Module

OModulePresheaf_isCoherent_ker.lean

Nyll.AlgebraicCurves.Cohomology.Sheaves.OModulePresheaf_isCoherent_ker
2 declarations indexed from sourceView source ↗

OModule Presheaf is Coherent ker

Declarations

AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_isCoherent_ker.AlgebraicGeometry.OModulePresheaf.G4T.isCoherent_ker
Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_isCoherent_ker
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.TopologicalSpace
open _root_.Opposite
noncomputable section
namespace AlgebraicGeometry
open _root_.AlgebraicGeometry
namespace OModulePresheaf
open _root_.AlgebraicGeometry.OModulePresheaf
namespace G4T
open _root_.AlgebraicGeometry.OModulePresheaf
open _root_.AlgebraicGeometry
universe u
section PModAlg
variable {R : Type u} [CommRing R] {V : Scheme.{u}} {π : V ⟶ Spec (.of R)}
variable {F G : OModulePresheaf π} (φ : OModulePresheaf.Hom F G)
variable [IsLocallyNoetherian V]
theorem isCoherent_ker (hF : OModulePresheaf.IsCoherent F) :
    OModulePresheaf.IsCoherent (OModulePresheaf.ker φ)
AlgebraicGeometry.OModulePresheaf.isCoherent_ker
Source context 8 commands

Surrounding namespaces, section variables, and local commands at this declaration.

section ProofImplementation
namespace AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_isCoherent_ker
open _root_.AlgebraicGeometry
open _root_.CategoryTheory
open _root_.TopologicalSpace
open _root_.Opposite
universe u
open _root_.AlgebraicGeometry
  _root_.AlgebraicCurves.Cohomology.Sheaves.Proofs.OModulePresheaf_isCoherent_ker.AlgebraicGeometry
  in
theorem _root_.AlgebraicGeometry.OModulePresheaf.isCoherent_ker {R : Type u} [CommRing R] {V :
  Scheme.{u}} {π : V ⟶ Spec (.of R)} [IsLocallyNoetherian V] {F G : OModulePresheaf π} (φ :
  OModulePresheaf.Hom F G) (hF : F.IsCoherent) : (OModulePresheaf.ker φ).IsCoherent