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L3 Lab 3

R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F'✝ : Filtration I M h : Stable F inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M hF : Submodule.FG (Filtration.submodule F) F' : Filtration I M hf : F' ≤ F this : IsNoetherian ↥(reesAlgebra I) ↥(Filtration.submodule F) ⊢ 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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' ≤ F) : F'.Stable

Mathlib_RingTheory_Filtration

R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F'✝ : Filtration I M h : Stable F inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M hF : Submodule.FG (Filtration.submodule F) F' : Filtration I M hf : F' ≤ F this : ∀ s ≤ Filtration.submodule F, Submodule.FG s ⊢ 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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' ≤ F) : F'.Stable

Mathlib_RingTheory_Filtration

R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M x : M ⊢ x ∈ ⨅ i, I ^ i • ⊤ ↔ ∃ r, ↑r • 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by

let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M)

theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) : x ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • 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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x

Mathlib_RingTheory_Filtration

R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M x : M N : Submodule R M := ⨅ i, I ^ i • ⊤ ⊢ x ∈ ⨅ i, I ^ i • ⊤ ↔ ∃ r, ↑r • 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M)

have hN : ∀ k, (I.stableFiltration ⊤ ⊓ 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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : 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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x

Mathlib_RingTheory_Filtration

R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M x : M N : Submodule R M := ⨅ i, I ^ i • ⊤ k : ℕ ⊢ I ^ k • ⊤ = Filtration.N (stableFiltration I ⊤) 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ 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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ 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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x

Mathlib_RingTheory_Filtration

R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M x : M N : Submodule R M := ⨅ i, I ^ i • ⊤ hN : ∀ (k : ℕ), Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) k = N ⊢ x ∈ ⨅ i, I ^ i • ⊤ ↔ ∃ r, ↑r • 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ 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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ 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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x

Mathlib_RingTheory_Filtration

case mp R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M x : M N : Submodule R M := ⨅ i, I ^ i • ⊤ hN : ∀ (k : ℕ), Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) k = N ⊢ x ∈ ⨅ i, I ^ i • ⊤ → ∃ r, ↑r • 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor ·

obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor ·

Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213

theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) : x ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x

Mathlib_RingTheory_Filtration

R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M x : M N : Submodule R M := ⨅ i, I ^ i • ⊤ hN : ∀ (k : ℕ), Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) k = N ⊢ N ≤ I • 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by

obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N)

theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) : x ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := 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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x

Mathlib_RingTheory_Filtration

case intro R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M x : M N : Submodule R M := ⨅ i, I ^ i • ⊤ hN : ∀ (k : ℕ), Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) k = N k : ℕ hk : ∀ n ≥ k, I • Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) n = Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) (n + 1) ⊢ N ≤ I • 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x

Mathlib_RingTheory_Filtration

case intro R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M x : M N : Submodule R M := ⨅ i, I ^ i • ⊤ hN : ∀ (k : ℕ), Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) k = N k : ℕ hk : ∀ n ≥ k, I • Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) n = Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) (n + 1) this : I • Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) k = Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) (k + 1) ⊢ N ≤ I • 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x

Mathlib_RingTheory_Filtration

case intro R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M x : M N : Submodule R M := ⨅ i, I ^ i • ⊤ hN : ∀ (k : ℕ), Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) k = N k : ℕ hk : ∀ n ≥ k, I • Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) n = Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) (n + 1) this : I • N = N ⊢ N ≤ I • 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x

Mathlib_RingTheory_Filtration

case mp.intro.intro R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M x : M N : Submodule R M := ⨅ i, I ^ i • ⊤ hN : ∀ (k : ℕ), Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) k = N r : R hr₁ : r ∈ I hr₂ : ∀ n ∈ N, r • n = n ⊢ x ∈ ⨅ i, I ^ i • ⊤ → ∃ r, ↑r • 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x

Mathlib_RingTheory_Filtration

case mp.intro.intro R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M x : M N : Submodule R M := ⨅ i, I ^ i • ⊤ hN : ∀ (k : ℕ), Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) k = N r : R hr₁ : r ∈ I hr₂ : ∀ n ∈ N, r • n = n H : x ∈ ⨅ i, I ^ i • ⊤ ⊢ ∃ r, ↑r • 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H

exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩

theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) : x ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x

Mathlib_RingTheory_Filtration

case mpr R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M x : M N : Submodule R M := ⨅ i, I ^ i • ⊤ hN : ∀ (k : ℕ), Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) k = N ⊢ (∃ r, ↑r • x = x) → x ∈ ⨅ i, I ^ i • ⊤

/- 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ ·

rintro ⟨r, eq⟩

theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) : x ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ 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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x

Mathlib_RingTheory_Filtration

case mpr.intro R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M x : M N : Submodule R M := ⨅ i, I ^ i • ⊤ hN : ∀ (k : ℕ), Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) k = N r : ↥I eq : ↑r • x = x ⊢ x ∈ ⨅ i, I ^ i • ⊤

/- 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩

rw [Submodule.mem_iInf]

theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) : x ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, 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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x

Mathlib_RingTheory_Filtration

case mpr.intro R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M x : M N : Submodule R M := ⨅ i, I ^ i • ⊤ hN : ∀ (k : ℕ), Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) k = N r : ↥I eq : ↑r • x = x ⊢ ∀ (i : ℕ), x ∈ I ^ i • ⊤

/- 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf]

intro i

theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) : x ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ 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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x

Mathlib_RingTheory_Filtration

case mpr.intro R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M x : M N : Submodule R M := ⨅ i, I ^ i • ⊤ hN : ∀ (k : ℕ), Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) k = N r : ↥I eq : ↑r • x = x i : ℕ ⊢ x ∈ I ^ i • ⊤

/- 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ 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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ 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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x

Mathlib_RingTheory_Filtration

case mpr.intro.zero R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M x : M N : Submodule R M := ⨅ i, I ^ i • ⊤ hN : ∀ (k : ℕ), Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) k = N r : ↥I eq : ↑r • x = x ⊢ x ∈ I ^ Nat.zero • ⊤

/- 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi ·

simp

theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) : x ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi ·

Mathlib.RingTheory.Filtration.434_0.wQ6WBws0g3n9213

theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) : x ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x

Mathlib_RingTheory_Filtration

case mpr.intro.succ R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M x : M N : Submodule R M := ⨅ i, I ^ i • ⊤ hN : ∀ (k : ℕ), Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) k = N r : ↥I eq : ↑r • x = x i : ℕ hi : x ∈ I ^ i • ⊤ ⊢ x ∈ I ^ Nat.succ i • ⊤

/- 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp ·

rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← eq]

theorem Ideal.mem_iInf_smul_pow_eq_bot_iff [IsNoetherianRing R] [Module.Finite R M] (x : M) : x ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · 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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x

Mathlib_RingTheory_Filtration

case mpr.intro.succ R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : Module.Finite R M x : M N : Submodule R M := ⨅ i, I ^ i • ⊤ hN : ∀ (k : ℕ), Filtration.N (stableFiltration I ⊤ ⊓ trivialFiltration I N) k = N r : ↥I eq : ↑r • x = x i : ℕ hi : x ∈ I ^ i • ⊤ ⊢ ↑r • x ∈ I • I ^ i • ⊤

/- 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp · rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← 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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp · rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← 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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x

Mathlib_RingTheory_Filtration

R M : Type u inst✝⁵ : CommRing R inst✝⁴ : AddCommGroup M inst✝³ : Module R M I : Ideal R F F' : Filtration I M inst✝² : IsNoetherianRing R inst✝¹ : LocalRing R inst✝ : Module.Finite R M h : I ≠ ⊤ ⊢ ⨅ i, I ^ i • ⊤ = ⊥

/- 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp · rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := by

rw [eq_bot_iff]

theorem Ideal.iInf_pow_smul_eq_bot_of_localRing [IsNoetherianRing R] [LocalRing R] [Module.Finite R M] (h : I ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥

Mathlib_RingTheory_Filtration

R M : Type u inst✝⁵ : CommRing R inst✝⁴ : AddCommGroup M inst✝³ : Module R M I : Ideal R F F' : Filtration I M inst✝² : IsNoetherianRing R inst✝¹ : LocalRing R inst✝ : Module.Finite R M h : I ≠ ⊤ ⊢ ⨅ i, I ^ i • ⊤ ≤ ⊥

