# 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, I • (N i) ≤ N (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.
theorem Ideal.Filtration.ext_iff {R : Type u} :
∀ {inst : } {I : } {M : Type u} {inst_1 : } {inst_2 : Module R M} (x y : I.Filtration M), x = y x.N = y.N
theorem Ideal.Filtration.ext {R : Type u} :
∀ {inst : } {I : } {M : Type u} {inst_1 : } {inst_2 : Module R M} (x y : I.Filtration M), x.N = y.Nx = y
structure Ideal.Filtration {R : Type u} [] (I : ) (M : Type u) [] [Module R M] :

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 ⊤.

• N :
• mono : ∀ (i : ), self.N (i + 1) self.N i
• smul_le : ∀ (i : ), I self.N i self.N (i + 1)
Instances For
theorem Ideal.Filtration.mono {R : Type u} [] {I : } {M : Type u} [] [Module R M] (self : I.Filtration M) (i : ) :
self.N (i + 1) self.N i
theorem Ideal.Filtration.smul_le {R : Type u} [] {I : } {M : Type u} [] [Module R M] (self : I.Filtration M) (i : ) :
I self.N i self.N (i + 1)
theorem Ideal.Filtration.pow_smul_le {R : Type u} {M : Type u} [] [] [Module R M] {I : } (F : I.Filtration M) (i : ) (j : ) :
I ^ i F.N j F.N (i + j)
theorem Ideal.Filtration.pow_smul_le_pow_smul {R : Type u} {M : Type u} [] [] [Module R M] {I : } (F : I.Filtration M) (i : ) (j : ) (k : ) :
I ^ (i + k) F.N j I ^ k F.N (i + j)
theorem Ideal.Filtration.antitone {R : Type u} {M : Type u} [] [] [Module R M] {I : } (F : I.Filtration M) :
@[simp]
theorem Ideal.trivialFiltration_N {R : Type u} {M : Type u} [] [] [Module R M] (I : ) (N : ) :
∀ (x : ), (I.trivialFiltration N).N x = N
def Ideal.trivialFiltration {R : Type u} {M : Type u} [] [] [Module R M] (I : ) (N : ) :
I.Filtration M

The trivial I-filtration of N.

Equations
• I.trivialFiltration N = { N := fun (x : ) => N, mono := , smul_le := }
Instances For
instance Ideal.Filtration.instSup {R : Type u} {M : Type u} [] [] [Module R M] {I : } :
Sup (I.Filtration M)

The sup of two I.Filtrations is an I.Filtration.

Equations
• Ideal.Filtration.instSup = { sup := fun (F F' : I.Filtration M) => { N := F.N F'.N, mono := , smul_le := } }
instance Ideal.Filtration.instSupSet {R : Type u} {M : Type u} [] [] [Module R M] {I : } :
SupSet (I.Filtration M)

The sSup of a family of I.Filtrations is an I.Filtration.

Equations
• Ideal.Filtration.instSupSet = { sSup := fun (S : Set (I.Filtration M)) => { N := sSup (Ideal.Filtration.N '' S), mono := , smul_le := } }
instance Ideal.Filtration.instInf {R : Type u} {M : Type u} [] [] [Module R M] {I : } :
Inf (I.Filtration M)

The inf of two I.Filtrations is an I.Filtration.

Equations
• Ideal.Filtration.instInf = { inf := fun (F F' : I.Filtration M) => { N := F.N F'.N, mono := , smul_le := } }
instance Ideal.Filtration.instInfSet {R : Type u} {M : Type u} [] [] [Module R M] {I : } :
InfSet (I.Filtration M)

The sInf of a family of I.Filtrations is an I.Filtration.

