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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 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | import Mathlib
import OAI.AlgebraicGeometry.Seshadri.Sheaves.CartierModule
namespace OAI
section
noncomputable section
section
namespace MaximalSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace
variable {X Y : Scheme.{0}}
lemma ideal_map_comap_closed (f : Y ⟶ X) [IsClosedImmersion f]
(J : X.IdealSheafData) : (J.comap f).map f = J ⊔ f.ker := by
apply Scheme.IdealSheafData.ext
funext U
rw [Scheme.IdealSheafData.ideal_map_of_isAffineHom]
rw [IdealPullback.comap_ideal J f ⟨f ⁻¹ᵁ U.1,U.2.preimage f⟩ U le_rfl]
have he : (f.appLE U.1 (f ⁻¹ᵁ U.1) le_rfl).hom = (f.app U.1).hom := by
simp [Scheme.Hom.appLE]
rw [he,Ideal.comap_map_of_surjective _ (f.app_surjective U.1 U.2)]
change J.ideal U ⊔ Ideal.comap (f.app U.1).hom ⊥ = J.ideal U ⊔ f.ker.ideal U
rw [Scheme.Hom.ker_apply]
rfl
def intrinsicIntersectionIso (I J : X.IdealSheafData) :
(J.comap I.subschemeι).subscheme ≅ (I ⊔ J).subscheme := by
let Q := J.comap I.subschemeι
have hQ : Q.map I.subschemeι = I ⊔ J := by
rw [ideal_map_comap_closed,Scheme.IdealSheafData.ker_subschemeι,sup_comm]
let u := Scheme.IdealSheafData.subschemeMap Q (I ⊔ J) I.subschemeι hQ.ge
have hu : u ≫ (I ⊔ J).subschemeι = Q.subschemeι ≫ I.subschemeι :=
Scheme.IdealSheafData.subschemeMap_subschemeι Q (I ⊔ J) I.subschemeι hQ.ge
have hk : (Q.subschemeι ≫ I.subschemeι).ker = (I ⊔ J).subschemeι.ker := by
rw [Scheme.Hom.ker_comp,Scheme.IdealSheafData.ker_subschemeι,
Scheme.IdealSheafData.ker_subschemeι,hQ]
letI : IsIso u := IsClosedImmersion.isIso_of_ker_eq _ _ u hu hk
exact asIso u
@[reassoc] lemma intrinsicIntersectionIso_hom_ι (I J : X.IdealSheafData) :
(intrinsicIntersectionIso I J).hom ≫ (I ⊔ J).subschemeι =
(J.comap I.subschemeι).subschemeι ≫ I.subschemeι :=
by
have hQ : I ⊔ J ≤ (J.comap I.subschemeι).map I.subschemeι := by
rw [ideal_map_comap_closed,Scheme.IdealSheafData.ker_subschemeι,sup_comm]
change Scheme.IdealSheafData.subschemeMap _ _ _ hQ ≫ _ = _
exact Scheme.IdealSheafData.subschemeMap_subschemeι _ _ _ hQ
variable {K : Type} [Field K]
def structuralGlobalMap (g : X ⟶ Spec (CommRingCat.of K)) (f : Y ⟶ X) :
letI : Algebra K Γ(X,⊤) :=
(g.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom).toAlgebra
letI : Algebra K Γ(Y,⊤) :=
((f ≫ g).appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom).toAlgebra
Γ(X,⊤) →ₐ[K] Γ(Y,⊤) := by
letI : Algebra K Γ(X,⊤) :=
(g.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom).toAlgebra
letI : Algebra K Γ(Y,⊤) :=
((f ≫ g).appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom).toAlgebra
refine { f.appTop.hom with commutes' := ?_ }
intro c
change f.appTop (g.appTop ((Scheme.ΓSpecIso (CommRingCat.of K)).inv c)) =
(f ≫ g).appTop ((Scheme.ΓSpecIso (CommRingCat.of K)).inv c)
rw [Scheme.Hom.comp_appTop]
rfl
lemma structuralGlobal_finrank_iso (g : X ⟶ Spec (CommRingCat.of K))
(f : Y ⟶ X) [IsIso f] :
letI : Algebra K Γ(X,⊤) :=
(g.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom).toAlgebra
letI : Algebra K Γ(Y,⊤) :=
((f ≫ g).appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom).toAlgebra
Module.finrank K Γ(X,⊤) = Module.finrank K Γ(Y,⊤) := by
let : Algebra K Γ(X,⊤) :=
(g.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom).toAlgebra
let : Algebra K Γ(Y,⊤) :=
((f ≫ g).appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom).toAlgebra
have : IsIso f.appTop := inferInstanceAs (IsIso (Scheme.Γ.map f.op))
exact (AlgEquiv.ofBijective (structuralGlobalMap g f)
(ConcreteCategory.bijective_of_isIso f.appTop)).toLinearEquiv.finrank_eq
lemma intrinsicIntersection_length (g : X ⟶ Spec (CommRingCat.of K))
(I J : X.IdealSheafData) :
letI : Algebra K Γ((I ⊔ J).subscheme,⊤) :=
(((I ⊔ J).subschemeι ≫ g).appTop.hom.comp
(Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom).toAlgebra
letI : Algebra K Γ((J.comap I.subschemeι).subscheme,⊤) :=
(((J.comap I.subschemeι).subschemeι ≫ (I.subschemeι ≫ g)).appTop.hom.comp
(Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom).toAlgebra
Module.finrank K Γ((I ⊔ J).subscheme,⊤) =
Module.finrank K Γ((J.comap I.subschemeι).subscheme,⊤) := by
have h := structuralGlobal_finrank_iso ((I ⊔ J).subschemeι ≫ g)
(intrinsicIntersectionIso I J).hom
rw [← Category.assoc,intrinsicIntersectionIso_hom_ι,Category.assoc] at h
exact h
end
end MaximalSeshadri.Geometry
end
end
end
end OAI
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