Differential operators and finite-dimensional algebras (Q1895549)

The paper investigates the structure of \({\mathcal D}(M)\) the ring of differential operators in \(M\), where \(M\) is a rank one torsionfree module over a Dedekind domain which is an affine \(k\)-algebra, \(k\) an algebraically closed field of characteristic 0. It is shown that \({\mathcal D}(M)\) has a unique minimal non-zero ideal, \(J(M)\), and that the factor ring, \({\mathcal D}(M)/J(M)\), is a finite dimensional \(k\)-algebra. This factor ring 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 \(A\cong{\mathcal D}(M)/J(M)\).