/- 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp · rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥

Mathlib_RingTheory_Filtration

R M : Type u inst✝⁵ : CommRing R inst✝⁴ : AddCommGroup M inst✝³ : Module R M I : Ideal R F F' : Filtration I M inst✝² : IsNoetherianRing R inst✝¹ : LocalRing R inst✝ : Module.Finite R M h : I ≠ ⊤ x : M hx : x ∈ ⨅ i, I ^ i • ⊤ ⊢ 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp · rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := by rw [eq_bot_iff] intro x hx

obtain ⟨r, hr⟩ := (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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥

Mathlib_RingTheory_Filtration

case intro R M : Type u inst✝⁵ : CommRing R inst✝⁴ : AddCommGroup M inst✝³ : Module R M I : Ideal R F F' : Filtration I M inst✝² : IsNoetherianRing R inst✝¹ : LocalRing R inst✝ : Module.Finite R M h : I ≠ ⊤ x : M hx : x ∈ ⨅ i, I ^ i • ⊤ r : ↥I hr : ↑r • x = 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp · rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := by rw [eq_bot_iff] intro x hx obtain ⟨r, hr⟩ := (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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := by rw [eq_bot_iff] intro x hx obtain ⟨r, hr⟩ := (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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥

Mathlib_RingTheory_Filtration

case intro R M : Type u inst✝⁵ : CommRing R inst✝⁴ : AddCommGroup M inst✝³ : Module R M I : Ideal R F F' : Filtration I M inst✝² : IsNoetherianRing R inst✝¹ : LocalRing R inst✝ : Module.Finite R M h : I ≠ ⊤ x : M hx : x ∈ ⨅ i, I ^ i • ⊤ r : ↥I hr : ↑r • x = x this : IsUnit (1 - ↑r) ⊢ 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp · rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := by rw [eq_bot_iff] intro x hx obtain ⟨r, hr⟩ := (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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := by rw [eq_bot_iff] intro x hx obtain ⟨r, hr⟩ := (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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥

Mathlib_RingTheory_Filtration

case intro R M : Type u inst✝⁵ : CommRing R inst✝⁴ : AddCommGroup M inst✝³ : Module R M I : Ideal R F F' : Filtration I M inst✝² : IsNoetherianRing R inst✝¹ : LocalRing R inst✝ : Module.Finite R M h : I ≠ ⊤ x : M hx : x ∈ ⨅ i, I ^ i • ⊤ r : ↥I hr : ↑r • x = x this : IsUnit (1 - ↑r) ⊢ (1 - ↑r) • x = (1 - ↑r) • 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp · rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := by rw [eq_bot_iff] intro x hx obtain ⟨r, hr⟩ := (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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := by rw [eq_bot_iff] intro x hx obtain ⟨r, hr⟩ := (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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥

Mathlib_RingTheory_Filtration

R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : LocalRing R h : I ≠ ⊤ ⊢ ⨅ i, I ^ i = ⊥

/- 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp · rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := by rw [eq_bot_iff] intro x hx obtain ⟨r, hr⟩ := (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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥

Mathlib_RingTheory_Filtration

case h.e'_2.h.e'_4.h R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : LocalRing R h : I ≠ ⊤ x✝ : ℕ ⊢ I ^ x✝ = I ^ 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp · rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := by rw [eq_bot_iff] intro x hx obtain ⟨r, hr⟩ := (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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥

Mathlib_RingTheory_Filtration

case h.e'_2.h.e'_4.h.h R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : LocalRing R h : I ≠ ⊤ x✝ : ℕ i : R ⊢ i ∈ I ^ x✝ ↔ i ∈ I ^ 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp · rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := by rw [eq_bot_iff] intro x hx obtain ⟨r, hr⟩ := (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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := by convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h ext i

rw [smul_eq_mul, ← 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥

Mathlib_RingTheory_Filtration

R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : IsDomain R h : I ≠ ⊤ ⊢ ⨅ i, I ^ i = ⊥

/- 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp · rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := by rw [eq_bot_iff] intro x hx obtain ⟨r, hr⟩ := (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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := by convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h ext i rw [smul_eq_mul, ← 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥

Mathlib_RingTheory_Filtration

R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : IsDomain R h : I ≠ ⊤ ⊢ ⨅ i, I ^ i ≤ ⊥

/- 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp · rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := by rw [eq_bot_iff] intro x hx obtain ⟨r, hr⟩ := (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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := by convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h ext i rw [smul_eq_mul, ← 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥

Mathlib_RingTheory_Filtration

R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : IsDomain R h : I ≠ ⊤ x : R hx : x ∈ ⨅ i, I ^ i ⊢ 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp · rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := by rw [eq_bot_iff] intro x hx obtain ⟨r, hr⟩ := (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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := by convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h ext i rw [smul_eq_mul, ← 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥

Mathlib_RingTheory_Filtration

R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : IsDomain R h : I ≠ ⊤ x : R hx : x ∈ ⨅ i, I ^ i hx' : x ∉ ⊥ ⊢ 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp · rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := by rw [eq_bot_iff] intro x hx obtain ⟨r, hr⟩ := (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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := by convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h ext i rw [smul_eq_mul, ← 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥

Mathlib_RingTheory_Filtration

R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : IsDomain R h : I ≠ ⊤ x : R hx : x ∈ ⨅ i, I ^ i hx' : x ∉ ⊥ this : x ∈ ⨅ i, I ^ i • ⊤ ↔ ∃ r, ↑r • x = x ⊢ 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp · rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := by rw [eq_bot_iff] intro x hx obtain ⟨r, hr⟩ := (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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := by convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h ext i rw [smul_eq_mul, ← 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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, ← 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥

Mathlib_RingTheory_Filtration

R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : IsDomain R h : I ≠ ⊤ x : R hx : x ∈ ⨅ i, I ^ i hx' : x ∉ ⊥ this : x ∈ ⨅ i, I ^ i ↔ ∃ r, ↑r * x = x ⊢ 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp · rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := by rw [eq_bot_iff] intro x hx obtain ⟨r, hr⟩ := (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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := by convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h ext i rw [smul_eq_mul, ← 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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, ← Ideal.one_eq_top, mul_one] at this

obtain ⟨r, hr⟩ := this.mp hx

/-- **Krull's intersection theorem** for noetherian domains. -/ theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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, ← 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥

Mathlib_RingTheory_Filtration

case intro R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : IsDomain R h : I ≠ ⊤ x : R hx : x ∈ ⨅ i, I ^ i hx' : x ∉ ⊥ this : x ∈ ⨅ i, I ^ i ↔ ∃ r, ↑r * x = x r : ↥I hr : ↑r * x = x ⊢ 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp · rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := by rw [eq_bot_iff] intro x hx obtain ⟨r, hr⟩ := (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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := by convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h ext i rw [smul_eq_mul, ← 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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, ← Ideal.one_eq_top, mul_one] at this obtain ⟨r, hr⟩ := this.mp hx

have := mul_right_cancel₀ 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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, ← Ideal.one_eq_top, mul_one] at this obtain ⟨r, hr⟩ := 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥

