发表机构
Saint Petersburg University（圣彼得堡国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明半局部环上各向同性秩至少为2且排除两个特殊Tits指标时，$ \mathrm{K}_1 $-函子可解，并推出初等子群为最大完美子群。
AI 中文摘要
我们证明，若各向同性秩至少为 $ 2 $，且 Tits 指标既不是 $ {}^{2} \mathsf{E}_{6, 2}^{16'} $ 也不是 $ \mathsf{E}_{8, 2}^{78} $，则基于半局部环上单纯约化群的 $ \mathrm{K}_1 $-函子是可解的。对于这两个 Tits 指标，结果已知，但需假设基环包含一个域。我们的结果蕴含初等子群（或其导出子群）是约化群的最大完美子群。
英文摘要
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.
Comments2 figures. This is a part of arXiv:2608.17488v1 concerning only semilocal rings. The argument for $ {}^{2} \mathsf{E}_{6, 2}^{16'} $ was invalid, so for this Tits index we refer to the known result about algebras over a field
