fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
fg_iff_compact (s : Submodule R M) : s.FG ↔ CompleteLattice.IsCompactElement s := by
classical
let sp : M → Submodule R M := fun a => span R {a}
have supr_rw : ∀ t : Finset M, ⨆ x ∈ t, sp x = ⨆ x ∈ (↑t : Set M), sp x := fun t => by rfl
constructor
· rintro ⟨t, rfl⟩
rw [span_eq_iSup_of_singleton_spans, ← supr_rw, ← Finset.sup_eq_iSup t sp]
apply CompleteLattice.isCompactElement_finsetSup
exact fun n _ => singleton_span_isCompactElement n
· intro h
have sSup' : s = sSup (sp '' ↑s) := by
rw [sSup_eq_iSup, iSup_image, ← span_eq_iSup_of_singleton_spans, eq_comm, span_eq]
obtain ⟨u, ⟨huspan, husup⟩⟩ := h (sp '' ↑s) (le_of_eq sSup')
have ssup : s = u.sup id := by
suffices u.sup id ≤ s from le_antisymm husup this
rw [sSup', Finset.sup_id_eq_sSup]
exact sSup_le_sSup huspan
obtain ⟨t, -, rfl⟩ := Finset.subset_set_image_iff.mp huspan
rw [Finset.sup_image, Function.id_comp, Finset.sup_eq_iSup, supr_rw, ←
span_eq_iSup_of_singleton_spans, eq_comm] at ssup
exact ⟨t, ssup⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | fg_iff_compact | Finitely generated submodules are precisely compact elements in the submodule lattice. |
of_surjective [hM : Module.Finite R M] (f : M →ₛₗ[σ] P) (hf : Surjective f) :
Module.Finite S P :=
⟨by
rw [← LinearMap.range_eq_top.2 hf, ← Submodule.map_top]
exact hM.1.map f⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | of_surjective | null |
_root_.LinearMap.finite_iff_of_bijective (f : M →ₛₗ[σ] P) (hf : Function.Bijective f) :
Module.Finite R M ↔ Module.Finite S P :=
⟨fun _ ↦ of_surjective f hf.surjective, fun _ ↦ ⟨fg_of_fg_map_injective f hf.injective <| by
rwa [Submodule.map_top, LinearMap.range_eq_top.2 hf.surjective, ← Module.finite_def]⟩⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | _root_.LinearMap.finite_iff_of_bijective | null |
quotient (R) {A M} [Semiring R] [AddCommGroup M] [Ring A] [Module A M] [Module R M]
[SMul R A] [IsScalarTower R A M] [Module.Finite R M]
(N : Submodule A M) : Module.Finite R (M ⧸ N) :=
Module.Finite.of_surjective (N.mkQ.restrictScalars R) N.mkQ_surjective | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | quotient | null |
range {F : Type*} [FunLike F M N] [SemilinearMapClass F (RingHom.id R) M N]
[Module.Finite R M] (f : F) : Module.Finite R (LinearMap.range f) :=
of_surjective (SemilinearMapClass.semilinearMap f).rangeRestrict
fun ⟨_, y, hy⟩ => ⟨y, Subtype.ext hy⟩ | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | range | The range of a linear map from a finite module is finite. |
map (p : Submodule R M) [Module.Finite R p] (f : M →ₗ[R] N) : Module.Finite R (p.map f) :=
of_surjective (f.restrict fun _ => Submodule.mem_map_of_mem) fun ⟨_, _, hy, hy'⟩ =>
⟨⟨_, hy⟩, Subtype.ext hy'⟩ | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | map | Pushforwards of finite submodules are finite. |
pi {ι : Type*} {M : ι → Type*} [_root_.Finite ι] [∀ i, AddCommMonoid (M i)]
[∀ i, Module R (M i)] [h : ∀ i, Module.Finite R (M i)] : Module.Finite R (∀ i, M i) :=
⟨by
rw [← Submodule.pi_top]
exact Submodule.fg_pi fun i => (h i).1⟩ | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | pi | null |
of_pi {ι : Type*} (M : ι → Type*) [∀ i, AddCommMonoid (M i)] [∀ i, Module R (M i)]
[Module.Finite R (∀ i, M i)] (i : ι) : Module.Finite R (M i) :=
Module.Finite.of_surjective _ <| LinearMap.proj_surjective i | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | of_pi | null |
pi_iff {ι : Type*} {M : ι → Type*} [_root_.Finite ι] [∀ i, AddCommMonoid (M i)]
[∀ i, Module R (M i)] : Module.Finite R (∀ i, M i) ↔ ∀ i, Module.Finite R (M i) :=
⟨fun _ i => of_pi M i, fun _ => inferInstance⟩
variable (R) | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | pi_iff | null |
self : Module.Finite R R :=
⟨⟨{1}, by simpa only [Finset.coe_singleton] using Ideal.span_singleton_one⟩⟩
variable (M) | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | self | null |
of_restrictScalars_finite (R A M : Type*) [Semiring R] [Semiring A] [AddCommMonoid M]
[Module R M] [Module A M] [SMul R A] [IsScalarTower R A M] [hM : Module.Finite R M] :
Module.Finite A M := by
rw [finite_def, Submodule.fg_def] at hM ⊢
obtain ⟨S, hSfin, hSgen⟩ := hM
refine ⟨S, hSfin, eq_top_iff.2 ?_⟩
have := Submodule.span_le_restrictScalars R A S
rwa [hSgen] at this
variable {R M} | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | of_restrictScalars_finite | null |
equiv [Module.Finite R M] (e : M ≃ₗ[R] N) : Module.Finite R N :=
of_surjective (e : M →ₗ[R] N) e.surjective | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | equiv | null |
equiv_iff (e : M ≃ₗ[R] N) : Module.Finite R M ↔ Module.Finite R N :=
⟨fun _ ↦ equiv e, fun _ ↦ equiv e.symm⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | equiv_iff | null |
ulift [Module.Finite R M] : Module.Finite R (ULift M) := equiv ULift.moduleEquiv.symm | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | ulift | null |
iff_fg {N : Submodule R M} : Module.Finite R N ↔ N.FG := Module.finite_def.trans N.fg_top
variable (R M) | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | iff_fg | null |
bot : Module.Finite R (⊥ : Submodule R M) := iff_fg.mpr fg_bot | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | bot | null |
top [Module.Finite R M] : Module.Finite R (⊤ : Submodule R M) := iff_fg.mpr fg_top
variable {M} | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | top | null |
span_of_finite {A : Set M} (hA : Set.Finite A) :
Module.Finite R (Submodule.span R A) :=
⟨(Submodule.fg_top _).mpr ⟨hA.toFinset, hA.coe_toFinset.symm ▸ rfl⟩⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | span_of_finite | The submodule generated by a finite set is `R`-finite. |
span_singleton (x : M) : Module.Finite R (R ∙ x) :=
Module.Finite.span_of_finite R <| Set.finite_singleton _ | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | span_singleton | The submodule generated by a single element is `R`-finite. |
span_finset (s : Finset M) : Module.Finite R (span R (s : Set M)) :=
⟨(Submodule.fg_top _).mpr ⟨s, rfl⟩⟩
variable {R} | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | span_finset | The submodule generated by a finset is `R`-finite. |
trans {R : Type*} (A M : Type*) [Semiring R] [Semiring A] [Module R A]
[AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] :
