{
  "id": "1007.3873",
  "version": "v3",
  "published": "2010-07-22T12:56:24.000Z",
  "updated": "2011-10-23T09:48:25.000Z",
  "title": "Ramification and cleanliness",
  "authors": [
    "Ahmed Abbes",
    "Takeshi Saito"
  ],
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "This article is devoted to studying the ramification of Galois torsors and of $\\ell$-adic sheaves in characteristic $p>0$ (with $\\ell\\not=p$). Let $k$ be a perfect field of characteristic $p>0$, $X$ be a smooth, separated and quasi-compact $k$-scheme, $D$ be a simple normal crossing divisor on $X$, $U=X-D$, $\\Lambda$ be a finite local ${\\mathbb Z}_\\ell$-algebra, $F$ be a locally constant constructible sheaf of $\\Lambda$-modules on $U$. We introduce a boundedness condition on the ramification of $F$ along $D$, and study its main properties, in particular, some specialization properties that lead to the fundamental notion of cleanliness and to the definition of the characteristic cycle of $F$. The cleanliness condition extends the one introduced by Kato for rank one sheaves. Roughly speaking, it means that the ramification of $F$ along $D$ is controlled by its ramification at the generic points of $D$. Under this condition, we propose a conjectural Riemann-Roch type formula for $F$. Some cases of this formula have been previously proved by Kato and by the second author (T.S.).",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-10-23T09:48:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ramification",
      "conjectural riemann-roch type formula",
      "cleanliness condition extends",
      "simple normal crossing divisor",
      "characteristic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.3873A"
    }
  }
}