{
  "abstract": "Let $A$ be a finite dimensional algebra and let $\\rmHH^*(A)$ be its Hochschild cohomology ring, which is a Gerstenhaber algebra. Denote by $\\calN$ (resp. $G$, $\\calG$) the ideal (resp. weak Gerstenhaber ideal, Gerstenhaber ideal) generated by all homogeneous nilpotent elements. Motivated by their work on support varieties via Hochschild cohomology, Snashall and Solberg conjectured that $\\rmHH^*(A)/\\calN$ is a finitely generated algebra. Xu constructed a counterexample to the Snashall-Solberg conjecture over a base field of characteristic two, and Snashall generalized this example to arbitrary characteristic. Hermann further asked whether $\\rmHH^*(A)/G$ is a finitely generated algebra and suggested considering first the Xu--Snashall algebra. In this paper, we answer this question for the Xu--Snashall algebra. In fact, by explicitly computing the Gerstenhaber algebra structure on the Hochschild cohomology ring, we show that $G=\\calN$; hence $\\rmHH^*(A)/G=\\rmHH^*(A)/\\calN$ is not a finitely generated algebra. Furthermore, we show that $\\rmHH^*(A)/\\calG\\cong K$. Therefore, one may ask whether, for a finite dimensional algebra $A$, $\\rmHH^*(A)/\\calG$ is always a finitely generated algebra. Our main tools are two-sided Anick resolutions and weak self-homotopies.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19560",
  "author_total": 3,
  "authors": [
    {
      "name": "Qi Long"
    },
    {
      "name": "Ziyang Shi"
    },
    {
      "name": "Guodong Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e3202ft0x152w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19560",
  "fulltext_url": null,
  "id": "e3202ft0x152w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19560",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.KT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7213595675",
  "pdf_url": "https://arxiv.org/pdf/2609.19560",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e3202ft0x152w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19560",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19560"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.KT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Gerstenhaber algebra structure on the Hochschild cohomology ring of the Xu--Snashall algebra",
  "title_text": "Gerstenhaber algebra structure on the Hochschild cohomology ring of the Xu--Snashall algebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/e3202ft0x152w",
  "venue": "arXiv",
  "versions": []
}