{
  "abstract": "The stable derived category $D_{sg}(R)$ of a Gorenstein ring $R$ is the Verdier quotient of the bounded derived category $D^b(\\mathsf{mod}\\, R)$ by the thick subcategory of perfect complexes, and was introduced by Buchweitz as a homological measure of the singularities of $R$. This thesis contributes to its study, centered around representation-theoretic and Koszul duality aspects. In Part I, we complete (over $\\mathbb{C}$) the classification of homogeneous complete intersection isolated singularities $R$ for which $D^{\\mathbb{Z}}_{sg}(R)$ (respectively, $D^b(\\mathsf{coh}\\, X)$ for $X = \\mathsf{proj}\\, R$) contains a tilting object. We then use results of Buchweitz-Iyama-Yamaura to describe the indecomposable objects in $D_{sg}^{\\mathbb{Z}}(R_Y)$ for $R_Y$ the coordinate rings of $4$ points on $\\mathbb{P}^1$ and $4$ points on $\\mathbb{P}^2$ in general position, and classify their Betti tables. In Part II, for $R$ a Koszul Gorenstein algebra, we study a natural pair of subcategories whose intersection $\\mathcal{H}^{\\mathsf{lin}}(R) \\subseteq D_{sg}^{\\mathbb{Z}}(R)$ consists of modules with eventually linear projective resolutions. We prove that such a pair forms a bounded t-structure if and only if $R$ is absolutely Koszul in the sense of Herzog-Iyengar, in which case $D^b(\\mathcal{H}^{\\mathsf{lin}}(R)) \\cong D_{sg}^{\\mathbb{Z}}(R)$. We then relate the heart to modules over the Koszul dual $R^!$. This extends the BGG correspondence to any pair of Koszul dual algebras $(R, R^!)$ with $R$ absolutely Koszul Gorenstein, applying e.g. to elliptic normal curves and del Pezzo surfaces of degree $\\geq 4$. Finally, we relate our results to conjectures of Bondal and Minamoto on the graded coherence of AS-regular algebras and higher preprojective algebras; we characterise when these conjectures hold in a restricted setting, and give counterexamples to both in all dimensions $\\geq 4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32074",
  "author_total": 1,
  "authors": [
    {
      "name": "Vincent Gélinas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bx26wg0e6gjzh.bib",
  "cited_by": 2,
  "doi": "10.48550/arxiv.2610.32074",
  "fulltext_url": null,
  "id": "bx26wg0e6gjzh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32074",
  "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.AG"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W2941846604",
  "pdf_url": "https://arxiv.org/pdf/2610.32074",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bx26wg0e6gjzh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32074",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32074"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Contributions to the Stable Derived Categories of Gorenstein Rings",
  "title_text": "Contributions to the Stable Derived Categories of Gorenstein Rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/bx26wg0e6gjzh",
  "venue": "arXiv",
  "versions": []
}