Regular bi-interpretability of Chevalley groups over local rings

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 $R^{\prime }\equiv R$ such that $H\cong G_{\pi }(\Phi ,R^{\prime })$.