The differential operator ring of an affine curve
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- by Jerry L. Muhasky PDF
- Trans. Amer. Math. Soc. 307 (1988), 705-723 Request permission
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 $k$.References
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Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 307 (1988), 705-723
- MSC: Primary 16A05; Secondary 13E15, 14H99, 16A33
- DOI: https://doi.org/10.1090/S0002-9947-1988-0940223-5
- MathSciNet review: 940223