{
  "id": "1006.5808",
  "version": "v2",
  "published": "2010-06-30T08:55:46.000Z",
  "updated": "2010-11-30T11:11:28.000Z",
  "title": "Exactness of the reduction on étale modules",
  "authors": [
    "Gergely Zábrádi"
  ],
  "comment": "18 pages; some typos corrected and proof of Lemma 1 rewritten, to appear in Journal of Algebra",
  "journal": "Journal of Algebra 331 (2011) pp. 400-415",
  "doi": "10.1016/j.jalgebra.2010.11.011",
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "We prove the exactness of the reduction map from \\'etale $(\\phi,\\Gamma)$-modules over completed localized group rings of compact open subgroups of unipotent $p$-adic algebraic groups to usual \\'etale $(\\phi,\\Gamma)$-modules over Fontaine's ring. This reduction map is a component of a functor from smooth $p$-power torsion representations of $p$-adic reductive groups (or more generally of Borel subgroups of these) to $(\\phi,\\Gamma)$-modules. Therefore this gives evidence for this functor---which is intended as some kind of $p$-adic Langlands correspondence for reductive groups---to be exact. We also show that the corresponding higher $\\Tor$-functors vanish. Moreover, we give the example of the Steinberg representation as an illustration and show that it is acyclic for this functor to $(\\phi,\\Gamma)$-modules whenever our reductive group is $\\GL_{d+1}(\\mathbb{Q}_p)$ for some $d\\geq 1$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-11-30T11:11:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50"
    ],
    "keywords": [
      "reduction map",
      "power torsion representations",
      "adic algebraic groups",
      "adic langlands correspondence",
      "compact open subgroups"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1006.5808Z"
    }
  }
}