Constructive Computation of Bases of Free Modules over the Weyl Algebras
Résumé
A well-known result due to J. T. Stafford asserts that a stably free left module $M$ over the Weyl algebras $D=A_n(k)$ or $B_n(k)$ $-$ where $k$ is a field of characteristic $0$ $-$ with $rank_D(M) \geq2$ 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 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 $rank_D(M) \geqsr(D)$ is free, where $sr(D)$ denotes the stable range 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.
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