# mathlibdocumentation

ring_theory.filtration

# I-filtrations of modules #

This file contains the definitions and basic results around (stable) I-filtrations of modules.

theorem ideal.filtration.ext {R : Type u} {_inst_1 : comm_ring R} {I : ideal R} {M : Type u} {_inst_4 : add_comm_group M} {_inst_5 : M} (x y : I.filtration M) (h : x.N = y.N) :
x = y
@[ext]
structure ideal.filtration {R : Type u} [comm_ring R] (I : ideal R) (M : Type u) [ M] :
Type u
• N : M
• mono : ∀ (i : ), self.N (i + 1) self.N i
• smul_le : ∀ (i : ), I self.N i self.N (i + 1)

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 ⊤.

Instances for ideal.filtration
theorem ideal.filtration.ext_iff {R : Type u} {_inst_1 : comm_ring R} {I : ideal R} {M : Type u} {_inst_4 : add_comm_group M} {_inst_5 : M} (x y : I.filtration M) :
x = y x.N = y.N
theorem ideal.filtration.pow_smul_le {R M : Type u} [comm_ring R] [ M] {I : ideal R} (F : I.filtration M) (i j : ) :
I ^ i F.N j F.N (i + j)
theorem ideal.filtration.pow_smul_le_pow_smul {R M : Type u} [comm_ring R] [ M] {I : ideal R} (F : I.filtration M) (i j k : ) :
I ^ (i + k) F.N j I ^ k F.N (i + j)
@[protected]
theorem ideal.filtration.antitone {R M : Type u} [comm_ring R] [ M] {I : ideal R} (F : I.filtration M) :
@[simp]
theorem ideal.trivial_filtration_N {R M : Type u} [comm_ring R] [ M] (I : ideal R) (N : M) (i : ) :
def ideal.trivial_filtration {R M : Type u} [comm_ring R] [ M] (I : ideal R) (N : M) :

The trivial I-filtration of N.

Equations
@[protected, instance]
def ideal.filtration.has_sup {R M : Type u} [comm_ring R] [ M] {I : ideal R} :

The sup of two I.filtrations is an I.filtration.

Equations
@[protected, instance]
def ideal.filtration.has_Sup {R M : Type u} [comm_ring R] [ M] {I : ideal R} :

The Sup of a family of I.filtrations is an I.filtration.

Equations
@[protected, instance]
def ideal.filtration.has_inf {R M : Type u} [comm_ring R] [ M] {I : ideal R} :

The inf of two I.filtrations is an I.filtration.

Equations
@[protected, instance]
def ideal.filtration.has_Inf {R M : Type u} [comm_ring R] [ M] {I : ideal R} :

The Inf of a family of I.filtrations is an I.filtration.

Equations
@[protected, instance]
def ideal.filtration.has_top {R M : Type u} [comm_ring R] [ M] {I : ideal R} :
Equations
@[protected, instance]
def ideal.filtration.has_bot {R M : Type u} [comm_ring R] [ M] {I : ideal R} :
Equations
@[simp]
theorem ideal.filtration.sup_N {R M : Type u} [comm_ring R] [ M] {I : ideal R} (F F' : I.filtration M) :
(F F').N = F.N F'.N
@[simp]
theorem ideal.filtration.Sup_N {R M : Type u} [comm_ring R] [ M] {I : ideal R} (S : set (I.filtration M)) :
@[simp]
theorem ideal.filtration.inf_N {R M : Type u} [comm_ring R] [ M] {I : ideal R} (F F' : I.filtration M) :
(F F').N = F.N F'.N
@[simp]
theorem ideal.filtration.Inf_N {R M : Type u} [comm_ring R] [ M] {I : ideal R} (S : set (I.filtration M)) :
@[simp]
theorem ideal.filtration.top_N {R M : Type u} [comm_ring R] [ M] {I : ideal R} :
@[simp]
theorem ideal.filtration.bot_N {R M : Type u} [comm_ring R] [ M] {I : ideal R} :
@[simp]
theorem ideal.filtration.supr_N {R M : Type u} [comm_ring R] [ M] {I : ideal R} {ι : Sort u_1} (f : ι → I.filtration M) :
(supr f).N = ⨆ (i : ι), (f i).N
@[simp]
theorem ideal.filtration.infi_N {R M : Type u} [comm_ring R] [ M] {I : ideal R} {ι : Sort u_1} (f : ι → I.filtration M) :
(infi f).N = ⨅ (i : ι), (f i).N
@[protected, instance]
def ideal.filtration.complete_lattice {R M : Type u} [comm_ring R] [ M] {I : ideal R} :
Equations
@[protected, instance]
def ideal.filtration.inhabited {R M : Type u} [comm_ring R] [ M] {I : ideal R} :
Equations
def ideal.filtration.stable {R M : Type u} [comm_ring R] [ M] {I : ideal R} (F : I.filtration M) :
Prop

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

Equations
def ideal.stable_filtration {R M : Type u} [comm_ring R] [ M] (I : ideal R) (N : M) :

The trivial stable I-filtration of N.

Equations
@[simp]
theorem ideal.stable_filtration_N {R M : Type u} [comm_ring R] [ M] (I : ideal R) (N : M) (i : ) :
(I.stable_filtration N).N i = I ^ i N
theorem ideal.stable_filtration_stable {R M : Type u} [comm_ring R] [ M] (I : ideal R) (N : M) :
theorem ideal.filtration.stable.exists_pow_smul_eq {R M : Type u} [comm_ring R] [ M] {I : ideal R} {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 M : Type u} [comm_ring R] [ M] {I : ideal R} {F : I.filtration M} (h : F.stable) :
∃ (n₀ : ), ∀ (n : ), n n₀F.N n = I ^ (n - n₀) F.N n₀
theorem ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge {R M : Type u} [comm_ring R] [ M] {I : ideal R} {F : I.filtration M} :
F.stable ∃ (n₀ : ), ∀ (n : ), n n₀F.N n = I ^ (n - n₀) F.N n₀
theorem ideal.filtration.stable.exists_forall_le {R M : Type u} [comm_ring R] [ M] {I : ideal R} {F 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 M : Type u} [comm_ring R] [ M] {I : ideal R} {F 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