{
  "abstract": "Let $G$ be a group and $k$ be a commutative ring. The Gorenstein homological dimension $\\text{Ghd}_k G$ and the Gorenstein cohomological dimension $\\text{Gcd}_k G$ of $G$ over $k$ are defined as the Gorenstein flat and the Gorenstein projective dimension, respectively, of the trivial $kG$-module $k$. We prove that the Gorenstein cohomological dimension of a countable group $G$ is at most one more than its Gorenstein homological dimension and, more generally, that $\\text{Gcd}_k G\\le\\text{Ghd}_k G+m+1$ for every $m\\ge0$ and every group $G$ of cardinality at most $\\aleph_m$. When the supremum sfli$k$ of the flat dimensions of the injective $k$-modules is finite, this gives $\\text{Ghd}_k G\\le\\text{Gcd}_k G\\le\\text{Ghd}_k G+1$ for every countable group $G$, so that the two dimensions are finite simultaneously. These results are Gorenstein analogues of Bieri's inequality for the homological and the cohomological dimension of countable groups. They rest on a module-theoretic theorem of independent interest, which is a Gorenstein version of results of Jensen and Osofsky on the projective dimension of flat modules: over an arbitrary ring, every $\\aleph_m$-presented Gorenstein flat module has projectively coresolved Gorenstein flat dimension at most $m+1$. An appendix characterizes the modules of PGF-dimension at most $n$ as the direct summands of the strongly $n$-PGF modules.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24512",
  "author_total": 1,
  "authors": [
    {
      "name": "Dimitra-Dionysia Stergiopoulou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ajjvfej5r579d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24512",
  "fulltext_url": null,
  "id": "ajjvfej5r579d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24512",
  "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.RA"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7214055422",
  "pdf_url": "https://arxiv.org/pdf/2609.24512",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ajjvfej5r579d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24512",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24512"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "$\\aleph_m$-presented Gorenstein flat modules and Gorenstein dimensions of groups",
  "title_text": "ℵ_m-presented Gorenstein flat modules and Gorenstein dimensions of groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/ajjvfej5r579d",
  "venue": "arXiv",
  "versions": []
}