Muhasky — The Differential Operator Ring Of An Affine Curve

The purpose of this paper is to investigate the structure of the ring D(R) of all linear differential operators on the coordinate ring of an affine algebraic variety X (possibly reducible) over a field k (not necessarily algebraically closed) of characteristic zero, concentrating on the case that dimX {\(<\)} 1. In this case, it is proved that D(R) is a (left and right) noetherian ring with (left and right) Krull dimension equal to dim X, that the endomorphism ring of any simple (left or right) D(iJ)-module is finite dimensional over fc, that D(R) has a unique smallest ideal L essential as a left or right ideal, and that D(R)/L is finite dimensional over fc. The following ring-theoretic tool is developed for use in deriving the above results. Let D be a subalgebra of a left noetherian fc-algebra E such that E is finitely generated as a left Dmodule and all simple left E-modules have finite dimensional endomorphism rings (over fc), and assume that D contains a left ideal I of E such that E/I has finite length. Then it is proved that D is left noetherian and that the endomorphism ring of any simple left D-module is finite dimensional over fc.