## Abstract

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 K_{2}(A) when A is semilocal.

Original language | English (US) |
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Pages (from-to) | 165-191 |

Number of pages | 27 |

Journal | Transactions of the American Mathematical Society |

Volume | 178 |

DOIs | |

State | Published - Jan 1 1973 |

## Keywords

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

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics