{
  "abstract": "We prove that every commutative Gaussian ring has finitistic weak dimension at most two. The proof works with a fixed module of finite flat dimension and does not assume that the weak global dimension of the ring is finite. For a local Gaussian ring $A$, with nilradical $N$, the kernel $I$ of $A\\to A_N$ satisfies $NI=I^2=0$. A hypothetical module $M$ of flat dimension three gives a nonzero torsion module $H=\\Tor_2^A(A/N,M)$ over the valuation domain $A/N$, with $H\\cong\\Tor_1^A(I,M)$ and $I\\otimes_{A/N}H=0$. Two carefully controlled localizations, a propagation argument in the chain ring $A/D$, where $D=\\{x:x^2=0\\}$, and an annihilator factorization in a filtered system of cyclic modules yield a contradiction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38241",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiaolei Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1e8cqqg44e42b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38241",
  "fulltext_url": null,
  "id": "1e8cqqg44e42b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38241",
  "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",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7215049165",
  "pdf_url": "https://arxiv.org/pdf/2609.38241",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1e8cqqg44e42b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38241",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38241"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The finitistic weak dimensions of Gaussian rings",
  "title_text": "The finitistic weak dimensions of Gaussian rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/1e8cqqg44e42b",
  "venue": "arXiv",
  "versions": []
}