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 $Φ$ 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 $G(Φ,R)$ of type $Φ$ over $R$. The unrelativised elementary subgroup $E(Φ,I)$ of level $I$ is generated (as a group) by the elementary unipotents $x_α(ξ)$, $α\inΦ$, $ξ\in I$, of level $I$. Obviously, in general $E(Φ,I)$ has no chances to be normal in $E(Φ,R)$, its normal closure in the absolute elementary subgroup $E(Φ,R)$ is denoted by $E(Φ,R,I)$. The main results of [12] implied that the commutator $\big[E(Φ,I),E(Φ,J)]$ is in fact normal in $E(Φ,R)$. In the present paper we prove an unexpected result that in fact $\big[E(Φ,I),E(Φ,J)]=\big[E(Φ,R,I),E(Φ,R,J)\big]$. It follows that the standard commutator formula also holds in the unrelativised form, namely $\big[E(Φ,I),C(Φ,R,J)]=\big[E(Φ,I),E(Φ,J)\big]$, where $C(Φ,R,I)$ is the full congruence subgroup of level $I$. In particular, $E(Φ,I)$ is normal in $C(Φ,R,I)$.