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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The differential operator ring of an affine curve


Author: Jerry L. Muhasky
Journal: Trans. Amer. Math. Soc. 307 (1988), 705-723
MSC: Primary 16A05; Secondary 13E15, 14H99, 16A33
MathSciNet review: 940223
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Abstract: The purpose of this paper is to investigate the structure of the ring $ D(R)$ of all linear differential operators on the coordinate ring of an affine algebraic variety $ X$ (possibly reducible) over a field $ k$ (not necessarily algebraically closed) of characteristic zero, concentrating on the case that dim $ X \leqslant 1$. In this case, it is proved that $ D(R)$ is a (left and right) noetherian ring with (left and right) Krull dimension equal to dim $ X$, that the endomorphism ring of any simple (left or right) $ D(R)$-module is finite dimensional over $ k$, that $ D(R)$ has a unique smallest ideal $ L$ essential as a left or right ideal, and that $ D(R)/L$ is finite dimensional over $ k$. The following ring-theoretic tool is developed for use in deriving the above results. Let $ D$ be a subalgebra of a left noetherian $ k$-algebra $ E$ such that $ E$ is finitely generated as a left $ D$-module and all simple left $ E$-modules have finite dimensional endomorphism rings (over $ k$), and assume that $ D$ contains a left ideal $ I$ of $ E$ such that $ E/I$ has finite length. Then it is proved that $ D$ is left noetherian and that the endomorphism ring of any simple left $ D$-module is finite dimensional over $ k$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0940223-5
PII: S 0002-9947(1988)0940223-5
Article copyright: © Copyright 1988 American Mathematical Society