Regular bi-interpretability of Chevalley groups over local rings

Abstract

In this paper we prove that if) 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 ≡ G π (Φ, R), than there exists a ring R ′ ≡ R such that H ∼ = G π (Φ, R ′ ).

Regular bi-interpretability of Chevalley groups over local rings

“…As it was observed in Section 1, Problem ( * ) has a positive solution for the Chevalley groups G P (Φ, R), when R is an algebraically closed field [68], when R is a Dedekind ring of the arithmetic type (can be deduced from [55]), and when is R is a local ring [14]. In the later two cases some mild invertability conditions on R are assumed.…”

“…D. Segal and K. Tent (see [55]) showed that for Chevalley groups G P (Φ, R) of rank > 1 over an integral domain R if G P (Φ, R) has finite elementary width or is adjoint, then G P (Φ, R) and R are bi-interpretable. In [14] it was proved that over local rings Chevalley groups G P (Φ, R) of rank > 1 are regularly bi-interpretable with the corresponding rings.…”

“…The existence of regular bi-interpretation plays the decisive role for establishing elementary definability of given classes of groups. If linear/Chevalley groups (or another derivative structures) of any concrete type over some classes of fields/rings are regularly bi-interpretable with the corresponding rings, then this class of groups (structures) is elementary definable (see [14]).…”