{
  "id": "1412.7301",
  "version": "v1",
  "published": "2014-12-23T10:00:22.000Z",
  "updated": "2014-12-23T10:00:22.000Z",
  "title": "Derived equivalences and stable equivalences of Morita type, II",
  "authors": [
    "Wei Hu",
    "Changchang Xi"
  ],
  "comment": "This is the second paper of the series: Derived equivalences and stable equivalences of Morita type. The first paper appeared in Nagoya Math. J. 200(2010) 107-152",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "Motivated by understanding the Brou\\'e's abelian defect group conjecture from algebraic point of view, we consider the question of how to lift a stable equivalence of Morita type between arbitrary finite dimensional algebras to a derived equivalence. In this paper, we present a machinery to solve this question for a class of stable equivalences of Morita type. In particular, we show that every stable equivalence of Morita type between Frobenius-finite algebras over an algebraically closed field can be lifted to a derived equivalence. %Thus Frobenius-finite algebras share many common %invariants of both derived equivalences and stable equivalences. Especially, Auslander-Reiten conjecrure is true for stable equivalences of Morita type between Frobenius-finite algebras without semisimple direct summands. Examples of such a class of algebras are abundant, including Auslander algebras, cluster-tilted algebras and certain Frobenius extensions. As a byproduct of our methods, we further show that, for a Nakayama-stable idempotent element $e$ in an algebra $A$ over an arbitrary field, each tilting complex over $eAe$ can be extended to a tilting complex over $A$ that induces an almost $\\nu$-stable derived equivalence studied in the first paper of this series. Moreover, we demonstrate that our techniques are applicable to verify the Brou\\'{e}'s abelian defect group conjecture for several cases mentioned by Okuyama.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-23T10:00:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G10",
      "18E30",
      "20C20",
      "18G35",
      "16L60",
      "20K40"
    ],
    "keywords": [
      "stable equivalence",
      "morita type",
      "derived equivalence",
      "frobenius-finite algebras",
      "broues abelian defect group conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.7301H"
    }
  }
}