{
  "id": "1411.1246",
  "version": "v1",
  "published": "2014-11-05T11:59:14.000Z",
  "updated": "2014-11-05T11:59:14.000Z",
  "title": "On The Cohomology of $\\mathrm{SL}_2$ with Coefficients in a Simple Module",
  "authors": [
    "John Rizkallah"
  ],
  "comment": "9 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be the simple algebraic group $\\mathrm{SL}_2$ defined over an algebraically closed field $k$ of characteristic $p > 0$. Using results of A. Parker, we develop a method which gives, for any $q \\in \\mathbb{N}$, a closed form description of all simple modules $M$ such that $\\mathrm{H}^q(G,M) \\neq 0$, together with the associated dimensions $\\mathrm{dim}\\mathrm{H}^q(G,M)$. We apply this method for arbitrary primes $p$ and for $q \\leq 3$, confirming results of Cline and Stewart along the way. Furthermore, we show that under the hypothesis $p > q$, the dimension of the cohomology $\\mathrm{H}^q(G,M)$ is at most 1, for any simple module $M$. Based on this evidence we discuss a conjecture for general semisimple algebraic groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-05T11:59:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "simple module",
      "cohomology",
      "general semisimple algebraic groups",
      "coefficients",
      "arbitrary primes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}