ON STAFFORD-SMITHS PROBLEM OF RING OF DIFFERENTIAL-OPERATORS OVER CURVES

Let X be an affine algebraic curve over an algebraically closed field k of characteristic zero. We use O(X) to denote the regular function r ing of X, and D(X) to denote the ring of differential operator of X, H (X) to denote its induced Artin algebra. Stafford-Smith proposed the f ollowing two problems([1]). Problem I. May D(X) have infinite or arbit rarily large finite global homological dimension? Problem II. Are ther e restrictions on structure of H(X) - Can any finite-dimensional algeb ra occur? Moreover, Brown proposed the following problem in ref.[2]: I s H(X) always a quasi-hereditary algebra? In this note, we give answer s to these problems.