Mathlib_RingTheory_Filtration

case intro R M : Type u inst✝⁴ : CommRing R inst✝³ : AddCommGroup M inst✝² : Module R M I : Ideal R F F' : Filtration I M inst✝¹ : IsNoetherianRing R inst✝ : IsDomain R h : I ≠ ⊤ x : R hx : x ∈ ⨅ i, I ^ i hx' : x ∉ ⊥ this✝ : x ∈ ⨅ i, I ^ i ↔ ∃ r, ↑r * x = x r : ↥I hr : ↑r * x = x this : ↑r = 1 ⊢ 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 • N ≤ I (i + 1)`. Note that we do not require the filtration to start from `⊤`. - `Ideal.Filtration.Stable`: An `I`-filtration is stable if `I • (N i) = N (i + 1)` for large enough `i`. - `Ideal.Filtration.submodule`: The associated module `⨁ Nᵢ` 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' ≤ F`, then `F.Stable → F'.Stable`. - `Ideal.exists_pow_inf_eq_pow_smul`: **Artin-Rees lemma**. given `N ≤ M`, there exists a `k` such that `IⁿM ⊓ N = Iⁿ⁻ᵏ(IᵏM ⊓ N)` for all `n ≥ k`. - `Ideal.iInf_pow_eq_bot_of_localRing`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) for noetherian local rings. - `Ideal.iInf_pow_eq_bot_of_isDomain`: **Krull's intersection theorem** (`⨅ i, I ^ i = ⊥`) 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 • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. -/ @[ext] structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where N : ℕ → Submodule R M mono : ∀ i, N (i + 1) ≤ N i smul_le : ∀ i, I • N i ≤ N (i + 1) #align ideal.filtration Ideal.Filtration variable (F F' : I.Filtration M) {I} namespace Ideal.Filtration theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by induction' i with _ ih · simp · rw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, ← 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 : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • 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) := ⟨fun F F' => ⟨F.N ⊔ 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)⟩⟩ /-- The `sSup` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : SupSet (I.Filtration M) := ⟨fun S => { N := sSup (Ideal.Filtration.N '' S) mono := fun i => by apply sSup_le_sSup_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply] apply iSup_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ /-- The `inf` of two `I.Filtration`s is an `I.Filtration`. -/ instance : Inf (I.Filtration M) := ⟨fun F F' => ⟨F.N ⊓ 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)⟩⟩ /-- The `sInf` of a family of `I.Filtration`s is an `I.Filtration`. -/ instance : InfSet (I.Filtration M) := ⟨fun S => { N := sInf (Ideal.Filtration.N '' S) mono := fun i => by apply sInf_le_sInf_of_forall_exists_le _ rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩ exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩ smul_le := fun i => by rw [sInf_eq_iInf', iInf_apply, iInf_apply] refine' Submodule.smul_iInf_le.trans _ apply iInf_mono _ rintro ⟨_, F, hF, rfl⟩ exact F.smul_le i }⟩ instance : Top (I.Filtration M) := ⟨I.trivialFiltration ⊤⟩ instance : Bot (I.Filtration M) := ⟨I.trivialFiltration ⊥⟩ @[simp] theorem sup_N : (F ⊔ F').N = F.N ⊔ 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 ⊓ F').N = F.N ⊓ 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 : (⊤ : I.Filtration M).N = ⊤ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.top_N Ideal.Filtration.top_N @[simp] theorem bot_N : (⊥ : I.Filtration M).N = ⊥ := rfl set_option linter.uppercaseLean3 false in #align ideal.filtration.bot_N Ideal.Filtration.bot_N @[simp] theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ 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 {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ 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) := ⟨⊥⟩ /-- An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. -/ def Stable : Prop := ∃ n₀, ∀ n ≥ n₀, I • 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 • 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 : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by obtain ⟨n₀, hn⟩ := h use n₀ intro k induction' k with _ ih · simp · rw [Nat.succ_eq_add_one, ← add_assoc, ← 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 : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq use n₀ intro n hn convert hn₀ (n - n₀) 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 ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by refine' ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => _⟩⟩ rw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, ← 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 ≤ F'.N 0) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by obtain ⟨n₀, hF⟩ := h use n₀ intro n induction' n with n hn · refine' (F.antitone _).trans e; simp · rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← 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) : ∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e) obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm) use max n₁ n₂ intro n refine' ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩ <;> 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 | ∀ i, f i ∈ 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 ⟨j, k⟩ 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 ∈ F.submodule ↔ ∀ i, f i ∈ F.N i := Iff.rfl #align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ 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 (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by apply le_antisymm · rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff] rintro i _ ⟨m, hm, rfl⟩ j rw [single_apply] split_ifs with h · rwa [← h] · exact (F.N j).zero_mem · intro f hf rw [← 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) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by rw [← 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 (n₀ : ℕ) : F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔ ∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff] swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _) constructor · intro H n hn refine' (F.smul_le n).antisymm _ intro x hx obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩) replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl dsimp only at hl erw [Finsupp.single_eq_same] at hl rw [← hl, Finsupp.total_apply, Finsupp.sum_apply] apply Submodule.sum_mem _ _ rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ - dsimp only [Subtype.coe_mk] rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)] have e : n' ≤ 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, ← add_tsub_assoc_of_le e, add_comm] at this exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm) · let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) intro hF i have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by -- Porting note: Original proof was -- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span` intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi induction' i with j hj · exact this _ (zero_le _) by_cases hj' : j.succ ≤ n₀ · exact this _ hj' simp only [not_le, Nat.lt_succ_iff] at hj' rw [Nat.succ_eq_add_one, ← hF _ hj'] rintro _ ⟨m, hm, rfl⟩ refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _) · rw [add_comm, ← monomial_smul_single] exact F'.smul_mem ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm') · 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' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by classical delta Ideal.Filtration.Stable simp_rw [← F.submodule_eq_span_le_iff_stable_ge] constructor · rintro H refine H.stablizes_of_iSup_eq ⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_ · intro n m e rw [Submodule.span_le, Set.iUnion₂_subset_iff] intro i hi refine Set.Subset.trans ?_ Submodule.subset_span refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_ exact hi.trans e · dsimp rw [← Submodule.span_iUnion, ← submodule_span_single] congr 1 ext simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop] constructor · rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩ · rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩ · rintro ⟨n, hn⟩ rw [hn] simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype'] apply Submodule.fg_iSup rintro ⟨i, hi⟩ obtain ⟨s, hs⟩ := 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, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl rw [Subtype.coe_mk, ← this] exact ⟨_, rfl⟩ #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' ≤ F) : F'.Stable := by rw [← submodule_fg_iff_stable] at hF ⊢ 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 ⊓ 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' ⊓ 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) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) := ((I.stableFiltration_stable ⊤).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 ∈ (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) ↔ ∃ r : I, (r : R) • x = x := by let N := (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) have hN : ∀ k, (I.stableFiltration ⊤ ⊓ I.trivialFiltration N).N k = N := fun k => inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp) constructor · obtain ⟨r, hr₁, hr₂⟩ := Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N) (by obtain ⟨k, hk⟩ := (I.stableFiltration_stable ⊤).inter_right (I.trivialFiltration N) have := hk k (le_refl _) rw [hN, hN] at this exact le_of_eq this.symm) intro H exact ⟨⟨r, hr₁⟩, hr₂ _ H⟩ · rintro ⟨r, eq⟩ rw [Submodule.mem_iInf] intro i induction' i with i hi · simp · rw [Nat.succ_eq_one_add, pow_add, ← smul_smul, pow_one, ← 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 ≠ ⊤) : (⨅ i : ℕ, I ^ i • ⊤ : Submodule R M) = ⊥ := by rw [eq_bot_iff] intro x hx obtain ⟨r, hr⟩ := (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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := by convert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h ext i rw [smul_eq_mul, ← 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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, ← Ideal.one_eq_top, mul_one] at this obtain ⟨r, hr⟩ := this.mp hx have := mul_right_cancel₀ hx' (hr.trans (one_mul x).symm)

exact I.eq_top_iff_one.not.mp h (this ▸ r.prop)

/-- **Krull's intersection theorem** for noetherian domains. -/ theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥ := 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, ← Ideal.one_eq_top, mul_one] at this obtain ⟨r, hr⟩ := this.mp hx have := mul_right_cancel₀ 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 ≠ ⊤) : ⨅ i : ℕ, I ^ i = ⊥

Mathlib_RingTheory_Filtration

x : ℚ ⊢ 0 ≤ x ↔ x ∈ 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 ℚ Here we put the trivial `star` operation on `ℚ` for convenience and show that it is a `StarOrderedRing`. In particular, this means that every element of `ℚ` is a sum of squares. -/ namespace Rat instance : StarRing ℚ where star := id star_involutive _ := rfl star_mul _ _ := mul_comm _ _ star_add _ _ := rfl instance : TrivialStar ℚ where star_trivial _ := rfl instance : StarOrderedRing ℚ := StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by

refine' ⟨fun hx => _, fun hx => AddSubmonoid.closure_induction hx (by rintro - ⟨s, rfl⟩; exact mul_self_nonneg s) le_rfl fun _ _ => add_nonneg⟩

instance : StarOrderedRing ℚ := StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by

