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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: Michael R. Stein
Journal: Trans. Amer. Math. Soc. 178 (1973), 165-191
MSC: Primary 20G35; Secondary 14L15
MathSciNet review: 0327925
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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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Keywords: Chevalley group, universal central extension, stability theorems, Steinberg group, commutators in Chevalley groups, <IMG WIDTH="32" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img13.gif" ALT="${K_2}$">, second homology group, Bruhat decomposition
Article copyright: © Copyright 1973 American Mathematical Society