Global dimension of differential operator rings

Abstract:

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