Mathlib.Data.Rat.Star.30_0.tRpObn3UXJ6ySXp

instance : StarOrderedRing ℚ

Mathlib_Data_Rat_Star

x : ℚ hx : x ∈ AddSubmonoid.closure (Set.range fun s => star s * s) ⊢ ∀ x ∈ Set.range fun s => star s * s, 0 ≤ 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 ℚ Here we put the trivial `star` operation on `ℚ` for convenience and show that it is a `StarOrderedRing`. In particular, this means that every element of `ℚ` is a sum of squares. -/ namespace Rat instance : StarRing ℚ where star := id star_involutive _ := rfl star_mul _ _ := mul_comm _ _ star_add _ _ := rfl instance : TrivialStar ℚ where star_trivial _ := rfl instance : StarOrderedRing ℚ := StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by refine' ⟨fun hx => _, fun hx => AddSubmonoid.closure_induction hx (by

rintro - ⟨s, rfl⟩

instance : StarOrderedRing ℚ := StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by refine' ⟨fun hx => _, fun hx => AddSubmonoid.closure_induction hx (by

Mathlib.Data.Rat.Star.30_0.tRpObn3UXJ6ySXp

instance : StarOrderedRing ℚ

Mathlib_Data_Rat_Star

case intro x : ℚ hx : x ∈ AddSubmonoid.closure (Set.range fun s => star s * s) s : ℚ ⊢ 0 ≤ (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 ℚ Here we put the trivial `star` operation on `ℚ` for convenience and show that it is a `StarOrderedRing`. In particular, this means that every element of `ℚ` is a sum of squares. -/ namespace Rat instance : StarRing ℚ where star := id star_involutive _ := rfl star_mul _ _ := mul_comm _ _ star_add _ _ := rfl instance : TrivialStar ℚ where star_trivial _ := rfl instance : StarOrderedRing ℚ := StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by refine' ⟨fun hx => _, fun hx => AddSubmonoid.closure_induction hx (by rintro - ⟨s, rfl⟩;

exact mul_self_nonneg s

instance : StarOrderedRing ℚ := StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by refine' ⟨fun hx => _, fun hx => AddSubmonoid.closure_induction hx (by rintro - ⟨s, rfl⟩;

Mathlib.Data.Rat.Star.30_0.tRpObn3UXJ6ySXp

instance : StarOrderedRing ℚ

Mathlib_Data_Rat_Star

x : ℚ hx : 0 ≤ x ⊢ x ∈ 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 ℚ Here we put the trivial `star` operation on `ℚ` for convenience and show that it is a `StarOrderedRing`. In particular, this means that every element of `ℚ` is a sum of squares. -/ namespace Rat instance : StarRing ℚ where star := id star_involutive _ := rfl star_mul _ _ := mul_comm _ _ star_add _ _ := rfl instance : TrivialStar ℚ where star_trivial _ := rfl instance : StarOrderedRing ℚ := StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by refine' ⟨fun hx => _, fun hx => AddSubmonoid.closure_induction hx (by rintro - ⟨s, rfl⟩; exact mul_self_nonneg s) le_rfl fun _ _ => add_nonneg⟩ /- If `x = p / q`, then, since `0 ≤ x`, we have `p q : ℕ`, 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) • (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 ℕ ((1 : ℚ) / x.den * ((1 : ℚ) / x.den))) = x by exact this ▸ sum_mem fun n _ => AddSubmonoid.subset_closure ⟨_, rfl⟩

instance : StarOrderedRing ℚ := StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by refine' ⟨fun hx => _, fun hx => AddSubmonoid.closure_induction hx (by rintro - ⟨s, rfl⟩; exact mul_self_nonneg s) le_rfl fun _ _ => add_nonneg⟩ /- If `x = p / q`, then, since `0 ≤ x`, we have `p q : ℕ`, 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) • (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 ℚ

Mathlib_Data_Rat_Star

x : ℚ hx : 0 ≤ x this : Finset.sum (Finset.range (Int.natAbs x.num * x.den)) (Function.const ℕ (1 / ↑x.den * (1 / ↑x.den))) = x ⊢ x ∈ 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 ℚ Here we put the trivial `star` operation on `ℚ` for convenience and show that it is a `StarOrderedRing`. In particular, this means that every element of `ℚ` is a sum of squares. -/ namespace Rat instance : StarRing ℚ where star := id star_involutive _ := rfl star_mul _ _ := mul_comm _ _ star_add _ _ := rfl instance : TrivialStar ℚ where star_trivial _ := rfl instance : StarOrderedRing ℚ := StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by refine' ⟨fun hx => _, fun hx => AddSubmonoid.closure_induction hx (by rintro - ⟨s, rfl⟩; exact mul_self_nonneg s) le_rfl fun _ _ => add_nonneg⟩ /- If `x = p / q`, then, since `0 ≤ x`, we have `p q : ℕ`, 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) • (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 ℕ ((1 : ℚ) / x.den * ((1 : ℚ) / x.den))) = x by

exact this ▸ sum_mem fun n _ => AddSubmonoid.subset_closure ⟨_, rfl⟩

instance : StarOrderedRing ℚ := StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by refine' ⟨fun hx => _, fun hx => AddSubmonoid.closure_induction hx (by rintro - ⟨s, rfl⟩; exact mul_self_nonneg s) le_rfl fun _ _ => add_nonneg⟩ /- If `x = p / q`, then, since `0 ≤ x`, we have `p q : ℕ`, 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) • (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 ℕ ((1 : ℚ) / x.den * ((1 : ℚ) / x.den))) = x by

Mathlib.Data.Rat.Star.30_0.tRpObn3UXJ6ySXp

instance : StarOrderedRing ℚ

Mathlib_Data_Rat_Star

x : ℚ hx : 0 ≤ x ⊢ Finset.sum (Finset.range (Int.natAbs x.num * x.den)) (Function.const ℕ (1 / ↑x.den * (1 / ↑x.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 ℚ Here we put the trivial `star` operation on `ℚ` for convenience and show that it is a `StarOrderedRing`. In particular, this means that every element of `ℚ` is a sum of squares. -/ namespace Rat instance : StarRing ℚ where star := id star_involutive _ := rfl star_mul _ _ := mul_comm _ _ star_add _ _ := rfl instance : TrivialStar ℚ where star_trivial _ := rfl instance : StarOrderedRing ℚ := StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by refine' ⟨fun hx => _, fun hx => AddSubmonoid.closure_induction hx (by rintro - ⟨s, rfl⟩; exact mul_self_nonneg s) le_rfl fun _ _ => add_nonneg⟩ /- If `x = p / q`, then, since `0 ≤ x`, we have `p q : ℕ`, 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) • (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 ℕ ((1 : ℚ) / x.den * ((1 : ℚ) / x.den))) = x by exact this ▸ sum_mem fun n _ => AddSubmonoid.subset_closure ⟨_, rfl⟩

simp only [Function.const_apply, Finset.sum_const, Finset.card_range, nsmul_eq_mul]

instance : StarOrderedRing ℚ := StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by refine' ⟨fun hx => _, fun hx => AddSubmonoid.closure_induction hx (by rintro - ⟨s, rfl⟩; exact mul_self_nonneg s) le_rfl fun _ _ => add_nonneg⟩ /- If `x = p / q`, then, since `0 ≤ x`, we have `p q : ℕ`, 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) • (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 ℕ ((1 : ℚ) / x.den * ((1 : ℚ) / x.den))) = x by exact this ▸ sum_mem fun n _ => AddSubmonoid.subset_closure ⟨_, rfl⟩

