Critical modules over the second Weyl algebra
type: Item
instance of: scholarly article
title: Critical modules over the second Weyl algebra
zbMATH Open document ID: 0944.16024
DOI: 10.1016/S0022-4049(97)00116-3
author: S. C. Coutinho
published in: Journal of Pure and Applied Algebra
publication date: 2000-06-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 16S32
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 16D30
Mathematics Subject Classification ID: 16W25
zbMATH DE Number: 1308281
zbMATH Keywords: Weyl algebras
zbMATH Keywords: critical modules
zbMATH Keywords: finitely generated modules
zbMATH Keywords: irreducible modules
zbMATH Keywords: maximal submodules
zbMATH Keywords: derivations
reviewed by: David A. Jordan
MaRDI profile type: Publication
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full work available at URL: s0022-4049(97)00116-3
OpenAlex ID: W1972772273
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description: scientific article; zbMATH DE number 1308281