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

Abstract

Let G G be a universal Chevalley-Demazure group scheme associated to a reduced irreducible root system of rank > 1. >1. For a commutative ring R R , we let Γ = E ( R ) \Gamma = E(R) denote the elementary subgroup of the group of R R -points G ( R ) . G(R). The congruence kernel C ( Γ ) C(\Gamma ) is then defined to be the kernel of the natural homomorphism Γ ^ → Γ ¯ , \widehat {\Gamma } \to \overline {\Gamma }, where Γ ^ \widehat {\Gamma } is the profinite completion of Γ \Gamma and Γ ¯ \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 R (with some minor restrictions if G G is of type C n C_n or G 2 G_2 ), the congruence kernel C ( Γ ) C(\Gamma ) is central in Γ ^ . \widehat {\Gamma }.