openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /ClosedPowerSectionLifting.lean
| 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 | |