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

1. If S is f.p., then M is R-f.p. implies M is S-f.p.
2. 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.
3. If S is f.p. as a module, then S is f.p. as an algebra. In particular,
4. 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.

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

@[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

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.

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

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

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

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

@[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.

instance instFinitePresentation {R : Type u_1} [Ring R] : Module.FinitePresentation R R

instance instFinitePresentationFinsupp {R : Type u_1} [Ring R] {ι : Type u_2} [Finite ι] : Module.FinitePresentation R (ι →₀ R)

instance instFinitePresentationForall {R : Type u_1} [Ring R] {ι : Type u_2} [Finite ι] : Module.FinitePresentation R (ι → R)

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

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

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

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

instance instFinitePresentationTensorProduct {R : Type u_2} {M : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] {A : Type u_1} [CommRing A] [Algebra R A] [Module.FinitePresentation R M] : Module.FinitePresentation A (TensorProduct R A M)

theorem FinitePresentation.of_isBaseChange {R : Type u_2} {M : Type u_4} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {A : Type u_1} [CommRing A] [Algebra R A] [Module A N] [IsScalarTower R A N] (f : M →ₗ[R] N) (h : IsBaseChange A f) [Module.FinitePresentation R M] : Module.FinitePresentation A N

instance instFinitePresentationLocalizationLocalizedModule {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (S : Submonoid R) [Module.FinitePresentation R M] : Module.FinitePresentation (Localization S) (LocalizedModule S M)

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

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

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.

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.

instance Module.FinitePresentation.isLocalizedModule_map {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] : IsLocalizedModule S (IsLocalizedModule.map S f g)

instance Module.FinitePresentation.isLocalizedModule_mapExtendScalars {R : Type u_4} {M : Type u_5} {N : Type u_6} [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] (Rₛ : Type u_3) [CommRing Rₛ] [Algebra R Rₛ] [Module Rₛ M'] [Module Rₛ N'] [IsScalarTower R Rₛ M'] [IsScalarTower R Rₛ N'] [IsLocalization S Rₛ] [Module.FinitePresentation R M] : IsLocalizedModule S (IsLocalizedModule.mapExtendScalars S f g Rₛ)

instance instIsLocalizedModuleLinearMapIdLocalizationLocalizedModuleMapOfFinitePresentation {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) [Module.FinitePresentation R M] : IsLocalizedModule S (LocalizedModule.map S)