Elementary subgroups of isotropic reductive groups

V. Petrov ; A. Stavrova

Algebra i analiz, Tome 20 (2008) no. 4, pp. 160-188.

Résumé

Let $G$ be a not necessarily split reductive group scheme over a commutative ring $R$ with $1$. Given a parabolic subgroup $P$ of $G$, the elementary group $E_P(R)$ is defined to be the subgroup of $G(R)$ generated by $U_P(R)$ and $U_{P^-}(R)$, where $U_P$ and $U_{P^-}$ are the unipotent radicals of $P$ and its opposite $P^-$ respectively. It is proved that if $G$ contains a Zariski locally split torus of rank 2, then the group $E_P(R)=E(R)$ does not depend on $P$, and, in particular, is normal in $G(R)$.

Détail

Keywords: Reductive group scheme, elementary subgroup, Whitehead group, parabolic subgroup.

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