{
  "id": "2401.09291",
  "version": "v1",
  "published": "2024-01-17T15:44:06.000Z",
  "updated": "2024-01-17T15:44:06.000Z",
  "title": "The index with respect to a contravariantly finite subcategory",
  "authors": [
    "Francesca Fedele",
    "Peter Jorgensen",
    "Amit Shah"
  ],
  "comment": "13 pages",
  "categories": [
    "math.RT",
    "math.CT"
  ],
  "abstract": "The index in triangulated categories plays a key role in the categorification of cluster algebras; specifically, it categorifies the notion of $g$-vectors. For an object $C$ in a triangulated category $\\mathcal{C}$, the index of $C$ was originally defined as $[X_1] - [X_0]$ when $X_1 \\to X_0 \\to C \\to \\Sigma X_1$ is a triangle with the $X_i$ in a cluster tilting subcategory $\\mathcal{X}$ of $\\mathcal{C}$. Using the theory of extriangulated categories, the index was later generalised to the case where $\\mathcal{X}$ is a contravariantly finite subcategory which is rigid, that is, satisfies $\\mathcal{C}( \\mathcal{X},\\Sigma \\mathcal{X} ) = 0$. This paper generalises further by dropping the assumption that $\\mathcal{X}$ is rigid, vastly increasing the potential choice of $\\mathcal{X}$. We show that this version of the index still has the key property of being additive on triangles up to an error term.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-17T15:44:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16E20",
      "18E05",
      "18G80"
    ],
    "keywords": [
      "contravariantly finite subcategory",
      "triangulated category",
      "error term",
      "cluster algebras",
      "triangulated categories plays"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}