∀ [Module.Finite R A] [Module.Finite A M], Module.Finite R M
| ⟨⟨s, hs⟩⟩, ⟨⟨t, ht⟩⟩ =>
⟨Submodule.fg_def.2
⟨Set.image2 (· • ·) (↑s : Set A) (↑t : Set M),
Set.Finite.image2 _ s.finite_toSet t.finite_toSet, by
rw [Set.image2_smul, Submodule.span_smul_of_span_eq_top hs (↑t : Set M), ht,
Submodule.restrictScalars_top]⟩⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | trans | null |
of_equiv_equiv {A₁ B₁ A₂ B₂ : Type*} [CommSemiring A₁] [CommSemiring B₁]
[CommSemiring A₂] [Semiring B₂] [Algebra A₁ B₁] [Algebra A₂ B₂] (e₁ : A₁ ≃+* A₂)
(e₂ : B₁ ≃+* B₂)
(he : RingHom.comp (algebraMap A₂ B₂) ↑e₁ = RingHom.comp ↑e₂ (algebraMap A₁ B₁))
[Module.Finite A₁ B₁] : Module.Finite A₂ B₂ := by
letI := e₁.toRingHom.toAlgebra
letI := ((algebraMap A₁ B₁).comp e₁.symm.toRingHom).toAlgebra
haveI : IsScalarTower A₁ A₂ B₁ := IsScalarTower.of_algebraMap_eq
(fun x ↦ by simp [RingHom.algebraMap_toAlgebra])
let e : B₁ ≃ₐ[A₂] B₂ :=
{ e₂ with
commutes' := fun r ↦ by
simpa [RingHom.algebraMap_toAlgebra] using DFunLike.congr_fun he.symm (e₁.symm r) }
haveI := Module.Finite.of_restrictScalars_finite A₁ A₂ B₁
exact Module.Finite.equiv e.toLinearEquiv | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | of_equiv_equiv | null |
finite_sup (S₁ S₂ : Submodule R V) [h₁ : Module.Finite R S₁]
[h₂ : Module.Finite R S₂] : Module.Finite R (S₁ ⊔ S₂ : Submodule R V) := by
rw [finite_def] at *
exact (fg_top _).2 (((fg_top S₁).1 h₁).sup ((fg_top S₂).1 h₂)) | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | finite_sup | The sup of two fg submodules is finite. Also see `Submodule.FG.sup`. |
finite_finset_sup {ι : Type*} (s : Finset ι) (S : ι → Submodule R V)
[∀ i, Module.Finite R (S i)] : Module.Finite R (s.sup S : Submodule R V) := by
refine
@Finset.sup_induction _ _ _ _ s S (fun i => Module.Finite R ↑i) (Module.Finite.bot R V)
?_ fun i _ => by infer_instance
intro S₁ hS₁ S₂ hS₂
exact Submodule.finite_sup S₁ S₂ | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | finite_finset_sup | The submodule generated by a finite supremum of finite-dimensional submodules is
finite-dimensional.
Note that strictly this only needs `∀ i ∈ s, FiniteDimensional K (S i)`, but that doesn't
work well with typeclass search. |
id : Finite (RingHom.id A) :=
Module.Finite.self A | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | id | null |
of_surjective (f : A →+* B) (hf : Surjective f) : f.Finite :=
letI := f.toAlgebra
Module.Finite.of_surjective (Algebra.linearMap A B) hf | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | of_surjective | null |
comp {g : B →+* C} {f : A →+* B} (hg : g.Finite) (hf : f.Finite) : (g.comp f).Finite := by
algebraize [f, g, g.comp f]
exact Module.Finite.trans B C | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | comp | null |
of_comp_finite {f : A →+* B} {g : B →+* C} (h : (g.comp f).Finite) : g.Finite := by
algebraize [f, g, g.comp f]
exact Module.Finite.of_restrictScalars_finite A B C | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | of_comp_finite | null |
id : Finite (AlgHom.id R A) :=
RingHom.Finite.id A
variable {R A} | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | id | null |
comp {g : B →ₐ[R] C} {f : A →ₐ[R] B} (hg : g.Finite) (hf : f.Finite) : (g.comp f).Finite :=
RingHom.Finite.comp hg hf | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | comp | null |
of_surjective (f : A →ₐ[R] B) (hf : Surjective f) : f.Finite :=
RingHom.Finite.of_surjective f.toRingHom hf | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | of_surjective | null |
of_comp_finite {f : A →ₐ[R] B} {g : B →ₐ[R] C} (h : (g.comp f).Finite) : g.Finite :=
RingHom.Finite.of_comp_finite h | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | of_comp_finite | null |
private instModuleFiniteAux : Module.Finite R≥0 R := by
simp_rw [Module.finite_def, Submodule.fg_def, Submodule.eq_top_iff']
refine ⟨{1, -1}, by simp, fun x ↦ ?_⟩
obtain hx | hx := le_total 0 x
· simpa using Submodule.smul_mem (M := R) (.span R≥0 {1, -1}) ⟨x, hx⟩ (x := 1)
(Submodule.subset_span <| by simp)
· simpa using Submodule.smul_mem (M := R) (.span R≥0 {1, -1}) ⟨-x, neg_nonneg.2 hx⟩ (x := -1)
(Submodule.subset_span <| by simp) | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | instModuleFiniteAux | null |
instModuleFinite [Module.Finite R E] : Module.Finite R≥0 E := .trans R E | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Order.Nonneg.Module",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Quotient.Defs",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Basic.lean | instModuleFinite | If a module is finite over a linearly ordered ring, then it is also finite over the non-negative
scalars. |
FG.map₂ (f : M →ₗ[R] N →ₗ[R] P) {p : Submodule R M} {q : Submodule R N} (hp : p.FG)
(hq : q.FG) : (map₂ f p q).FG :=
let ⟨sm, hfm, hm⟩ := fg_def.1 hp
let ⟨sn, hfn, hn⟩ := fg_def.1 hq
fg_def.2
⟨Set.image2 (fun m n => f m n) sm sn, hfm.image2 _ hfn,
map₂_span_span R f sm sn ▸ hm ▸ hn ▸ rfl⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.Finiteness.Defs",
"Mathlib.Algebra.Module.Submodule.Bilinear"
] | Mathlib/RingTheory/Finiteness/Bilinear.lean | FG.map₂ | null |
Submodule.fg_iff_exists_fin_linearMap {N : Submodule R M} :
N.FG ↔ ∃ (n : ℕ) (f : (Fin n → R) →ₗ[R] M), LinearMap.range f = N := by
simp_rw [fg_iff_exists_fin_generating_family, ← ((Pi.basisFun R _).constr ℕ).exists_congr_right]
simp [Basis.constr_range] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Cardinality.lean | Submodule.fg_iff_exists_fin_linearMap | null |
AddSubmonoid.fg_iff_exists_fin_addMonoidHom {M : Type*} [AddCommMonoid M]
{S : AddSubmonoid M} : S.FG ↔ ∃ (n : ℕ) (f : (Fin n → ℕ) →+ M), AddMonoidHom.mrange f = S := by
rw [← S.toNatSubmodule_toAddSubmonoid, ← Submodule.fg_iff_addSubmonoid_fg,
Submodule.fg_iff_exists_fin_linearMap]
exact exists_congr fun n => ⟨fun ⟨f, hf⟩ => ⟨f, hf ▸ LinearMap.range_toAddSubmonoid _⟩,
fun ⟨f, hf⟩ => ⟨f.toNatLinearMap, Submodule.toAddSubmonoid_inj.mp <|
hf ▸ LinearMap.range_toAddSubmonoid _⟩⟩ | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Cardinality.lean | AddSubmonoid.fg_iff_exists_fin_addMonoidHom | null |
AddSubgroup.fg_iff_exists_fin_addMonoidHom {M : Type*} [AddCommGroup M]
{H : AddSubgroup M} : H.FG ↔ ∃ (n : ℕ) (f : (Fin n → ℤ) →+ M), AddMonoidHom.range f = H := by
rw [← H.toIntSubmodule_toAddSubgroup, ← Submodule.fg_iff_addSubgroup_fg,
