Module

OModulePresheafEulerChar.lean

Nyll.AlgebraicCurves.Cohomology.Sheaves.OModulePresheafEulerChar
5 declarations indexed from sourceView source ↗

OModule Presheaf Euler Char

Imports2
Imported by27

Declarations

AlgebraicGeometry.OModulePresheaf.cechFinrank

The rank of Čech cohomology in the specified degree.

Source context 6 commands

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

noncomputable section
universe u
namespace AlgebraicGeometry.OModulePresheaf
open CategoryTheory
variable {R : Type u} [CommRing R] {V : Scheme.{u}} {π : V ⟶ Spec (.of R)}
variable (F : OModulePresheaf π) (K : V.OrderedAffineCover)
def cechFinrank : ℕ → ℕ
  | 0 => Module.finrank R (F.H0 K)
  | i + 1 => Module.finrank R (F.HSucc K i)
AlgebraicGeometry.OModulePresheaf.cechFinrank_zero
Source context 6 commands

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

noncomputable section
universe u
namespace AlgebraicGeometry.OModulePresheaf
open CategoryTheory
variable {R : Type u} [CommRing R] {V : Scheme.{u}} {π : V ⟶ Spec (.of R)}
variable (F : OModulePresheaf π) (K : V.OrderedAffineCover)
theorem cechFinrank_zero : F.cechFinrank K 0 = Module.finrank R (F.H0 K)
AlgebraicGeometry.OModulePresheaf.cechFinrank_succ
Source context 6 commands

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

noncomputable section
universe u
namespace AlgebraicGeometry.OModulePresheaf
open CategoryTheory
variable {R : Type u} [CommRing R] {V : Scheme.{u}} {π : V ⟶ Spec (.of R)}
variable (F : OModulePresheaf π) (K : V.OrderedAffineCover)
theorem cechFinrank_succ (i : ℕ) : F.cechFinrank K (i + 1) = Module.finrank R (F.HSucc K i)
AlgebraicGeometry.OModulePresheaf.eulerChar

The alternating sum of the ranks of Čech cohomology for the finite cover.

Source context 6 commands

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

noncomputable section
universe u
namespace AlgebraicGeometry.OModulePresheaf
open CategoryTheory
variable {R : Type u} [CommRing R] {V : Scheme.{u}} {π : V ⟶ Spec (.of R)}
variable (F : OModulePresheaf π) (K : V.OrderedAffineCover)
def eulerChar : ℤ
AlgebraicGeometry.OModulePresheaf.eulerChar_def
Source context 6 commands

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

noncomputable section
universe u
namespace AlgebraicGeometry.OModulePresheaf
open CategoryTheory
variable {R : Type u} [CommRing R] {V : Scheme.{u}} {π : V ⟶ Spec (.of R)}
variable (F : OModulePresheaf π) (K : V.OrderedAffineCover)
theorem eulerChar_def :
    F.eulerChar K = ∑ i ∈ Finset.range (Fintype.card K.ι), (-1 : ℤ) ^ i * (F.cechFinrank K i : ℤ)