{
  "id": "1210.4521",
  "version": "v2",
  "published": "2012-10-16T18:36:45.000Z",
  "updated": "2014-01-27T17:33:17.000Z",
  "title": "Cohomology of locally-closed semi-algebraic subsets",
  "authors": [
    "Florent Martin"
  ],
  "comment": "We obtain a more general result using a recent cohomological finiteness result for affinoid spaces proved by Vladimir Berkovich",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let k be a non archimedean field. If X is a k-algebraic variety and U a locally closed semi-algebraic subset of X^{an} -- the Berkovich space associated to X -- we show that for l \\neq char(\\tilde{k}), the cohomology groups H^i_c (\\bar{U}, Q_l) behave like H^i_c(\\bar{X}, Q_l), where \\bar{U} = U \\otimes \\hat{\\bar{k}}. In particular, they are finite-dimensional vector spaces. This result has been used by E. Hrushovski and F. Loeser. Moreover, we prove analogous finiteness properties concerning rigid semi-analytic subsets of compact Berkovich spaces (resp. adic spaces associated to quasi-compact quasi-separated k-rigid spaces) when char(\\tilde{k}) \\neq 0 (resp in any characteristic).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-01-27T17:33:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F20",
      "14G22"
    ],
    "keywords": [
      "locally-closed semi-algebraic subsets",
      "cohomology",
      "berkovich space",
      "finiteness properties concerning rigid semi-analytic",
      "properties concerning rigid semi-analytic subsets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.4521M"
    }
  }
}