Submodule.fg_iff_exists_fin_linearMap]
refine exists_congr fun n => ⟨fun ⟨f, hf⟩ => ⟨f, hf ▸ LinearMap.range_toAddSubgroup _⟩,
fun ⟨f, hf⟩ => ⟨f.toIntLinearMap, Submodule.toAddSubmonoid_inj.mp ?_⟩⟩
simp [hf] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Cardinality.lean | AddSubgroup.fg_iff_exists_fin_addMonoidHom | null |
exists_fin' [Module.Finite R M] : ∃ (n : ℕ) (f : (Fin n → R) →ₗ[R] M), Surjective f :=
have ⟨n, f, hf⟩ := (Submodule.fg_iff_exists_fin_linearMap R M).mp fg_top
⟨n, f, by rw [← LinearMap.range_eq_top, hf]⟩ | lemma | RingTheory | [
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Cardinality.lean | exists_fin' | A finite module admits a surjective linear map from a finite free module. |
exists_fin_quot_equiv (R M : Type*) [Ring R] [AddCommGroup M] [Module R M]
[Module.Finite R M] :
∃ (n : ℕ) (S : Submodule R (Fin n → R)), Nonempty ((_ ⧸ S) ≃ₗ[R] M) :=
let ⟨n, f, hf⟩ := Module.Finite.exists_fin' R M
⟨n, LinearMap.ker f, ⟨f.quotKerEquivOfSurjective hf⟩⟩
variable {M} | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Cardinality.lean | exists_fin_quot_equiv | A finite module can be realised as a quotient of `Fin n → R` (i.e. `R^n`). |
_root_.Module.finite_of_finite [Finite R] [Module.Finite R M] : Finite M := by
obtain ⟨n, f, hf⟩ := exists_fin' R M; exact .of_surjective f hf
variable {R} | lemma | RingTheory | [
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Cardinality.lean | _root_.Module.finite_of_finite | null |
_root_.Module.finite_iff_finite [Finite R] : Module.Finite R M ↔ Finite M :=
⟨fun _ ↦ finite_of_finite R, fun _ ↦ .of_finite⟩
variable (R) in | lemma | RingTheory | [
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Cardinality.lean | _root_.Module.finite_iff_finite | A module over a finite ring has finite dimension iff it is finite. |
_root_.Set.Finite.submoduleSpan [Finite R] {s : Set M} (hs : s.Finite) :
(Submodule.span R s : Set M).Finite := by
lift s to Finset M using hs
rw [Set.Finite, ← Module.finite_iff_finite (R := R)]
dsimp
infer_instance | lemma | RingTheory | [
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Cardinality.lean | _root_.Set.Finite.submoduleSpan | null |
finite_basis [Nontrivial R] {ι} [Module.Finite R M]
(b : Basis ι R M) :
_root_.Finite ι :=
let ⟨s, hs⟩ := ‹Module.Finite R M›
basis_finite_of_finite_spans s.finite_toSet hs b | lemma | RingTheory | [
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Cardinality.lean | finite_basis | If a free module is finite, then any arbitrary basis is finite. |
not_finite_of_infinite_basis [Nontrivial R] {ι} [Infinite ι] (b : Basis ι R M) :
¬ Module.Finite R M :=
fun _ ↦ (Finite.finite_basis b).not_infinite ‹_› | lemma | RingTheory | [
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Cardinality.lean | not_finite_of_infinite_basis | null |
noncomputable kerRepr := LinearMap.ker (Finite.exists_fin' R M).choose_spec.choose | def | RingTheory | [
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Cardinality.lean | kerRepr | The kernel of a random surjective linear map from a finite free module
to a given finite module. |
protected repr : Type u := _ ⧸ kerRepr R M | abbrev | RingTheory | [
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Cardinality.lean | repr | A representative of a finite module in the same universe as the ring. |
noncomputable reprEquiv : Finite.repr R M ≃ₗ[R] M :=
LinearMap.quotKerEquivOfSurjective _ (Finite.exists_fin' R M).choose_spec.choose_spec | def | RingTheory | [
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Cardinality.lean | reprEquiv | The representative is isomorphic to the original module. |
FG (N : Submodule R M) : Prop :=
∃ S : Finset M, Submodule.span R ↑S = N | def | RingTheory | [
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Data.Set.Finite.Lemmas",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.RingTheory.Ideal.Span",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/Finiteness/Defs.lean | FG | A submodule of `M` is finitely generated if it is the span of a finite subset of `M`. |
fg_def {N : Submodule R M} : N.FG ↔ ∃ S : Set M, S.Finite ∧ span R S = N :=
⟨fun ⟨t, h⟩ => ⟨_, Finset.finite_toSet t, h⟩, by
rintro ⟨t', h, rfl⟩
rcases Finite.exists_finset_coe h with ⟨t, rfl⟩
exact ⟨t, rfl⟩⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Data.Set.Finite.Lemmas",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.RingTheory.Ideal.Span",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/Finiteness/Defs.lean | fg_def | null |
fg_iff_addSubmonoid_fg (P : Submodule ℕ M) : P.FG ↔ P.toAddSubmonoid.FG :=
⟨fun ⟨S, hS⟩ => ⟨S, by simpa [← span_nat_eq_addSubmonoidClosure] using hS⟩, fun ⟨S, hS⟩ =>
⟨S, by simpa [← span_nat_eq_addSubmonoidClosure] using hS⟩⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Data.Set.Finite.Lemmas",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.RingTheory.Ideal.Span",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/Finiteness/Defs.lean | fg_iff_addSubmonoid_fg | null |
fg_iff_addSubgroup_fg {G : Type*} [AddCommGroup G] (P : Submodule ℤ G) :
P.FG ↔ P.toAddSubgroup.FG :=
⟨fun ⟨S, hS⟩ => ⟨S, by simpa [← span_int_eq_addSubgroupClosure] using hS⟩, fun ⟨S, hS⟩ =>
⟨S, by simpa [← span_int_eq_addSubgroupClosure] using hS⟩⟩
@[deprecated (since := "2025-08-20")] alias fg_iff_add_subgroup_fg := fg_iff_addSubgroup_fg | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Data.Set.Finite.Lemmas",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.RingTheory.Ideal.Span",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/Finiteness/Defs.lean | fg_iff_addSubgroup_fg | null |
fg_iff_exists_fin_generating_family {N : Submodule R M} :
N.FG ↔ ∃ (n : ℕ) (s : Fin n → M), span R (range s) = N := by
rw [fg_def]
constructor
· rintro ⟨S, Sfin, hS⟩
obtain ⟨n, f, rfl⟩ := Sfin.fin_embedding
exact ⟨n, f, hS⟩
· rintro ⟨n, s, hs⟩
exact ⟨range s, finite_range s, hs⟩
universe w v u in | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Data.Set.Finite.Lemmas",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.RingTheory.Ideal.Span",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/Finiteness/Defs.lean | fg_iff_exists_fin_generating_family | null |
fg_iff_exists_finite_generating_family {A : Type u} [Semiring A] {M : Type v}
[AddCommMonoid M] [Module A M] {N : Submodule A M} :
