The Artin algebras associated with differential operators on singular affine curves
type: Item
instance of: scholarly article
title: The Artin algebras associated with differential operators on singular affine curves
zbMATH Open document ID: 0731.14020
DOI: 10.1007/BF02571353
published in: Mathematische Zeitschrift
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174235
review text: 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.
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 13N10
Mathematics Subject Classification ID: 13E10
zbMATH DE Number: 4208258
zbMATH Keywords: Artin algebras
zbMATH Keywords: differential operator
zbMATH Keywords: Krull dimension
zbMATH Keywords: Grothendieck groups
zbMATH Keywords: differential operators
zbMATH Keywords: Krull dimension 1
zbMATH Keywords: normalization.
author: Kenneth A. Brown
reviewed by: Nicolae Manolache
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4208258