Surjective stability in dimension $0$ for $K_{2}$ and related functors

Abstract:

This paper continues the investigation of generators and relations for Chevalley groups over commutative rings initiated in [14]. The main result is that if A is a semilocal ring generated by its units, the groups $L({\mathbf {\Phi }},A)$ of [14] are generated by the values of certain cocycles on ${A^\ast } \times {A^\ast }$. From this follows a surjective stability theorem for the groups $L({\mathbf {\Phi }},A)$, as well as the result that $L({\mathbf {\Phi }},A)$ is the Schur multiplier of the elementary subgroup of the points in A of the universal Chevalley-Demazure group scheme with root system ${\mathbf {\Phi }}$, if ${\mathbf {\Phi }}$ has large enough rank. These results are proved via a Bruhat-type decomposition for a suitably defined relative group associated to a radical ideal. These theorems generalize to semilocal rings results of Steinberg for Chevalley groups over fields, and they give an effective tool for computing Milnor’s groups ${K_2}(A)$ when A is semilocal.