N.FG ↔ ∃ (G : Type w) (_ : Finite G) (g : G → M), Submodule.span A (Set.range g) = N := by
constructor
· intro hN
obtain ⟨n, f, h⟩ := Submodule.fg_iff_exists_fin_generating_family.1 hN
refine ⟨ULift (Fin n), inferInstance, f ∘ ULift.down, ?_⟩
convert h
ext x
simp only [Set.mem_range, Function.comp_apply, ULift.exists]
· rintro ⟨G, _, g, hg⟩
have := Fintype.ofFinite (range g)
exact ⟨(range g).toFinset, by simpa using hg⟩ | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Data.Set.Finite.Lemmas",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.RingTheory.Ideal.Span",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/Finiteness/Defs.lean | fg_iff_exists_finite_generating_family | null |
fg_span_iff_fg_span_finset_subset (s : Set M) :
(span R s).FG ↔ ∃ s' : Finset M, ↑s' ⊆ s ∧ span R s = span R s' := by
unfold FG
constructor
· intro ⟨s'', hs''⟩
obtain ⟨s', hs's, hss'⟩ := subset_span_finite_of_subset_span <| hs'' ▸ subset_span
refine ⟨s', hs's, ?_⟩
apply le_antisymm
· rwa [← hs'', Submodule.span_le]
· rw [Submodule.span_le]
exact le_trans hs's subset_span
· intro ⟨s', _, h⟩
exact ⟨s', h.symm⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Data.Set.Finite.Lemmas",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.RingTheory.Ideal.Span",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/Finiteness/Defs.lean | fg_span_iff_fg_span_finset_subset | null |
FG (I : Ideal R) : Prop :=
∃ S : Finset R, Ideal.span ↑S = I | def | RingTheory | [
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Data.Set.Finite.Lemmas",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.RingTheory.Ideal.Span",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/Finiteness/Defs.lean | FG | An ideal of `R` is finitely generated if it is the span of a finite subset of `R`.
This is defeq to `Submodule.FG`, but unfolds more nicely. |
protected Module.Finite [Semiring R] [AddCommMonoid M] [Module R M] : Prop where
fg_top : (⊤ : Submodule R M).FG
attribute [inherit_doc Module.Finite] Module.Finite.fg_top | class | RingTheory | [
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Data.Set.Finite.Lemmas",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.RingTheory.Ideal.Span",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/Finiteness/Defs.lean | Module.Finite | A module over a semiring is `Module.Finite` if it is finitely generated as a module. |
finite_def {R M} [Semiring R] [AddCommMonoid M] [Module R M] :
Module.Finite R M ↔ (⊤ : Submodule R M).FG :=
⟨fun h => h.1, fun h => ⟨h⟩⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Data.Set.Finite.Lemmas",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.RingTheory.Ideal.Span",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/Finiteness/Defs.lean | finite_def | null |
iff_addMonoid_fg {M : Type*} [AddCommMonoid M] : Module.Finite ℕ M ↔ AddMonoid.FG M :=
⟨fun h => AddMonoid.fg_def.2 <| (Submodule.fg_iff_addSubmonoid_fg ⊤).1 (finite_def.1 h), fun h =>
finite_def.2 <| (Submodule.fg_iff_addSubmonoid_fg ⊤).2 (AddMonoid.fg_def.1 h)⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Data.Set.Finite.Lemmas",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.RingTheory.Ideal.Span",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/Finiteness/Defs.lean | iff_addMonoid_fg | null |
iff_addGroup_fg {G : Type*} [AddCommGroup G] : Module.Finite ℤ G ↔ AddGroup.FG G :=
⟨fun h => AddGroup.fg_def.2 <| (Submodule.fg_iff_addSubgroup_fg ⊤).1 (finite_def.1 h), fun h =>
finite_def.2 <| (Submodule.fg_iff_addSubgroup_fg ⊤).2 (AddGroup.fg_def.1 h)⟩
variable {R M N} | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Data.Set.Finite.Lemmas",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.RingTheory.Ideal.Span",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/Finiteness/Defs.lean | iff_addGroup_fg | null |
exists_fin [Module.Finite R M] : ∃ (n : ℕ) (s : Fin n → M), Submodule.span R (range s) = ⊤ :=
Submodule.fg_iff_exists_fin_generating_family.mp fg_top | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Data.Set.Finite.Lemmas",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.RingTheory.Ideal.Span",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/Finiteness/Defs.lean | exists_fin | See also `Module.Finite.exists_fin'`. |
AddMonoid.FG.to_moduleFinite_nat {M : Type*} [AddCommMonoid M] [FG M] :
Module.Finite ℕ M :=
Module.Finite.iff_addMonoid_fg.mpr ‹_› | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Data.Set.Finite.Lemmas",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.RingTheory.Ideal.Span",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/Finiteness/Defs.lean | AddMonoid.FG.to_moduleFinite_nat | null |
AddMonoid.FG.to_moduleFinite_int {G : Type*} [AddCommGroup G] [FG G] :
Module.Finite ℤ G :=
Module.Finite.iff_addGroup_fg.mpr <| AddGroup.fg_iff_addMonoid_fg.mpr ‹_› | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Data.Set.Finite.Lemmas",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.RingTheory.Ideal.Span",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/Finiteness/Defs.lean | AddMonoid.FG.to_moduleFinite_int | null |
@[algebraize Module.Finite, stacks 0563]
Finite (f : A →+* B) : Prop :=
letI : Algebra A B := f.toAlgebra
Module.Finite A B
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Data.Set.Finite.Lemmas",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.RingTheory.Ideal.Span",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/Finiteness/Defs.lean | Finite | A ring morphism `A →+* B` is `RingHom.Finite` if `B` is finitely generated as `A`-module. |
finite_algebraMap [Algebra A B] :
(algebraMap A B).Finite ↔ Module.Finite A B := by
rw [RingHom.Finite, toAlgebra_algebraMap] | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Data.Set.Finite.Lemmas",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.RingTheory.Ideal.Span",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/Finiteness/Defs.lean | finite_algebraMap | null |
Finite (f : A →ₐ[R] B) : Prop :=
f.toRingHom.Finite | def | RingTheory | [
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Data.Set.Finite.Lemmas",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.RingTheory.Ideal.Span",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/Finiteness/Defs.lean | Finite | An algebra morphism `A →ₐ[R] B` is finite if it is finite as ring morphism.
