{
  "id": "0903.0761",
  "version": "v2",
  "published": "2009-03-04T13:53:58.000Z",
  "updated": "2009-03-24T02:59:31.000Z",
  "title": "Higher Auslander Algebras Admitting Trivial Maximal Orthogonal Subcategories",
  "authors": [
    "Zhaoyong Huang",
    "Xiaojin Zhang"
  ],
  "comment": "25 pages. This version is a combination of the orginal version of this paper with \"From Auslander Algebras to Tilted Algebras\" (arXiv:0903.0760). The latter paper has been withdrawn",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "For an Artinian $(n-1)$-Auslander algebra $\\Lambda$ with global dimension $n(\\geq 2)$, we show that if $\\Lambda$ admits a trivial maximal $(n-1)$-orthogonal subcategory of $\\mod\\Lambda$, then $\\Lambda$ is a Nakayama algebra and the projective or injective dimension of any indecomposable module in $\\mod\\Lambda$ is at most $n-1$. As a result, for an Artinian Auslander algebra with global dimension 2, if $\\Lambda$ admits a trivial maximal 1-orthogonal subcategory of $\\mod\\Lambda$, then $\\Lambda$ is a tilted algebra of finite representation type. Further, for a finite-dimensional algebra $\\Lambda$ over an algebraically closed field $K$, we show that $\\Lambda$ is a basic and connected $(n-1)$-Auslander algebra $\\Lambda$ with global dimension $n(\\geq 2)$ admitting a trivial maximal $(n-1)$-orthogonal subcategory of $\\mod\\Lambda$ if and only if $\\Lambda$ is given by the quiver: $$\\xymatrix{1 & \\ar[l]_{\\beta_{1}} 2 & \\ar[l]_{\\beta_{2}} 3 & \\ar[l]_{\\beta_{3}} ... & \\ar[l]_{\\beta_{n}} n+1} $$ modulo the ideal generated by $\\{\\beta_{i}\\beta_{i+1}| 1\\leq i\\leq n-1 \\}$. As a consequence, we get that a finite-dimensional algebra over an algebraically closed field $K$ is an $(n-1)$-Auslander algebra with global dimension $n(\\geq 2)$ admitting a trivial maximal $(n-1)$-orthogonal subcategory if and only if it is a finite direct product of $K$ and $\\Lambda$ as above. Moreover, we give some necessary condition for an Artinian Auslander algebra admitting a non-trivial maximal 1-orthogonal subcategory.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-03-24T02:59:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G10",
      "16G70",
      "16E10"
    ],
    "keywords": [
      "orthogonal subcategory",
      "auslander algebras admitting trivial maximal",
      "admitting trivial maximal orthogonal subcategories",
      "algebras admitting trivial maximal orthogonal",
      "higher auslander algebras admitting trivial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.0761H"
    }
  }
}