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 of all linear differential operators on the coordinate ring of an affine algebraic variety (possibly reducible) over a field (not necessarily algebraically closed) of characteristic zero, concentrating on the case that dim . In this case, it is proved that is a (left and right) noetherian ring with (left and right) Krull dimension equal to dim , that the endomorphism ring of any simple (left or right) -module is finite dimensional over , that has a unique smallest ideal essential as a left or right ideal, and that is finite dimensional over . The following ring-theoretic tool is developed for use in deriving the above results. Let be a subalgebra of a left noetherian -algebra such that is finitely generated as a left -module and all simple left -modules have finite dimensional endomorphism rings (over ), and assume that contains a left ideal of such that has finite length. Then it is proved that is left noetherian and that the endomorphism ring of any simple left -module is finite dimensional over .

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DOI:
https://doi.org/10.1090/S0002-9947-1988-0940223-5

Article copyright:
© Copyright 1988
American Mathematical Society