{
  "id": "1801.02245",
  "version": "v1",
  "published": "2018-01-07T20:29:09.000Z",
  "updated": "2018-01-07T20:29:09.000Z",
  "title": "Essential dimension in mixed characteristic",
  "authors": [
    "Patrick Brosnan",
    "Zinovy Reichstein",
    "Angelo Vistoli"
  ],
  "comment": "12 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Suppose $G$ is a finite group and $p$ is either a prime number or $0$. For $p$ positive, we say that $G$ is weakly tame at $p$ if $G$ has no non-trivial normal $p$-subgroups. By convention we say that every finite group is weakly tame at $0$. Now suppose that $G$ is a finite group which is weakly tame at the residue characteristic of a discrete valuation ring $R$. Our main result shows that the essential dimension of $G$ over the fraction field $K$ of $R$ is at least as large as the essential dimension of $G$ over the residue field $k$. We also prove a more general statement of this type for a class of \\'etale gerbes over $R$. As a corollary, we show that, if $G$ is weakly tame at $p$ and $k$ is any field of characteristic $p >0$ containing the algebraic closure of $\\mathbb{F}_p$, then the essential dimension of $G$ over $k$ is less than or equal to the essential dimension of $G$ over any characteristic $0$ field. A conjecture of A. Ledet asserts that the essential dimension, $\\mathrm{ed}_k(\\mathbb{Z}/p^n\\mathbb{Z})$, of the cyclic group of order $p^n$ over a field $k$ is equal to $n$ whenever $k$ is a field of characteristic $p$. We show that this conjecture implies that $\\mathrm{ed}_{\\mathbb{C}}(G) \\geq n$ for any finite group $G$ which is weakly tame at $p$ and contains an element of order $p^n$. To the best of our knowledge, an unconditional proof of the last inequality is out of the reach of all presently known techniques.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-07T20:29:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14A20",
      "13A18",
      "13A50"
    ],
    "keywords": [
      "essential dimension",
      "weakly tame",
      "finite group",
      "mixed characteristic",
      "conjecture implies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}