In other words, if `B` is finitely generated as `A`-module. |
fg_of_fg_map_of_fg_inf_ker (f : M →ₗ[R] P) {s : Submodule R M}
(hs1 : (s.map f).FG)
(hs2 : (s ⊓ LinearMap.ker f).FG) : s.FG := by
haveI := Classical.decEq R
haveI := Classical.decEq M
haveI := Classical.decEq P
obtain ⟨t1, ht1⟩ := hs1
obtain ⟨t2, ht2⟩ := hs2
have : ∀ y ∈ t1, ∃ x ∈ s, f x = y := by
intro y hy
have : y ∈ s.map f := by
rw [← ht1]
exact subset_span hy
rcases mem_map.1 this with ⟨x, hx1, hx2⟩
exact ⟨x, hx1, hx2⟩
have : ∃ g : P → M, ∀ y ∈ t1, g y ∈ s ∧ f (g y) = y := by
choose g hg1 hg2 using this
exists fun y => if H : y ∈ t1 then g y H else 0
intro y H
constructor
· simp only [dif_pos H]
apply hg1
· simp only [dif_pos H]
apply hg2
obtain ⟨g, hg⟩ := this
clear this
exists t1.image g ∪ t2
rw [Finset.coe_union, span_union, Finset.coe_image]
apply le_antisymm
· refine sup_le (span_le.2 <| image_subset_iff.2 ?_) (span_le.2 ?_)
· intro y hy
exact (hg y hy).1
· intro x hx
have : x ∈ span R t2 := subset_span hx
rw [ht2] at this
exact this.1
intro x hx
have : f x ∈ s.map f := by
rw [mem_map]
exact ⟨x, hx, rfl⟩
rw [← ht1, ← Set.image_id (t1 : Set P), Finsupp.mem_span_image_iff_linearCombination] at this
rcases this with ⟨l, hl1, hl2⟩
refine
mem_sup.2
⟨(linearCombination R id).toFun ((lmapDomain R R g : (P →₀ R) → M →₀ R) l), ?_,
x - linearCombination R id ((lmapDomain R R g : (P →₀ R) → M →₀ R) l), ?_,
add_sub_cancel _ _⟩
· rw [← Set.image_id (g '' ↑t1), Finsupp.mem_span_image_iff_linearCombination]
refine ⟨_, ?_, rfl⟩
haveI : Inhabited P := ⟨0⟩
rw [← Finsupp.lmapDomain_supported _ _ g, mem_map]
... | theorem | RingTheory | [
"Mathlib.Algebra.FreeAbelianGroup.Finsupp",
"Mathlib.Algebra.MonoidAlgebra.Module",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Finsupp.lean | fg_of_fg_map_of_fg_inf_ker | If 0 → M' → M → M'' → 0 is exact and M' and M'' are
finitely generated then so is M. |
fg_ker_comp (f : M →ₗ[R] N) (g : N →ₗ[R] P)
(hf1 : (LinearMap.ker f).FG) (hf2 : (LinearMap.ker g).FG)
(hsur : Function.Surjective f) : (LinearMap.ker (g.comp f)).FG := by
rw [LinearMap.ker_comp]
apply fg_of_fg_map_of_fg_inf_ker f
· rwa [Submodule.map_comap_eq, LinearMap.range_eq_top.2 hsur, top_inf_eq]
· rwa [inf_of_le_right (show (LinearMap.ker f) ≤
(LinearMap.ker g).comap f from comap_mono bot_le)] | theorem | RingTheory | [
"Mathlib.Algebra.FreeAbelianGroup.Finsupp",
"Mathlib.Algebra.MonoidAlgebra.Module",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Finsupp.lean | fg_ker_comp | The kernel of the composition of two linear maps is finitely generated if both kernels are and
the first morphism is surjective. |
_root_.Module.Finite.of_submodule_quotient (N : Submodule R M) [Module.Finite R N]
[Module.Finite R (M ⧸ N)] : Module.Finite R M where
fg_top := fg_of_fg_map_of_fg_inf_ker N.mkQ
(by simpa only [map_top, range_mkQ] using Module.finite_def.mp ‹_›) <| by
simpa only [top_inf_eq, ker_mkQ] using Module.Finite.iff_fg.mp ‹_› | theorem | RingTheory | [
"Mathlib.Algebra.FreeAbelianGroup.Finsupp",
"Mathlib.Algebra.MonoidAlgebra.Module",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Finsupp.lean | _root_.Module.Finite.of_submodule_quotient | null |
Module.Finite.finsupp {ι : Type*} [_root_.Finite ι] [Module.Finite R V] :
Module.Finite R (ι →₀ V) :=
Module.Finite.equiv (Finsupp.linearEquivFunOnFinite R V ι).symm | instance | RingTheory | [
"Mathlib.Algebra.FreeAbelianGroup.Finsupp",
"Mathlib.Algebra.MonoidAlgebra.Module",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Finsupp.lean | Module.Finite.finsupp | null |
moduleFinite : Module.Finite R S[ι] := .finsupp | instance | RingTheory | [
"Mathlib.Algebra.FreeAbelianGroup.Finsupp",
"Mathlib.Algebra.MonoidAlgebra.Module",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Finsupp.lean | moduleFinite | null |
moduleFinite : Module.Finite R (MonoidAlgebra S ι) := .finsupp | instance | RingTheory | [
"Mathlib.Algebra.FreeAbelianGroup.Finsupp",
"Mathlib.Algebra.MonoidAlgebra.Module",
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Finsupp.lean | moduleFinite | null |
FG.map {R S : Type*} [Semiring R] [Semiring S] {I : Ideal R} (h : I.FG) (f : R →+* S) :
(I.map f).FG := by
classical
obtain ⟨s, hs⟩ := h
refine ⟨s.image f, ?_⟩
rw [Finset.coe_image, ← Ideal.map_span, hs] | theorem | RingTheory | [
"Mathlib.RingTheory.Finiteness.Finsupp",
"Mathlib.RingTheory.Ideal.Maps"
] | Mathlib/RingTheory/Finiteness/Ideal.lean | FG.map | The image of a finitely generated ideal is finitely generated.
