| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Sheaves.ClosedModuleExact |
| import OAI.AlgebraicGeometry.Seshadri.Sheaves.SectionPullback |
| import OAI.AlgebraicGeometry.Seshadri.Sheaves.OpenPullbackUnit |
| import OAI.AlgebraicGeometry.Seshadri.Sheaves.ModuleLocal |
|
|
| namespace OAI |
|
|
|
|
| |
| section |
|
|
| namespace MaximalSeshadri.LineClosedUnit |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace Opposite |
| open MaximalSeshadri.Geometry MaximalSeshadri.Frames |
|
|
| variable {X Y : Scheme.{0}} (f : X ⟶ Y) |
|
|
| def map (L : LineBundle Y) : L.sheaf ⟶ |
| (Scheme.Modules.pushforward f).obj ((Scheme.Modules.pullback f).obj L.sheaf) := |
| (Scheme.Modules.pullbackPushforwardAdjunction f).unit.app L.sheaf |
|
|
| def frame (L : LineBundle Y) (U : Y.Opens) (e : L.sheaf.restrict U.ι ≅ O U.toScheme) : |
| ((Scheme.Modules.pushforward f).obj ((Scheme.Modules.pullback f).obj L.sheaf)).restrict U.ι ≅ |
| (Scheme.Modules.pushforward (f ∣_ U)).obj (O (f ⁻¹ᵁ U).toScheme) := |
| OpenBaseChange.iso f U _ ≪≫ |
| (Scheme.Modules.pushforward (f ∣_ U)).mapIso |
| ((OpenBaseChange.leftSquare f U).app L.sheaf).symm ≪≫ |
| (Scheme.Modules.pushforward (f ∣_ U)).mapIso (pullbackFrame (f ∣_ U) e) |
|
|
| lemma framed_map (L : LineBundle Y) (U : Y.Opens) |
| (e : L.sheaf.restrict U.ι ≅ O U.toScheme) : |
| e.inv ≫ (Scheme.Modules.restrictFunctor U.ι).map (map f L) ≫ |
| (frame f L U e).hom = IdealModule.structureMap (f ∣_ U) := by |
| dsimp only [map, IdealModule.structureMap] |
| have hu := OpenBaseChange.unit_compatibility f U L.sheaf |
| have hn := (Scheme.Modules.pullbackPushforwardAdjunction (f ∣_ U)).unit_naturality e.hom |
| have hb := pullbackUnit_adjunction (f ∣_ U) |
| rw [Adjunction.homEquiv_unit] at hb |
| dsimp only [frame, Iso.trans_hom, Functor.mapIso_hom, Iso.symm_hom] |
| simp only [← Category.assoc] |
| rw [Category.assoc e.inv, hu] |
| simp only [Category.assoc] |
| have hec : (OpenBaseChange.leftSquare f U).hom.app L.sheaf ≫ |
| ((OpenBaseChange.leftSquare f U).app L.sheaf).inv = 𝟙 _ := |
| ((OpenBaseChange.leftSquare f U).app L.sheaf).hom_inv_id |
| rw [← Functor.map_comp_assoc, hec] |
| erw [CategoryTheory.Functor.map_id, Category.id_comp] |
| change e.inv ≫ (Scheme.Modules.pullbackPushforwardAdjunction (f ∣_ U)).unit.app _ ≫ |
| (Scheme.Modules.pushforward (f ∣_ U)).map |
| ((Scheme.Modules.pullback (f ∣_ U)).map e.hom ≫ (pullbackUnitIso (f ∣_ U)).hom) = _ |
| rw [Functor.map_comp] |
| have hn' := congrArg (fun t => e.inv ≫ t ≫ |
| (Scheme.Modules.pushforward (f ∣_ U)).map (pullbackUnitIso (f ∣_ U)).hom) hn |
| simp only [Category.assoc, hb] at hn' |
| exact hn'.trans (e.inv_hom_id_assoc _) |
|
|
| theorem epi [IsClosedImmersion f] (L : LineBundle Y) : Epi (map f L) := by |
| apply ModuleLocal.epi_of_local |
| intro y |
| obtain ⟨U,hy,⟨e⟩⟩ := L.locallyRankOne y |
| refine ⟨U,hy,?_⟩ |
| have hg : Epi (IdealModule.structureMap (f ∣_ U)) := IdealModule.structureMap_epi _ |
| have he := framed_map f L U e |
| have : Epi (e.inv ≫ (Scheme.Modules.restrictFunctor U.ι).map (map f L) ≫ |
| (frame f L U e).hom) := he.symm ▸ hg |
| have h1 : Epi ((Scheme.Modules.restrictFunctor U.ι).map (map f L) ≫ |
| (frame f L U e).hom) := epi_of_epi e.inv _ |
| exact (epi_comp_iff_of_isIso _ _).mp h1 |
|
|
| end |
| end MaximalSeshadri.LineClosedUnit |
|
|
| end |
|
|
|
|
| end OAI |
|
|
