{
  "abstract": "For a finitely generated module $M$ of finite Gorenstein dimension over a commutative noetherian local ring $(R,\\mathfrak{m},\\Bbbk)$, we prove that the natural map $\\mathrm{Ext}_R(\\Bbbk,M)\\longrightarrow\\mathrm{Ext}_R(\\Bbbk,M/\\mathfrak{m} M)$ is zero unless $R$ is Gorenstein. More generally, the same conclusion holds whenever $\\mathrm{Ext}_R^i(M,R)=0$ for $i\\gg0$. This gives a Gorenstein analogue of a regularity criterion of Avramov and Iyengar. The proof uses the right $\\mathrm{Ext}_R(\\Bbbk,\\Bbbk)$-module structure of bounded cohomology studied by the author in a previous work. As an application, we show that $R$ is Gorenstein if a nonzero quotient of a finite direct sum of syzygies of $\\Bbbk$ has finite Gorenstein dimension. In particular, this answers affirmatively a question of Takahashi concerning direct summands of syzygies of the residue field.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01630",
  "author_total": 1,
  "authors": [
    {
      "name": "Luigi Ferraro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b5xysbhdc3dw7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01630",
  "fulltext_url": null,
  "id": "b5xysbhdc3dw7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01630",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01630",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b5xysbhdc3dw7.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01630",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01630"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On a question of Takahashi concerning direct summands of syzygies",
  "title_text": "On a question of Takahashi concerning direct summands of syzygies",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/b5xysbhdc3dw7",
  "venue": "arXiv",
  "versions": []
}