class
SheafOfModules.IsCoherent
{C : Type u_1}
[CategoryTheory.Category.{u_2, u_1} C]
{J : CategoryTheory.GrothendieckTopology C}
{R : CategoryTheory.Sheaf J RingCat}
[∀ (X : C), (J.over X).HasSheafCompose (CategoryTheory.forget₂ RingCat AddCommGrpCat)]
[∀ (X : C), CategoryTheory.HasWeakSheafify (J.over X) AddCommGrpCat]
[∀ (X : C), (J.over X).WEqualsLocallyBijective AddCommGrpCat]
(M : SheafOfModules R)
:
A sheaf of modules M is coherent if it is quasi-coherent and of finite type
(locally generated by finitely many sections).
- isQuasicoherent : M.IsQuasicoherent
- isFiniteType : M.IsFiniteType
Instances
instance
SheafOfModules.instIsCoherentOfIsFinitePresentation
{C : Type u_1}
[CategoryTheory.Category.{u_2, u_1} C]
{J : CategoryTheory.GrothendieckTopology C}
{R : CategoryTheory.Sheaf J RingCat}
[∀ (X : C), (J.over X).HasSheafCompose (CategoryTheory.forget₂ RingCat AddCommGrpCat)]
[∀ (X : C), CategoryTheory.HasWeakSheafify (J.over X) AddCommGrpCat]
[∀ (X : C), (J.over X).WEqualsLocallyBijective AddCommGrpCat]
(M : SheafOfModules R)
[M.IsFinitePresentation]
:
Algebraic counterpart: an R-module M is "coherent" in our sense iff it is
finitely generated as an R-module.
Instances For
theorem
isCoherentModule_iff_finite
{R : Type u_1}
[CommRing R]
(M : Type u_2)
[AddCommGroup M]
[Module R M]
:
Unfolds the definition: IsCoherentModule R M is by definition Module.Finite R M.
theorem
fg_noetherian_finitePresentation
{R : Type u_1}
[CommRing R]
[IsNoetherianRing R]
(M : Type u_2)
[AddCommGroup M]
[Module R M]
[Module.Finite R M]
:
Over a Noetherian ring, every finitely generated module is finitely presented.
theorem
isCoherentModule_iff_finitePresentation
{R : Type u_1}
[CommRing R]
[IsNoetherianRing R]
(M : Type u_2)
[AddCommGroup M]
[Module R M]
:
Over a Noetherian ring, coherent (= finitely generated) and finitely presented coincide.