File size: 3,471 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 69 70 71 | import OAI.NumberTheory.PiExponent.Approximation.ClosedSectionLifting
import OAI.NumberTheory.PiExponent.Approximation.ProjectionFormula
import OAI.NumberTheory.PiExponent.Approximation.SerrePowerDescent
namespace OAI
noncomputable section
namespace PiExponentSeshadri.Geometry
open AlgebraicGeometry CategoryTheory CategoryTheory.Limits CategoryTheory.Abelian Opposite
open PiExponentSeshadri.TensorPure
variable {X Y : Scheme.{0}}
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
end PiExponentSeshadri.Geometry
end
end OAI
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