Finitely Presented Modules #

Main definition #

Module.FinitePresentation: A module is finitely presented if it is generated by some finite set s and the kernel of the presentation Rˢ → M is also finitely generated.

Main results #

Module.finitePresentation_iff_finite: If R is noetherian, then f.p. iff f.g. on R-modules.

Suppose 0 → K → M → N → 0 is an exact sequence of R-modules.

Module.finitePresentation_of_surjective: If M is f.p., K is f.g., then N is f.p.
Module.FinitePresentation.fg_ker: If M is f.g., N is f.p., then K is f.g.
Module.finitePresentation_of_ker: If N and K is f.p., then M is also f.p.
Module.FinitePresentation.isLocalizedModule_map: If M and N are R-modules and M is f.p., and S is a submonoid of R, then Hom(Mₛ, Nₛ) is the localization of Hom(M, N).

Also the instances finite + free => f.p. => finite are also provided

TODO #

Suppose S is an R-algebra, M is an S-module. Then

If S is f.p., then M is R-f.p. implies M is S-f.p.
If S is both f.p. (as an algebra) and finite (as a module), then M is S-fp implies that M is R-f.p.
If S is f.p. as a module, then S is f.p. as an algebra. In particular,
S is f.p. as an R-module iff it is f.p. as an algebra and is finite as a module.

For finitely presented algebras, see Algebra.FinitePresentation in file Mathlib.RingTheory.FinitePresentation.

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class Module.FinitePresentation (R : Type u_1) (M : Type u_2) [Semiring R] [AddCommMonoid M] [Module R M] :

Prop

A module is finitely presented if it is finitely generated by some set s and the kernel of the presentation Rˢ → M is also finitely generated.

out : ∃ (s : Finset M), Submodule.span R ↑s = ⊤ ∧ (LinearMap.ker (Finsupp.linearCombination R Subtype.val)).FG

Instances

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@[instance 100]

instance instFiniteOfFinitePresentation (R : Type u_1) (M : Type u_2) [Semiring R] [AddCommMonoid M] [Module R M] [h : Module.FinitePresentation R M] :

Module.Finite R M

Equations

⋯ = ⋯

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theorem Module.FinitePresentation.equiv_quotient (R M : Type u) [Ring R] [AddCommGroup M] [Module R M] [fp : Module.FinitePresentation R M] :

∃ (L : Type u) (x : AddCommGroup L) (x_1 : Module R L) (K : Submodule R L) (x_2 : M ≃ₗ[R] L ⧸ K), Module.Free R L ∧ Module.Finite R L ∧ K.FG

A finitely presented module is isomorphic to the quotient of a finite free module by a finitely generated submodule.

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theorem Module.finitePresentation_of_finite (R : Type u_1) (M : Type u_2) [Ring R] [AddCommGroup M] [Module R M] [IsNoetherianRing R] [h : Module.Finite R M] :

Module.FinitePresentation R M

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theorem Module.finitePresentation_iff_finite (R : Type u_1) (M : Type u_2) [Ring R] [AddCommGroup M] [Module R M] [IsNoetherianRing R] :

Module.FinitePresentation R M ↔ Module.Finite R M

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theorem Module.finitePresentation_of_free_of_surjective {R : Type u_1} {M : Type u_2} {N : Type u_3} [Ring R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module.Free R M] [Module.Finite R M] (l : M →ₗ[R] N) (hl : Function.Surjective ⇑l) (hl' : (LinearMap.ker l).FG) :

Module.FinitePresentation R N

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theorem Module.finitePresentation_of_projective (R : Type u_1) (M : Type u_2) [Ring R] [AddCommGroup M] [Module R M] [Module.Projective R M] [Module.Finite R M] :

Module.FinitePresentation R M

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@[deprecated Module.finitePresentation_of_projective]

theorem Module.finitePresentation_of_free (R : Type u_1) (M : Type u_2) [Ring R] [AddCommGroup M] [Module R M] [Module.Projective R M] [Module.Finite R M] :

Module.FinitePresentation R M

Alias of Module.finitePresentation_of_projective.

