Finite-dimensional cohomology of line bundles on projective curves

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OAI.MaximalSeshadri.Geometry.projective_curve_cohomology_finite

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

maximal-seshadrioai-math

With $K$ an infinite field, $\sigma$ finite, $X$ integral Noetherian, $p:X\to\operatorname{Spec}K$ proper, $\dim X=1$ , and sections giving a closed immersion to $\operatorname{Proj}K[\sigma]$ : for a line bundle $L$ and every $n$ , the $K$ -module $\operatorname{Ext}^n(\mathcal O_X,L)$ is finite.

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
noncomputable section

section

namespace MaximalSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open MaximalSeshadri.Projective MaximalSeshadri.Frames

variable {K σ : Type} [Field K] [Infinite K] [Fintype σ]
    {X : Scheme.{0}} [IsIntegral X] [IsNoetherian X]

Formal statement

theorem projective_curve_cohomology_finite
    (p : X ⟶ Spec (CommRingCat.of K)) [IsProper p]
    (hd : topologicalKrullDim X = 1) {M : X.Modules}
    (s : σ → (O X ⟶ M)) (hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤)
    [IsClosedImmersion (sectionsMorphism
      (p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom) s hs)]
    (L : LineBundle X) (n : ℕ) :
    letI := Module.compHom (cohomology L.sheaf n)
      (p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom)
    Module.Finite K (cohomology L.sheaf n) := by
  sorry

end
end MaximalSeshadri.Geometry
end
end
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.projective_curve_cohomology_finite`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Cohomology/ProjectiveCurveCohomology.lean#L23

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