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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 | import OAI.NumberTheory.PiExponent.Approximation.ClosedImmersionSerreTransfer
import OAI.NumberTheory.PiExponent.Approximation.ClosedRestrictionTensor
import OAI.NumberTheory.PiExponent.Approximation.ClosedThickeningSections
namespace OAI
namespace PiExponentSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory CategoryTheory.Limits CategoryTheory.Abelian
open PiExponentSeshadri.Frames
variable {X Y : Scheme.{0}}
local instance : HasExt.{1} X.Modules := HasExt.standard _
theorem pullbackPowerSection_surjective_of_ideal_ext_zero (i : Y ⟶ X)
[IsClosedImmersion i] (L : LineBundle X) (n : ℕ)
(hzero : ∀ x : Abelian.Ext.{1} (structureSheaf X)
((moduleTwistFunctor L n).obj (IdealModule.idealModule i)) 1, x = 0) :
Function.Surjective (pullbackPowerSection L i n) := by
let S := ShortComplex.mk (IdealModule.inclusion i) (IdealModule.structureMap i)
(kernel.condition _)
have hS := moduleTwistFunctor_shortExact L n S (IdealModule.closedSequence_exact i)
have hsurj := globalSections_surjective_of_ext_one_zero (S.map (moduleTwistFunctor L n)) hS hzero
intro σ
let e := PiExponent.ClosedRestrictionTensor.powerProjectionIso i L n
let b : GlobalSections X ((Scheme.Modules.pushforward i).obj ((L.pullback i).pow n).sheaf) :=
IdealModule.structureMap i ≫ (Scheme.Modules.pushforward i).map σ
obtain ⟨a,ha⟩ := hsurj (b ≫ e.inv)
let t : GlobalSections X (L.pow n).sheaf := a ≫ (moduleTwistUnitIso L n).hom
refine ⟨t, ?_⟩
apply (PiExponent.ClosedImmersionSerreTransfer.globalHom_push_bijective i
((L.pullback i).pow n).sheaf).injective
have ht := PiExponent.ClosedRestrictionTensor.section_restriction_eq i L n t
change t ≫ (moduleTwistUnitIso L n).inv ≫
(moduleTwistFunctor L n).map (IdealModule.structureMap i) ≫ e.hom =
IdealModule.structureMap i ≫ (Scheme.Modules.pushforward i).map
(pullbackPowerSection L i n t) at ht
change IdealModule.structureMap i ≫ (Scheme.Modules.pushforward i).map
(pullbackPowerSection L i n t) = b
rw [← ht]
dsimp only [t]
erw [Category.assoc a (moduleTwistUnitIso L n).hom,
(moduleTwistUnitIso L n).hom_inv_id_assoc]
change (a ≫ (moduleTwistFunctor L n).map (IdealModule.structureMap i)) ≫ e.hom = b
have ha' : a ≫ (moduleTwistFunctor L n).map (IdealModule.structureMap i) = b ≫ e.inv := ha
exact (congrArg (fun z => z ≫ e.hom) ha').trans (by simp only [Category.assoc,
Iso.inv_hom_id, Category.comp_id])
end
end PiExponentSeshadri.Geometry
end OAI
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