| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Sheaves.DiscreteLineBundle |
| import OAI.AlgebraicGeometry.Seshadri.Cohomology.CohomologyIso |
| import OAI.AlgebraicGeometry.Seshadri.Intersection.IntersectionEuler |
| import OAI.AlgebraicGeometry.Seshadri.LocalAlgebra.FiniteStalkSum |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| |
| section |
|
|
| namespace MaximalSeshadri.FiniteSupport |
| noncomputable section |
| open CategoryTheory AlgebraicGeometry MaximalSeshadri.Geometry |
|
|
| variable {X Y : Scheme.{0}} |
|
|
| theorem finite_line_pushforward_euler (f : X ⟶ Y) |
| (g : Y ⟶ Spec (CommRingCat.of ℂ)) [IsProper (f ≫ g)] [Finite X] |
| (L : LineBundle X) (d : ℕ) : |
| eulerCharacteristic g d ((Scheme.Modules.pushforward f).obj L.sheaf) = |
| let _ : Algebra ℂ Γ(X,⊤) := (baseScalars (f ≫ g)).toAlgebra |
| (Module.finrank ℂ Γ(X,⊤) : ℤ) := by |
| let : DiscreteTopology X := proper_finite_discrete (f ≫ g) |
| obtain ⟨e⟩ := lineBundle_trivial L |
| rw [eulerCharacteristic_iso g ((Scheme.Modules.pushforward f).mapIso e) d] |
| exact finite_pushforward_euler f g d |
|
|
| theorem finite_line_pushforward_finiteDimensional (f : X ⟶ Y) |
| (g : Y ⟶ Spec (CommRingCat.of ℂ)) [IsProper (f ≫ g)] [Finite X] |
| (L : LineBundle X) (n : ℕ) : |
| let M := (Scheme.Modules.pushforward f).obj L.sheaf |
| let _ := Module.compHom (cohomology M n) (baseScalars g) |
| FiniteDimensional ℂ (cohomology M n) := by |
| let : DiscreteTopology X := proper_finite_discrete (f ≫ g) |
| obtain ⟨e⟩ := lineBundle_trivial L |
| let M := (Scheme.Modules.pushforward f).obj L.sheaf |
| let N := (Scheme.Modules.pushforward f).obj (structureSheaf X) |
| let := Module.compHom (cohomology M n) (baseScalars g) |
| let := Module.compHom (cohomology N n) (baseScalars g) |
| let : FiniteDimensional ℂ (cohomology N n) := finite_pushforward_finiteDimensional f g n |
| exact Module.Finite.of_surjective |
| (cohomologyIso g ((Scheme.Modules.pushforward f).mapIso e) n).symm.toLinearMap |
| (cohomologyIso g ((Scheme.Modules.pushforward f).mapIso e) n).symm.surjective |
|
|
| theorem finite_line_euler_eq_sum_stalks (f : X ⟶ Y) |
| (g : Y ⟶ Spec (CommRingCat.of ℂ)) [IsProper (f ≫ g)] [Fintype X] |
| (L : LineBundle X) (d : ℕ) : |
| eulerCharacteristic g d ((Scheme.Modules.pushforward f).obj L.sheaf) = |
| ∑ x : X, let _ := structuralStalkAlgebra (f ≫ g) x |
| (Module.finrank ℂ (X.presheaf.stalk x) : ℤ) := by |
| rw [finite_line_pushforward_euler] |
| dsimp only |
| have h := proper_finite_finrank_eq_sum_stalks (f ≫ g) |
| dsimp only at h |
| exact_mod_cast h |
|
|
| end |
| end MaximalSeshadri.FiniteSupport |
| end |
|
|
|
|
| end |
| end |
|
|
| end OAI |
|
|
