{
  "id": "math/0504301",
  "version": "v2",
  "published": "2005-04-14T17:54:52.000Z",
  "updated": "2005-09-30T11:15:59.000Z",
  "title": "The Auslander-Reiten Translation in Submodule Categories",
  "authors": [
    "Claus Michael Ringel",
    "Markus Schmidmeier"
  ],
  "comment": "Dedicated to Idun Reiten",
  "categories": [
    "math.RT",
    "math.CT"
  ],
  "abstract": "Let $\\Lambda$ be an artin algebra and $S(\\Lambda)$ the category of all embeddings $(A\\subseteq B)$ where $B$ is a finitely generated $\\Lambda$-module and $A$ is a submodule of $B$. Then $S(\\Lambda)$ is an exact Krull-Schmidt category which has Auslander-Reiten sequences. In this manuscript we show that the Auslander-Reiten translation in $S(\\Lambda)$ can be computed within the category of $\\Lambda$-modules by using our construction of minimal monomorphisms. If in addition $\\Lambda$ is uniserial then any nonprojective indecomposable object in $\\Cal S(\\Lambda)$ is invariant under the sixth power of the Auslander-Reiten translation.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-09-30T11:15:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G70",
      "18E30"
    ],
    "keywords": [
      "auslander-reiten translation",
      "submodule categories",
      "exact krull-schmidt category",
      "minimal monomorphisms",
      "auslander-reiten sequences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......4301R"
    }
  }
}