Differential Operators on an Affine Curve

Abstract

Let X denote an irreducible affine algebraic curve over an algebraically closed field k of characteristic zero. Denote by Dx the sheaf of differential operators on X, and D(X)=Γ(X,Dx), the ring of global differential operators on X. The following is established: THEOREM. D(X) is a finitely generated k-algebra, and a noetherian ring. Furthermore, D(X) has a unique minimal non-zero ideal J, and D(X)/J is a finite-dimensional k-algebra. Let X ˜ denoted the normalisation of X, and π: X ˜ →X the projection map. The main technique is to compare D(X) with D( X ˜ ). THEOREM. The following are equivalent: (i) π is injective, (ii) D(X) is a simple ring, (iii) D(X) is Morita equivalent to D( X ˜ ), (iv) the categories DX-Mod and D X ˜ -Mod are equivalent, (v) gr D(X) is noetherian, (vi) the global homological dimension of D(X) is 1. For higher-dimensional varieties the techniques produce examples of varieties X for which D(X) is right but not left noetherian.