Critical modules over the second Weyl algebra
Let \(A\) be the second Weyl algebra over \(\mathbb{C}\). Examples are presented of GK-critical \(A\)-modules of length \(2\). The first such example was given by \textit{G. S. Perets} [Isr. J. Math. 83, No. 3, 361-368 (1993; Zbl 0797.16035)]. The paper under review is motivated by a comment in the paper of Perets that it should be possible to explain the existence of such examples using Ext groups rather than direct computation. The author offers such an explanation in terms of duality. Let \(M\) be a finitely generated left \(A\)-module and suppose that \(k\) is such that \(\text{Ext}^j(M,A)=0\) whenever \(j\neq k\). The dual of \(M\) is defined to be the left module obtained from the right module \(\text{Ext}^k(M,A)\) using a standard anti-automorphism of \(A\). The relevant case for this paper is where \(k=1\) and \(M=A/Aa\) for some \(a\in A\). It is shown that there are examples where \(M\) is irreducible but the dual \(M^*\) is not irreducible. Critical modules of length \(2\) then occur as maximal submodules of \(M^*\). The elements \(a\) which are used to form \(M\) as above have the form \(d+f\), where \(d\) is a derivation of \(\mathbb{C}[x_1,x_2]\). The elements \(d\) and \(f\) are required to satisfy various geometric conditions and the paper includes an interesting discussion of derivations for which these may be satisfied.
scientific article; zbMATH DE number 1308281