{
  "abstract": "We propose a local-cohomology obstruction to the existence of small Cohen--Macaulay modules over completed section rings of surfaces. Let $Y$ be a smooth connected complex projective surface, and let $H$ be an ample globally generated divisor with $K_Y\\equiv4H$ numerically. Put $c(Y,H)=15H^2/8-χ(\\OO_Y)$. The main claim is that, when $c(Y,H)>0$, every nonzero finite reflexive module $M$ over the completed vertex local ring of $\\bigoplus_{n\\geq0}H^0(Y,\\OO_Y(nH))$ satisfies $\\dim_\\C H^2_\\mm(M)\\geq c(Y,H)\\rk M$. The argument combines elementary transformations of reflexive extensions, Harder--Narasimhan slopes, Bogomolov's inequality, and a two-section Koszul estimate. A fixed-source kernel estimate transfers the resulting cohomology to the punctured spectrum without requiring a grading on $M$. For the degree-six member of the six-line Hirzebruch--Kummer family, $H^2=1080$ and $χ(\\OO_Y)=1926$, so the bound is $99\\rk M$. We give an explicit complete-intersection model for this application and spell out the argument leading to the claimed nonexistence of nonzero finite maximal Cohen--Macaulay modules.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24142",
  "author_total": 1,
  "authors": [
    {
      "name": "Liang Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bqh3wzqw4e04v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24142",
  "fulltext_url": null,
  "id": "bqh3wzqw4e04v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24142",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7214001118",
  "pdf_url": "https://arxiv.org/pdf/2609.24142",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bqh3wzqw4e04v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24142",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24142"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A local cohomology obstruction to small Cohen--Macaulay modules",
  "title_text": "A local cohomology obstruction to small Cohen--Macaulay modules",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/bqh3wzqw4e04v",
  "venue": "arXiv",
  "versions": []
}