Irreducible components of characteristic varieties

✍ Scribed by Gregory G. Smith

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 158 KB
• Volume: 165
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(01)00109-8

✦ Synopsis

We give a dimension bound on the irreducible components of the characteristic variety of a system of linear partial di erential equations deÿned from a suitable ÿltration of the Weyl algebra An. This generalizes an important consequence of the fact that a characteristic variety deÿned from the order ÿltration is involutive. More explicitly, we consider a ÿltration of An induced by any vector (u; v) ∈ Z n × Z n such that the associated graded algebra is a commutative polynomial ring. Any ÿnitely generated left An-module M has a good ÿltration with respect to (u; v) and this gives rise to a characteristic variety Ch (u; v) (M ) which depends only on (u; v) and M . When (u; v) = (0; 1), the characteristic variety is involutive and this implies that its irreducible components have dimension at least n. In general, the characteristic variety may fail to be involutive, but we are still able to prove that each irreducible component of Ch (u; v) (M ) has dimension at least n.