{
  "id": "1903.11307",
  "version": "v1",
  "published": "2019-03-27T09:36:38.000Z",
  "updated": "2019-03-27T09:36:38.000Z",
  "title": "Pure semisimple $n$-cluster tilting subcategories",
  "authors": [
    "Ramin Ebrahimi",
    "Alireza Nasr-Isfahani"
  ],
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "From the viewpoint of higher homological algebra, we introduce pure semisimple $n$-abelian category, which is analogs of pure semisimple abelian category. Let $\\Lambda$ be an Artin algebra and $\\mathcal{M}$ be an $n$-cluster tilting subcategory of $Mod$-$\\Lambda$. We show that $\\mathcal{M}$ is pure semisimple if and only if each module in $\\mathcal{M}$ is a direct sum of finitely generated modules. Let $\\mathfrak{m}$ be an $n$-cluster tilting subcategory of $mod$-$\\Lambda$. We show that $Add(\\mathfrak{m})$ is an $n$-cluster tilting subcategory of $Mod$-$\\Lambda$ if and only if $\\mathfrak{m}$ has an additive generator if and only if $Mod(\\mathfrak{m})$ is locally finite. This generalizes Auslander's classical results on pure semisimplicity of Artin algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-27T09:36:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16E30"
    ],
    "keywords": [
      "cluster tilting subcategory",
      "artin algebra",
      "pure semisimple abelian category",
      "generalizes auslanders classical results",
      "pure semisimplicity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}