K1 of Chevalley groups are nilpotent

✍ Scribed by Roozbeh Hazrat; Nikolai Vavilov

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 195 KB
• Volume: 179
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(02)00292-x

✦ Synopsis

Let be a reduced irreducible root system and R be a commutative ring. Further, let G( ; R) be a Chevalley group of type over R and E( ; R) be its elementary subgroup. We prove that if the rank of is at least 2 and the Bass-Serre dimension of R is ÿnite, then the quotient G( ; R)=E( ; R) is nilpotent by abelian. In particular, when G( ; R) is simply connected the quotient K1( ; R) = G( ; R)=E( ; R) is nilpotent. This result was previously established by Bak for the series A1 and by Hazrat for C1 and D1. As in the above papers we use the localisation-completion method of Bak, with some technical simpliÿcations.