Finite Length and Pure-Injective Modules over a Ring of Differential Operators

Let k be an algebraically closed field of characteristic zero, n = k[[x1,…,xn]] the ring of formal power series over k, and n the ring of differential operators over n. Suppose that ρ is a prime ideal of n of height n − 1; i.e., A = n/ρ is a curve. We prove that every indecomposable finite length module over n with support on ρ is uniserial with isomorphic or alternating composition factors. For the ring (A) of differential operators over A we also classify indecomposable pure-injective modules and show that the Cantor–Bendixson rank of the Ziegler spectrum over (A) is equal to 2.