Local-global principle for congruence subgroups of Chevalley groups

TL;DR

This paper proves a local-global principle for principal congruence subgroups of Chevalley groups, extending previous results and applying to rings with certain restrictions, thereby advancing the understanding of algebraic group structures.

Contribution

It generalizes Suslin's local-global principle to principal congruence subgroups of Chevalley groups over rings with specific conditions, unifying and extending prior work.

Findings

Proves the local-global principle for Chevalley groups with ideal conditions.

Extends previous results to a broader class of rings and groups.

Provides a unified framework for classical and absolute cases.

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.