The Artin algebras associated with differential operators on singular affine curves (Q807695)

It was shown by \textit{S. P. Smith} and \textit{J. T. Stafford} [Proc. Lond. Math. Soc., III. Ser. 56, No.2, 229-259 (1988; Zbl 0672.14017)] that \((1)\quad the\) ring of differential operators \({\mathcal D}(X)\) of an irreducible affine curve X over \({\mathbb{C}}\) is a noetherian affine domain of Krull dimension 1 which has only one minimal nonzero ideal J(X) and \((2)\quad if\) we denote by H(X) the quotient \({\mathcal D}(X)/J(X)\) and define by analogy \({\mathcal D},J,H\) for any local ring \({\mathfrak O}_{X,x}\), then H(X) is the direct sum of all H(X,x), x singular points of X. In this paper one studies (1) the relationship between H(X,x) and the nature of the singularity x and (2) the relation between the structure of H(X) and that of \({\mathcal D}(X)\), in particular one gives a description of the Grothendieck groups of \({\mathcal D}(X)\) with the help of those of H(X) and of the ring of regular functions on the normalization of X.