{
  "id": "2209.00898",
  "version": "v1",
  "published": "2022-09-02T09:12:11.000Z",
  "updated": "2022-09-02T09:12:11.000Z",
  "title": "A functorial approach to rank functions on triangulated categories",
  "authors": [
    "Teresa Conde",
    "Mikhail Gorsky",
    "Frederik Marks",
    "Alexandra Zvonareva"
  ],
  "comment": "31 pages",
  "categories": [
    "math.RT",
    "math.CT",
    "math.RA"
  ],
  "abstract": "We study rank functions on a triangulated category $\\mathcal{C}$ via its abelianisation $\\operatorname{mod}\\mathcal{C}$. We prove that every rank function on $\\mathcal{C}$ can be interpreted as an additive function on $\\operatorname{mod}\\mathcal{C}$. As a consequence, every integral rank function has a unique decomposition into irreducible ones. Furthermore, we relate integral rank functions to a number of important concepts in the functor category $\\operatorname{Mod}\\mathcal{C}$. We study the connection between rank functions and functors from $\\mathcal{C}$ to locally finite triangulated categories, generalising results by Chuang and Lazarev. In the special case $\\mathcal{C}=\\mathcal{T}^c$ for a compactly generated triangulated category $\\mathcal{T}$, this connection becomes particularly nice, providing a link between rank functions on $\\mathcal{C}$ and smashing localisations of $\\mathcal{T}$. In this context, any integral rank function can be described using the composition length with respect to certain endofinite objects in $\\mathcal{T}$. Finally, if $\\mathcal{C}=\\operatorname{per}(A)$ for a differential graded algebra $A$, we classify homological epimorphisms $A\\to B$ with $\\operatorname{per}(B)$ locally finite via special rank functions which we call idempotent.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-02T09:12:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "triangulated category",
      "functorial approach",
      "relate integral rank functions",
      "special rank functions",
      "study rank functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}