{
  "id": "2408.01359",
  "version": "v1",
  "published": "2024-08-02T16:10:07.000Z",
  "updated": "2024-08-02T16:10:07.000Z",
  "title": "Auslander-Reiten translations in the monomorphism categories of exact categories",
  "authors": [
    "Xiu-Hua Luo",
    "Shijie Zhu"
  ],
  "comment": "26 pages",
  "categories": [
    "math.RT",
    "math.CT"
  ],
  "abstract": "Let $\\Lambda$ be a finite dimensional algebra. Let $\\mathcal C$ be a functorially finite exact subcategory of $\\Lambda$-mod with enough projective and injective objects and $\\mathcal S (\\mathcal C)$ be its monomorphism category. It turns out that the category $\\mathcal S (\\mathcal C)$ has almost split sequences. We show an explicit formula for the Auslander-Reiten translation in $\\mathcal S (\\mathcal C)$. Furthermore, if $\\mathcal C$ is a stably $d$-Calabi-Yau Frobenius category, we calculate objects under powers of Auslander-Reiten translation in the triangulated category $\\overline{\\mathcal S(\\mathcal C)}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-02T16:10:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G70",
      "18A20",
      "18A25",
      "16B50"
    ],
    "keywords": [
      "auslander-reiten translation",
      "monomorphism category",
      "exact categories",
      "functorially finite exact subcategory",
      "calabi-yau frobenius category"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}