Mathlib.Data.Rat.Star.30_0.tRpObn3UXJ6ySXp

instance : StarOrderedRing ℚ

Mathlib_Data_Rat_Star

x : ℚ hx : 0 ≤ x ⊢ ↑(Int.natAbs x.num * x.den) * (1 / ↑x.den * (1 / ↑x.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 ℚ Here we put the trivial `star` operation on `ℚ` for convenience and show that it is a `StarOrderedRing`. In particular, this means that every element of `ℚ` is a sum of squares. -/ namespace Rat instance : StarRing ℚ where star := id star_involutive _ := rfl star_mul _ _ := mul_comm _ _ star_add _ _ := rfl instance : TrivialStar ℚ where star_trivial _ := rfl instance : StarOrderedRing ℚ := StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by refine' ⟨fun hx => _, fun hx => AddSubmonoid.closure_induction hx (by rintro - ⟨s, rfl⟩; exact mul_self_nonneg s) le_rfl fun _ _ => add_nonneg⟩ /- If `x = p / q`, then, since `0 ≤ x`, we have `p q : ℕ`, 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) • (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 ℕ ((1 : ℚ) / x.den * ((1 : ℚ) / x.den))) = x by exact this ▸ sum_mem fun n _ => AddSubmonoid.subset_closure ⟨_, rfl⟩ simp only [Function.const_apply, Finset.sum_const, Finset.card_range, nsmul_eq_mul]

rw [← 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 ℚ := StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by refine' ⟨fun hx => _, fun hx => AddSubmonoid.closure_induction hx (by rintro - ⟨s, rfl⟩; exact mul_self_nonneg s) le_rfl fun _ _ => add_nonneg⟩ /- If `x = p / q`, then, since `0 ≤ x`, we have `p q : ℕ`, 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) • (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 ℕ ((1 : ℚ) / x.den * ((1 : ℚ) / x.den))) = x by exact this ▸ sum_mem fun n _ => AddSubmonoid.subset_closure ⟨_, rfl⟩ simp only [Function.const_apply, Finset.sum_const, Finset.card_range, nsmul_eq_mul]

Mathlib.Data.Rat.Star.30_0.tRpObn3UXJ6ySXp

instance : StarOrderedRing ℚ

Mathlib_Data_Rat_Star

x : ℚ hx : 0 ≤ x ⊢ ↑x.num * ↑x.den * (1 / ↑x.den * (1 / ↑x.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 ℚ Here we put the trivial `star` operation on `ℚ` for convenience and show that it is a `StarOrderedRing`. In particular, this means that every element of `ℚ` is a sum of squares. -/ namespace Rat instance : StarRing ℚ where star := id star_involutive _ := rfl star_mul _ _ := mul_comm _ _ star_add _ _ := rfl instance : TrivialStar ℚ where star_trivial _ := rfl instance : StarOrderedRing ℚ := StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by refine' ⟨fun hx => _, fun hx => AddSubmonoid.closure_induction hx (by rintro - ⟨s, rfl⟩; exact mul_self_nonneg s) le_rfl fun _ _ => add_nonneg⟩ /- If `x = p / q`, then, since `0 ≤ x`, we have `p q : ℕ`, 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) • (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 ℕ ((1 : ℚ) / x.den * ((1 : ℚ) / x.den))) = x by exact this ▸ sum_mem fun n _ => AddSubmonoid.subset_closure ⟨_, rfl⟩ simp only [Function.const_apply, Finset.sum_const, Finset.card_range, nsmul_eq_mul] rw [← 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 [← mul_assoc, mul_assoc (x.num : ℚ), mul_one_div_cancel (Nat.cast_ne_zero.mpr x.pos.ne'), mul_one, mul_one_div, Rat.num_div_den]

instance : StarOrderedRing ℚ := StarOrderedRing.ofNonnegIff (fun {_ _} => add_le_add_left) fun x => by refine' ⟨fun hx => _, fun hx => AddSubmonoid.closure_induction hx (by rintro - ⟨s, rfl⟩; exact mul_self_nonneg s) le_rfl fun _ _ => add_nonneg⟩ /- If `x = p / q`, then, since `0 ≤ x`, we have `p q : ℕ`, 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) • (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 ℕ ((1 : ℚ) / x.den * ((1 : ℚ) / x.den))) = x by exact this ▸ sum_mem fun n _ => AddSubmonoid.subset_closure ⟨_, rfl⟩ simp only [Function.const_apply, Finset.sum_const, Finset.card_range, nsmul_eq_mul] rw [← 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 ℚ

Mathlib_Data_Rat_Star

f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ ⊢ fourierCoeff (Periodic.lift (_ : Periodic (⇑(∑' (n : ℤ), ContinuousMap.comp f (ContinuousMap.addRight (n • 1)))) 1)) m = 𝓕 ⇑f ↑m

/- 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) ⊢ fourierCoeff (Periodic.lift (_ : Periodic (⇑(∑' (n : ℤ), ContinuousMap.comp f (ContinuousMap.addRight (n • 1)))) 1)) m = 𝓕 ⇑f ↑m

/- 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩

have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩

Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) ⊢ ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖

/- 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by

have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x))

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by

Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) this : ∀ (x : ℝ), ‖e x‖ = 1 ⊢ ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖

/- 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x))

intro K g

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x))

Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) this : ∀ (x : ℝ), ‖e x‖ = 1 K : Compacts ℝ g : C(ℝ, ℂ) ⊢ ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖

/- 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ ⊢ fourierCoeff (Periodic.lift (_ : Periodic (⇑(∑' (n : ℤ), ContinuousMap.comp f (ContinuousMap.addRight (n • 1)))) 1)) m = 𝓕 ⇑f ↑m

/- 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]

have eadd : ∀ (n : ℤ), 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ ⊢ ∀ (n : ℤ), ContinuousMap.comp e (ContinuousMap.addRight ↑n) = 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), e.comp (ContinuousMap.addRight n) = e := by

intro n

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ n : ℤ ⊢ ContinuousMap.comp e (ContinuousMap.addRight ↑n) = 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

case h f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ n : ℤ x : ℝ ⊢ (ContinuousMap.comp e (ContinuousMap.addRight ↑n)) 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

case h f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ n : ℤ x : ℝ this : Periodic (⇑e) 1 ⊢ (ContinuousMap.comp e (ContinuousMap.addRight ↑n)) 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp e (ContinuousMap.addRight ↑n) = e ⊢ fourierCoeff (Periodic.lift (_ : Periodic (⇑(∑' (n : ℤ), ContinuousMap.comp f (ContinuousMap.addRight (n • 1)))) 1)) m = 𝓕 ⇑f ↑m

/- 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp e (ContinuousMap.addRight ↑n) = e ⊢ fourierCoeff (Periodic.lift (_ : Periodic (⇑(∑' (n : ℤ), ContinuousMap.comp f (ContinuousMap.addRight (n • 1)))) 1)) m = ∫ (x : ℝ) in 0 ..1, e x * (∑' (n : ℤ), ContinuousMap.comp f (ContinuousMap.addRight ↑n)) 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp e (ContinuousMap.addRight ↑n) = e ⊢ ∫ (x : ℝ) in 0 ..1, e x * (∑' (n : ℤ), ContinuousMap.comp f (ContinuousMap.addRight ↑n)) x = ∫ (x : ℝ) in 0 ..1, ∑' (n : ℤ), (e * ContinuousMap.comp f (ContinuousMap.addRight ↑n)) 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by

simp_rw [coe_mul, Pi.mul_apply, ← 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp e (ContinuousMap.addRight ↑n) = e ⊢ ∫ (x : ℝ) in 0 ..1, ∑' (n : ℤ), (e * ContinuousMap.comp f (ContinuousMap.addRight ↑n)) x = ∑' (n : ℤ), ∫ (x : ℝ) in 0 ..1, (e * ContinuousMap.comp f (ContinuousMap.addRight ↑n)) 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp e (ContinuousMap.addRight ↑n) = e ⊢ Summable fun i => ‖ContinuousMap.restrict (↑{ carrier := uIcc 0 1, isCompact' := (_ : IsCompact (uIcc 0 1)) }) (e * ContinuousMap.comp f (ContinuousMap.addRight ↑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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm

convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

case h.e'_5 f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp e (ContinuousMap.addRight ↑n) = e ⊢ (fun i => ‖ContinuousMap.restrict (↑{ carrier := uIcc 0 1, isCompact' := (_ : IsCompact (uIcc 0 1)) }) (e * ContinuousMap.comp f (ContinuousMap.addRight ↑i))‖) = fun n => ‖ContinuousMap.restrict (↑{ carrier := uIcc 0 1, isCompact' := (_ : IsCompact (uIcc 0 1)) }) (ContinuousMap.comp f (ContinuousMap.addRight ↑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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1

exact funext fun n => neK _ _

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1

Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp e (ContinuousMap.addRight ↑n) = e ⊢ ∑' (n : ℤ), ∫ (x : ℝ) in 0 ..1, (e * ContinuousMap.comp f (ContinuousMap.addRight ↑n)) x = ∑' (n : ℤ), ∫ (x : ℝ) in 0 ..1, (ContinuousMap.comp (e * f) (ContinuousMap.addRight ↑n)) 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by

simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp (ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)) (ContinuousMap.addRight ↑n) = ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) ⊢ ∑' (n : ℤ), ∫ (x : ℝ) in 0 ..1, (ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) * ContinuousMap.comp f (ContinuousMap.addRight ↑n)) x = ∑' (n : ℤ), ∫ (x : ℝ) in 0 ..1, (ContinuousMap.comp (ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)) (ContinuousMap.addRight ↑n) * ContinuousMap.comp f (ContinuousMap.addRight ↑n)) 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢

simp_rw [eadd]

