{
  "id": "1701.03325",
  "version": "v1",
  "published": "2017-01-12T12:45:36.000Z",
  "updated": "2017-01-12T12:45:36.000Z",
  "title": "Tensor products of n-complete algebras",
  "authors": [
    "Andrea Pasquali"
  ],
  "comment": "16 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "If $A$ and $B$ are $n$- and $m$-representation finite $k$-algebras, then their tensor product $\\Lambda = A\\otimes_k B$ is not in general $(n+m)$-representation finite. However, we prove that if $A$ and $B$ are acyclic and satisfy the weaker assumption of $n$- and $m$-completeness, then $\\Lambda$ is $(n+m)$-complete. This mirrors the fact that taking higher Auslander algebra does not preserve $d$-representation finiteness in general, but it does preserve $d$-completeness. As a corollary, we get the necessary condition for $\\Lambda$ to be $(n+m)$-representation finite which was found by Herschend and Iyama by using a certain twisted fractionally Calabi-Yau property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-12T12:45:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tensor product",
      "n-complete algebras",
      "higher auslander algebra",
      "representation finiteness",
      "completeness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}