Surjective stability in dimension 0 for k2 and related functors

Michael R Stein*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

68 Scopus citations


This paper continues the investigation of generators and relations for Chevalley groups over commutative rings initiated in [l4]. The main result is that if A is a semilocal ring generated by its units, the groups L(Ø, A) of [l4] are generated by the values of certain cocycles on A* x A*. From this follows a surjective stability theorem for the groups L(Ø, A), as well as the result that L(Ø, A) is the Schur multiplier of the elementary subgroup of the points in A of the universal Chevalley-Demazure group scheme with root system Ø, if Ø 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 K2(A) when A is semilocal.

Original languageEnglish (US)
Pages (from-to)165-191
Number of pages27
JournalTransactions of the American Mathematical Society
StatePublished - Jan 1 1973


  • Bru- hat decomposition
  • Chevalley group
  • Commutators in Chevalley groups
  • K2
  • Second homology group
  • Stability theorems
  • Steinberg group
  • Universal central extension

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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