The differential operator ring of an affine curve

Muhasky, Jerry L.

doi:10.1090/S0002-9947-1988-0940223-5

The differential operator ring of an affine curve
HTML articles powered by AMS MathViewer

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

Allen Altman and Steven Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. MR 0274461
Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, Graduate Texts in Mathematics, Vol. 13, Springer-Verlag, New York-Heidelberg, 1974. MR 0417223

Differential operators on the cubic cone

27

William C. Brown, A note on higher derivations and ordinary points of curves, Rocky Mountain J. Math. 14 (1984), no. 2, 397–402. MR 747286, DOI 10.1216/RMJ-1984-14-2-397
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), no. 1, 15–20. MR 760868, DOI 10.1112/jlms/s2-30.1.15
R. Hart, Differential operators on affine algebras, J. London Math. Soc. (2) 28 (1983), no. 3, 470–476. MR 724716, DOI 10.1112/jlms/s2-28.3.470
A. Joseph, A generalization of Quillen’s lemma and its application to the Weyl algebras, Israel J. Math. 28 (1977), no. 3, 177–192. MR 573083, DOI 10.1007/BF02759808
J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1987. With the cooperation of L. W. Small; A Wiley-Interscience Publication. MR 934572
Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
Ian M. Musson, Some rings of differential operators which are Morita equivalent to the Weyl algebra $A_1$, Proc. Amer. Math. Soc. 98 (1986), no. 1, 29–30. MR 848868, DOI 10.1090/S0002-9939-1986-0848868-1
J. C. Robson and L. W. Small, Orders equivalent to the first Weyl algebra, Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 148, 475–482. MR 868622, DOI 10.1093/qmath/37.4.475
S. P. Smith and J. T. Stafford, Differential operators on an affine curve, Proc. London Math. Soc. (3) 56 (1988), no. 2, 229–259. MR 922654, DOI 10.1112/plms/s3-56.2.229
Bo Stenström, Rings of quotients, Die Grundlehren der mathematischen Wissenschaften, Band 217, Springer-Verlag, New York-Heidelberg, 1975. An introduction to methods of ring theory. MR 0389953
Mark L. Teply, On the transfer of properties to subidealizer rings, Comm. Algebra 5 (1977), no. 7, 743–758. MR 437602, DOI 10.1080/00927877708822191