AI 中文总结
该研究证明了基环满足特定条件时局部各向同性K₁函子的幂零性，给出经典或整体各向同性情形下K₁的可解性结果，明确了约化群极大完全子群的结构。
AI 中文摘要
我们证明，当基环具有有限Bass–Serre维数且对于Tits指数𝖤₈,₂⁷⁸包含一个域时，以局部各向同性约化群为模型的K₁函子是次阿贝尔的；对于经典或整体各向同性约化群概型，K₁实际上是可解的。这表明初等子群（或其导出子群）是约化群的极大完全子群。
英文摘要
We show that the $ \mathrm{K}_1 $-functor modeled on locally isotropic reductive groups is hypoabelian if the base ring has finite Bass--Serre dimension and, for the Tits index $ \mathsf{E}_{8, 2}^{78} $, contains a field. For classical or globally isotropic reductive groups schemes $ \mathrm{K}_1 $ is actually solvable. This implies that the elementary subgroup (or its derived subgroup) is the maximal perfect subgroup of the reductive group.
