{
  "id": "math/0607589",
  "version": "v1",
  "published": "2006-07-24T10:10:20.000Z",
  "updated": "2006-07-24T10:10:20.000Z",
  "title": "Some homological properties of the category $\\mathcal{O}$",
  "authors": [
    "Volodymyr Mazorchuk"
  ],
  "comment": "30 pages",
  "journal": "Pacific J. Math. 232 (2007), no. 2, 313--341.",
  "categories": [
    "math.RT"
  ],
  "abstract": "In the first part of this paper the projective dimension of the structural modules in the BGG category $\\mathcal{O}$ is studied. This dimension is computed for simple, standard and costandard modules. For tilting and injective modules an explicit conjecture relating the result to Lusztig's $\\mathbf{a}$-function is formulated (and proved for type $A$). The second part deals with the extension algebra of Verma modules. It is shown that this algebra is in a natural way $\\mathbb{Z}^2$-graded and that it has two $\\mathbb{Z}$-graded Koszul subalgebras. The dimension of the space $\\mathrm{Ext}^1$ into the projective Verma module is determined. In the last part several new classes of Koszul modules and modules, represented by linear complexes of tilting modules, are constructed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-07-24T10:10:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10"
    ],
    "keywords": [
      "homological properties",
      "second part deals",
      "explicit conjecture",
      "first part",
      "structural modules"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......7589M"
    }
  }
}