Krull Dimension of Differential Operator Rings IV-Multiple Derivations

Abstract

This paper is concerned with the Krull dimension (in the sense of Gabriel and Rentschler) of a differential operator ring T = R[θ1,…,θu], where R is a commutative noetherian ring with finite Krull dimension, equipped with u commuting derivations. The main theorem states that K.dim(T) is the maximum of all the values height(M)+u and height(P)+differential-dimension(P), where M ranges over those maximal ideals of R with char(R/M) > 0 and P ranges over those prime ideals of R with char(R/P)=0. As applications, the Krull dimension of the Weyl algebra Au(S) is computed for any commutative noetherian ring S with finite Krull dimension, and when T has finite global dimension, gl.dim(T) is shown to coincide with K.dim(T). In the case where R is a finitely generated algebra over a field of characteristic zero, the formula for K.dim(T) is reduced to the maximum of the values height(M)+differential−dimension(M), where M ranges over just the maximal ideals of R. This formula is also proved for arbitrary commutative noetherian 2-differential Q-algebras, and a slightly weaker formula, where M ranges over the prime ideals of R of depth at most 1, is proved for the case when R is a localization of a finitely generated algebra over a field of characteristic zero.