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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
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