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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 | import Mathlib
import OAI.AlgebraicGeometry.Seshadri.Geometry.SmoothAffineRefinement
import OAI.AlgebraicGeometry.Seshadri.Intersection.SchematicIntersection
import OAI.AlgebraicGeometry.Seshadri.LocalAlgebra.ClosedColength
namespace OAI
section
noncomputable section
section
namespace MaximalSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open MaximalSeshadri.ProjectiveBertini
variable {K : Type} [Field K] {X : Scheme}
lemma structuralStalkAlgebra_eq_open (g : X ⟶ Spec (CommRingCat.of K))
(U : X.Opens) (x : X) (hx : x ∈ U) :
structuralStalkAlgebra g x =
((X.presheaf.germ U x hx).hom.comp (openScalars g U)).toAlgebra := by
apply congrArg RingHom.toAlgebra
ext k
change (X.presheaf.germ ⊤ x trivial) (g.appTop ((Scheme.ΓSpecIso (CommRingCat.of K)).inv k)) = _
unfold openScalars Scheme.Hom.appLE
change _ = (g.app ⊤ ≫ X.presheaf.map _ ≫ X.presheaf.germ U x hx) _
rw [TopCat.Presheaf.germ_res]
rfl
theorem intersection_local_colength_le (g : X ⟶ Spec (CommRingCat.of K)) [IsProper g]
(I J : X.IdealSheafData) [IsIntegral I.subscheme]
(hd : topologicalKrullDim I.subscheme ≤ 1) (hJ : ¬ J ≤ I)
(U : X.affineOpens) (y : (I ⊔ J).subscheme)
(hy : (I ⊔ J).subschemeι y ∈ U.1) (t f : Γ(X,U.1))
(hI : I.ideal U = Ideal.span {t}) (hF : J.ideal U = Ideal.span {f}) :
let _ : Algebra K (X.presheaf.stalk ((I ⊔ J).subschemeι y)) :=
((X.presheaf.germ U.1 ((I ⊔ J).subschemeι y) hy).hom.comp (openScalars g U.1)).toAlgebra
let _ : Algebra K Γ((I ⊔ J).subscheme,⊤) :=
(((I ⊔ J).subschemeι ≫ g).appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom).toAlgebra
Module.finrank K ((X.presheaf.stalk ((I ⊔ J).subschemeι y)) ⧸ Ideal.span
{X.presheaf.germ U.1 ((I ⊔ J).subschemeι y) hy f,
X.presheaf.germ U.1 ((I ⊔ J).subschemeι y) hy t}) ≤
Module.finrank K Γ((I ⊔ J).subscheme,⊤) := by
let : Finite (I ⊔ J).subscheme := finite_schematic_intersection g I J hd hJ
have h := closed_local_colength_le g (I ⊔ J).subschemeι U y hy
rw [structuralStalkAlgebra_eq_open g U.1 _ hy] at h
have he : (((I ⊔ J).subschemeι.ker).ideal U).map
(X.presheaf.germ U.1 ((I ⊔ J).subschemeι y) hy).hom = Ideal.span
{X.presheaf.germ U.1 ((I ⊔ J).subschemeι y) hy f,
X.presheaf.germ U.1 ((I ⊔ J).subschemeι y) hy t} := by
rw [Scheme.IdealSheafData.ker_subschemeι]
change Ideal.map _ (I.ideal U ⊔ J.ideal U) = _
rw [hI,hF,Ideal.map_sup,Ideal.map_span,Ideal.map_span]
simp only [Set.image_singleton]
rw [← Ideal.span_union]
simp only [Set.singleton_union]
rw [Set.pair_comm]
rw [he] at h
exact h
end
end MaximalSeshadri.Geometry
end
end
end
end OAI
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