The short exact sequence of a Cartier divisor

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OAI.MaximalSeshadri.CartierSequence.exact

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

maximal-seshadrioai-math

Let $M,N$ be line bundles on $X$ and $\varphi:M\to N$ a monomorphism. Let $I$ be an ideal sheaf that is locally on affine opens generated by the coefficient of $\varphi$ in trivializations of $M$ and $N$ . Then $\varphi$ composed with $N\to(\iota_I)_*\iota_I^*N$ is zero, and $0\to M\to N\to(\iota_I)_*\iota_I^*N\to0$ is short exact.

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.CartierSequence
noncomputable section
open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace Opposite
open MaximalSeshadri.Geometry MaximalSeshadri.Frames MaximalSeshadri.ProjectiveBertini

variable {X : Scheme.{0}}

Formal statement

theorem exact (p : X ⟶ Spec (CommRingCat.of ℂ))
    (M N : LineBundle X) (φ : M.sheaf ⟶ N.sheaf) [Mono φ]
    (I : X.IdealSheafData)
    (heq : ∀ x : X, ∃ U : X.affineOpens, x ∈ U.1 ∧
      ∃ e : M.sheaf.restrict U.1.ι ≅ O U.1.toScheme,
      ∃ d : N.sheaf.restrict U.1.ι ≅ O U.1.toScheme,
        I.ideal U = Ideal.span {U.1.topIso.hom
          (endValue (e.inv ≫ (Scheme.Modules.restrictFunctor U.1.ι).map φ ≫ d.hom))}) :
    ∃ hz : φ ≫ LineClosedUnit.map I.subschemeι N = 0,
      (ShortComplex.mk φ (LineClosedUnit.map I.subschemeι N) hz).ShortExact := by
  sorry

end
end MaximalSeshadri.CartierSequence
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.CartierSequence.exact`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Divisors/CartierSequence.lean#L45

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