Krull Dimension of Differential Operator Rings IV—Multiple Derivations

This paper is concerned with the Krull dimension (in the sense of Gabriel and Rentschler) of a differential operator ring T = R [θ 1 ,…,θ u ], 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 A u ( 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.