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢

Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp e (ContinuousMap.addRight ↑n) = e ⊢ ∑' (n : ℤ), ∫ (x : ℝ) in 0 ..1, (ContinuousMap.comp (e * f) (ContinuousMap.addRight ↑n)) x = ∫ (x : ℝ), 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp e (ContinuousMap.addRight ↑n) = e ⊢ Integrable ⇑(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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

case hf f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp e (ContinuousMap.addRight ↑n) = e ⊢ Summable fun n => ‖ContinuousMap.restrict (Icc 0 1) (ContinuousMap.comp (e * f) (ContinuousMap.addRight ↑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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

case h.e'_5 f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp e (ContinuousMap.addRight ↑n) = e ⊢ (fun n => ‖ContinuousMap.restrict (Icc 0 1) (ContinuousMap.comp (e * f) (ContinuousMap.addRight ↑n))‖) = fun n => ‖ContinuousMap.restrict (↑{ carrier := Icc 0 1, isCompact' := (_ : IsCompact (Icc 0 1)) }) (ContinuousMap.comp f (ContinuousMap.addRight ↑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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1

simp_rw [mul_comp] at eadd ⊢

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1

Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

case h.e'_5 f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp (ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)) (ContinuousMap.addRight ↑n) = ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) ⊢ (fun n => ‖ContinuousMap.restrict (Icc 0 1) (ContinuousMap.comp (ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)) (ContinuousMap.addRight ↑n) * ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖) = fun n => ‖ContinuousMap.restrict (↑{ carrier := Icc 0 1, isCompact' := (_ : IsCompact (Icc 0 1)) }) (ContinuousMap.comp f (ContinuousMap.addRight ↑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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢

simp_rw [eadd]

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢

Mathlib.Analysis.Fourier.PoissonSummation.56_0.1MbUAOzT9Ye0D34

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

case h.e'_5 f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp (ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk)) (ContinuousMap.addRight ↑n) = ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) ⊢ (fun n => ‖ContinuousMap.restrict (Icc 0 1) (ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) * ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖) = fun n => ‖ContinuousMap.restrict (↑{ carrier := Icc 0 1, isCompact' := (_ : IsCompact (Icc 0 1)) }) (ContinuousMap.comp f (ContinuousMap.addRight ↑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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd]

exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp e (ContinuousMap.addRight ↑n) = e ⊢ ∫ (x : ℝ), e x * f x = 𝓕 ⇑f ↑m

/- 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 f m := by

rw [fourierIntegral_eq_integral_exp_smul]

/-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp e (ContinuousMap.addRight ↑n) = e ⊢ ∫ (x : ℝ), e x * f x = ∫ (v : ℝ), cexp (↑(-2 * π * v * ↑m) * I) • 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

case e_f.h f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp e (ContinuousMap.addRight ↑n) = e x : ℝ ⊢ e x * f x = cexp (↑(-2 * π * x * ↑m) * 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

case e_f.h f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp e (ContinuousMap.addRight ↑n) = e x : ℝ ⊢ cexp (2 * ↑π * I * ↑(-m) * ↑x / ↑1) * f x = cexp (↑(-2 * π * x * ↑m) * 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

case e_f.h.e_a.e_z f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp e (ContinuousMap.addRight ↑n) = e x : ℝ ⊢ 2 * ↑π * I * ↑(-m) * ↑x / ↑1 = ↑(-2 * π * x * ↑m) * 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

case e_f.h.e_a.e_z f : C(ℝ, ℂ) hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ m : ℤ e : C(ℝ, ℂ) := ContinuousMap.comp (fourier (-m)) (ContinuousMap.mk QuotientAddGroup.mk) neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restrict (↑K) g‖ eadd : ∀ (n : ℤ), ContinuousMap.comp e (ContinuousMap.addRight ↑n) = e x : ℝ ⊢ 2 * ↑π * I * -↑m * ↑x / 1 = -2 * ↑π * ↑x * ↑m * 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) h_norm : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ h_sum : Summable fun n => 𝓕 ⇑f ↑n ⊢ ∑' (n : ℤ), f ↑n = ∑' (n : ℤ), 𝓕 ⇑f ↑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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by

let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩

/-- **Poisson's summation formula**, most general form. -/ theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n)

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) h_norm : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ h_sum : Summable fun n => 𝓕 ⇑f ↑n F : C(UnitAddCircle, ℂ) := ContinuousMap.mk (Periodic.lift (_ : Periodic (⇑(∑' (n : ℤ), ContinuousMap.comp f (ContinuousMap.addRight (n • 1)))) 1)) ⊢ ∑' (n : ℤ), f ↑n = ∑' (n : ℤ), 𝓕 ⇑f ↑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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩

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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩

Mathlib.Analysis.Fourier.PoissonSummation.108_0.1MbUAOzT9Ye0D34

/-- **Poisson's summation formula**, most general form. -/ theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n)

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) h_norm : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ h_sum : Summable fun n => 𝓕 ⇑f ↑n F : C(UnitAddCircle, ℂ) := ContinuousMap.mk (Periodic.lift (_ : Periodic (⇑(∑' (n : ℤ), ContinuousMap.comp f (ContinuousMap.addRight (n • 1)))) 1)) ⊢ Summable (fourierCoeff ⇑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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ have : Summable (fourierCoeff F) := by

convert h_sum

/-- **Poisson's summation formula**, most general form. -/ theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n)

Mathlib_Analysis_Fourier_PoissonSummation

case h.e'_5.h f : C(ℝ, ℂ) h_norm : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ h_sum : Summable fun n => 𝓕 ⇑f ↑n F : C(UnitAddCircle, ℂ) := ContinuousMap.mk (Periodic.lift (_ : Periodic (⇑(∑' (n : ℤ), ContinuousMap.comp f (ContinuousMap.addRight (n • 1)))) 1)) x✝ : ℤ ⊢ fourierCoeff (⇑F) x✝ = 𝓕 ⇑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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n)

Mathlib_Analysis_Fourier_PoissonSummation

f : C(ℝ, ℂ) h_norm : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ h_sum : Summable fun n => 𝓕 ⇑f ↑n F : C(UnitAddCircle, ℂ) := ContinuousMap.mk (Periodic.lift (_ : Periodic (⇑(∑' (n : ℤ), ContinuousMap.comp f (ContinuousMap.addRight (n • 1)))) 1)) this : Summable (fourierCoeff ⇑F) ⊢ ∑' (n : ℤ), f ↑n = ∑' (n : ℤ), 𝓕 ⇑f ↑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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n)

Mathlib_Analysis_Fourier_PoissonSummation

case h.e'_2 f : C(ℝ, ℂ) h_norm : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ h_sum : Summable fun n => 𝓕 ⇑f ↑n F : C(UnitAddCircle, ℂ) := ContinuousMap.mk (Periodic.lift (_ : Periodic (⇑(∑' (n : ℤ), ContinuousMap.comp f (ContinuousMap.addRight (n • 1)))) 1)) this : Summable (fourierCoeff ⇑F) ⊢ ∑' (n : ℤ), f ↑n = 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 ·

