Net subgroups of Chevalley groups. II. Gauss decomposition

Abstract

This paper is a continuation of RZhMat 1980, 5A439, where there was introduced the subgroup γ(δ) of the Chevalley group G(Φ,R) of type Φ over a commutative ring R that corresponds to a net δ, i.e., to a set б=(б∝),∝∈Φ, of ideals б∝ of R such that б∝бβ⊑б∝+β whenever α,Β,α+Β ∃Φ. It is proved that if the ring R is semilocal, then Γ(б) coincides with the group γ0δ considered earlier in RZhMat 1976, 10A151; 1977, 10A301; 1978, 6A476. For this purpose there is constructed a decomposition of γ(δ) into a product of unipotent subgroups and a torus. Analogous results are obtained for sub-radical nets over an arbitrary commutative ring.