{
  "id": "1911.07666",
  "version": "v1",
  "published": "2019-11-18T14:46:23.000Z",
  "updated": "2019-11-18T14:46:23.000Z",
  "title": "On the Grothendieck-Serre Conjecture for Classical Groups",
  "authors": [
    "Eva Bayer-Fluckiger",
    "Uriya A. First",
    "Raman Parimala"
  ],
  "comment": "29+10 pages (paper and appendix); comments are welcome",
  "categories": [
    "math.AG",
    "math.KT",
    "math.NT"
  ],
  "abstract": "Let $(A,\\sigma)$ be an Azumaya algebra with involution over a regular semilocal ring $R$ and let $\\varepsilon\\in \\{\\pm 1\\}$. We prove that the Gersten-Witt complex associated to $(A,\\sigma)$ and $\\varepsilon$ is exact if (i) $\\dim R\\leq 2$, or (ii) $\\dim R\\leq 3$, $\\mathrm{ind}\\, A\\leq 2$ and $\\sigma$ is of the first kind, or (iii) $\\dim R\\leq 4$ and $\\mathrm{ind}\\, A$ is odd. As a corollary, we prove the Grothendieck-Serre conjecture on principal homogeneous spaces for all forms of $\\mathbf{GL}_n$, $\\mathbf{Sp}_{2n}$ and $\\mathbf{SO}_n$ when $\\dim R\\leq 2$, and all outer forms of $\\mathbf{GL}_{2n+1}$ when $\\dim R\\leq 4$. In the process, we present a new construction of the Gersten-Witt complex, defined using explicit second-residue maps.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-18T14:46:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "grothendieck-serre conjecture",
      "classical groups",
      "gersten-witt complex",
      "explicit second-residue maps",
      "azumaya algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}