{
  "id": "0912.0647",
  "version": "v2",
  "published": "2009-12-03T13:06:44.000Z",
  "updated": "2010-02-23T15:25:52.000Z",
  "title": "Derived equivalences for $Φ$-Auslander-Yoneda algebras",
  "authors": [
    "Wei Hu",
    "Changchang Xi"
  ],
  "comment": "27 pages",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "In this paper, we introduce $\\Phi$-Auslander-Yoneda algebras in a triangulated category with $\\Phi$ a parameter set in $\\mathbb N$, and provide a method to construct new derived equivalences between these $\\Phi$-Auslander-Yoneda algebras (not necessarily Artin algebras), or their quotient algebras, from a given almost $\\nu$-stable derived equivalence. As consequences of our method, we have: (1) Suppose that $A$ and $B$ are representation-finite, self-injective Artin algebras with $_AX$ and $_BY$ additive generators for $A$ and $B$, respectively. If $A$ and $B$ are derived-equivalent, then the $\\Phi$-Auslander-Yoneda algebras of $X$ and $Y$ are derived-equivalent for every admissible set $\\Phi$. In particular, the Auslander algebras of $A$ and $B$ are both derived-equivalent and stably equivalent. (2) For a self-injective Artin algeba $A$ and an $A$-module $X$, the $\\Phi$-Auslander-Yoneda algebras of $A\\oplus X$ and $A\\oplus \\Omega_A(X)$ are derived-equivalent for every admissible set $\\Phi$, where $\\Omega$ is the Heller loop operator. Motivated by these derived equivalences between $\\Phi$-Auslander-Yoneda algebras, we consider constructions of derived equivalences for quotient algebras, and show, among others, that a derived equivalence between two basic self-injective algebras may transfer to a derived equivalence between their quotient algebras obtained by factorizing out socles.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-02-23T15:25:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "18E30",
      "16G10",
      "16S50",
      "18G15"
    ],
    "keywords": [
      "derived equivalence",
      "auslander-yoneda algebras",
      "quotient algebras",
      "derived-equivalent",
      "heller loop operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.0647H"
    }
  }
}