2025/10/22 by Gwyn Bellamy, Bellamy, Gwyn
Mathematics · #01-02 #13N10 #14F10 #16S32 #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2510.19344
openalex publication_date 2025/10/22 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
The seminal paper "J.T. Stafford, Module structure of Weyl algebras, J. London Math. Soc. (2) 18 (1978), no. 3, 429--442" was a major step forward in our understanding of Weyl algebras. Beginning with Serre's Theorem on free summands of projective modules and Bass' Stable Range Theorem in commutative algebra, we attempt to trace the origins of this work and explain how it led to Stafford's construction of non-holonomic simple modules over Weyl algebras. We also describe Bernstein-Lunt's geometric construction of infinite families of non-holonomic simple modules. We recall more recent developments related to Weyl algebras, especially that of parametrizing right ideals in the first Weyl algebra and its relation to Calogero-Moser spaces. Finally, we revisit Stafford's results in the context of quantized symplectic singularities, where they lead naturally to open problems on the behaviour of simple modules.