{
  "id": "2101.12460",
  "version": "v1",
  "published": "2021-01-29T08:13:56.000Z",
  "updated": "2021-01-29T08:13:56.000Z",
  "title": "Homomorphisms of algebraic groups: representability and rigidity",
  "authors": [
    "Michel Brion"
  ],
  "comment": "25 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Consider two group schemes $G$, $H$, locally of finite type over a field $k$. We show that the functor of morphisms (of schemes) $\\mathbf{Hom}(G,H)$ is represented by a group scheme, locally of finite type, if the $k$-vector space $\\mathcal{O}(G)$ is finite-dimensional; the converse holds if $H$ is not \\'etale. When $G$ is algebraic and the unipotent radical of $G_{\\bar{k}}$ is trivial, we show that the functor of group homomorphisms $\\mathbf{Hom}_{\\rm{gp}}(G,H)$ is represented by a sum of schemes of finite type. Along the way, we obtain rigidity properties for morphisms in the above settings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-29T08:13:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14L15",
      "14D20",
      "14K05",
      "14L30",
      "20G15"
    ],
    "keywords": [
      "algebraic groups",
      "finite type",
      "representability",
      "group scheme",
      "vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}