On completing unimodular polynomial vectors of length three

Abstract

It is shown that if R R is a local ring of dimension three, with 1 2 ∈ R \frac {1} {2} \in R , then a polynomial three vector ( v 0 ( X ) , v 1 ( X ) , v 2 ( X ) ) ({v_0}(X),{v_1}(X),{v_2}(X)) over R [ X ] R[X] can be completed to an invertible matrix if and only if it is unimodular. In particular, if 1 / 3 ! ∈ R 1/3! \in R , then every stably free projective R [ X 1 , … , X n ] R[{X_1}, \ldots ,{X_n}] -module is free.