{
  "id": "1408.3360",
  "version": "v1",
  "published": "2014-08-14T17:39:12.000Z",
  "updated": "2014-08-14T17:39:12.000Z",
  "title": "On the crystalline cohomology of Deligne-Lusztig varieties",
  "authors": [
    "Elmar Grosse-Klönne"
  ],
  "journal": "Finite Fields and their Applications 13, No. 4, 896-921 (2007)",
  "categories": [
    "math.RT",
    "math.AG"
  ],
  "abstract": "Let $X\\to Y^0$ be an abelian prime-to-$p$ Galois covering of smooth schemes over a perfect field $k$ of characteristic $p>0$. Let $Y$ be a smooth compactification of $Y^0$ such that $Y-Y^0$ is a normal crossings divisor on $Y$. We describe a logarithmic $F$-crystal on $Y$ whose rational crystalline cohomology is the rigid cohomology of $X$, in particular provides a natural $W[F]$-lattice inside the latter; here $W$ is the Witt vector ring of $k$. If a finite group $G$ acts compatibly on $X$, $Y^0$ and $Y$ then our construction is $G$-equivariant. As an example we apply it to Deligne-Lusztig varieties. For a finite field $k$, if ${\\mathbb G}$ is a connected reductive algebraic group defined over $k$ and ${\\mathbb L}$ a $k$-rational torus satisfying a certain standard condition, we obtain a meaningful equivariant $W[F]$-lattice in the cohomology ($\\ell$-adic or rigid) of the corresponding Deligne-Lusztig variety and an expression of its reduction modulo $p$ in terms of equivariant Hodge cohomology groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-14T17:39:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "reductive algebraic group",
      "equivariant hodge cohomology groups",
      "normal crossings divisor",
      "rational crystalline cohomology",
      "corresponding deligne-lusztig variety"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.3360G"
    }
  }
}