{
  "id": "math/0504543",
  "version": "v1",
  "published": "2005-04-26T21:30:27.000Z",
  "updated": "2005-04-26T21:30:27.000Z",
  "title": "Noncommutative Deformations of Type A Kleinian Singularities and Hilbert Schemes",
  "authors": [
    "Ian M. Musson"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $H_{\\mathbf{k}}$ be a symplectic reflection algebra corresponding to a cyclic subgroup $\\Gamma \\subseteq SL_2 \\C$ of order $n$ and $U_{\\mathbf{k}} = eH_{\\mathbf{k}} e$ the spherical subalgebra of $H_{\\mathbf{k}}$. We show that for suitable ${\\mathbf{k}}$ there is a filtered $\\Z^{n-1}$-algebra $R$ such that (1) there is an equivalence of categories $R-\\mathrm{qgr} \\simeq U_{\\bf k}$-mod, (2) there is an equivalence of categories $gr R-\\mathrm{qgr} \\simeq \\ttt{Coh}(Hilb_\\Gamma \\mathbb{C}^2)$. Here $ \\ttt{Coh}(Hilb_\\Gamma \\mathbb{C}^2)$ is the category of coherent sheaves on the $\\Gamma$-Hilbert scheme. and for a graded algebra $\\mathcal{R},$ we write $ \\mathcal{R}-\\mathrm{qgr}$ for the quotient category of finitely generated graded $\\mathcal{R}$-modules modulo torsion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-04-26T21:30:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hilbert scheme",
      "kleinian singularities",
      "noncommutative deformations",
      "modules modulo torsion",
      "equivalence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......4543M"
    }
  }
}