{
  "id": "2208.13623",
  "version": "v1",
  "published": "2022-08-29T14:12:56.000Z",
  "updated": "2022-08-29T14:12:56.000Z",
  "title": "Regular bi-interpretability of Chevalley groups over local rings",
  "authors": [
    "Elena Bunina"
  ],
  "categories": [
    "math.GR",
    "math.LO"
  ],
  "abstract": "In this paper we prove that if $G(R)=G_\\pi (\\Phi,R)$ $(E(R)=E_{\\pi}(\\Phi, R))$ is an (elementary) Chevalley group of rank $> 1$, $R$ is a local ring (with $\\frac{1}{2}$ for the root systems ${\\mathbf A}_2, {\\mathbf B}_l, {\\mathbf C}_l, {\\mathbf F}_4, {\\mathbf G}_2$ and with $\\frac{1}{3}$ for ${\\mathbf G}_{2})$, then the group $G(R)$ (or $(E(R)$) is regularly bi-interpretable with the ring~$R$. As a consequence of this theorem, we show that the class of all Chevalley groups over local rings (with the listed restrictions) is elementary definable, i.\\,e., if for an arbitrary group~$H$ we have $H\\equiv G_\\pi(\\Phi, R)$, than there exists a ring $R'\\equiv R$ such that $H\\cong G_\\pi(\\Phi,R')$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-29T14:12:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "chevalley group",
      "local ring",
      "regular bi-interpretability",
      "root systems",
      "elementary"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}