Improved stability for SK1 and WMSd of a non-singular affine algebra

In [MS, Theorem 1], M. P. Murthy–R. G. Swan have shown that a stably free projective module over a two-dimensional affine variety A over an algebraically closed field k is free. (The example of the tangent bundle over the real two sphere shows that the condition that the base field is algebraically closed is necessary.) They showed that every unimodular vector (a, b, c) ∈ Um3(A) could be transformed to a unimodular vector of the form (x, y, z), and that a unimodular vector of the form (x, y, z) over any commutative ring A can always be completed to an invertible matrix. In [Su1] A. Suslin generalised this by showing that a unimodular vector of the form (a0, a1, a 2 2, . . . , a r r) over any commutative ring A can always be completed to an invertible matrix; from which he deduced that a stably free projective A-module of rank ≥ dim(A) is free where A is an affine algebra over an algebraically closed field k. In [Su2] Suslin generalised this further by proving that stably free projective A-modules of rank ≥ dim(A) are free when A is an affine algebra over a perfect C1 field k. (Whenever we speak of “perfect C1 field”, which is admittedly not a very useful combination outside characteristic 0, the more technical conditions in 3.1 actually suffice.) In this note we prove a K1 analogue of Suslin’s result. We prove that: