| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.LocalAlgebra.FiniteStalkSum |
| import OAI.AlgebraicGeometry.Seshadri.LocalAlgebra.ClosedStalk |
|
|
| namespace OAI |
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| section |
| noncomputable section |
| |
| section |
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|
| namespace MaximalSeshadri.Geometry |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
|
|
| variable {K : Type} [Field K] {X Y : Scheme} |
|
|
| def structuralStalkMap (g : X ⟶ Spec (CommRingCat.of K)) (f : Y ⟶ X) (y : Y) : |
| let _ := structuralStalkAlgebra g (f y) |
| let _ := structuralStalkAlgebra (f ≫ g) y |
| (X.presheaf.stalk (f y)) →ₐ[K] (Y.presheaf.stalk y) := by |
| dsimp only |
| letI := structuralStalkAlgebra g (f y) |
| letI := structuralStalkAlgebra (f ≫ g) y |
| refine { (f.stalkMap y).hom with commutes' := ?_ } |
| intro scalar |
| have image_mem : f y ∈ (⊤ : X.Opens) := trivial |
| have point_mem : y ∈ (⊤ : Y.Opens) := trivial |
| change f.stalkMap y (X.presheaf.germ ⊤ (f y) image_mem |
| (g.appTop ((Scheme.ΓSpecIso (CommRingCat.of K)).inv scalar))) = |
| Y.presheaf.germ ⊤ y point_mem |
| ((f ≫ g).appTop ((Scheme.ΓSpecIso (CommRingCat.of K)).inv scalar)) |
| rw [Scheme.Hom.comp_appTop, f.germ_stalkMap_apply] |
| rfl |
|
|
| def closedStalkQuotientEquiv (g : X ⟶ Spec (CommRingCat.of K)) |
| (f : Y ⟶ X) [IsClosedImmersion f] (U : X.affineOpens) (y : Y) (hy : f y ∈ U.1) : |
| let _ := structuralStalkAlgebra g (f y) |
| let _ := structuralStalkAlgebra (f ≫ g) y |
| ((X.presheaf.stalk (f y)) ⧸ |
| ((f.ker.ideal U).map (X.presheaf.germ U.1 (f y) hy).hom)) ≃ₐ[K] |
| (Y.presheaf.stalk y) := by |
| dsimp only |
| letI := structuralStalkAlgebra g (f y) |
| letI := structuralStalkAlgebra (f ≫ g) y |
| have kernel_eq : RingHom.ker (structuralStalkMap g f y) = |
| (f.ker.ideal U).map (X.presheaf.germ U.1 (f y) hy).hom := by |
| change RingHom.ker (f.stalkMap y).hom = _ |
| exact closedImmersion_stalk_kernel f U y hy |
| exact (Ideal.quotientEquivAlgOfEq K kernel_eq.symm).trans |
| (Ideal.quotientKerAlgEquivOfSurjective (f := structuralStalkMap g f y) |
| (by |
| change Function.Surjective (f.stalkMap y) |
| exact f.stalkMap_surjective y)) |
|
|
| theorem closed_local_colength_le (g : X ⟶ Spec (CommRingCat.of K)) |
| (f : Y ⟶ X) [IsClosedImmersion f] [IsProper (f ≫ g)] [Finite Y] |
| (U : X.affineOpens) (y : Y) (hy : f y ∈ U.1) : |
| let _ := structuralStalkAlgebra g (f y) |
| let _ : Algebra K Γ(Y,⊤) := |
| ((f ≫ g).appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom).toAlgebra |
| Module.finrank K ((X.presheaf.stalk (f y)) ⧸ |
| ((f.ker.ideal U).map (X.presheaf.germ U.1 (f y) hy).hom)) ≤ |
| Module.finrank K Γ(Y,⊤) := by |
| dsimp only |
| let := structuralStalkAlgebra g (f y) |
| let := structuralStalkAlgebra (f ≫ g) y |
| let : Algebra K Γ(Y,⊤) := |
| ((f ≫ g).appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom).toAlgebra |
| rw [(closedStalkQuotientEquiv g f U y hy).toLinearEquiv.finrank_eq] |
| exact (proper_finite_stalk_finrank_le (f ≫ g) y).2 |
|
|
| end |
| end MaximalSeshadri.Geometry |
| end |
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| end |
| end |
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| end OAI |
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