TY - JOUR
T1 - K 1 of Chevalley groups are nilpotent
AU - Hazrat, Roozbeh
AU - Vavilov, Nikolai
PY - 2003/4/1
Y1 - 2003/4/1
N2 - Let Φ be a reduced irreducible root system and R be a commutative ring. Further, let G(Φ, R) be a Chevalley group of type Φ over R and E(Φ, R) be its elementary subgroup. We prove that if the rank of Φ is at least 2 and the Bass-Serre dimension of R is finite, then the quotient G(Φ, R)/E(Φ, R) is nilpotent by abelian. In particular, when G(Φ, R) is simply connected the quotient K 1 (Φ, R) = G(Φ, R)/E(Φ, R) is nilpotent. This result was previously established by Bak for the series A1 and by Hazrat for C 1 and D 1 . As in the above papers we use the localisation-completion method of Bak, with some technical simplifications.
AB - Let Φ be a reduced irreducible root system and R be a commutative ring. Further, let G(Φ, R) be a Chevalley group of type Φ over R and E(Φ, R) be its elementary subgroup. We prove that if the rank of Φ is at least 2 and the Bass-Serre dimension of R is finite, then the quotient G(Φ, R)/E(Φ, R) is nilpotent by abelian. In particular, when G(Φ, R) is simply connected the quotient K 1 (Φ, R) = G(Φ, R)/E(Φ, R) is nilpotent. This result was previously established by Bak for the series A1 and by Hazrat for C 1 and D 1 . As in the above papers we use the localisation-completion method of Bak, with some technical simplifications.
UR - http://www.scopus.com/inward/record.url?scp=0037398185&partnerID=8YFLogxK
U2 - 10.1016/S0022-4049(02)00292-X
DO - 10.1016/S0022-4049(02)00292-X
M3 - Article
SN - 0022-4049
VL - 179
SP - 99
EP - 116
JO - Journal of Pure and Applied Algebra
JF - Journal of Pure and Applied Algebra
IS - 1-2
ER -