Differential Operators and Finite-Dimensional Algebras

AbstractLet 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)