{
  "id": "1512.03613",
  "version": "v1",
  "published": "2015-12-11T12:11:41.000Z",
  "updated": "2015-12-11T12:11:41.000Z",
  "title": "Some applications of $τ$-tilting theory",
  "authors": [
    "Shen Li",
    "Shunhua Zhang"
  ],
  "comment": "19 pages",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "Let $A$ be a finite dimensional algebra over an algebraically closed field $k$, and $M$ be a partial tilting $A$-module. We prove that the Bongartz $\\tau$-tilting complement of $M$ coincides with its Bongartz complement, and then we give a new proof of that every almost complete tilting $A$-module has at most two complements. Let $A=kQ$ be a path algebra. We prove that the support $\\tau$-tilting quiver $\\overrightarrow{Q}({\\rm s}\\tau$-${\\rm tilt} A)$ of $A$ is connected. As an application, we investigate the conjecture of Happel and Unger in [9] which claims that each connected component of the tilting quiver $\\overrightarrow{Q}({\\rm tilt} A)$ contains only finitely many non-saturated vertices. We prove that this conjecture is true for $Q$ being all Dynkin and Euclidean quivers and wild quivers with two or three vertices, and we also give an example to indicates that this conjecture is not true if $Q$ is a wild quiver with four vertices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-11T12:11:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tilting theory",
      "application",
      "wild quiver",
      "tilting quiver",
      "finite dimensional algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}