H^2 is finite for a finite projection to the plane

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OAI.MaximalSeshadri.Geometry.finite_projection_H2_finite

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

maximal-seshadrioai-math

Let $X$ be separated Noetherian with sections $s_0,s_1,s_2$ giving a finite morphism to the plane. For a line bundle $L$ , $\operatorname{Ext}^2(\mathcal O_X,L)$ is a finite $K$ -module.

Notation. The iso-open of $s$ (SectionOpens.isoOpen s) is the union of the opens on which the morphism $s$ restricts to an isomorphism. sectionsMorphism k s hs is the morphism from $X$ to $\operatorname{Proj}$ of the graded polynomial ring in variables indexed by the sections, defined by sections whose iso-opens cover $X$ , with $k$ the ring map from the base into $\Gamma(X,\top)$ .

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 TopologicalSpace
open MaximalSeshadri.Projective MaximalSeshadri.Frames BaseSections

variable {K : Type} [Field K] {X : Scheme.{0}}

Formal statement

theorem finite_projection_H2_finite [X.IsSeparated] [IsNoetherian X]
    (k : K →+* Γ(X,⊤)) {M : X.Modules}
    (s : Fin 3 → (O X ⟶ M)) (hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤)
    [IsFinite (sectionsMorphism k s hs)] (L : LineBundle X) :
    letI := Module.compHom (cohomology L.sheaf 2) k
    Module.Finite K (cohomology L.sheaf 2) := 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.finite_projection_H2_finite`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Cohomology/SurfaceH2Finite.lean#L20

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