{
  "id": "2306.07006",
  "version": "v1",
  "published": "2023-06-12T10:19:51.000Z",
  "updated": "2023-06-12T10:19:51.000Z",
  "title": "Singularity Categories of Higher Nakayama Algebras",
  "authors": [
    "Wei Xing"
  ],
  "comment": "23 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "For a higher Nakayama algebra $A$ in the sense of Jasso-K\\\"{u}lshammer, we show that the singularity category of $A$ is triangulated equivalent to the stable module category of a self-injective higher Nakayama algebra. This generalizes a similar result for usual Nakayama algebras due to Shen. Our proof relies on the existence of $d\\mathbb{Z}$-cluster tilting subcategories in the module category of $A$ and the result of Kvamme that each $d\\mathbb{Z}$-cluster tilting subcategory of $A$ induces a $d\\mathbb{Z}$-cluster tilting subcategory in its singularity category. Moreover, our result provides many concrete examples of the triangulated Auslander-Iyama correspondence introduced by Jasso-Muro, namely, there is a bijective correspondence between the equivalence classes of the singularity categories of $d$-Nakayama algebras with its basic $d\\mathbb{Z}$-cluster tilting object and the isomorphism classes of self-injective $(d+1)$-Nakayama algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-12T10:19:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "singularity category",
      "cluster tilting subcategory",
      "module category",
      "usual nakayama algebras",
      "self-injective higher nakayama algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}