{
  "id": "1705.06241",
  "version": "v1",
  "published": "2017-05-17T16:20:00.000Z",
  "updated": "2017-05-17T16:20:00.000Z",
  "title": "Lifting the Cartier transform of Ogus-Vologodsky modulo $p^n$",
  "authors": [
    "Daxin Xu"
  ],
  "comment": "95 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $W$ be the ring of the Witt vectors of a perfect field of characteristic $p$, $\\mathfrak{X}$ a smooth formal scheme over $W$, $\\mathfrak{X}'$ the base change of $\\mathfrak{X}$ by the Frobenius morphism of $W$, $\\mathfrak{X}_{2}'$ the reduction modulo $p^{2}$ of $\\mathfrak{X}'$ and $X$ the special fiber of $\\mathfrak{X}$. We lift the Cartier transform of Ogus-Vologodsky defined by $\\mathfrak{X}_{2}'$ modulo $p^{n}$. More precisely, we construct a functor from the category of $p^{n}$-torsion $\\mathscr{O}_{\\mathfrak{X}'}$-modules with integrable $p$-connection to the category of $p^{n}$-torsion $\\mathscr{O}_{\\mathfrak{X}}$-modules with integrable connection, each subject to suitable nilpotence conditions. Our construction is based on Oyama's reformulation of the Cartier transform of Ogus-Vologodsky in characteristic $p$. If there exists a lifting $F:\\mathfrak{X}\\to \\mathfrak{X}'$ of the relative Frobenius morphism of $X$, our functor is compatible with a functor constructed by Shiho from $F$. As an application, we give a new interpretation of Faltings' relative Fontaine modules and of the computation of their cohomology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-17T16:20:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cartier transform",
      "ogus-vologodsky modulo",
      "smooth formal scheme",
      "characteristic",
      "relative frobenius morphism"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 95,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}