{
  "id": "1410.1732",
  "version": "v1",
  "published": "2014-10-07T14:00:58.000Z",
  "updated": "2014-10-07T14:00:58.000Z",
  "title": "Induced and Coinduced Modules in Cluster-Tilted Algebras",
  "authors": [
    "Ralf Schiffler",
    "Khrystyna Serhiyenko"
  ],
  "comment": "38 pages, 1 figure",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "We propose a new approach to study the relation between the module categories of a tilted algebra $C$ and the corresponding cluster-tilted algebra $B=C\\ltimes E$. This new approach consists of using the induction functor $-\\otimes_C B$ as well as the coinduction functor $D(B\\otimes_C D-)$. We give an explicit construction of injective resolutions of projective $B$-modules, and as a consequence, we obtain a new proof of the 1-Gorenstein property for cluster-tilted algebras. We show that $DE$ is a partial tilting and a $\\tau$-rigid $C$-module and that the induced module $DE\\otimes_C B$ is a partial tilting and a $\\tau$-rigid $B$-module. Furthermore, if $C=\\text{End}_A T$ for a tilting module $T$ over a hereditary algebra $A$, we compare the induction and coinduction functors to the Buan-Marsh-Reiten functor $\\text{Hom}_{\\mathcal{C}_A}(T,-)$ from the cluster-category of $A$ to the module category of $B$. We also study the question which $B$-modules are actually induced or coinduced from a module over a tilted algebra.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-07T14:00:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "16G70",
      "13F60"
    ],
    "keywords": [
      "cluster-tilted algebra",
      "coinduced modules",
      "module category",
      "coinduction functor",
      "approach consists"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.1732S"
    }
  }
}