Global dimension of differential operator rings

Abstract

This paper is concerned with finding the global homological dimension of the ring of differential operators R [ θ ] R[\theta ] over a differential ring R R with a single derivation. Examples are constructed to show that R [ θ ] R[\theta ] may have finite dimension even when R R has infinite dimension. For a commutative noetherian differential algebra R R over the rationals, with finite global dimension n n , it is shown that the global dimension of R [ θ ] R[\theta ] is the supremum of n n and one plus the projective dimensions of the modules R / P R/P , where P P ranges over all prime differential ideals of R R . One application derives the global dimension of the Weyl algebra over a commutative noetherian ring S S of finite global dimension, where S S either is an algebra over the rationals or else has positive characteristic.