{
  "id": "1710.04420",
  "version": "v1",
  "published": "2017-10-12T09:32:39.000Z",
  "updated": "2017-10-12T09:32:39.000Z",
  "title": "On bounds of homological dimensions in Nakayama algebras",
  "authors": [
    "Dag Oskar Madsen",
    "Rene Marczinzik"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $A$ be a Nakayama algebra with $n$ simple modules and a simple module $S$ of even projective dimension $m$. Choose $m$ minimal such that a simple $A$-module with projective dimension $2m$ exists, then we show that the global dimension of $A$ is bounded by $n+m-1$. This gives a combined generalisation of results of Gustafson \\cite{Gus} and Madsen \\cite{Mad}. In \\cite{Bro}, Brown proved that the global dimension of quasi-hereditary Nakayama algebras with $n$ simple modules is bounded by $n$. Using our result on the bounds of global dimensions of Nakayama algebras, we give a short new proof of this result and generalise Brown's result from quasi-hereditary to standardly stratified Nakayama algebras, where the global dimension is replaced with the finitistic dimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-12T09:32:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "global dimension",
      "homological dimensions",
      "simple module",
      "projective dimension",
      "generalise browns result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}