{
  "id": "1504.08282",
  "version": "v1",
  "published": "2015-04-30T15:54:17.000Z",
  "updated": "2015-04-30T15:54:17.000Z",
  "title": "Approximations of injective modules and finitistic dimension",
  "authors": [
    "François Huard",
    "David Smith"
  ],
  "comment": "4 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\Lambda$ be an artin algebra and let $\\mathcal{P}^{<\\infty}_\\Lambda$ the category of finitely generated right $\\Lambda$-modules of finite projective dimension. We show that $\\mathcal{P}^{<\\infty}_\\Lambda$ is contravariantly finite in $\\rm mod\\,\\Lambda$ if and only if the direct sum $E$ of the indecomposable Ext-injective modules in $\\mathcal{P}^{<\\infty}_\\Lambda$ form a tilting module in $\\rm mod\\,\\Lambda$. Moreover, we show that in this case $E$ coincides with the direct sum of the minimal right $\\mathcal{P}^{<\\infty}_\\Lambda$-approximations of the indecomposable $\\Lambda$-injective modules and that the projective dimension of $E$ equal to the finitistic dimension of $\\Lambda$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-30T15:54:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G10"
    ],
    "keywords": [
      "finitistic dimension",
      "approximations",
      "direct sum",
      "finite projective dimension",
      "artin algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150408282H"
    }
  }
}