| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Cohomology.FiniteSupportCohomology |
| import OAI.AlgebraicGeometry.Seshadri.Sheaves.ClosedModuleExact |
| import OAI.AlgebraicGeometry.Seshadri.Intersection.EulerExact |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| |
| section |
|
|
| namespace MaximalSeshadri.FiniteSupport |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits AlgebraicGeometry MaximalSeshadri.Geometry |
|
|
| variable {X Y : Scheme.{0}} |
|
|
| theorem finite_pushforward_finiteDimensional (f : X ⟶ Y) |
| (g : Y ⟶ Spec (CommRingCat.of ℂ)) [IsProper (f ≫ g)] [Finite X] (n : ℕ) : |
| let M := (Scheme.Modules.pushforward f).obj (structureSheaf X) |
| let _ := Module.compHom (cohomology M n) (baseScalars g) |
| FiniteDimensional ℂ (cohomology M n) := by |
| let M := (Scheme.Modules.pushforward f).obj (structureSheaf X) |
| let := Module.compHom (cohomology M n) (baseScalars g) |
| cases n with |
| | zero => |
| let : Module ℂ Γ(M,⊤) := Module.compHom _ (baseScalars g) |
| let : Algebra ℂ Γ(X,⊤) := (baseScalars (f ≫ g)).toAlgebra |
| let : FiniteDimensional ℂ Γ(X,⊤) := proper_finite_functions (f ≫ g) |
| let e := (cohomologyZeroSections g M).trans (pushforwardUnitSections f g) |
| exact Module.Finite.of_surjective e.symm.toLinearMap e.symm.surjective |
| | succ n => |
| let : Subsingleton (cohomology M (n+1)) := |
| ⟨fun x y => (finite_pushforward_ext_zero (f ≫ g) f _ n x).trans |
| (finite_pushforward_ext_zero (f ≫ g) f _ n y).symm⟩ |
| infer_instance |
|
|
| theorem closed_ideal_euler_difference (f : X ⟶ Y) |
| (g : Y ⟶ Spec (CommRingCat.of ℂ)) [IsClosedImmersion f] |
| [IsProper (f ≫ g)] [Finite X] (d : ℕ) |
| (hfiniteI : ∀ n ≤ d, |
| let _ := Module.compHom (cohomology (IdealModule.idealModule f) n) (baseScalars g) |
| FiniteDimensional ℂ (cohomology (IdealModule.idealModule f) n)) |
| (hfiniteO : ∀ n ≤ d, |
| let _ := Module.compHom (cohomology (structureSheaf Y) n) (baseScalars g) |
| FiniteDimensional ℂ (cohomology (structureSheaf Y) n)) |
| [Subsingleton (cohomology (IdealModule.idealModule f) (d+1))] : |
| eulerCharacteristic g d (structureSheaf Y) - |
| eulerCharacteristic g d (IdealModule.idealModule f) = |
| let _ : Algebra ℂ Γ(X,⊤) := (baseScalars (f ≫ g)).toAlgebra |
| (Module.finrank ℂ Γ(X,⊤) : ℤ) := by |
| have h := eulerCharacteristic_add g (IdealModule.closedSequence_exact f) d |
| hfiniteI hfiniteO (fun n _ => finite_pushforward_finiteDimensional f g n) |
| change eulerCharacteristic g d (structureSheaf Y) = |
| eulerCharacteristic g d (IdealModule.idealModule f) + |
| eulerCharacteristic g d ((Scheme.Modules.pushforward f).obj (structureSheaf X)) at h |
| rw [h, add_sub_cancel_left, finite_pushforward_euler] |
|
|
| end |
| end MaximalSeshadri.FiniteSupport |
| end |
|
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|
|
| end |
| end |
|
|
| end OAI |
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|
