Regular bi-interpretability of Chevalley groups over local rings

TL;DR

This paper proves that Chevalley groups over certain local rings are bi-interpretable with the rings themselves, leading to their elementary definability within the class of all such groups.

Contribution

It establishes regular bi-interpretability between Chevalley groups over local rings and the rings, extending model-theoretic understanding of these algebraic structures.

Findings

Chevalley groups over local rings are bi-interpretable with the rings.

Elementary class of Chevalley groups over local rings is definable.

Group isomorphism corresponds to ring elementary equivalence.

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 $H\cong G_\pi(\Phi,R')$.