Generation of relative commutator subgroups in Chevalley groups. II

Abstract

In the present paper, which is a direct sequel of our paper [12] joint with Roozbeh Hazrat, we prove unrelativised version of the standard commutator formula in the setting of Chevalley groups. Namely, let $\Phi$ be a reduced irreducible root system of rank $\ge 2$, let $R$ be a commutative ring and let $I,J$ be two ideals of $R$. We consider subgroups of the Chevalley group $G(\Phi,R)$ of type $\Phi$ over $R$. The unrelativised elementary subgroup $E(\Phi,I)$ of level $I$ is generated (as a group) by the elementary unipotents $x_{\alpha}(\xi)$, $\alpha\in\Phi$, $\xi\in I$, of level $I$. Obviously, in general $E(\Phi,I)$ has no chances to be normal in $E(\Phi,R)$, its normal closure in the absolute elementary subgroup $E(\Phi,R)$ is denoted by $E(\Phi,R,I)$. The main results of [12] implied that the commutator $\big[E(\Phi,I),E(\Phi,J)]$ is in fact normal in $E(\Phi,R)$. In the present paper we prove an unexpected result that in fact $\big[E(\Phi,I),E(\Phi,J)]=\big[E(\Phi,R,I),E(\Phi,R,J)\big]$. It follows that the standard commutator formula also holds in the unrelativised form, namely $\big[E(\Phi,I),C(\Phi,R,J)]=\big[E(\Phi,I),E(\Phi,J)\big]$, where $C(\Phi,R,I)$ is the full congruence subgroup of level $I$. In particular, $E(\Phi,I)$ is normal in $C(\Phi,R,I)$.