Prime ideals in differential operator rings. Catenarity

Abstract:

Let $R$ be a commutative algebra over the commutative ring $k$, and let $\Delta = \{ {\delta _1}, \ldots ,{\delta _n}\}$ be a finite set of commuting $k$-linear derivations from $R$ to $R$. Let $T = R[{\theta _1}, \ldots ,{\theta _n};{\delta _1}, \ldots ,{\delta _n}]$ be the corresponding ring of differential operators. We define and study an isomorphism of left $R$-modules between $T$ and its associated graded ring $R[{x_1}, \ldots ,{x_n}]$, a polynomial ring over $R$. This isomorphism is used to study the prime ideals of $T$, with emphasis on the question of catenarity. We prove that $T$ is catenary when $R$ is a commutative noetherian universally catenary $k$-algebra and one of the following cases occurs: (A) $k$ is a field of characteristic zero and $\Delta$ acts locally finitely; (B) $k$ is a field of positive characteristic; (C) $k$ is the ring of integers, $R$ is affine over $k$, and $\Delta$ acts locally finitely.