{
  "id": "0808.1629",
  "version": "v2",
  "published": "2008-08-12T09:54:13.000Z",
  "updated": "2010-01-23T11:41:59.000Z",
  "title": "Purity of level m stratifications",
  "authors": [
    "Marc-Hubert Nicole",
    "Adrian Vasiu",
    "Torsten Wedhorn"
  ],
  "comment": "Final version 38 pages. To appear in Ann. Sci. \\'Ec. Norm. Sup",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let $k$ be a field of characteristic $p>0$. Let $D_m$ be a $\\BT_m$ over $k$ (i.e., an $m$-truncated Barsotti--Tate group over $k$). Let $S$ be a\\break $k$-scheme and let $X$ be a $\\BT_m$ over $S$. Let $S_{D_m}(X)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $p\\ge 5$, we show that $S_{D_m}(X)$ is pure in $S$ i.e., the immersion $S_{D_m}(X) \\hookrightarrow S$ is affine. For $p\\in\\{2,3\\}$, we prove purity if $D_m$ satisfies a certain property depending only on its $p$-torsion $D_m[p]$. For $p\\ge 5$, we apply the developed techniques to show that all level $m$ stratifications associated to Shimura varieties of Hodge type are pure.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-01-23T11:41:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E57",
      "11G10",
      "11G18",
      "11G25",
      "14F30",
      "14G35",
      "14L05",
      "14L15",
      "14L30",
      "14R20",
      "20G25"
    ],
    "keywords": [
      "stratifications",
      "fppf topology isomorphic",
      "truncated barsotti-tate group",
      "shimura varieties",
      "hodge type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}