{
  "id": "1405.2009",
  "version": "v3",
  "published": "2014-05-08T16:40:01.000Z",
  "updated": "2015-08-06T20:18:12.000Z",
  "title": "Topology on cohomology of local fields",
  "authors": [
    "Kestutis Cesnavicius"
  ],
  "comment": "36 pages; final version, to appear in Forum of Mathematics, Sigma",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Arithmetic duality theorems over a local field $k$ are delicate to prove if $\\mathrm{char} k > 0$. In this case, the proofs often exploit topologies carried by the cohomology groups $H^n(k, G)$ for commutative finite type $k$-group schemes $G$. These \"\\v{C}ech topologies\", defined using \\v{C}ech cohomology, are impractical due to the lack of proofs of their basic properties, such as continuity of connecting maps in long exact sequences. We propose another way to topologize $H^n(k, G)$: in the key case $n = 1$, identify $H^1(k, G)$ with the set of isomorphism classes of objects of the groupoid of $k$-points of the classifying stack $\\mathbf{B} G$ and invoke Moret-Bailly's general method of topologizing $k$-points of locally of finite type $k$-algebraic stacks. Geometric arguments prove that these \"classifying stack topologies\" enjoy the properties expected from the \\v{C}ech topologies. With this as the key input, we prove that the \\v{C}ech and the classifying stack topologies actually agree. The expected properties of the \\v{C}ech topologies follow, which streamlines a number of arithmetic duality proofs given elsewhere.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-07-03T02:59:18.000Z",
      "comment": "27 pages; included an appendix on lifting points of non-quasi-separated algebraic stacks and made several improvements",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-08-06T20:18:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11S99",
      "11S25",
      "14A20"
    ],
    "keywords": [
      "local field",
      "cohomology",
      "classifying stack topologies",
      "invoke moret-baillys general method",
      "properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.2009C"
    }
  }
}