Differential Operators and Finite-Dimensional Algebras

Abstract

Let R be a Dedekind domain that is finitely generated over k, an algebraically closed field of characteristic zero. Let M be a torsionfree module of rank one over a subalgebra of R with integral closure R. This paper investigates the structure of D ( M), the ring of differential operators on M. It is shown that D ( M) has a unique minimal non-zero ideal, J( M), and that the factor, D ( M)/ J( M), is a finite-dimensional k-algebra. This factor is realised as the algebra of all endomorphisms of an associated vector space that preserve certain subspaces. The main result is that given any finite-dimensional k-algebra A there exists such an M with A ≅ D ( M)/ J( M).