{
  "id": "2201.00740",
  "version": "v3",
  "published": "2022-01-03T16:35:00.000Z",
  "updated": "2023-10-16T14:52:47.000Z",
  "title": "The index with respect to a rigid subcategory of a triangulated category",
  "authors": [
    "Peter Jørgensen",
    "Amit Shah"
  ],
  "comment": "V1: 26 pages. V2: 26 pages, minor title change, added to Remark 4.13. V3: Several changes made following a review. We now use the notation add(X*Y) for the extension subcategory previously denoted X*Y",
  "doi": "10.1093/imrn/rnad130",
  "categories": [
    "math.RT",
    "math.CT",
    "math.KT"
  ],
  "abstract": "Palu defined the index with respect to a cluster tilting object in a suitable triangulated category, in order to better understand the Caldero-Chapoton map that exhibits the connection between cluster algebras and representation theory. We push this further by proposing an index with respect to a contravariantly finite, rigid subcategory, and we show this index behaves similarly to the classical index. Let $\\mathcal{C}$ be a skeletally small triangulated category with split idempotents, which is thus an extriangulated category $(\\mathcal{C},\\mathbb{E},\\mathfrak{s})$. Suppose $\\mathcal{X}$ is a contravariantly finite, rigid subcategory in $\\mathcal{C}$. We define the index $\\mathrm{ind}_{\\mathcal{X}}(C)$ of an object $C\\in\\mathcal{C}$ with respect to $\\mathcal{X}$ as the $K_{0}$-class $[C]_{\\mathcal{X}}$ in Grothendieck group $K_{0}(\\mathcal{C},\\mathbb{E}_{\\mathcal{X}},\\mathfrak{s}_{\\mathcal{X}})$ of the relative extriangulated category $(\\mathcal{C},\\mathbb{E}_{\\mathcal{X}},\\mathfrak{s}_{\\mathcal{X}})$. By analogy to the classical case, we give an additivity formula with error term for $\\mathrm{ind}_{\\mathcal{X}}$ on triangles in $\\mathcal{C}$. In case $\\mathcal{X}$ is contained in another suitable subcategory $\\mathcal{T}$ of $\\mathcal{C}$, there is a surjection $Q\\colon K_{0}(\\mathcal{C},\\mathbb{E}_{\\mathcal{T}},\\mathfrak{s}_{\\mathcal{T}}) \\twoheadrightarrow K_{0}(\\mathcal{C},\\mathbb{E}_{\\mathcal{X}},\\mathfrak{s}_{\\mathcal{X}})$. Thus, in order to describe $K_{0}(\\mathcal{C},\\mathbb{E}_{\\mathcal{X}},\\mathfrak{s}_{\\mathcal{X}})$, it suffices to determine $K_{0}(\\mathcal{C},\\mathbb{E}_{\\mathcal{T}},\\mathfrak{s}_{\\mathcal{T}})$ and $\\operatorname{Ker} Q$. We do this under certain assumptions.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2023-10-16T14:52:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16E20",
      "18E05",
      "18G80"
    ],
    "keywords": [
      "rigid subcategory",
      "contravariantly finite",
      "extriangulated category",
      "better understand",
      "caldero-chapoton map"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}