Finite length and pure-injective modules over a ring of differential operators
Let \({\mathcal O}_n=kx_1,\dots,x_n\) and let \({\mathcal D}_n={\mathcal O}_n[\tfrac{\partial}{\partial x_1},\dots,\tfrac{\partial}{\partial x_n}]\) be the ring of differential operators on \({\mathcal O}_n\), where \(k\) is an algebraically closed field of characteristic 0. The author classifies the finite length \({\mathcal D}_1\)-modules, showing that every indecomposable finite-length module is uniserial and either homogeneous or with two alternating simple composition factors. This classification uses the classification [\textit{G. Puninski}, Pure-injective and finite length modules over certain rings of differential polynomials, Proc. Moscow-Taiwan Conf. (preprint)] of finite length modules over the ring of differential operators of the quotient ring \(k( (x))\) of \(kx\) which, in turn, uses results of \textit{W. Zimmermann} [J. Pure Appl. Algebra 74, No. 3, 317-332 (1991; Zbl 0742.16008)]. Using Morita duality the classification for \({\mathcal D}_1\) is extended to \({\mathcal D}_n\)-modules supported on a curve. The classification of finite length \({\mathcal D}_1\)-modules is used to describe the Ziegler spectrum of \({\mathcal D}_1\), which turns out to be very similar to that of finite dimensional tame hereditary algebras [\textit{M. Prest}, J. Algebra 207, No.~1, 146-164 (1998; Zbl 0936.16014) and \textit{C. M. Ringel}, Colloq. Math. 76, No.~1, 105-115 (1998; Zbl 0901.16006)].
finite dimensional tame hereditary algebras
scientific article; zbMATH DE number 1526423