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