Localization and Catenarity in Iterated Differential Operator Rings

ABSTRACT

Let k be a field, R an associative k-algebra with identity, Δ a finite set of derivations of R, and R[Θ₁, δ₁] ··· [Θ _n , δ _n ] an iterated differential operator k-algebra over R such that δ _j (Θ _i ) ∈ R[Θ₁, δ₁] ··· [Θ _i−1, δ _i−1]; 1 ≤ i < j ≤ n. If R is Noetherian Δ-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 Δ-prime ideals of R, the local ring A _P is regular in the sense of Robert Walker. We use this result to study the catenarity of A and to compute the numbers μ _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.

Key Words:

• AR property
• Catenarity
• Classical Krull dimension
• Classically localizable prime ideals
• Crossed products
• Derivations
• Differential operator rings
• Height of a prime ideal
• Homological dimensions
• Invariant of Bass
• Krull dimension
• Lie algebra
• Noetherian rings
• Regular local rings.

Mathematics Subject Classification:

• Primary 16A10
• Secondary 17B30

ACKNOWLEDGMENTS

I would like to thank the Atlantic Association for Research in the Mathematical Sciences (AARMS) for partially supporting my research. I would also like to thank the Department of Mathematics and Computer Science of Mount Allison University of Sackville for its warm hospitality. Special thanks are due to Margaret Beattie for help and encouragement. Finally, I'm very grateful to the referee for several helpful suggestions.

Communicated by E. Zelmanov.