This is the `Ideal` version of `Submodule.FG.map`. |
fg_ker_comp {R S A : Type*} [CommRing R] [CommRing S] [CommRing A] (f : R →+* S)
(g : S →+* A) (hf : (RingHom.ker f).FG) (hg : (RingHom.ker g).FG)
(hsur : Function.Surjective f) :
(RingHom.ker (g.comp f)).FG := by
letI : Algebra R S := RingHom.toAlgebra f
letI : Algebra R A := RingHom.toAlgebra (g.comp f)
letI : Algebra S A := RingHom.toAlgebra g
letI : IsScalarTower R S A := IsScalarTower.of_algebraMap_eq fun _ => rfl
let f₁ := Algebra.linearMap R S
let g₁ := (IsScalarTower.toAlgHom R S A).toLinearMap
exact Submodule.fg_ker_comp f₁ g₁ hf (Submodule.fg_restrictScalars (RingHom.ker g) hg hsur) hsur | theorem | RingTheory | [
"Mathlib.RingTheory.Finiteness.Finsupp",
"Mathlib.RingTheory.Ideal.Maps"
] | Mathlib/RingTheory/Finiteness/Ideal.lean | fg_ker_comp | null |
exists_radical_pow_le_of_fg {R : Type*} [CommSemiring R] (I : Ideal R) (h : I.radical.FG) :
∃ n : ℕ, I.radical ^ n ≤ I := by
have := le_refl I.radical; revert this
refine Submodule.fg_induction _ _ (fun J => J ≤ I.radical → ∃ n : ℕ, J ^ n ≤ I) ?_ ?_ _ h
· intro x hx
obtain ⟨n, hn⟩ := hx (subset_span (Set.mem_singleton x))
exact ⟨n, by rwa [← Ideal.span, span_singleton_pow, span_le, Set.singleton_subset_iff]⟩
· intro J K hJ hK hJK
obtain ⟨n, hn⟩ := hJ fun x hx => hJK <| Ideal.mem_sup_left hx
obtain ⟨m, hm⟩ := hK fun x hx => hJK <| Ideal.mem_sup_right hx
use n + m
rw [← Ideal.add_eq_sup, add_pow, Ideal.sum_eq_sup, Finset.sup_le_iff]
refine fun i _ => Ideal.mul_le_right.trans ?_
obtain h | h := le_or_gt n i
· apply Ideal.mul_le_right.trans ((Ideal.pow_le_pow_right h).trans hn)
· apply Ideal.mul_le_left.trans
refine (Ideal.pow_le_pow_right ?_).trans hm
rw [add_comm, Nat.add_sub_assoc h.le]
apply Nat.le_add_right | theorem | RingTheory | [
"Mathlib.RingTheory.Finiteness.Finsupp",
"Mathlib.RingTheory.Ideal.Maps"
] | Mathlib/RingTheory/Finiteness/Ideal.lean | exists_radical_pow_le_of_fg | null |
exists_pow_le_of_le_radical_of_fg_radical {R : Type*} [CommSemiring R] {I J : Ideal R}
(hIJ : I ≤ J.radical) (hJ : J.radical.FG) :
∃ k : ℕ, I ^ k ≤ J := by
obtain ⟨k, hk⟩ := J.exists_radical_pow_le_of_fg hJ
use k
calc
I ^ k ≤ J.radical ^ k := Ideal.pow_right_mono hIJ _
_ ≤ J := hk | theorem | RingTheory | [
"Mathlib.RingTheory.Finiteness.Finsupp",
"Mathlib.RingTheory.Ideal.Maps"
] | Mathlib/RingTheory/Finiteness/Ideal.lean | exists_pow_le_of_le_radical_of_fg_radical | null |
exists_pow_le_of_le_radical_of_fg {R : Type*} [CommSemiring R] {I J : Ideal R}
(h' : I ≤ J.radical) (h : I.FG) :
∃ n : ℕ, I ^ n ≤ J := by
revert h'
apply Submodule.fg_induction _ _ _ _ _ I h
· intro x hJ
simp only [Ideal.submodule_span_eq, Ideal.span_le,
Set.singleton_subset_iff, SetLike.mem_coe] at hJ
obtain ⟨n, hn⟩ := hJ
refine ⟨n, by simpa [Ideal.span_singleton_pow, Ideal.span_le]⟩
· intro I₁ I₂ h₁ h₂ hJ
obtain ⟨n₁, hn₁⟩ := h₁ (le_sup_left.trans hJ)
obtain ⟨n₂, hn₂⟩ := h₂ (le_sup_right.trans hJ)
use n₁ + n₂
exact Ideal.sup_pow_add_le_pow_sup_pow.trans (sup_le hn₁ hn₂) | lemma | RingTheory | [
"Mathlib.RingTheory.Finiteness.Finsupp",
"Mathlib.RingTheory.Ideal.Maps"
] | Mathlib/RingTheory/Finiteness/Ideal.lean | exists_pow_le_of_le_radical_of_fg | null |
finite_iSup {ι : Sort*} [Finite ι] (S : ι → Submodule R V)
[∀ i, Module.Finite R (S i)] : Module.Finite R ↑(⨆ i, S i) := by
cases nonempty_fintype (PLift ι)
rw [← iSup_plift_down, ← Finset.sup_univ_eq_iSup]
exact Submodule.finite_finset_sup _ _ | instance | RingTheory | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Finiteness.Basic"
] | Mathlib/RingTheory/Finiteness/Lattice.lean | finite_iSup | The submodule generated by a supremum of finite-dimensional submodules, indexed by a finite
sort is finite-dimensional. |
@[stacks 00DV]
exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
[AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
∃ r : R, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = (0 : M) := by
rw [fg_def] at hn
rcases hn with ⟨s, hfs, hs⟩
have H : ∃ r : R, r - 1 ∈ I ∧ N ≤ (I • span R s).comap (LinearMap.lsmul R M r) ∧ s ⊆ N := by
refine ⟨1, ?_, ?_, ?_⟩
· rw [sub_self]
exact I.zero_mem
· rw [hs]
intro n hn
rw [mem_comap]
change (1 : R) • n ∈ I • N
rw [one_smul]
exact hin hn
· rw [← span_le, hs]
clear hin hs
induction s, hfs using Set.Finite.induction_on with
| empty =>
rcases H with ⟨r, hr1, hrn, _⟩
refine ⟨r, hr1, fun n hn => ?_⟩
specialize hrn hn
rwa [mem_comap, span_empty, smul_bot, mem_bot] at hrn
| @insert i s _ _ ih =>
apply ih
rcases H with ⟨r, hr1, hrn, hs⟩
rw [← Set.singleton_union, span_union, smul_sup] at hrn
rw [Set.insert_subset_iff] at hs
have : ∃ c : R, c - 1 ∈ I ∧ c • i ∈ I • span R s := by
specialize hrn hs.1
rw [mem_comap, mem_sup] at hrn
rcases hrn with ⟨y, hy, z, hz, hyz⟩
dsimp at hyz
rw [mem_smul_span_singleton] at hy
rcases hy with ⟨c, hci, rfl⟩
use r - c
constructor
· rw [sub_right_comm]
exact I.sub_mem hr1 hci
· rw [sub_smul, ← hyz, add_sub_cancel_left]
exact hz
rcases this with ⟨c, hc1, hci⟩
refine ⟨c * r, ?_, ?_, hs.2⟩
· simpa only [mul_sub, mul_one, sub_add_sub_cancel] using I.add_mem (I.mul_mem_left c hr1) hc1
