Computation of bases of free modules over the Weyl algebras

A well-known result due to J. T. Stafford asserts that a stably free left module M over the Weyl algebras D = An(k) or Bn(k) − where k is a field of characteristic 0 − with rankD(M) ≥ 2 is free. The purpose of this paper is to present a new constructive proof of this result as well as an effective algorithm for the computation of bases of M . This algorithm, based on the new constructive proofs (Hillebrand and Schmale, 2001; Leykin, 2004) of J. T. Stafford’s result on the number of generators of left ideals of D, performs Gaussian elimination on the formal adjoint of the presentation matrix of M . We show that J. T. Stafford’s result is a particular case of a more general one asserting that a stably free left D-module M with rankD(M) ≥ sr(D) is free, where sr(D) denotes the stable rank of a ring D. This result is constructive if the stability of unimodular vectors with entries in D can be tested. Finally, an algorithm which computes the left projective dimension of a general left D-module M defined by means of a finite free resolution is presented. It allows us to check whether or not the left D-module M is stably free.