{
  "id": "1012.5367",
  "version": "v2",
  "published": "2010-12-24T09:12:17.000Z",
  "updated": "2014-06-05T06:14:38.000Z",
  "title": "Degree three cohomology of function fields of surfaces",
  "authors": [
    "R. Parimala",
    "V. Suresh"
  ],
  "comment": "26 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let F be a finite field and l a prime not equal to the characteristic of F. Let K be the function field of a surface over F. Assume that K contains a primitive lth root of unity. In the paper we prove a certain local-global principle for elements of H^3(K, {\\mu}_l) in terms of symbols in H^2(K, {\\mu}_l) with respect to discrete valuations of K. We also show that this local global principle is equivalent to the vanishing of certain unramified cohomology groups of 3-folds over finite fields. Using this local-global principle we show that every element in H^3(F, {\\mu}_l) is a symbol. The vanishing of the unramified cohomology groups has consequences in the study of integral Tate conjecture and Brauer-Manin obstruction for existence of zero-cycles.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-06-05T06:14:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "function field",
      "unramified cohomology groups",
      "local-global principle",
      "finite field",
      "integral tate conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.5367P"
    }
  }
}