{
  "id": "0811.1168",
  "version": "v1",
  "published": "2008-11-07T16:19:05.000Z",
  "updated": "2008-11-07T16:19:05.000Z",
  "title": "A Simpson correspondence in positive characteristic",
  "authors": [
    "Michel Gros",
    "Bernard Le Stum",
    "Adolfo Quirós"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "We define the $p^m$-curvature map on the sheaf of differential operators of level $m$ on a scheme of positive characteristic $p$ as dual to some divided power map on infinitesimal neighborhhods. This leads to the notion of $p^m$-curvature on differential modules of level $m$. We use this construction to recover Kaneda's description of a semi-linear Azumaya splitting of the sheaf of differential operators of level $m$. Then, using a lifting modulo $p^2$ of Frobenius, we are able to define a Frobenius map on differential operators of level $m$ as dual to some divided Frobenius on infinitesimal neighborhhods. We use this map to build a true Azumaya splitting of the completed sheaf of differential operators of level $m$ (up to an automorphism of the center). From this, we derive the fact that Frobenius pull back gives, when restricted to quasi-nilpotent objects, an equivalence between Higgs-modules and differential modules of level $m$. We end by explaining the relation with related work of Ogus-Vologodski and van der Put in level zero as well as Berthelot's Frobenius descent.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-11-07T16:19:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive characteristic",
      "differential operators",
      "simpson correspondence",
      "infinitesimal neighborhhods",
      "differential modules"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0811.1168G"
    }
  }
}