Vaserstein's pre-stabilization theorem over commutative rings

Abstract

We give a new proof of Vaserstein’s Pre-stabilization theorem. This theorem describes GLn (A) ⋂ E(A) when n is just below the stable range for GLm (A)/Em (A) The new proof works only for commutative rings (or ideals in such rings) but it does not need assumptions on Krull dimension, like the old proofs did. All one needs is the relevant stable range con-dition. The new ideas in the proof come from Vaserstein’s recent treatment of the case n = 2. (See preceding paper).