_le hy
have : y ^ (-b) β€ (x + R) ^ (-b) := by
rw [rpow_neg, rpow_neg, inv_le_inv]
Β· gcongr
all_goals | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case hb
E : Type u_1
instβ : NormedAddCommGroup E
f : C(β, E)
b : β
hb : 0 < b
hf : βf =O[atTop] fun x => |x| ^ (-b)
R S x y : β
hy : x + R β€ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
β’ 0 < (x + R) ^ b | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
/-!
# Poisson's summation formula
We prove Poisson's summation formula `β (n : β€), f n = β (n : β€), π f n`, where `π f` is the
Fourier transform of `f`, under the following hypotheses:
* `f` is a continuous function `β β β`.
* The sum `β (n : β€), π f n` is convergent.
* For all compacts `K β β`, the sum `β (n : β€), sup { βf(x + n)β | x β K }` is convergent.
See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
These hypotheses are potentially a little awkward to apply, so we also provide the less general but
easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
`π f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `π f` are Schwartz
functions.
## TODO
At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
and `π f` are Schwartz functions. In fact, `π f` is automatically Schwartz if `f` is; and once
we have this lemma in the library, we should adjust the hypotheses here accordingly.
-/
noncomputable section
open Function hiding comp_apply
open Set hiding restrict_apply
open Complex hiding abs_of_nonneg
open Real
open TopologicalSpace Filter MeasureTheory Asymptotics
open scoped Real BigOperators Filter FourierTransform
attribute [local instance] Real.fact_zero_lt_one
open ContinuousMap
/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`β' n : β€, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
theorem Real.fourierCoeff_tsum_comp_add {f : C(β, β)}
(hf : β K : Compacts β, Summable fun n : β€ => β(f.comp (ContinuousMap.addRight n)).restrict Kβ)
(m : β€) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = π f m := by
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(β, β) := (fourier (-m)).comp β¨((β) : β β UnitAddCircle), continuous_quotient_mk'β©
have neK : β (K : Compacts β) (g : C(β, β)), β(e * g).restrict Kβ = βg.restrict Kβ := by
have : β x : β, βe xβ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m β’ x))
intro K g
simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
have eadd : β (n : β€), e.comp (ContinuousMap.addRight n) = e := by
intro n; ext1 x
have : Periodic e 1 := Periodic.comp (fun x => AddCircle.coe_add_period 1 x) (fourier (-m))
simpa only [mul_one] using this.int_mul n x
-- Now the main argument. First unwind some definitions.
calc
fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
β« x in (0 : β)..1, e x * (β' n : β€, f.comp (ContinuousMap.addRight n)) x := by
simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
-- Transform sum in C(β, β) evaluated at x into pointwise sum of values.
_ = β« x in (0:β)..1, β' n : β€, (e * f.comp (ContinuousMap.addRight n)) x := by
simp_rw [coe_mul, Pi.mul_apply,
β ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
-- Swap sum and integral.
_ = β' n : β€, β« x in (0:β)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
convert hf β¨uIcc 0 1, isCompact_uIccβ© using 1
exact funext fun n => neK _ _
_ = β' n : β€, β« x in (0:β)..1, (e * f).comp (ContinuousMap.addRight n) x := by
simp only [ContinuousMap.comp_apply, mul_comp] at eadd β’
simp_rw [eadd]
-- Rearrange sum of interval integrals into an integral over `β`.
_ = β« x, e x * f x := by
suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
apply integrable_of_summable_norm_Icc
convert hf β¨Icc 0 1, isCompact_Iccβ© using 1
simp_rw [mul_comp] at eadd β’
simp_rw [eadd]
exact funext fun n => neK β¨Icc 0 1, isCompact_Iccβ© _
-- Minor tidying to finish
_ = π f m := by
rw [fourierIntegral_eq_integral_exp_smul]
congr 1 with x : 1
rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
congr 2
push_cast
ring
#align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
/-- **Poisson's summation formula**, most general form. -/
theorem Real.tsum_eq_tsum_fourierIntegral {f : C(β, β)}
(h_norm :
β K : Compacts β, Summable fun n : β€ => β(f.comp <| ContinuousMap.addRight n).restrict Kβ)
(h_sum : Summable fun n : β€ => π f n) : β' n : β€, f n = (β' n : β€, π f n) := by
let F : C(UnitAddCircle, β) :=
β¨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)β©
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
Β· have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
simpa only [coe_mk, β QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add] using this
Β· congr 1 with n : 1
rw [β Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]
rfl
#align real.tsum_eq_tsum_fourier_integral Real.tsum_eq_tsum_fourierIntegral
section RpowDecay
variable {E : Type*} [NormedAddCommGroup E]
/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· positivity
Β· linarith
have hy' : 0 < y := hxR.trans_le hy
have : y ^ (-b) β€ (x + R) ^ (-b) := by
rw [rpow_neg, rpow_neg, inv_le_inv]
Β· gcongr
all_goals | positivity | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) := by
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim :
β x : β, max 0 (-2 * R) < x β β y : β, x + R β€ y β y ^ (-b) β€ (1 / 2) ^ (-b) * x ^ (-b) := by
intro x hx y hy
rw [max_lt_iff] at hx
obtain β¨hx1, hx2β© := hx
have hxR : 0 < x + R := by
rcases le_or_lt 0 R with (h | _)
Β· positivity
Β· linarith
have hy' : 0 < y := hxR.trans_le hy
have : y ^ (-b) β€ (x + R) ^ (-b) := by
rw [rpow_neg, rpow_neg, inv_le_inv]
Β· gcongr
all_goals | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`Ξ» x, βf.restrict (Icc (x + R) (x + S))β` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(β, E)} {b : β} (hb : 0 < b)
(hf : IsBigO atTop f fun x : β => |x| ^ (-b)) (R S : β) :
IsBigO atTop (fun x : β => βf.restrict (Icc (x + R) (x + S))β) fun x : β => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |