{
  "id": "2008.12175",
  "version": "v1",
  "published": "2020-08-27T15:04:53.000Z",
  "updated": "2020-08-27T15:04:53.000Z",
  "title": "A functorial presentation of units of Burnside rings",
  "authors": [
    "Serge Bouc"
  ],
  "categories": [
    "math.GR",
    "math.CT",
    "math.RA",
    "math.RT"
  ],
  "abstract": "Let $B^\\times$ be the biset functor over $\\mathbb{F}_2$ sending a finite group~$G$ to the group $B^\\times(G)$ of units of its Burnside ring $B(G)$, and let $\\widehat{B^\\times}$ be its dual functor. The main theorem of this paper gives a characterization of the cokernel of the natural injection from $B^\\times$ in the dual Burnside functor $\\widehat{\\mathbb{F}_2B}$, or equivalently, an explicit set of generators $\\mathcal{G}_S$ of the kernel $L$ of the natural surjection $\\mathbb{F}_2B\\to \\widehat{B^\\times}$. This yields a two terms projective resolution of $\\widehat{B^\\times}$, leading to some information on the extension functors $\\mathrm{Ext}^1(-,B^\\times)$. For a finite group $G$, this also allows for a description of $B^\\times(G)$ as a limit of groups $B^\\times(T/S)$ over sections $(T,S)$ of $G$ such that $T/S$ is cyclic of odd prime order, Klein four, dihedral of order 8, or a Roquette 2-group. Another consequence is that the biset functor $B^\\times$ is not finitely generated, and that its dual $\\widehat{B^\\times}$ is finitely generated, but not finitely presented. The last result of the paper shows in addition that $\\mathcal{G}_S$ is a minimal set of generators of $L$, and it follows that the lattice of subfunctors of $L$ is uncountable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-27T15:04:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "19A22",
      "16U60",
      "20J15"
    ],
    "keywords": [
      "functorial presentation",
      "burnside ring",
      "biset functor",
      "odd prime order",
      "dual burnside functor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}