Localization and Catenarity in Iterated Differential Operator Rings

Let k be a field, R an associative k-algebra with identity, a finite set of derivations of R, and R 1 1 · · · n n an iterated differential operator k-algebra over R such that j i ∈ R 1 1 · · · i−1 i−1 1 ≤ i < j ≤ n. If R is Noetherian -hypercentral, then every prime ideal P of A is classically localizable. The aim of this article is to show that under some additional hypotheses on the -prime ideals of R, the local ring AP is regular in the sense of Robert Walker. We use this result to study the catenarity of A and to compute the numbers i of Bass. Let g be a nilpotent Lie algebra of finite dimension n acting on R by derivations and U g the enveloping algebra of g. Then the crossed product of R by U g is an iterated differential operator k-algebra as above. In this particular case, our results are known if k has characteristic zero.