×

Critical modules over the second Weyl algebra. (English) Zbl 0944.16024

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 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.

MSC:

16S32 Rings of differential operators (associative algebraic aspects)
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16P90 Growth rate, Gelfand-Kirillov dimension
16D30 Infinite-dimensional simple rings (except as in 16Kxx)
16W25 Derivations, actions of Lie algebras

Citations:

Zbl 0797.16035
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bernstein, J.; Lunts, V., On non-holonomic irreducible \(D\)-modules, Invent. Math., 94, 223-243 (1988) · Zbl 0658.32009
[2] Björk, J.-E., Rings of Differential Operators (1979), North-Holland: North-Holland Amsterdam
[3] Borel, A., Algebraic \(D\)-modules, (Perspectives in Mathematics, vol. 2 (1987), Academic Press: Academic Press Orlando) · Zbl 0642.32001
[4] Cerveau, D.; Ncto, A. Lins, Holomorphic foliations in CP(2) having an invariant algebraic curve, Ann. Inst. Fourier, Grenoble, 41, 883-903 (1991) · Zbl 0734.34007
[5] Coutinho, S. C., A primer of algebraic \(D\)-modules, (London Mathematical Society Student Texts, vol. 33 (1995), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0848.16019
[6] Gordon, R.; Robson, J. C., Krull Dimension, (Mem. Amer. Math. Soc., 133 (1973), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0269.16017
[7] Jouanolou, J. C., Equations de Pfaff algébriques, (Lecture Notes in Mathematics, vol. 708 (1979), Springer: Springer Berlin) · Zbl 0477.58002
[8] Ncto, A. Lins, Algebraic solutions of polynomial differential equations and foliations in dimension two, (Holomorphic Dynamics. Holomorphic Dynamics, Lecture Notes in Mathematics, vol. 1345 (1988), Springer), 192-232, Berlin
[9] Perets, G. S., \(d\)-Critical modules of length 2 over Weyl algebras, Israel J. Math., 83, 361-368 (1993) · Zbl 0797.16035
[10] Stafford, J. T., Non-holonomic modules over Weyl algebras and enveloping algebras, Invent. Math., 79, 619-638 (1985) · Zbl 0558.17011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.