openAI-math-dataset / lean /OAI /AlgebraicGeometry /Seshadri /Intersection /GlobalIntersectionLength.lean
| 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 | |