| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Model |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.FiniteScheme |
| import OAI.AlgebraicGeometry.Seshadri.Cohomology.AffineVanishing |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| |
| section |
|
|
| namespace MaximalSeshadri.Geometry |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Abelian Opposite |
|
|
| variable {X : Scheme} |
|
|
| def globalHomLinearEquiv (M : X.Modules) : |
| GlobalSections X M ≃ₗ[Γ(X,⊤)] Γ(M,⊤) where |
| toEquiv := FlasqueCohomology.globalHomEquiv X.ringCatSheaf M |
| map_add' := by intro f g; rfl |
| map_smul' := by |
| intro r f |
| change restrictScalar X ⊤ r • f.app ⊤ (1 : Γ(X,⊤)) = r • f.app ⊤ (1 : Γ(X,⊤)) |
| congr 1 |
| change (X.presheaf.map (𝟙 (op ⊤))) r = r |
| rw [X.presheaf.map_id] |
| rfl |
|
|
| def cohomologyZeroSections (f : X ⟶ Spec (CommRingCat.of ℂ)) (M : X.Modules) : |
| let _ : Module ℂ (cohomology M 0) := Module.compHom _ (baseScalars f) |
| let _ : Module ℂ Γ(M,⊤) := Module.compHom _ (baseScalars f) |
| cohomology M 0 ≃ₗ[ℂ] Γ(M,⊤) := by |
| dsimp only |
| letI : Module ℂ (cohomology M 0) := Module.compHom _ (baseScalars f) |
| letI : Module ℂ Γ(M,⊤) := Module.compHom _ (baseScalars f) |
| let e := (Ext.linearEquiv₀ (R := Γ(X,⊤)) (X := structureSheaf X) (Y := M)).trans |
| (globalHomLinearEquiv M) |
| exact { e.toAddEquiv with map_smul' := fun r x => e.map_smul (baseScalars f r) x } |
|
|
| lemma cohomologyDimension_zero (f : X ⟶ Spec (CommRingCat.of ℂ)) (M : X.Modules) : |
| cohomologyDimension f M 0 = |
| let _ : Module ℂ Γ(M,⊤) := Module.compHom _ (baseScalars f) |
| Module.finrank ℂ Γ(M,⊤) := by |
| let : Module ℂ (cohomology M 0) := Module.compHom _ (baseScalars f) |
| let : Module ℂ Γ(M,⊤) := Module.compHom _ (baseScalars f) |
| exact (cohomologyZeroSections f M).finrank_eq |
|
|
| theorem affine_eulerCharacteristic (f : X ⟶ Spec (CommRingCat.of ℂ)) |
| [IsAffine X] [IsNoetherian X] (M : X.Modules) [M.IsQuasicoherent] (d : ℕ) : |
| eulerCharacteristic f d M = |
| let _ : Module ℂ Γ(M,⊤) := Module.compHom _ (baseScalars f) |
| (Module.finrank ℂ Γ(M,⊤) : ℤ) := by |
| unfold eulerCharacteristic |
| rw [Finset.sum_eq_single 0] |
| · simp only [pow_zero, one_mul, cohomologyDimension_zero] |
| · intro n hn hn0 |
| obtain ⟨m,rfl⟩ := Nat.exists_eq_succ_of_ne_zero hn0 |
| have : Subsingleton (cohomology M (m+1)) := |
| ⟨fun x y => (AffineSchemeCohomology.affine_ext_zero X M m x).trans |
| (AffineSchemeCohomology.affine_ext_zero X M m y).symm⟩ |
| simp [cohomologyDimension, Module.finrank_zero_of_subsingleton] |
| · simp |
|
|
| theorem proper_finite_euler (f : X ⟶ Spec (CommRingCat.of ℂ)) |
| [IsProper f] [Finite X] (d : ℕ) : |
| eulerCharacteristic f d (structureSheaf X) = |
| let _ : Algebra ℂ Γ(X,⊤) := (baseScalars f).toAlgebra |
| (Module.finrank ℂ Γ(X,⊤) : ℤ) := by |
| let : IsAffine X := proper_finite_scheme_isAffine f |
| let : IsLocallyNoetherian X := LocallyOfFiniteType.isLocallyNoetherian f |
| let : IsNoetherian X := ⟨⟩ |
| let : (structureSheaf X).IsQuasicoherent := |
| (SheafOfModules.isQuasicoherent X.ringCatSheaf).prop_of_iso |
| (CategoryTheory.Limits.coproductUniqueIso |
| (fun (_ : PUnit) => SheafOfModules.unit X.ringCatSheaf)) |
| (inferInstanceAs (SheafOfModules.free (R := X.ringCatSheaf) PUnit).IsQuasicoherent) |
| exact affine_eulerCharacteristic f (structureSheaf X) d |
|
|
| end |
| end MaximalSeshadri.Geometry |
| end |
|
|
|
|
| end |
| end |
|
|
| end OAI |
|
|
