{
  "id": "1112.3089",
  "version": "v1",
  "published": "2011-12-14T01:10:40.000Z",
  "updated": "2011-12-14T01:10:40.000Z",
  "title": "The Brauer group and the Brauer-Manin set of products of varieties",
  "authors": [
    "Alexei N. Skorobogatov",
    "Yuri G. Zarhin"
  ],
  "comment": "20 pages",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $\\mathbb Q$, and let $\\ov X$ and $\\ov Y$ be the varieties over an algebraic closure of $k$ obtained from $X$ and $Y$, respectively, by extension of the ground field. We show that the Galois invariant subgroup of $\\Br(\\ov X)\\oplus \\Br(\\ov Y)$ has finite index in the Galois invariant subgroup of $\\Br(\\ov X\\times\\ov Y)$. This implies that the cokernel of the natural map $\\Br(X)\\oplus\\Br(Y)\\to\\Br(X\\times Y)$ is finite when $k$ is a number field. In this case we prove that the Brauer-Manin set of the product of varieties is the product of their Brauer-Manin sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-12-14T01:10:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F22",
      "14G25"
    ],
    "keywords": [
      "brauer-manin set",
      "brauer group",
      "galois invariant subgroup",
      "number field",
      "natural map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1112.3089S"
    }
  }
}