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arXiv · 2609.25124

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

Egor Voronetsky
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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'} $ 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.

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Egor Voronetsky. 2026-09-20. Solvability of isotropic $ \mathrm{K}_1 $-functor over semilocal rings. https://arxiv.org/abs/2609.25124

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