| import OAI.NumberTheory.PiExponent.Approximation.ClosedSectionLifting |
| import OAI.NumberTheory.PiExponent.Approximation.ProjectionFormula |
| import OAI.NumberTheory.PiExponent.Approximation.SerrePowerDescent |
|
|
| namespace OAI |
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|
| noncomputable section |
| namespace PiExponentSeshadri.Geometry |
|
|
| open AlgebraicGeometry CategoryTheory CategoryTheory.Limits CategoryTheory.Abelian Opposite |
| open PiExponentSeshadri.TensorPure |
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|
| variable {X Y : Scheme.{0}} |
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| def closedLineQuotientIso (f : Y ⟶ X) (M : LineBundle X) : |
| moduleTensor X ((Scheme.Modules.pushforward f).obj (structureSheaf Y)) M.sheaf ≅ |
| (Scheme.Modules.pushforward f).obj (M.pullback f).sheaf := |
| PiExponent.ProjectionFormula.iso f (structureSheaf Y) M ≪≫ |
| (Scheme.Modules.pushforward f).mapIso (moduleTensorUnit (M.pullback f).sheaf) |
|
|
| theorem closedLineQuotientIso_map (f : Y ⟶ X) (M : LineBundle X) : |
| moduleTensorMap (PiExponentSeshadri.IdealModule.structureMap f) (𝟙 M.sheaf) ≫ |
| (closedLineQuotientIso f M).hom = |
| (moduleTensorUnit M.sheaf).hom ≫ |
| (Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M.sheaf := by |
| apply PiExponentSeshadri.TensorPure.hom_ext |
| intro U a m |
| let η := ((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M.sheaf).val.app (op U) |
| have ht := map_pure (PiExponentSeshadri.IdealModule.structureMap f) (𝟙 M.sheaf) U a m |
| have hp := PiExponent.ProjectionFormula.hom_pure f (structureSheaf Y) M.sheaf U |
| ((PiExponentSeshadri.IdealModule.structureMap f).app U a) m |
| have hu := unit_pure (M.pullback f).sheaf (f ⁻¹ᵁ U) |
| ((PiExponentSeshadri.IdealModule.structureMap f).app U a) (η m) |
| have hv := unit_pure M.sheaf U a m |
| have hs := η.hom.map_smul a m |
| exact (congrArg (fun z => (moduleTensorUnit (M.pullback f).sheaf).hom.app (f ⁻¹ᵁ U) |
| ((PiExponent.ProjectionFormula.hom f (structureSheaf Y) M.sheaf).app U z)) ht).trans |
| ((congrArg (fun z => (moduleTensorUnit (M.pullback f).sheaf).hom.app (f ⁻¹ᵁ U) z) hp).trans |
| (hu.trans (hs.symm.trans (congrArg (fun z => η z) hv).symm))) |
|
|
| def closedLineUnitKernelIso (f : Y ⟶ X) [IsClosedImmersion f] (M : LineBundle X) : |
| moduleTensor X (PiExponentSeshadri.IdealModule.idealModule f) M.sheaf ≅ |
| kernel ((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M.sheaf) := by |
| let S := ShortComplex.mk (PiExponentSeshadri.IdealModule.inclusion f) |
| (PiExponentSeshadri.IdealModule.structureMap f) (kernel.condition _) |
| let T := S.map (moduleTwistFunctor M 1) |
| have hT : T.ShortExact := moduleTwistFunctor_shortExact M 1 S |
| (PiExponentSeshadri.IdealModule.closedSequence_exact f) |
| let e : T.X₁ ≅ kernel T.g := |
| hT.fIsKernel.conePointUniqueUpToIso (limit.isLimit _) |
| exact e ≪≫ kernel.mapIso T.g |
| ((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M.sheaf) |
| (moduleTensorUnit M.sheaf) (closedLineQuotientIso f M) |
| (closedLineQuotientIso_map f M) |
|
|
| theorem pullbackSection_surjective_of_idealTensor_cohomology_zero |
| (f : Y ⟶ X) [IsClosedImmersion f] (M : LineBundle X) |
| (hzero : ∀ z : cohomology |
| (moduleTensor X (PiExponentSeshadri.IdealModule.idealModule f) M.sheaf) 1, z = 0) : |
| Function.Surjective (fun s : GlobalSections X M.sheaf => pullbackSection f s) := by |
| apply pullbackSection_surjective_of_kernel_cohomology_zero f M |
| exact PiExponent.GeometrySupport.SerrePowerDescent.ext_zero_of_iso |
| (closedLineUnitKernelIso f M) 1 hzero |
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|
| end PiExponentSeshadri.Geometry |
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| end |
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| end OAI |
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|
