{
  "id": "1801.04738",
  "version": "v1",
  "published": "2018-01-15T11:11:41.000Z",
  "updated": "2018-01-15T11:11:41.000Z",
  "title": "Tilting modules over Auslander-Gorenstein Algebras",
  "authors": [
    "Osamu Iyama",
    "Xiaojin Zhang"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "For a finite dimensional algebra $\\Lambda$ and a non-negative integer $n$, we characterize when the set $\\tilt_n\\Lambda$ of additive equivalence classes of tilting modules with projective dimension at most $n$ has a minimal (or equivalently, minimum) element. This generalize results of Happel-Unger. Moreover, for an $n$-Gorenstein algebra $\\Lambda$ with $n\\geq 1$, we construct a minimal element in $\\tilt_{n}\\Lambda$. As a result, we give equivalent conditions for a $k$-Gorenstein algebra to be Iwanaga-Gorenstein. Moreover, for an $1$-Gorenstein algebra $\\Lambda$ and its factor algebra $\\Gamma=\\Lambda/(e)$, we show that there is a bijection between $\\tilt_1\\Lambda$ and the set $\\sttilt\\Gamma$ of isomorphism classes of basic support $\\tau$-tilting $\\Gamma$-modules, where $e$ is an idempotent such that $e\\Lambda $ is the additive generator of projective-injective $\\Lambda$-modules.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-15T11:11:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G10",
      "16E10"
    ],
    "keywords": [
      "tilting modules",
      "auslander-gorenstein algebras",
      "finite dimensional algebra",
      "basic support",
      "factor algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}