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7f5fab8 | 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 73 74 75 76 77 78 79 | 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
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