{
  "id": "1411.4856",
  "version": "v1",
  "published": "2014-11-18T14:54:30.000Z",
  "updated": "2014-11-18T14:54:30.000Z",
  "title": "On the Enlargement by Prüfer Objects of the Cluster Category of type $A_\\infty$",
  "authors": [
    "Thomas A. Fisher"
  ],
  "comment": "36 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "In a paper by Holm and Jorgensen, the cluster category $\\mathscr{D}$ of type $A_\\infty$, with Auslander-Reiten quiver $\\mathbb{Z} A_\\infty$, is introduced. Slices in the Auslander-Reiten quiver of $\\mathscr{D}$ give rise to direct systems; the homotopy colimit of such direct systems can be computed and these \"Pr\\\"ufer objects\" can be adjoined to form a larger category. It is this larger category, $\\overline{\\mathscr{D}},$ which is the main object of study in this paper. We show that $\\overline{\\mathscr{D}}$ inherits a nice geometrical structure from $\\mathscr{D}$; \"arcs\" between non-neighbouring integers on the number line correspond to indecomposable objects, and in the case of $\\overline{\\mathscr{D}}$ we also have arcs to infinity which correspond to the Pr\\\"ufer objects. During the course of this paper, we show that $\\overline{\\mathscr{D}}$ is triangulated, compute homs, investigate the geometric model, and we conclude by computing the cluster tilting subcategories of $\\overline{\\mathscr{D}}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-18T14:54:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13F60",
      "16E45",
      "16G70",
      "18E30"
    ],
    "keywords": [
      "cluster category",
      "prüfer objects",
      "auslander-reiten quiver",
      "direct systems",
      "larger category"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}