Localization and Catenarity in Iterated Differential Operator Rings |
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Authors: | T Guédénon |
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Institution: | 1. Department of Mathematics and Computer Science , Mount Allison University , Sackville, New Brunswick, Canada guedenon@caramail.com |
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Abstract: | ABSTRACT Let k be a field, R an associative k-algebra with identity, Δ a finite set of derivations of R, and RΘ1, δ1] ··· Θ n , δ n ] an iterated differential operator k-algebra over R such that δ j (Θ i ) ∈ RΘ1, 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. |
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Keywords: | 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 |
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