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instance instFinitePresentation {R : Type u_1} [Ring R] :

Module.FinitePresentation R R

Equations

⋯ = ⋯

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instance instFinitePresentationFinsupp {R : Type u_1} [Ring R] {ι : Type u_2} [Finite ι] :

Module.FinitePresentation R (ι →₀ R)

Equations

⋯ = ⋯

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instance instFinitePresentationForall {R : Type u_1} [Ring R] {ι : Type u_2} [Finite ι] :

Module.FinitePresentation R (ι → R)

Equations

⋯ = ⋯

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theorem Module.finitePresentation_of_surjective {R : Type u_1} {M : Type u_2} {N : Type u_3} [Ring R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [h : Module.FinitePresentation R M] (l : M →ₗ[R] N) (hl : Function.Surjective ⇑l) (hl' : (LinearMap.ker l).FG) :

Module.FinitePresentation R N

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theorem Module.FinitePresentation.fg_ker {R : Type u_1} {M : Type u_2} {N : Type u_3} [Ring R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module.Finite R M] [h : Module.FinitePresentation R N] (l : M →ₗ[R] N) (hl : Function.Surjective ⇑l) :

(LinearMap.ker l).FG

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theorem Module.FinitePresentation.fg_ker_iff {R : Type u_1} {M : Type u_2} {N : Type u_3} [Ring R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module.FinitePresentation R M] (l : M →ₗ[R] N) (hl : Function.Surjective ⇑l) :

(LinearMap.ker l).FG ↔ Module.FinitePresentation R N

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theorem Module.finitePresentation_of_ker {R : Type u_1} {M : Type u_3} {N : Type u_2} [Ring R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module.FinitePresentation R N] (l : M →ₗ[R] N) (hl : Function.Surjective ⇑l) [Module.FinitePresentation R ↥(LinearMap.ker l)] :

Module.FinitePresentation R M

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theorem Module.FinitePresentation.exists_lift_of_isLocalizedModule {R : Type u_1} {M : Type u_2} {N : Type u_4} {N' : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [AddCommGroup N'] [Module R N'] (S : Submonoid R) (f : N →ₗ[R] N') [IsLocalizedModule S f] [h : Module.FinitePresentation R M] (g : M →ₗ[R] N') :

∃ (h : M →ₗ[R] N) (s : ↥S), f ∘ₗ h = s • g

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theorem Module.Finite.exists_smul_of_comp_eq_of_isLocalizedModule {R : Type u_1} {M : Type u_2} {N : Type u_3} {N' : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [AddCommGroup N'] [Module R N'] (S : Submonoid R) (f : N →ₗ[R] N') [IsLocalizedModule S f] [hM : Module.Finite R M] (g₁ g₂ : M →ₗ[R] N) (h : f ∘ₗ g₁ = f ∘ₗ g₂) :

∃ (s : ↥S), s • g₁ = s • g₂

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theorem exists_bijective_map_powers {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (S : Submonoid R) {M' : Type u_1} [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') [IsLocalizedModule S f] {N' : Type u_2} [AddCommGroup N'] [Module R N'] (g : N →ₗ[R] N') [IsLocalizedModule S g] [Module.Finite R M] [Module.FinitePresentation R N] (l : M →ₗ[R] N) (hf : Function.Bijective ⇑((IsLocalizedModule.map S f g) l)) :

∃ r ∈ S, ∀ (t : R), r ∣ t → Function.Bijective ⇑((LocalizedModule.map (Submonoid.powers t)) l)

Let M be a finite R-module, and N be a finitely presented R-module. If l : M →ₗ[R] N is a linear map whose localization at S : Submonoid R is bijective, then l is already bijective under the localization at some r ∈ S.

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theorem Module.FinitePresentation.exists_lift_equiv_of_isLocalizedModule {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (S : Submonoid R) {M' : Type u_1} [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') [IsLocalizedModule S f] {N' : Type u_2} [AddCommGroup N'] [Module R N'] (g : N →ₗ[R] N') [IsLocalizedModule S g] [Module.FinitePresentation R M] [Module.FinitePresentation R N] (l : M' ≃ₗ[R] N') :

∃ (r : R) (hr : r ∈ S) (l' : LocalizedModule (Submonoid.powers r) M ≃ₗ[Localization (Submonoid.powers r)] LocalizedModule (Submonoid.powers r) N), LocalizedModule.lift (Submonoid.powers r) g ⋯ ∘ₗ ↑R ↑l' = ↑l ∘ₗ LocalizedModule.lift (Submonoid.powers r) f ⋯

Let M N be a finitely presented R-modules. Any Mₛ ≃ₗ[R] Nₛ between the localizations at S : Submonoid R can be lifted to an isomorphism between Mᵣ ≃ₗ[R] Nᵣ for some r ∈ S.

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