{
  "id": "1407.2326",
  "version": "v1",
  "published": "2014-07-09T00:53:12.000Z",
  "updated": "2014-07-09T00:53:12.000Z",
  "title": "A Homological Bridge Between Finite and Infinite Dimensional Representations of Algebras",
  "authors": [
    "B. Huisgen-Zimmermann",
    "S. O. Smalø"
  ],
  "journal": "Algebras and Representation Theory 1 (1998) 169-188",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "Given a finite dimensional algebra $\\Lambda$, we show that a frequently satisfied finiteness condition for the category ${\\cal P}^{\\infty}(\\Lambda\\rm{-mod})$ of all finitely generated (left) $\\Lambda$-modules of finite projective dimension, namely contravariant finiteness of ${\\cal P}^{\\infty}(\\Lambda\\rm{-mod})$ in $\\Lambda\\rm{-mod}$, forces arbitrary modules of finite projective dimension to be direct limits of objects in ${\\cal P}^{\\infty}(\\Lambda\\rm{-mod})$. Among numerous applications, this yields an encompassing sufficient condition for the validity of the first finitistic dimension conjecture, that is, for the little finitistic dimension of $\\Lambda$ to coincide with the big (this is well-known to fail over finite dimensional algebras in general).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-09T00:53:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16E10",
      "16D90",
      "16G10"
    ],
    "keywords": [
      "infinite dimensional representations",
      "homological bridge",
      "finite dimensional algebra",
      "finite projective dimension",
      "first finitistic dimension conjecture"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Represent. Theory"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.2326H"
    }
  }
}