{
  "id": "2410.04474",
  "version": "v2",
  "published": "2024-10-06T13:32:43.000Z",
  "updated": "2025-05-08T07:04:07.000Z",
  "title": "The period and index of a Galois cohomology class of a reductive group over a local or global field",
  "authors": [
    "Mikhail Borovoi"
  ],
  "comment": "Withdrawn because the text was included in the new version of arXiv:2403.07659",
  "categories": [
    "math.NT",
    "math.AG",
    "math.GR",
    "math.RT"
  ],
  "abstract": "Let $K$ be a local or global field. For a connected reductive group $G$ over $K$, in another preprint [5] we defined a power operation $$(\\xi,n)\\mapsto \\xi^{\\Diamond n}\\,\\colon\\, H^1(K,G)\\times {\\mathbb Z}\\to H^1(K,G)$$ of raising to power $n$ in the Galois cohomology pointed set $H^1(K,G)$. In this paper, for a cohomology class $\\xi$ in $H^1(K,G)$, we compare the period ${\\rm per}(\\xi)$ defined to be the least integer $n\\ge 1$ such that $\\xi^{\\Diamond n}=1$, and the index ${\\rm ind}(\\xi)$ defined to be the greatest common divisor of the degrees $[L:K]$ of finite separable extensions $L/K$ splitting $\\xi$. These period and index generalize the period and index a central simple algebra over $K$. For an arbitrary reductive $K$-group $G$, we proved in [5] that ${\\rm per}(\\xi)$ divides ${\\rm ind}(\\xi)$. In this paper we show that the index may be strictly greater than the period. In [5] we proved that for any $K$, $G$, and $\\xi\\in H^1(K,G)$ as above, the index ${\\rm ind}(\\xi)$ divides ${\\rm per}(\\xi)^d$ for some positive integer $d$, and we gave upper bounds for $d$ in the local case and in the case of a number field. Here we give a characteristic-free proof of the fact that ${\\rm ind}(\\xi)$ divides ${\\rm per}(\\xi)^d$ for some positive integer $d$ in the global field case, and our proof gives an upper bound for $d$ that is valid also in the case of a function field.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-05-08T07:04:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E72",
      "20G10",
      "20G25",
      "20G30"
    ],
    "keywords": [
      "galois cohomology class",
      "reductive group",
      "greatest common divisor",
      "galois cohomology pointed set",
      "gave upper bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}