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12dee50 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 | 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
end
end
end OAI
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