{
  "id": "1012.1290",
  "version": "v3",
  "published": "2010-12-06T19:23:45.000Z",
  "updated": "2012-04-05T18:13:13.000Z",
  "title": "Inverse Problems for deformation rings",
  "authors": [
    "Frauke M. Bleher",
    "Ted Chinburg",
    "Bart de Smit"
  ],
  "comment": "17 pages; this paper is closely related to arXiv:1003.3143. In the third version, we added a new Section 5 with further examples",
  "journal": "Trans. Amer. Math. Soc. 365 (2013), 6149-6165",
  "doi": "10.1090/S0002-9947-2013-05848-5",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\mathcal{W}$ be a complete local commutative Noetherian ring with residue field $k$ of positive characteristic $p$. We study the inverse problem for the versal deformation rings $R_{\\mathcal{W}}(\\Gamma,V)$ relative to $\\mathcal{W}$ of finite dimensional representations $V$ of a profinite group $\\Gamma$ over $k$. We show that for all $p$ and $n \\ge 1$, the ring $\\mathcal{W}[[t]]/(p^n t,t^2)$ arises as a universal deformation ring. This ring is not a complete intersection if $p^n\\mathcal{W}\\neq\\{0\\}$, so we obtain an answer to a question of M. Flach in all characteristics. We also study the `inverse inverse problem' for the ring $\\mathcal{W}[[t]]/(p^n t,t^2)$; this is to determine all pairs $(\\Gamma, V)$ such that $R_{\\mathcal{W}}(\\Gamma,V)$ is isomorphic to this ring.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-04-05T18:13:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F80"
    ],
    "keywords": [
      "local commutative noetherian ring",
      "inverse inverse problem",
      "complete local commutative noetherian",
      "versal deformation rings",
      "finite dimensional representations"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Trans. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.1290B"
    }
  }
}