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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 ·

Mathlib.Analysis.Fourier.PoissonSummation.108_0.1MbUAOzT9Ye0D34

/-- **Poisson's summation formula**, most general form. -/ theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n)

Mathlib_Analysis_Fourier_PoissonSummation

case h.e'_2 f : C(ℝ, ℂ) h_norm : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ h_sum : Summable fun n => 𝓕 ⇑f ↑n F : C(UnitAddCircle, ℂ) := ContinuousMap.mk (Periodic.lift (_ : Periodic (⇑(∑' (n : ℤ), ContinuousMap.comp f (ContinuousMap.addRight (n • 1)))) 1)) this✝ : Summable (fourierCoeff ⇑F) this : ∑' (b : ℤ), (ContinuousMap.comp f (ContinuousMap.addRight ↑b)) 0 = (∑' (b : ℤ), ContinuousMap.comp f (ContinuousMap.addRight ↑b)) 0 ⊢ ∑' (n : ℤ), f ↑n = 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq

simpa only [coe_mk, ← 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n)

Mathlib_Analysis_Fourier_PoissonSummation

case h.e'_3 f : C(ℝ, ℂ) h_norm : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ h_sum : Summable fun n => 𝓕 ⇑f ↑n F : C(UnitAddCircle, ℂ) := ContinuousMap.mk (Periodic.lift (_ : Periodic (⇑(∑' (n : ℤ), ContinuousMap.comp f (ContinuousMap.addRight (n • 1)))) 1)) this : Summable (fourierCoeff ⇑F) ⊢ ∑' (n : ℤ), 𝓕 ⇑f ↑n = ∑' (b : ℤ), fourierCoeff (⇑F) b • (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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this ·

congr 1 with n : 1

/-- **Poisson's summation formula**, most general form. -/ theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this ·

Mathlib.Analysis.Fourier.PoissonSummation.108_0.1MbUAOzT9Ye0D34

/-- **Poisson's summation formula**, most general form. -/ theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n)

Mathlib_Analysis_Fourier_PoissonSummation

case h.e'_3.e_f.h f : C(ℝ, ℂ) h_norm : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ h_sum : Summable fun n => 𝓕 ⇑f ↑n F : C(UnitAddCircle, ℂ) := ContinuousMap.mk (Periodic.lift (_ : Periodic (⇑(∑' (n : ℤ), ContinuousMap.comp f (ContinuousMap.addRight (n • 1)))) 1)) this : Summable (fourierCoeff ⇑F) n : ℤ ⊢ 𝓕 ⇑f ↑n = fourierCoeff (⇑F) n • (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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this · congr 1 with n : 1

rw [← 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this · 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n)

Mathlib_Analysis_Fourier_PoissonSummation

case h.e'_3.e_f.h f : C(ℝ, ℂ) h_norm : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight ↑n))‖ h_sum : Summable fun n => 𝓕 ⇑f ↑n F : C(UnitAddCircle, ℂ) := ContinuousMap.mk (Periodic.lift (_ : Periodic (⇑(∑' (n : ℤ), ContinuousMap.comp f (ContinuousMap.addRight (n • 1)))) 1)) this : Summable (fourierCoeff ⇑F) n : ℤ ⊢ fourierCoeff (Periodic.lift (_ : Periodic (⇑(∑' (n : ℤ), ContinuousMap.comp f (ContinuousMap.addRight (n • 1)))) 1)) n = fourierCoeff (⇑F) 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this · congr 1 with n : 1 rw [← 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this · congr 1 with n : 1 rw [← 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n)

Mathlib_Analysis_Fourier_PoissonSummation

E : Type u_1 inst✝ : NormedAddCommGroup E f : C(ℝ, E) b : ℝ hb : 0 < b hf : ⇑f =O[atTop] fun x => |x| ^ (-b) R S : ℝ ⊢ (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this · congr 1 with n : 1 rw [← 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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) · positivity · linarith have hy' : 0 < y := hxR.trans_le hy have : y ^ (-b) ≤ (x + R) ^ (-b) := by rw [rpow_neg, rpow_neg, inv_le_inv] · gcongr all_goals positivity refine' this.trans _ rw [← mul_rpow, rpow_neg, rpow_neg] · gcongr linarith all_goals positivity

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b)

Mathlib_Analysis_Fourier_PoissonSummation

E : Type u_1 inst✝ : NormedAddCommGroup E f : C(ℝ, E) b : ℝ hb : 0 < b hf : ⇑f =O[atTop] fun x => |x| ^ (-b) R S : ℝ ⊢ ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this · congr 1 with n : 1 rw [← 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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by

intro x hx y hy

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b)

Mathlib_Analysis_Fourier_PoissonSummation

E : Type u_1 inst✝ : NormedAddCommGroup E f : C(ℝ, E) b : ℝ hb : 0 < b hf : ⇑f =O[atTop] fun x => |x| ^ (-b) R S x : ℝ hx : max 0 (-2 * R) < x y : ℝ hy : x + R ≤ y ⊢ y ^ (-b) ≤ (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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this · congr 1 with n : 1 rw [← 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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b)

Mathlib_Analysis_Fourier_PoissonSummation

E : Type u_1 inst✝ : NormedAddCommGroup E f : C(ℝ, E) b : ℝ hb : 0 < b hf : ⇑f =O[atTop] fun x => |x| ^ (-b) R S x : ℝ hx : 0 < x ∧ -2 * R < x y : ℝ hy : x + R ≤ y ⊢ y ^ (-b) ≤ (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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this · congr 1 with n : 1 rw [← 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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx

obtain ⟨hx1, hx2⟩ := hx

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b)

Mathlib_Analysis_Fourier_PoissonSummation

case intro E : Type u_1 inst✝ : NormedAddCommGroup E f : C(ℝ, E) b : ℝ hb : 0 < b hf : ⇑f =O[atTop] fun x => |x| ^ (-b) R S x y : ℝ hy : x + R ≤ y hx1 : 0 < x hx2 : -2 * R < x ⊢ y ^ (-b) ≤ (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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this · congr 1 with n : 1 rw [← 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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx

have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) · positivity · linarith

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx

Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b)

Mathlib_Analysis_Fourier_PoissonSummation

E : Type u_1 inst✝ : NormedAddCommGroup E f : C(ℝ, E) b : ℝ hb : 0 < b hf : ⇑f =O[atTop] fun x => |x| ^ (-b) R S x y : ℝ hy : x + R ≤ y hx1 : 0 < x hx2 : -2 * R < x ⊢ 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this · congr 1 with n : 1 rw [← 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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := 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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := 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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b)

Mathlib_Analysis_Fourier_PoissonSummation

case inl E : Type u_1 inst✝ : NormedAddCommGroup E f : C(ℝ, E) b : ℝ hb : 0 < b hf : ⇑f =O[atTop] fun x => |x| ^ (-b) R S x y : ℝ hy : x + R ≤ y hx1 : 0 < x hx2 : -2 * R < x h : 0 ≤ R ⊢ 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this · congr 1 with n : 1 rw [← 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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) ·

positivity

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) ·

Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b)

Mathlib_Analysis_Fourier_PoissonSummation

case inr E : Type u_1 inst✝ : NormedAddCommGroup E f : C(ℝ, E) b : ℝ hb : 0 < b hf : ⇑f =O[atTop] fun x => |x| ^ (-b) R S x y : ℝ hy : x + R ≤ y hx1 : 0 < x hx2 : -2 * R < x h✝ : R < 0 ⊢ 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this · congr 1 with n : 1 rw [← 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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) · positivity ·

linarith

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) · positivity ·

Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b)

Mathlib_Analysis_Fourier_PoissonSummation

case intro E : Type u_1 inst✝ : NormedAddCommGroup E f : C(ℝ, E) b : ℝ hb : 0 < b hf : ⇑f =O[atTop] fun x => |x| ^ (-b) R S x y : ℝ hy : x + R ≤ y hx1 : 0 < x hx2 : -2 * R < x hxR : 0 < x + R ⊢ y ^ (-b) ≤ (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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this · congr 1 with n : 1 rw [← 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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) · positivity · linarith

have hy' : 0 < y := hxR.trans_le hy

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) · positivity · linarith

Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b)

