{
  "id": "2210.11798",
  "version": "v1",
  "published": "2022-10-21T08:23:07.000Z",
  "updated": "2022-10-21T08:23:07.000Z",
  "title": "On Krull-Gabriel dimension of cluster repetitive categories and cluster-tilted algebras",
  "authors": [
    "Alicja Jaworska-Pastuszak",
    "Grzegorz Pastuszak",
    "Grzegorz Bobiński"
  ],
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "Assume that $K$ is an algebraically closed field and denote by $KG(R)$ the Krull-Gabriel dimension of $R$, where $R$ is a locally bounded $K$-category (or a bound quiver $K$-algebra). Assume that $C$ is a tilted $K$-algebra and $\\widehat{C},\\check{C},\\widetilde{C}$ are the associated repetitive category, cluster repetitive category and cluster-tilted algebra, respectively. Our first result states that $KG(\\widetilde{C})=KG(\\check{C})\\leq KG(\\widehat{C})$. Since the Krull-Gabriel dimensions of tame locally support-finite repetitive categories are known, we further conclude that $KG(\\widetilde{C})=KG(\\check{C})=KG(\\widehat{C})\\in\\{0,2,\\infty\\}$. Finally, in the Appendix Grzegorz Bobi\\'nski presents a different way of determining the Krull-Gabriel dimension of the cluster-tilted algebras, by applying results of Geigle.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-21T08:23:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20"
    ],
    "keywords": [
      "krull-gabriel dimension",
      "cluster repetitive category",
      "cluster-tilted algebra",
      "appendix grzegorz bobinski",
      "first result states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}