Solvability of isotropic ${\mathrm{K}}_1$-functor over semilocal rings

Abstract

We show that the ${\mathrm{K}}_1$-functor modeled on simple reductive groups over semilocal rings is solvable if the isotropic rank is at least $2$ and that the Tits index is neither ${}^2{\mathsf{E}}_{6,2}^{16^{\prime }}$ nor ${\mathsf{E}}_{8,2}^{78}$. For these two Tits indices the result is already known, but assuming that the base ring contains a field. Our result implies that the elementary subgroup (or its derived subgroup) is the maximal perfect subgroup of the reductive group.

arXiv comment2 figures. This is a part of arXiv:2608.17488v1 concerning only semilocal rings. The argument for ${}^2{\mathsf{E}}_{6,2}^{16^{\prime }}$ was invalid, so for this Tits index we refer to the known result about algebras over a field