{
  "id": "1711.04290",
  "version": "v1",
  "published": "2017-11-12T13:21:14.000Z",
  "updated": "2017-11-12T13:21:14.000Z",
  "title": "Triangulated categories with cluster-tilting subcategories",
  "authors": [
    "Wuzhong Yang",
    "Panyue Zhou",
    "Bin Zhu"
  ],
  "comment": "All comments welcome",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\C$ be a triangulated category with a cluster tilting subcategory $\\T$. We introduce the notion of $\\T[1]$-cluster tilting subcategories (also called ghost cluster tilting subcategories) of $\\C$, which are a generalization of cluster tilting subcategories. We first develop a basic theory on ghost cluster tilting subcategories. Secondly, we study links between ghost cluster tilting theory and $\\tau$-tilting theory: Inspired by the work of Iyama, J{\\o}rgensen and Yang \\cite{ijy}, we introduce the notion of $\\tau$-tilting subcategories and tilting subcategories of $\\mod\\T$. We show that there exists a bijection between weak $\\T[1]$-cluster tilting subcategories of $\\C$ and support $\\tau$-tilting subcategories of $\\mod\\T$. Moreover, we figure out the subcategories of $\\mod\\T$ which correspond to cluster tilting subcategories of $\\C$. This generalizes and improves several results by Adachi-Iyama-Reiten \\cite{AIR}, Beligiannis \\cite{Be2}, and Yang-Zhu \\cite{YZ}. Finally, we prove that the definition of ghost cluster tilting objects is equivalent to the definition of relative cluster tilting objects introduced by the first and the third author in \\cite{YZ}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-12T13:21:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "18E30",
      "16G20",
      "16G70"
    ],
    "keywords": [
      "cluster tilting subcategory",
      "triangulated category",
      "ghost cluster tilting subcategories",
      "cluster-tilting subcategories",
      "ghost cluster tilting theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}