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