| import OAI.NumberTheory.PiExponent.Ampleness.BlowupJetSurjectivity |
| import OAI.NumberTheory.PiExponent.Approximation.ProjectionFormula |
| import OAI.NumberTheory.PiExponent.Approximation.TwistSections |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace PiExponent.AffineJetSupport |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite |
| open PiExponentSeshadri.Geometry PiExponentSeshadri.Frames |
| open PiExponent.BlowupJetSurjectivity |
| variable {X Y : Scheme} |
|
|
| private theorem map_bijective_of_eq {T : Type*} [Category T] [Quiver.IsThin T] |
| (F : Tᵒᵖ ⥤ AddCommGrpCat) {U V : T} |
| (h : U = V) (i : U ⟶ V) : Function.Bijective (F.map i.op) := by |
| subst V |
| have hi : i = 𝟙 U := Subsingleton.elim _ _ |
| subst i |
| rw [op_id] |
| erw [F.map_id] |
| exact Function.bijective_id |
|
|
| theorem restriction_surjective_of_iso {M N : X.Modules} (e : M ≅ N) (U : X.Opens) |
| (h : Function.Surjective (N.presheaf.map (homOfLE (le_top : U ≤ ⊤)).op)) : |
| Function.Surjective (M.presheaf.map (homOfLE (le_top : U ≤ ⊤)).op) := by |
| intro s |
| obtain ⟨t, ht⟩ := h (e.hom.app U s) |
| obtain ⟨r, hr⟩ := (ConcreteCategory.bijective_of_isIso (e.hom.app ⊤)).surjective t |
| refine ⟨r, (ConcreteCategory.bijective_of_isIso (e.hom.app U)).injective ?_⟩ |
| have hn := CategoryTheory.congr_fun |
| (e.hom.mapPresheaf.naturality (homOfLE (le_top : U ≤ ⊤)).op) r |
| change e.hom.app U (M.presheaf.map _ r) = N.presheaf.map _ (e.hom.app ⊤ r) at hn |
| rw [hn, hr, ht] |
|
|
| theorem pushforward_restriction_bijective (f : Y ⟶ X) (M : Y.Modules) (U : X.Opens) |
| (hU : f ⁻¹ᵁ U = ⊤) : |
| Function.Bijective (((Scheme.Modules.pushforward f).obj M).presheaf.map |
| (homOfLE (le_top : U ≤ ⊤)).op) := by |
| rw [Scheme.Modules.pushforward_obj_presheaf_map] |
| have he : f ⁻¹ᵁ U = f ⁻¹ᵁ (⊤ : X.Opens) := by simp [hU] |
| exact map_bijective_of_eq M.presheaf he _ |
|
|
| theorem subscheme_preimage_eq_top (I : X.IdealSheafData) (U : X.Opens) |
| (hs : (I.support : Set X) ⊆ U) : I.subschemeι ⁻¹ᵁ U = ⊤ := by |
| apply top_unique |
| intro x hx |
| apply hs |
| rw [← I.range_subschemeι] |
| exact ⟨x, rfl⟩ |
|
|
| theorem twistedQuotient_restriction_surjective (I : X.IdealSheafData) |
| (A : LineBundle X) (n : ℕ) (U : X.Opens) |
| (hs : (I.support : Set X) ⊆ U) : |
| Function.Surjective (((moduleTwistFunctor A n).obj (jetQuotient I)).presheaf.map |
| (homOfLE (le_top : U ≤ ⊤)).op) := by |
| apply restriction_surjective_of_iso |
| (ProjectionFormula.twistIso I.subschemeι (PiExponentSeshadri.IdealModule.unit I.subscheme) A n).symm U |
| exact (pushforward_restriction_bijective I.subschemeι _ U |
| (subscheme_preimage_eq_top I U hs)).surjective |
|
|
| def restrictionSectionsIso (M : X.Modules) (U : X.Opens) : |
| Γ(M.restrict U.ι, ⊤) ≅ Γ(M,U) := |
| M.restrictAppIso U.ι ⊤ ≪≫ M.presheaf.mapIso (eqToIso U.ι_image_top.symm).op |
|
|
| def sectionsIso {M N : X.Modules} (e : M ≅ N) (U : X.Opens) : Γ(M,U) ≅ Γ(N,U) := |
| ((Scheme.Modules.toPresheaf X).mapIso e).app (op U) |
|
|
| def localQuotientEquiv (I : X.IdealSheafData) (A : LineBundle X) (n : ℕ) |
| (U : X.affineOpens) (e : A.sheaf.restrict U.1.ι ≅ structureSheaf U.1.toScheme) : |
| Γ((moduleTwistFunctor A n).obj (jetQuotient I), U.1) ≃ |
| (Γ(X,U.1) ⧸ I.ideal U) := |
| (restrictionSectionsIso ((moduleTwistFunctor A n).obj (jetQuotient I)) U.1).symm.addCommGroupIsoToAddEquiv.toEquiv |>.trans |
| ((sectionsIso ((moduleTwistRestrictFrame A U.1 e n).app (jetQuotient I)) ⊤).addCommGroupIsoToAddEquiv.toEquiv.trans |
| ((restrictionSectionsIso (jetQuotient I) U.1).addCommGroupIsoToAddEquiv.toEquiv.trans |
| (I.subschemeObjIso U).commRingCatIsoToRingEquiv.toEquiv)) |
|
|
| def globalQuotientValue (I : X.IdealSheafData) (A : LineBundle X) (n : ℕ) |
| (U : X.affineOpens) (e : A.sheaf.restrict U.1.ι ≅ structureSheaf U.1.toScheme) |
| (s : GlobalSections X ((moduleTwistFunctor A n).obj (jetQuotient I))) : |
| Γ(X,U.1) ⧸ I.ideal U := |
| localQuotientEquiv I A n U e |
| (((moduleTwistFunctor A n).obj (jetQuotient I)).presheaf.map |
| (homOfLE (le_top : U.1 ≤ ⊤)).op |
| (moduleSectionEquiv ((moduleTwistFunctor A n).obj (jetQuotient I)) s)) |
|
|
| theorem globalQuotientValue_surjective (I : X.IdealSheafData) (A : LineBundle X) (n : ℕ) |
| (U : X.affineOpens) (e : A.sheaf.restrict U.1.ι ≅ structureSheaf U.1.toScheme) |
| (hs : (I.support : Set X) ⊆ U.1) : |
| Function.Surjective (globalQuotientValue I A n U e) := |
| (localQuotientEquiv I A n U e).surjective.comp |
| ((twistedQuotient_restriction_surjective I A n U.1 hs).comp |
| (moduleSectionEquiv ((moduleTwistFunctor A n).obj (jetQuotient I))).surjective) |
|
|
| end PiExponent.AffineJetSupport |
| end |
|
|
| end OAI |
|
|
