Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] A. Altman and S. Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Math., vol. 146, Springer-Verlag, Berlin and New York, 1970. MR 0274461 (43:224)
  • [2] F. W. Anderson and K. R. Fuller, Rings and categories of modules, Graduate Texts in Math., no. 13, Springer-Verlag, Berlin and New York, 1974. MR 0417223 (54:5281)
  • [3] J. N. Bernstein, I. M. Gelfand and S. I. Gelfand, Differential operators on the cubic cone, Russian Math. Surveys 27 (1972), 169-174.
  • [4] W. C. Brown, A note on higher derivations and ordinary points of curves, Rocky Mountain J. Math. 14 (1984), 397-402. MR 747286 (85k:13007)
  • [5] K. R. Goodearl, T. H. Lenagan and P. C. Roberts, Height plus differential dimension in commutative noetherian rings, J. London Math. Soc. (2) 30 (1984), 15-20. MR 760868 (86e:13004)
  • [6] R. Hart, Differential operators on affine algebras, J. London Math. Soc. (2) 28 (1983), 470-476. MR 724716 (85b:13040)
  • [7] A. Joseph, A generalization of Quillen's lemma and its applications to the Weyl algebra, Israel J. Math. 28 (1977), 177-192. MR 0573083 (58:28097)
  • [8] J. C. McConnell and J. C. Robson, Noncommutative noetherian rings, Wiley-Interscience, Chichester and New York, 1987. MR 934572 (89j:16023)
  • [9] H. Matsumura, Commutative algebra, 2nd ed., Benjamin/Cummings, Reading, Mass., 1980. MR 575344 (82i:13003)
  • [10] I. M. Musson, Some rings of differential operators which are Morita equivalent to the Weyl algebra $ {A_1}$, Proc. Amer. Math. Soc. 98 (1986), 29-30. MR 848868 (88a:16012)
  • [11] J. C. Robson and L. W. Small, Orders equivalent to the first Weyl algebra, Preprint, University of Leeds, 1985. MR 868622 (88a:16010)
  • [12] S. P. Smith and J. T. Stafford, Differential operators on an affine curve, Preprint, University of Warwick, 1985. MR 922654 (89d:14039)
  • [13] B. Stenström, Rings of quotients, Grundlehren Math. Wiss., Band 217, Springer-Verlag, Berlin and New York, 1975. MR 0389953 (52:10782)
  • [14] M. L. Teply, On the transfer of properties to subidealizer rings, Comm. Algebra 5 (1977), 743-758. MR 0437602 (55:10526)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16A05, 13E15, 14H99, 16A33

Retrieve articles in all journals with MSC: 16A05, 13E15, 14H99, 16A33

Additional Information

Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society