Differential Operators on an Affine Curve

Let X denote an irreducible affine algebraic curve over an algebraically closed field k of characteristic zero. Denote by D x the sheaf of differential operators on X , and D ( X )=Γ( X , D x ), 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 D X ‐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.