Global dimension of differential operator rings. II

Abstract

The aim of this paper is to find the global homological dimension of the ring of linear differential operators R [ θ 1 , … , θ u ] R[{\theta _1}, \ldots ,{\theta _u}] over a differential ring R R with u u commuting derivations. When R R is a commutative noetherian ring with finite global dimension, the main theorem of this paper (Theorem 21) shows that the global dimension of R [ θ 1 , … , θ u ] R[{\theta _1}, \ldots ,{\theta _u}] is the maximum of k k and q + u q + u , where q q is the supremum of the ranks of all maximal ideals M M of R R for which R / M R/M has positive characteristic, and k k is the supremum of the sums r a n k ( P ) + d i f f d i m ( P ) rank(P) + diff\;dim(P) for all prime ideals P P of R R such that R / P R/P has characteristic zero. [The value d i f f d i m ( P ) diff\;dim(P) is an invariant measuring the differentiability of P P in a manner defined in §3.] In case we are considering only a single derivation on R R , this theorem leads to the result that the global dimension of R [ θ ] R[\theta ] is the supremum of gl d i m ( R ) dim(R) together with one plus the projective dimensions of the modules R / J R/J , where J J is any primary differential ideal of R R . One application of these results derives the global dimension of the Weyl algebra in any degree over any commutative noetherian ring with finite global dimension.