Cyclicity of finitely generated torsion modules over the Weyl algebra A_n

Determine whether every finitely generated torsion module over the nth Weyl algebra A_n is cyclic, i.e., generated by a single element as an A_n-module.

Background

In Theorem 3.7 of the survey (summarizing Stafford’s results), it is shown that any finitely generated torsion module over A_n can be generated by two elements. Motivated by this bound and by the fact that all holonomic A_n-modules (which are torsion) are known to be cyclic, Stafford proposed strengthening the two-generator result to the assertion that every finitely generated torsion A_n-module is actually cyclic.

The survey notes that this conjecture appears in slightly stronger form in Stafford’s original paper and emphasizes that, to the author’s knowledge, it has not been resolved.

References

Stafford conjectures: Every finitely generated torsion A_n-module is cyclic. As far as the author is aware, this conjecture is still open.

— Module structure of Weyl algebras  (2510.19344 - Bellamy, 22 Oct 2025) in Section 3 (Module structure of Weyl algebras), following Theorem 3.7

A long-standing conjecture of [Conjecture 3.8] asks, if any finitely generated torsion right module is cyclic. If this is true, the result would extend the single-column bound of to all Weyl algebras $A_n(k)$, $n \ge 1$, i.e. also to PDEs with polynomial coefficients.

— Ghost tasking for parametrized Gaussian Processes solving linear differential equations  (2610.12009 - Moser et al., 8 Oct 2026) in Section 3.2, paragraph 'Stronger conjecture' (following Proposition 3.2)