Prime ideals in differential operator rings. Catenarity

Abstract

Let R R be a commutative algebra over the commutative ring k k , and let Δ = { δ 1 , … , δ n } \Delta = \{ {\delta _1}, \ldots ,{\delta _n}\} be a finite set of commuting k k -linear derivations from R R to R R . Let T = R [ θ 1 , … , θ n ; δ 1 , … , δ n ] 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 R -modules between T T and its associated graded ring R [ x 1 , … , x n ] R[{x_1}, \ldots ,{x_n}] , a polynomial ring over R R . This isomorphism is used to study the prime ideals of T T , with emphasis on the question of catenarity. We prove that T T is catenary when R R is a commutative noetherian universally catenary k k -algebra and one of the following cases occurs: (A) k k is a field of characteristic zero and Δ \Delta acts locally finitely; (B) k k is a field of positive characteristic; (C) k k is the ring of integers, R R is affine over k k , and Δ \Delta acts locally finitely.