Equations
• Ideal.Filtration.instInfSet = { sInf := fun (S : Set (I.Filtration M)) => { N := sInf (Ideal.Filtration.N '' S), mono := , smul_le := } }
instance Ideal.Filtration.instTop {R : Type u} {M : Type u} [] [] [Module R M] {I : } :
Top (I.Filtration M)
Equations
• Ideal.Filtration.instTop = { top := I.trivialFiltration }
instance Ideal.Filtration.instBot {R : Type u} {M : Type u} [] [] [Module R M] {I : } :
Bot (I.Filtration M)
Equations
• Ideal.Filtration.instBot = { bot := I.trivialFiltration }
@[simp]
theorem Ideal.Filtration.sup_N {R : Type u} {M : Type u} [] [] [Module R M] {I : } (F : I.Filtration M) (F' : I.Filtration M) :
(F F').N = F.N F'.N
@[simp]
theorem Ideal.Filtration.sSup_N {R : Type u} {M : Type u} [] [] [Module R M] {I : } (S : Set (I.Filtration M)) :
(sSup S).N = sSup (Ideal.Filtration.N '' S)
@[simp]
theorem Ideal.Filtration.inf_N {R : Type u} {M : Type u} [] [] [Module R M] {I : } (F : I.Filtration M) (F' : I.Filtration M) :
(F F').N = F.N F'.N
@[simp]
theorem Ideal.Filtration.sInf_N {R : Type u} {M : Type u} [] [] [Module R M] {I : } (S : Set (I.Filtration M)) :
(sInf S).N = sInf (Ideal.Filtration.N '' S)
@[simp]
theorem Ideal.Filtration.top_N {R : Type u} {M : Type u} [] [] [Module R M] {I : } :
@[simp]
theorem Ideal.Filtration.bot_N {R : Type u} {M : Type u} [] [] [Module R M] {I : } :
@[simp]
theorem Ideal.Filtration.iSup_N {R : Type u} {M : Type u} [] [] [Module R M] {I : } {ι : Sort u_1} (f : ιI.Filtration M) :
(iSup f).N = ⨆ (i : ι), (f i).N
@[simp]
theorem Ideal.Filtration.iInf_N {R : Type u} {M : Type u} [] [] [Module R M] {I : } {ι : Sort u_1} (f : ιI.Filtration M) :
(iInf f).N = ⨅ (i : ι), (f i).N
instance Ideal.Filtration.instCompleteLattice {R : Type u} {M : Type u} [] [] [Module R M] {I : } :
CompleteLattice (I.Filtration M)
Equations
instance Ideal.Filtration.instInhabited {R : Type u} {M : Type u} [] [] [Module R M] {I : } :
Inhabited (I.Filtration M)
Equations
• Ideal.Filtration.instInhabited = { default := }
def Ideal.Filtration.Stable {R : Type u} {M : Type u} [] [] [Module R M] {I : } (F : I.Filtration M) :

An I filtration is stable if I • F.N n = F.N (n+1) for large enough n.

Equations
• F.Stable = ∃ (n₀ : ), nn₀, I F.N n = F.N (n + 1)
Instances For
@[simp]
theorem Ideal.stableFiltration_N {R : Type u} {M : Type u} [] [] [Module R M] (I : ) (N : ) (i : ) :
(I.stableFiltration N).N i = I ^ i N
def Ideal.stableFiltration {R : Type u} {M : Type u} [] [] [Module R M] (I : ) (N : ) :
I.Filtration M

The trivial stable I-filtration of N.

Equations
• I.stableFiltration N = { N := fun (i : ) => I ^ i N, mono := , smul_le := }
Instances For
theorem Ideal.stableFiltration_stable {R : Type u} {M : Type u} [] [] [Module R M] (I : ) (N : ) :
(I.stableFiltration N).Stable
theorem Ideal.Filtration.Stable.exists_pow_smul_eq {R : Type u} {M : Type u} [] [] [Module R M] {I : } {F : I.Filtration M} (h : F.Stable) :
∃ (n₀ : ), ∀ (k : ), F.N (n₀ + k) = I ^ k F.N n₀
theorem Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge {R : Type u} {M : Type u} [] [] [Module R M] {I : } {F : I.Filtration M} (h : F.Stable) :
∃ (n₀ : ), nn₀, F.N n = I ^ (n - n₀) F.N n₀
theorem Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge {R : Type u} {M : Type u} [] [] [Module R M] {I : } {F : I.Filtration M} :
F.Stable ∃ (n₀ : ), nn₀, F.N n = I ^ (n - n₀) F.N n₀
theorem Ideal.Filtration.Stable.exists_forall_le {R : Type u} {M : Type u} [] [] [Module R M] {I : } {F : I.Filtration M} {F' : I.Filtration M} (h : F.Stable) (e : F.N 0 F'.N 0) :
∃ (n₀ : ), ∀ (n : ), F.N (n + n₀) F'.N n
theorem Ideal.Filtration.Stable.bounded_difference {R : Type u} {M : Type u} [] [] [Module R M] {I : } {F : I.Filtration M} {F' : I.Filtration M} (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
def Ideal.Filtration.submodule {R : Type u} {M : Type u} [] [] [Module R M] {I : } (F : I.Filtration M) :
Submodule (()) ()

The R[IX]-submodule of M[X] associated with an I-filtration.

