\(K_{1}\) of Chevalley groups are nilpotent
This paper extends the main result of \textit{A. Bak} [\(K\)-Theory 4, No. 4, 363-397 (1991; Zbl 0741.19001)] to the case of Chevalley groups. More precisely, let \(\Phi\) be a reduced irreducible root system of rank at least two and let \(R\) be a commutative ring of finite Bass-Serre dimension or finite Krull dimension. Let \(G(\Phi,R)\) be a Chevalley group of type \(\Phi\) over \(R\) and let \(E(\Phi,R)\) be its elementary subgroup. It is shown that \(G(\Phi,R)/E(\Phi,R)\) is nilpotent-by-abelian. If one takes \(G(\Phi,R)\) simply connected, then \(G(\Phi,R)/E(\Phi,R)\) is called \(K_1 (\Phi,R)\) and they show it to be nilpotent. The authors feel the method deserves better publicity and therefore make an effort to explain it well.
scientific article; zbMATH DE number 1895983
scientific article