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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 | import OAI.NumberTheory.PiExponent.Ampleness.NakaiSectionLifting
import OAI.NumberTheory.PiExponent.Approximation.ClosedThickeningSections
namespace OAI
noncomputable section
namespace PiExponentSeshadri.Geometry
open AlgebraicGeometry CategoryTheory CategoryTheory.Limits CategoryTheory.Abelian
variable {X Y : Scheme.{0}}
local instance (Z : Scheme.{0}) : HasExt.{1} Z.Modules := HasExt.standard _
theorem closed_line_unit_epi (f : Y ⟶ X) [IsClosedImmersion f] (M : LineBundle X) :
Epi ((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M.sheaf) :=
PiExponentSeshadri.LineClosedUnit.epi f M
theorem pullbackSection_surjective_of_kernel_cohomology_zero
(f : Y ⟶ X) [IsClosedImmersion f] (M : LineBundle X)
(hzero : ∀ z : cohomology
(kernel ((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M.sheaf)) 1,
z = 0) :
Function.Surjective (fun s : GlobalSections X M.sheaf => pullbackSection f s) := by
let u := (Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M.sheaf
have : Epi u := closed_line_unit_epi f M
let S := ShortComplex.mk (kernel.ι u) u (kernel.condition u)
have hS : S.ShortExact := { exact := ShortComplex.exact_kernel u }
exact PiExponent.NumericalAmpleness.pullbackSection_surjective_of_unit_surjective f M
(globalSections_surjective_of_ext_one_zero S hS hzero)
end PiExponentSeshadri.Geometry
end
end OAI
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