Centrality of the congruence kernel for elementary subgroups of Chevalley groups of rank $> 1$ over noetherian rings

Abstract:

Let $G$ be a universal Chevalley-Demazure group scheme associated to a reduced irreducible root system of rank $>1.$ For a commutative ring $R$, we let $\Gamma = E(R)$ denote the elementary subgroup of the group of $R$-points $G(R).$ The congruence kernel $C(\Gamma )$ is then defined to be the kernel of the natural homomorphism $\widehat {\Gamma } \to \overline {\Gamma },$ where $\widehat {\Gamma }$ is the profinite completion of $\Gamma$ and $\overline {\Gamma }$ is the congruence completion defined by ideals of finite index. The purpose of this paper is to show that for an arbitrary noetherian ring $R$ (with some minor restrictions if $G$ is of type $C_n$ or $G_2$), the congruence kernel $C(\Gamma )$ is central in $\widehat {\Gamma }.$