{
  "id": "math/9803113",
  "version": "v1",
  "published": "1998-03-24T16:27:53.000Z",
  "updated": "1998-03-24T16:27:53.000Z",
  "title": "Grothendieck's theorem on non-abelian H^2 and local-global principles",
  "authors": [
    "Yuval Z. Flicker",
    "Claus Scheiderer",
    "R. Sujatha"
  ],
  "comment": "22 pages, AMS-TeX; accepted for publication by the Journal of the AMS",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "A theorem of Grothendieck asserts that over a perfect field k of cohomological dimension one, all non-abelian H^2-cohomology sets of algebraic groups are trivial. The purpose of this paper is to establish a formally real generalization of this theorem. The generalization -- to the context of perfect fields of virtual cohomological dimension one -- takes the form of a local-global principle for the H^2-sets with respect to the orderings of the field. This principle asserts in particular that an element in H^2 is neutral precisely when it is neutral in the real closure with respect to every ordering in a dense subset of the real spectrum of k. Our techniques provide a new proof of Grothendieck's original theorem. An application to homogeneous spaces over k is also given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1998-03-24T16:27:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14L30",
      "11R34",
      "12G05"
    ],
    "keywords": [
      "local-global principle",
      "grothendiecks theorem",
      "non-abelian",
      "perfect field",
      "grothendiecks original theorem"
    ],
    "note": {
      "typesetting": "AMS-TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1998math......3113F"
    }
  }
}