· intro n hn
specialize hrn hn
rw [mem_comap, mem_sup] at hrn
rcases hrn with ⟨y, hy, z, hz, hyz⟩
dsimp at hyz
rw [mem_smul_span_singleton] at hy
... | theorem | RingTheory | [
"Mathlib.RingTheory.Finiteness.Defs",
"Mathlib.RingTheory.Ideal.Operations"
] | Mathlib/RingTheory/Finiteness/Nakayama.lean | exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul | **Nakayama's Lemma**. Atiyah-Macdonald 2.5, Eisenbud 4.7, Matsumura 2.2. |
exists_mem_and_smul_eq_self_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
[AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
∃ r ∈ I, ∀ n ∈ N, r • n = n := by
obtain ⟨r, hr, hr'⟩ := exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul I N hn hin
exact ⟨-(r - 1), I.neg_mem hr, fun n hn => by simpa [sub_smul] using hr' n hn⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.Finiteness.Defs",
"Mathlib.RingTheory.Ideal.Operations"
] | Mathlib/RingTheory/Finiteness/Nakayama.lean | exists_mem_and_smul_eq_self_of_fg_of_le_smul | null |
Module.End.isNilpotent_iff_of_finite [Module.Finite R M] {f : End R M} :
IsNilpotent f ↔ ∀ m : M, ∃ n : ℕ, (f ^ n) m = 0 := by
refine ⟨fun ⟨n, hn⟩ m ↦ ⟨n, by simp [hn]⟩, fun h ↦ ?_⟩
rcases Module.Finite.fg_top (R := R) (M := M) with ⟨S, hS⟩
choose g hg using h
use Finset.sup S g
ext m
have hm : m ∈ Submodule.span R S := by simp [hS]
induction hm using Submodule.span_induction with
| mem x hx => exact pow_map_zero_of_le (Finset.le_sup hx) (hg x)
| zero => simp
| add => simp_all
| smul => simp_all | theorem | RingTheory | [
"Mathlib.RingTheory.Finiteness.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas"
] | Mathlib/RingTheory/Finiteness/Nilpotent.lean | Module.End.isNilpotent_iff_of_finite | null |
@[simp]
isNilpotent_transpose_iff :
IsNilpotent Aᵀ ↔ IsNilpotent A := by
simp_rw [IsNilpotent, ← transpose_pow, transpose_eq_zero] | theorem | RingTheory | [
"Mathlib.RingTheory.Finiteness.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas"
] | Mathlib/RingTheory/Finiteness/Nilpotent.lean | isNilpotent_transpose_iff | null |
isNilpotent_iff :
IsNilpotent A ↔ ∀ v, ∃ n : ℕ, A ^ n *ᵥ v = 0 := by
simp_rw [← isNilpotent_toLin'_iff, Module.End.isNilpotent_iff_of_finite, ← toLin'_pow,
toLin'_apply] | theorem | RingTheory | [
"Mathlib.RingTheory.Finiteness.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas"
] | Mathlib/RingTheory/Finiteness/Nilpotent.lean | isNilpotent_iff | null |
isNilpotent_iff_forall_row :
IsNilpotent A ↔ ∀ i, ∃ n : ℕ, (A ^ n).row i = 0 := by
rw [← isNilpotent_transpose_iff, isNilpotent_iff]
refine ⟨fun h i ↦ ?_, fun h v ↦ ?_⟩
· obtain ⟨n, hn⟩ := h (Pi.single i 1)
exact ⟨n, by simpa [← transpose_pow] using hn⟩
· choose n hn using h
suffices ∀ i, (A ^ ⨆ j, n j) i = 0 from ⟨⨆ j, n j, by simp [mulVec_eq_sum, this]⟩
exact fun i ↦ pow_row_eq_zero_of_le (hn i) (Finite.le_ciSup n i) | theorem | RingTheory | [
"Mathlib.RingTheory.Finiteness.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas"
] | Mathlib/RingTheory/Finiteness/Nilpotent.lean | isNilpotent_iff_forall_row | null |
isNilpotent_iff_forall_col :
IsNilpotent A ↔ ∀ i, ∃ n : ℕ, (A ^ n).col i = 0 := by
rw [← isNilpotent_transpose_iff, isNilpotent_iff_forall_row]
simp_rw [← transpose_pow, row_transpose] | theorem | RingTheory | [
"Mathlib.RingTheory.Finiteness.Basic",
"Mathlib.RingTheory.Nilpotent.Lemmas"
] | Mathlib/RingTheory/Finiteness/Nilpotent.lean | isNilpotent_iff_forall_col | null |
FG.prod {sb : Submodule R M} {sc : Submodule R P} (hsb : sb.FG) (hsc : sc.FG) :
(sb.prod sc).FG :=
let ⟨tb, htb⟩ := fg_def.1 hsb
let ⟨tc, htc⟩ := fg_def.1 hsc
fg_def.2
⟨LinearMap.inl R M P '' tb ∪ LinearMap.inr R M P '' tc, (htb.1.image _).union (htc.1.image _),
by rw [LinearMap.span_inl_union_inr, htb.2, htc.2]⟩ | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Prod",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Prod.lean | FG.prod | null |
prod [hM : Module.Finite R M] [hN : Module.Finite R N] : Module.Finite R (M × N) :=
⟨by
rw [← Submodule.prod_top]
exact hM.1.prod hN.1⟩ | instance | RingTheory | [
"Mathlib.LinearAlgebra.Prod",
"Mathlib.RingTheory.Finiteness.Defs"
] | Mathlib/RingTheory/Finiteness/Prod.lean | prod | null |
exists_comp_eq_id_of_projective [Module.Finite R M] [Projective R M] :
∃ (n : ℕ) (f : (Fin n → R) →ₗ[R] M) (g : M →ₗ[R] Fin n → R),
Function.Surjective f ∧ Function.Injective g ∧ f ∘ₗ g = .id :=
have ⟨n, f, surj⟩ := exists_fin' R M
have ⟨g, hfg⟩ := Module.projective_lifting_property f .id surj
⟨n, f, g, surj, LinearMap.injective_of_comp_eq_id _ _ hfg, hfg⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Module.Projective",
"Mathlib.RingTheory.Finiteness.Cardinality"
] | Mathlib/RingTheory/Finiteness/Projective.lean | exists_comp_eq_id_of_projective | null |
module_finite_of_liesOver [Module.Finite A B] : Module.Finite (A ⧸ p) (B ⧸ P) :=
Module.Finite.of_restrictScalars_finite A (A ⧸ p) (B ⧸ P) | instance | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Actions",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Pointwise",
"Mathlib.RingTheory.Ideal.Over"
