{
  "id": "2408.16783",
  "version": "v1",
  "published": "2024-08-19T06:24:13.000Z",
  "updated": "2024-08-19T06:24:13.000Z",
  "title": "Is there a group structure on the Galois cohomology of a reductive group over a global field?",
  "authors": [
    "Mikhail Borovoi"
  ],
  "comment": "5 pages. This is a part of arXiv:2403.07659 to be published separately",
  "categories": [
    "math.NT",
    "math.AG",
    "math.GR"
  ],
  "abstract": "Let K be a global field, that is, a number field or a global function field. It is known that the answer to the question in the title over K is \"Yes\" when K has no real embeddings. We show that otherwise the answer is \"No\". Namely, we show that when K is a number field admitting a real embedding, it is impossible to define a group structure on the first Galois cohomology sets H^1(K,G) for all reductive K-groups G in a functorial way.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-19T06:24:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E72",
      "20G10",
      "20G20",
      "20G30"
    ],
    "keywords": [
      "group structure",
      "global field",
      "reductive group",
      "number field",
      "first galois cohomology sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}