Surjectivity onto H^1 from the plane relation quotient

Proved

OAI.MaximalSeshadri.Geometry.planeBaseCohomologyOneMap_surjective

by wurtle · Nov 4, 2026 · Mathlib 0df444a (Lean v4.33.1)

maximal-seshadrioai-math

For a Noetherian scheme $X$ and sections $s_0,s_1,s_2$ with affine iso-opens covering $X$ , and a line bundle $L$ , the map planeBaseCohomologyOneMap from the quotient of $\operatorname{Hom}$ from the relation module to $L$ by the relation boundaries to $\operatorname{Ext}^1(\mathcal O_X,L)$ is surjective.

Notation. The iso-open of $s$ (SectionOpens.isoOpen s) is the union of the opens on which the morphism $s$ restricts to an isomorphism.

Formalization Note The statement is verbatim from OpenAI's Lean development, with Definitions files of this development imported in place of the upstream modules; these are supporting code for the statement.

Preamble

import Mathlib
import Definitions.Def_MaximalSeshadriConstants
import Definitions.Def_SeshadriPrelim6

namespace OAI

section

namespace MaximalSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace
open MaximalSeshadri.Projective MaximalSeshadri.Frames BaseSections ModuleFlasque ModuleMayerVietoris
open MaximalSeshadri.PlaneCech

variable {K : Type} [Field K] {X : Scheme.{0}}
attribute [local instance] OAI.MaximalSeshadri.Geometry.instLinearCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopSheafOfModulesRingCatSheaf_definitions_1

attribute [local instance] OAI.MaximalSeshadri.Geometry.instHasExtSheafOfModulesOpensCarrierCarrierCommRingCatRingCatSheaf_definitions_1

attribute [local instance] baseHomModule

Formal statement

theorem planeBaseCohomologyOneMap_surjective [IsNoetherian X] (k : K →+* Γ(X,⊤)) {M : X.Modules}
    (s : Fin 3 → (O X ⟶ M)) (hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤)
    (h : ∀ i, IsAffineOpen (SectionOpens.isoOpen (s i))) (L : LineBundle X) :
    Function.Surjective (planeBaseCohomologyOneMap k s hs h L) := by
  sorry

end
end MaximalSeshadri.Geometry
end
end OAI

Source

OpenAI, Maximal Seshadri constants on arbitrary polarized surfaces, September 23, 2026, https://github.com/openai/math (paper: preprints/Maximal-Seshadri-Constants-on-Arbitrary-Polarized-Surfaces-September-23-2026/main.pdf; Lean: lean/OAI/AlgebraicGeometry/Seshadri, Apache License 2.0); Lean declaration `OAI.MaximalSeshadri.Geometry.planeBaseCohomologyOneMap_surjective`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Cohomology/SurfaceH1Finite.lean#L69

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