Computation of bases of free modules over the Weyl algebras

A. Quadrat, D. Robertz

Journal of Symbolic Computation, 42:11-12 (2007), pp. 1113-1141
 
[.pdf] (cf. also INRIA report No. 5786)

 

 
Abstract: 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) >= 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, 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 rank_D(M) >= sr(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.
 
A preliminary version of the Stafford package is available here.
 
Maple worksheets:
 
 
Description Maple Worksheet Text version References
Projective dimensions & Minimal free resolutions Example1.mws Example1.pdf Quadrat, Robertz (2007), Pommaret (2001), Janet (1929)
Bases & injective parametrizations of ordinary differential systems Example2.mws Example2.pdf Quadrat, Robertz (2007), Chyzak et al. (2005), Malrait (2001), Quadrat, Robertz (2005)
Bases & injective parametrizations of partial differential equations Example3.mws Example3.pdf Quadrat, Robertz (2007), Pommaret (2001)
Computation of two generators (Theorem of Stafford) Example4.mws Example4.pdf Quadrat, Robertz (2007), Pommaret (2001), Leykin (2004)

 
 
See also the Library of Examples.