{
  "abstract": "Let $(A,\\mathfrak{m}), (B,\\mathfrak{n}) $ be Gorenstein local rings and let $\\underline{CM}(A)$ be its stable category of finitely generated maximal Cohen-Macaulay $A$-modules. Suppose we have an equivalence $Φ\\colon\\underline{CM}(A) \\rightarrow \\underline{CM}(A)$ as triangulated categories. We show (1) If $A, B$ are not hypersurfaces then $dim A = dim B$. (2) If $M$ is a maximal \\CM \\ $A$-module then $$\\text{curv}_A(M) = \\text{curv}_B (Φ(M)),$$ here $\\text{curv}_A(M) = \\limsup_n \\sqrt[n]{\\ell(\\text{Tor}^A_n(M,k))}$. (3) $A$ satisfies Serre's condition $R_i$ if and only if $B$ satisfies $R_i$. (4) $A$ is a complete intersection on the punctured spectrum of $A$ if and only if $B$ is a complete intersection on the punctured spectrum of $B$. We also show that $Φ$ imposes constraints of residue field of $B$ in terms of residue field of $A$ and vice-versa. Finally if $I$ is an ideal in $A$ such that the extended Rees algebra $\\mathcal{R}(I) = A[It, t^{-1}]$ is Gorenstein then we construct a triangulated functor $Ψ\\colon \\underline{CM}^\\mathbb{Z}(\\mathcal{R}(I)) \\rightarrow \\underline{CM}(A)$ where $\\underline{CM}^\\mathbb{Z}(\\mathcal{R}(I))$ is the stable category of all graded maximal Cohen-Macaulay $\\mathcal{R}(I)$-modules. We show that $Ψ$ induces an equivalence $\\underline{CM}^\\mathbb{Z}(\\mathcal{R}(I))/\\ker Ψ\\rightarrow \\underline{CM}(A)$. We give some applications of this map.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18626",
  "author_total": 1,
  "authors": [
    {
      "name": "Tony J. Puthenpurakal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ayp00vjxwpey0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18626",
  "fulltext_url": null,
  "id": "ayp00vjxwpey0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18626",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7213500209",
  "pdf_url": "https://arxiv.org/pdf/2609.18626",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ayp00vjxwpey0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18626",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18626"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the Stable category of maximal Cohen-Macaulay modules over Gorenstein rings-II",
  "title_text": "On the Stable category of maximal Cohen-Macaulay modules over Gorenstein rings-II",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/ayp00vjxwpey0",
  "venue": "arXiv",
  "versions": []
}