{
  "id": "2011.15107",
  "version": "v1",
  "published": "2020-11-30T18:41:22.000Z",
  "updated": "2020-11-30T18:41:22.000Z",
  "title": "Auslander's formula and correspondence for exact categories",
  "authors": [
    "Ruben Henrard",
    "Sondre Kvamme",
    "Adam-Christiaan van Roosmalen"
  ],
  "comment": "36 pages, comments welcome!",
  "categories": [
    "math.RT"
  ],
  "abstract": "The Auslander correspondence is a fundamental result in Auslander-Reiten theory. In this paper we introduce the category $\\operatorname{mod_{\\mathsf{adm}}}(\\mathcal{E})$ of admissibly finitely presented functors and use it to give a version of Auslander correspondence for any exact category $\\mathcal{E}$. An important ingredient in the proof is the localization theory of exact categories. We also investigate how properties of $\\mathcal{E}$ are reflected in $\\operatorname{mod_{\\mathsf{adm}}}(\\mathcal{E})$, for example being (weakly) idempotent complete or having enough projectives or injectives. Furthermore, we describe $\\operatorname{mod_{\\mathsf{adm}}}(\\mathcal{E})$ as a subcategory of $\\operatorname{mod}(\\mathcal{E})$ when $\\mathcal{E}$ is a resolving subcategory of an abelian category. This includes the category of Gorenstein projective modules and the category of maximal Cohen-Macaulay modules as special cases. Finally, we use $\\operatorname{mod_{\\mathsf{adm}}}(\\mathcal{E})$ to give a bijection between exact structures on an idempotent complete additive category $\\mathcal{C}$ and certain resolving subcategories of $\\operatorname{mod}(\\mathcal{C})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-30T18:41:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "18E05",
      "16G50",
      "18E35"
    ],
    "keywords": [
      "exact category",
      "auslanders formula",
      "auslander correspondence",
      "resolving subcategory",
      "maximal cohen-macaulay modules"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}