For any ring A, the group K1A is filtered by the Whitehead determinants of invertible matrices over A of different sizes. We want to compute the corresponding graded group (especially the highest degree non-zero term) in terms of symbols which generalize Mennicke's symbol. In particular, we generalize the Bass-Milnor-Serre result which presents SK1A of a Dedekind ring A via the Mennicke symbol, to an arbitrary commutative ring A satisfying the Bass second stable range condition. As an application, SK1is computed for some rings of continuous functions. Some of our theorems are partially known, but we have often weakened hypotheses, using stable range conditions rather than Krull dimension (having in mind applications to rings of continuous functions).
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory