{
  "id": "1811.12588",
  "version": "v1",
  "published": "2018-11-30T02:41:34.000Z",
  "updated": "2018-11-30T02:41:34.000Z",
  "title": "Two-term relative cluster tilting subcategories, $τ-$tilting modules and silting subcategories",
  "authors": [
    "Panyue Zhou",
    "Bin Zhu"
  ],
  "comment": "24pages",
  "categories": [
    "math.RT",
    "math.CT",
    "math.RA"
  ],
  "abstract": "Let $\\mathcal{C}$ be a triangulated category with shift functor $[1]$ and $\\mathcal{R}$ a rigid subcategory of $\\mathcal{C}$. We introduce the notions of two-term $\\mathcal{R}[1]$-rigid subcategories, two-term (weak) $\\mathcal{R}[1]$-cluster tilting subcategories and two-term maximal $\\mathcal{R}[1]$-rigid subcategories, and discuss relationship between them. Our main result shows that there exists a bijection between the set of two-term $\\mathcal{R}[1]$-rigid subcategories of $\\mathcal{C}$ and the set of $\\tau$-rigid subcategories of $\\mod\\mathcal{R}$, which induces a one-to-one correspondence between the set of two-term weak $\\mathcal{R}[1]$-cluster tilting subcategories of $\\mathcal{C}$ and the set of support $\\tau$-tilting subcategories of $\\mod\\mathcal{R}$. This generalizes the main results in \\cite{YZZ} where $\\mathcal{R}$ is a cluster tilting subcategory. When $\\mathcal{R}$ is a silting subcategory, we prove that the two-term weak $\\mathcal{R}[1]$-cluster tilting subcategories are precisely two-term silting subcategories in \\cite{IJY}. Thus the bijection above induces the bijection given by Iyama-J{\\o}rgensen-Yang in \\cite{IJY}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-30T02:41:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "18E30",
      "16G20",
      "16G70"
    ],
    "keywords": [
      "cluster tilting subcategory",
      "two-term relative cluster tilting subcategories",
      "silting subcategory",
      "rigid subcategory",
      "tilting modules"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}