{
  "id": "2002.04183",
  "version": "v1",
  "published": "2020-02-11T03:07:43.000Z",
  "updated": "2020-02-11T03:07:43.000Z",
  "title": "Cohomology of $\\text{PSL}_2(q)$",
  "authors": [
    "Jack Saunders"
  ],
  "comment": "19 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "In 2011, Guralnick and Tiep proved that if $G$ was a Chevalley group and $V$ an irreducible $G$-module in cross characteristic, then if $V^B = 0$, the dimension of $H^1(G,V)$ is determined by the structure of the permutation module on a Borel subgroup $B$ of $G$. We generalise this theorem to higher cohomology and an arbitrary finite group, so that if $H \\leq G$ such that $O_{r'}(H) = O^r(H)$ then if $V^H = 0$ we show $\\dim H^1(G,V)$ is determined by the structure of the permutation module on $H$, and $H^n(G,V)$ by $\\text{Ext}_G^{n-1}(V^*,M)$ for some $M$ dependent on $H$. We also explicitly determine $\\text{Ext}_G^n(V,W)$ for all irreducible $kG$-modules $V$, $W$ for $G = \\text{PSL}_2(q)$ in cross characteristic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-11T03:07:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C20",
      "20C33"
    ],
    "keywords": [
      "cross characteristic",
      "permutation module",
      "arbitrary finite group",
      "chevalley group",
      "borel subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}