On the congruence kernel of isotropic groups over rings

Abstract:

Let $R$ be a connected noetherian commutative ring, and let $G$ be a simply connected reductive group over $R$ of isotropic rank $\ge 2$. The elementary subgroup $E(R)$ of $G(R)$ is the subgroup generated by $U_{P^+}(R)$ and $U_{P^-}(R)$, where $U_{P^\pm }$ are the unipotent radicals of two opposite parabolic subgroups $P^\pm$ of $G$. Assume that $2\in R^\times$ if $G$ is of type $B_n,C_n,F_4,G_2$ and $3\in R^\times$ if $G$ is of type $G_2$. We prove that the congruence kernel of $E(R)$, defined as the kernel of the natural homomorphism $\widehat {E(R)}\to \overline {E(R)}$ between the profinite completion of $E(R)$ and the congruence completion of $E(R)$ with respect to congruence subgroups of finite index, is central in $\widehat {E(R)}$. In the course of the proof, we construct Steinberg groups associated to isotropic reducive groups and show that they are central extensions of $E(R)$ if $R$ is a local ring.