Local-global principle for congruence subgroups of Chevalley groups

Abstract

We prove Suslin's local-global principle for principal congruence subgroups of Chevalley groups. Let $G$ be a Chevalley--Demazure group scheme with a root system $\Phi\ne A_1$ and $E$ its elementary subgroup. Let $R$ be a ring and $I$ an ideal of $R$. Assume additionally that $R$ has no residue fields of 2 elements if $\Phi=C_2$ or $G_2$. Theorem. Let $g\in G(R[X],XR[X])$. Suppose that for every maximal ideal $\m$ of $R$ the image of $g$ under the localization homomorphism at $\m$ belongs to $E(R_\m[X],IR_\m[X])$. Then, $g\in E(R[X],IR[X])$. The theorem is a common generalization of the result of E.Abe for the absolute case ($I=R$) and H.Apte--P.Chattopadhyay--R.Rao for classical groups. It is worth mentioning that for the absolute case the local-global principle was obtained by V.Petrov and A.Stavrova in more general settings of isotropic reductive groups.