File size: 3,402 Bytes
a72760f | 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 | import OAI.NumberTheory.PiExponent.Approximation.CocycleTensor
namespace OAI
namespace PiExponentSeshadri.LineBundleGluing
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite MonoidalCategory
open PiExponentSeshadri.Geometry PiExponentSeshadri.TensorPure
variable {X : Scheme} {ι : Type} {U : ι → X.Opens}
lemma tensorHom_restrict_isIso (W : X.Opens)
{P P' Q Q' : PresheafOfModules X.ringCatSheaf.obj} (f : P ⟶ P') (g : Q ⟶ Q')
[IsIso ((modulePresheafRestrict W.ι).map f)]
[IsIso ((modulePresheafRestrict W.ι).map g)] :
IsIso ((modulePresheafRestrict W.ι).map
(PresheafOfModulesOfCommRing.Monoidal.tensorHom (R := X.presheaf) f g)) := by
let R := modulePresheafRestrict W.ι
let : MonoidalCategory (PresheafOfModules W.toScheme.ringCatSheaf.obj) :=
PresheafOfModulesOfCommRing.monoidalCategory (R := W.toScheme.presheaf)
have hh : IsIso (R.map (PresheafOfModulesOfCommRing.Monoidal.tensorHom (R := X.presheaf) f g) ≫
(modulePresheafTensorRestrict W P' Q').hom) := by
rw [modulePresheafTensorRestrict_natural]
change IsIso (_ ≫ (R.map f ⊗ₘ R.map g))
infer_instance
let hc : IsIso (C := PresheafOfModules W.toScheme.ringCatSheaf.obj)
(modulePresheafTensorRestrict W P' Q').hom :=
(modulePresheafTensorRestrict W P' Q').isIso_hom
exact (@isIso_comp_right_iff (PresheafOfModules W.toScheme.ringCatSheaf.obj)
_ _ _ _ _ _ hc).mp hh
def cocycleTensorComparison (c d : Cocycle U) (hcover : ⊤ ≤ ⨆ i, U i) :
(PresheafOfModules.sheafification (𝟙 X.ringCatSheaf.obj)).obj
(PresheafOfModulesOfCommRing.Monoidal.tensorObj (R := X.presheaf)
(presheaf c) (presheaf d)) ≅ moduleTensor X (sheaf c) (sheaf d) := by
let f := PresheafOfModulesOfCommRing.Monoidal.tensorHom (R := X.presheaf)
((adj X).unit.app (presheaf c)) ((adj X).unit.app (presheaf d))
have ht : IsIso ((PresheafOfModules.sheafification (𝟙 X.ringCatSheaf.obj)).map f) := by
apply sheafification_isIso_of_chart f hcover
intro i
have := cocycleUnit_restrict_isIso c i
have := cocycleUnit_restrict_isIso d i
exact tensorHom_restrict_isIso (U i) _ _
exact @asIso _ _ _ _ ((PresheafOfModules.sheafification (𝟙 X.ringCatSheaf.obj)).map f) ht
def trivialSheafIso (hcover : ⊤ ≤ ⨆ i, U i) :
sheaf (Cocycle.one (U := U)) ≅ structureSheaf X := by
have ht : IsIso ((PresheafOfModules.sheafification (𝟙 X.ringCatSheaf.obj)).map
(trivialCoefficientMap (U := U))) :=
sheafification_isIso_of_chart _ hcover trivialCoefficientMap_restrict_isIso
exact (@asIso _ _ _ _ ((PresheafOfModules.sheafification (𝟙 X.ringCatSheaf.obj)).map
(trivialCoefficientMap (U := U))) ht).symm ≪≫
(asIso (adj X).counit).app (structureSheaf X)
def tensorInverseIso (c : Cocycle U) (hcover : ⊤ ≤ ⨆ i, U i) :
moduleTensor X (sheaf c) (sheaf c.inverse) ≅ structureSheaf X := by
have ht : IsIso ((PresheafOfModules.sheafification (𝟙 X.ringCatSheaf.obj)).map
(inverseProductMap c)) :=
sheafification_isIso_of_chart _ hcover (inverseProductMap_restrict_isIso c)
exact (cocycleTensorComparison c c.inverse hcover).symm ≪≫
@asIso _ _ _ _ ((PresheafOfModules.sheafification (𝟙 X.ringCatSheaf.obj)).map
(inverseProductMap c)) ht ≪≫
trivialSheafIso hcover
end
end PiExponentSeshadri.LineBundleGluing
end OAI
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