] | Mathlib/RingTheory/Finiteness/Quotient.lean | module_finite_of_liesOver | `B ⧸ P` is a finite `A ⧸ p`-module if `B` is a finite `A`-module. |
algebra_finiteType_of_liesOver [Algebra.FiniteType A B] :
Algebra.FiniteType (A ⧸ p) (B ⧸ P) :=
Algebra.FiniteType.of_restrictScalars_finiteType A (A ⧸ p) (B ⧸ P) | instance | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Actions",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Pointwise",
"Mathlib.RingTheory.Ideal.Over"
] | Mathlib/RingTheory/Finiteness/Quotient.lean | algebra_finiteType_of_liesOver | `B ⧸ P` is a finitely generated `A ⧸ p`-algebra if `B` is a finitely generated `A`-algebra. |
isNoetherian_of_liesOver [IsNoetherian A B] : IsNoetherian (A ⧸ p) (B ⧸ P) :=
isNoetherian_of_tower A inferInstance | instance | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Actions",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Pointwise",
"Mathlib.RingTheory.Ideal.Over"
] | Mathlib/RingTheory/Finiteness/Quotient.lean | isNoetherian_of_liesOver | `B ⧸ P` is a Noetherian `A ⧸ p`-module if `B` is a Noetherian `A`-module. |
QuotientMapQuotient.isNoetherian [IsNoetherian A B] :
IsNoetherian (A ⧸ p) (B ⧸ p.map (algebraMap A B)) :=
isNoetherian_of_tower A <|
isNoetherian_of_surjective B (Ideal.Quotient.mkₐ A _).toLinearMap <|
LinearMap.range_eq_top.mpr Ideal.Quotient.mk_surjective | instance | RingTheory | [
"Mathlib.Algebra.Group.Subgroup.Actions",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Pointwise",
"Mathlib.RingTheory.Ideal.Over"
] | Mathlib/RingTheory/Finiteness/Quotient.lean | QuotientMapQuotient.isNoetherian | null |
small_sup {P Q : Submodule R M} [smallP : Small.{u} P] [smallQ : Small.{u} Q] :
Small.{u} (P ⊔ Q : Submodule R M) := by
rw [Submodule.sup_eq_range]
exact small_range _ | instance | RingTheory | [
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.RingTheory.FiniteType",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic",
"Mathlib.RingTheory.MvPo... | Mathlib/RingTheory/Finiteness/Small.lean | small_sup | null |
small_iSup
{ι : Type*} {P : ι → Submodule R M} [Small.{u} ι] [∀ i, Small.{u} (P i)] :
Small.{u} (iSup P : Submodule R M) := by
classical
rw [iSup_eq_range_dfinsupp_lsum]
apply small_range | instance | RingTheory | [
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.RingTheory.FiniteType",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic",
"Mathlib.RingTheory.MvPo... | Mathlib/RingTheory/Finiteness/Small.lean | small_iSup | null |
FG.small [Small.{u} R] (P : Submodule R M) (hP : P.FG) : Small.{u} P := by
rw [fg_iff_exists_fin_generating_family] at hP
obtain ⟨n, s, rfl⟩ := hP
rw [← Fintype.range_linearCombination]
apply small_range
variable (R M) in | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.RingTheory.FiniteType",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic",
"Mathlib.RingTheory.MvPo... | Mathlib/RingTheory/Finiteness/Small.lean | FG.small | null |
_root_.Module.Finite.small [Small.{u} R] [Module.Finite R M] : Small.{u} M := by
have : Small.{u} (⊤ : Submodule R M) :=
FG.small _ (Module.finite_def.mp inferInstance)
rwa [← small_univ_iff] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.RingTheory.FiniteType",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic",
"Mathlib.RingTheory.MvPo... | Mathlib/RingTheory/Finiteness/Small.lean | _root_.Module.Finite.small | null |
small_span_singleton [Small.{u} R] (m : M) :
Small.{u} (span R {m}) := FG.small _ (fg_span_singleton _) | instance | RingTheory | [
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.RingTheory.FiniteType",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic",
"Mathlib.RingTheory.MvPo... | Mathlib/RingTheory/Finiteness/Small.lean | small_span_singleton | null |
small_span [Small.{u} R] (s : Set M) [Small.{u} s] :
Small.{u} (span R s) := by
suffices span R s = iSup (fun i : s ↦ span R ({(↑i : M)} : Set M)) by
rw [this]
apply small_iSup
simp [← Submodule.span_iUnion] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.RingTheory.FiniteType",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic",
"Mathlib.RingTheory.MvPo... | Mathlib/RingTheory/Finiteness/Small.lean | small_span | null |
small_adjoin [Small.{u} R] {s : Set S} [Small.{u} s] :
Small.{u} (adjoin R s : Subalgebra R S) := by
rw [Algebra.adjoin_eq_range]
apply small_range | instance | RingTheory | [
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.RingTheory.FiniteType",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic",
"Mathlib.RingTheory.MvPo... | Mathlib/RingTheory/Finiteness/Small.lean | small_adjoin | null |
_root_.Subalgebra.FG.small [Small.{u} R] {A : Subalgebra R S} (fgS : A.FG) :
Small.{u} A := by
obtain ⟨s, hs, rfl⟩ := fgS
exact small_adjoin | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.RingTheory.FiniteType",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Basis.Cardinality",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.RingTheory.Finiteness.Basic",
"Mathlib.RingTheory.MvPo... | Mathlib/RingTheory/Finiteness/Small.lean | _root_.Subalgebra.FG.small | null |
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