Extension of certain modules over the first Weyl algebra (Q1333971)

The authors consider certain rather special finite length modules over the first Weyl algebra \(A_1 = k[x,d/dx]\), over a field of characteristic zero. They make use of an old result of McConnell and Robson computing \(\dim \text{Ext}(A_1/ (f + d/dx) A_1, A_1/ (g + d/dx) A_ 1) = 1\), for \(f, g \in k[x]\) with \(f - g\) of degree one. They use methods from the theory of finite dimensional algebras to investigate the category of finite length modules with composition factors isomorphic to \(A_1 / (f + d/dx) A_1\) or \(A_1/ (g + d/dx) A_1\). Specifically, they show that it corresponds to a tube of \(\mathbb{Z}/2 \mathbb{Z}\) type and that it is serial and standard.