Mathlib_Analysis_Fourier_PoissonSummation

case intro E : Type u_1 inst✝ : NormedAddCommGroup E f : C(ℝ, E) b : ℝ hb : 0 < b hf : ⇑f =O[atTop] fun x => |x| ^ (-b) R S x y : ℝ hy : x + R ≤ y hx1 : 0 < x hx2 : -2 * R < x hxR : 0 < x + R hy' : 0 < y ⊢ y ^ (-b) ≤ (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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this · congr 1 with n : 1 rw [← 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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) · positivity · linarith have hy' : 0 < y := hxR.trans_le hy

have : y ^ (-b) ≤ (x + R) ^ (-b) := by rw [rpow_neg, rpow_neg, inv_le_inv] · gcongr all_goals positivity

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) · positivity · 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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b)

Mathlib_Analysis_Fourier_PoissonSummation

E : Type u_1 inst✝ : NormedAddCommGroup E f : C(ℝ, E) b : ℝ hb : 0 < b hf : ⇑f =O[atTop] fun x => |x| ^ (-b) R S x y : ℝ hy : x + R ≤ y hx1 : 0 < x hx2 : -2 * R < x hxR : 0 < x + R hy' : 0 < y ⊢ y ^ (-b) ≤ (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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this · congr 1 with n : 1 rw [← 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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) · positivity · linarith have hy' : 0 < y := hxR.trans_le hy have : y ^ (-b) ≤ (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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) · positivity · linarith have hy' : 0 < y := hxR.trans_le hy have : y ^ (-b) ≤ (x + R) ^ (-b) := by

Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b)

Mathlib_Analysis_Fourier_PoissonSummation

E : Type u_1 inst✝ : NormedAddCommGroup E f : C(ℝ, E) b : ℝ hb : 0 < b hf : ⇑f =O[atTop] fun x => |x| ^ (-b) R S x y : ℝ hy : x + R ≤ y hx1 : 0 < x hx2 : -2 * R < x hxR : 0 < x + R hy' : 0 < y ⊢ (x + R) ^ b ≤ 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this · congr 1 with n : 1 rw [← 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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) · positivity · linarith have hy' : 0 < y := hxR.trans_le hy have : y ^ (-b) ≤ (x + R) ^ (-b) := by rw [rpow_neg, rpow_neg, inv_le_inv] ·

gcongr

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) · positivity · linarith have hy' : 0 < y := hxR.trans_le hy have : y ^ (-b) ≤ (x + R) ^ (-b) := by rw [rpow_neg, rpow_neg, inv_le_inv] ·

Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b)

Mathlib_Analysis_Fourier_PoissonSummation

case ha E : Type u_1 inst✝ : NormedAddCommGroup E f : C(ℝ, E) b : ℝ hb : 0 < b hf : ⇑f =O[atTop] fun x => |x| ^ (-b) R S x y : ℝ hy : x + R ≤ y hx1 : 0 < x hx2 : -2 * R < x hxR : 0 < x + R hy' : 0 < y ⊢ 0 < y ^ b case hb E : Type u_1 inst✝ : NormedAddCommGroup E f : C(ℝ, E) b : ℝ hb : 0 < b hf : ⇑f =O[atTop] fun x => |x| ^ (-b) R S x y : ℝ hy : x + R ≤ y hx1 : 0 < x hx2 : -2 * R < x hxR : 0 < x + R hy' : 0 < y ⊢ 0 < (x + R) ^ b case hx E : Type u_1 inst✝ : NormedAddCommGroup E f : C(ℝ, E) b : ℝ hb : 0 < b hf : ⇑f =O[atTop] fun x => |x| ^ (-b) R S x y : ℝ hy : x + R ≤ y hx1 : 0 < x hx2 : -2 * R < x hxR : 0 < x + R hy' : 0 < y ⊢ 0 ≤ x + R case hx E : Type u_1 inst✝ : NormedAddCommGroup E f : C(ℝ, E) b : ℝ hb : 0 < b hf : ⇑f =O[atTop] fun x => |x| ^ (-b) R S x y : ℝ hy : x + R ≤ y hx1 : 0 < x hx2 : -2 * R < x hxR : 0 < x + R hy' : 0 < y ⊢ 0 ≤ 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this · congr 1 with n : 1 rw [← 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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) · positivity · linarith have hy' : 0 < y := hxR.trans_le hy have : y ^ (-b) ≤ (x + R) ^ (-b) := by rw [rpow_neg, rpow_neg, inv_le_inv] · gcongr

all_goals positivity

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) · positivity · linarith have hy' : 0 < y := hxR.trans_le hy have : y ^ (-b) ≤ (x + R) ^ (-b) := by rw [rpow_neg, rpow_neg, inv_le_inv] · gcongr

Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b)

Mathlib_Analysis_Fourier_PoissonSummation

case ha E : Type u_1 inst✝ : NormedAddCommGroup E f : C(ℝ, E) b : ℝ hb : 0 < b hf : ⇑f =O[atTop] fun x => |x| ^ (-b) R S x y : ℝ hy : x + R ≤ y hx1 : 0 < x hx2 : -2 * R < x hxR : 0 < x + R hy' : 0 < y ⊢ 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this · congr 1 with n : 1 rw [← 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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) · positivity · linarith have hy' : 0 < y := hxR.trans_le hy have : y ^ (-b) ≤ (x + R) ^ (-b) := by rw [rpow_neg, rpow_neg, inv_le_inv] · gcongr all_goals

positivity

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) · positivity · linarith have hy' : 0 < y := hxR.trans_le hy have : y ^ (-b) ≤ (x + R) ^ (-b) := by rw [rpow_neg, rpow_neg, inv_le_inv] · gcongr all_goals

Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b)

Mathlib_Analysis_Fourier_PoissonSummation

case hb E : Type u_1 inst✝ : NormedAddCommGroup E f : C(ℝ, E) b : ℝ hb : 0 < b hf : ⇑f =O[atTop] fun x => |x| ^ (-b) R S x y : ℝ hy : x + R ≤ y hx1 : 0 < x hx2 : -2 * R < x hxR : 0 < x + R hy' : 0 < y ⊢ 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 `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ 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 `𝓕 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 `𝓕 f` are Schwartz functions. ## TODO At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f` and `𝓕 f` are Schwartz functions. In fact, `𝓕 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 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)} (hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖) (m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 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(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩ have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x)) intro K g simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul] have eadd : ∀ (n : ℤ), 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 = ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, 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(ℝ, ℂ) evaluated at x into pointwise sum of values. _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by simp_rw [coe_mul, Pi.mul_apply, ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left] -- Swap sum and integral. _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1 exact funext fun n => neK _ _ _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f).comp (ContinuousMap.addRight n) x := by simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢ simp_rw [eadd] -- Rearrange sum of interval integrals into an integral over `ℝ`. _ = ∫ 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 ⟨Icc 0 1, isCompact_Icc⟩ using 1 simp_rw [mul_comp] at eadd ⊢ simp_rw [eadd] exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _ -- Minor tidying to finish _ = 𝓕 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(ℝ, ℂ)} (h_norm : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖) (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by let F : C(UnitAddCircle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩ 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 · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply, coe_addRight, zero_add] using this · congr 1 with n : 1 rw [← 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 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) · positivity · linarith have hy' : 0 < y := hxR.trans_le hy have : y ^ (-b) ≤ (x + R) ^ (-b) := by rw [rpow_neg, rpow_neg, inv_le_inv] · gcongr all_goals

positivity

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |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 : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by intro x hx y hy rw [max_lt_iff] at hx obtain ⟨hx1, hx2⟩ := hx have hxR : 0 < x + R := by rcases le_or_lt 0 R with (h | _) · positivity · linarith have hy' : 0 < y := hxR.trans_le hy have : y ^ (-b) ≤ (x + R) ^ (-b) := by rw [rpow_neg, rpow_neg, inv_le_inv] · gcongr all_goals

Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34

/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b) (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) : IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b)

Mathlib_Analysis_Fourier_PoissonSummation