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

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- by Michael R. Stein
- Trans. Amer. Math. Soc.
**178**(1973), 165-191 - DOI: https://doi.org/10.1090/S0002-9947-1973-0327925-8
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## 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.

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## Bibliographic Information

- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**178**(1973), 165-191 - MSC: Primary 20G35; Secondary 14L15
- DOI: https://doi.org/10.1090/S0002-9947-1973-0327925-8
- MathSciNet review: 0327925