Equations
• F.submodule = { carrier := {f : | ∀ (i : ), f i F.N i}, add_mem' := , zero_mem' := , smul_mem' := }
Instances For
@[simp]
theorem Ideal.Filtration.mem_submodule {R : Type u} {M : Type u} [] [] [Module R M] {I : } (F : I.Filtration M) (f : ) :
f F.submodule ∀ (i : ), f i F.N i
theorem Ideal.Filtration.inf_submodule {R : Type u} {M : Type u} [] [] [Module R M] {I : } (F : I.Filtration M) (F' : I.Filtration M) :
(F F').submodule = F.submodule F'.submodule
def Ideal.Filtration.submoduleInfHom {R : Type u} (M : Type u) [] [] [Module R M] (I : ) :
InfHom (I.Filtration M) (Submodule (()) ())

Ideal.Filtration.submodule as an InfHom.

Equations
• = { toFun := Ideal.Filtration.submodule, map_inf' := }
Instances For
theorem Ideal.Filtration.submodule_closure_single {R : Type u} {M : Type u} [] [] [Module R M] {I : } (F : I.Filtration M) :
theorem Ideal.Filtration.submodule_span_single {R : Type u} {M : Type u} [] [] [Module R M] {I : } (F : I.Filtration M) :
Submodule.span (()) (⋃ (i : ), () '' (F.N i)) = F.submodule
theorem Ideal.Filtration.submodule_eq_span_le_iff_stable_ge {R : Type u} {M : Type u} [] [] [Module R M] {I : } (F : I.Filtration M) (n₀ : ) :
F.submodule = Submodule.span (()) (⋃ (i : ), ⋃ (_ : i n₀), () '' (F.N i)) nn₀, I F.N n = F.N (n + 1)
theorem Ideal.Filtration.submodule_fg_iff_stable {R : Type u} {M : Type u} [] [] [Module R M] {I : } (F : I.Filtration M) (hF' : ∀ (i : ), (F.N i).FG) :
F.submodule.FG F.Stable

If the components of a filtration are finitely generated, then the filtration is stable iff its associated submodule of is finitely generated.

theorem Ideal.Filtration.Stable.of_le {R : Type u} {M : Type u} [] [] [Module R M] {I : } {F : I.Filtration M} [] [] (hF : F.Stable) {F' : I.Filtration M} (hf : F' F) :
F'.Stable
theorem Ideal.Filtration.Stable.inter_right {R : Type u} {M : Type u} [] [] [Module R M] {I : } {F : I.Filtration M} (F' : I.Filtration M) [] [] (hF : F.Stable) :
(F F').Stable
theorem Ideal.Filtration.Stable.inter_left {R : Type u} {M : Type u} [] [] [Module R M] {I : } {F : I.Filtration M} (F' : I.Filtration M) [] [] (hF : F.Stable) :
(F' F).Stable
theorem Ideal.exists_pow_inf_eq_pow_smul {R : Type u} {M : Type u} [] [] [Module R M] (I : ) [] [] (N : ) :
∃ (k : ), nk, I ^ n N = I ^ (n - k) (I ^ k N)

Artin-Rees lemma

theorem Ideal.mem_iInf_smul_pow_eq_bot_iff {R : Type u} {M : Type u} [] [] [Module R M] (I : ) [] [] (x : M) :
x ⨅ (i : ), I ^ i ∃ (r : I), r x = x
theorem Ideal.iInf_pow_smul_eq_bot_of_localRing {R : Type u} {M : Type u} [] [] [Module R M] (I : ) [] [] [] (h : I ) :
⨅ (i : ), I ^ i =
theorem Ideal.iInf_pow_eq_bot_of_localRing {R : Type u} [] (I : ) [] [] (h : I ) :
⨅ (i : ), I ^ i =

Krull's intersection theorem for noetherian local rings.

Also see Ideal.isIdempotentElem_iff_eq_bot_or_top for integral domains.

theorem Ideal.iInf_pow_eq_bot_of_isDomain {R : Type u} [] (I : ) [] [] (h : I ) :
⨅ (i : ), I ^ i =

Krull's intersection theorem for noetherian domains.