The differential operator ring of an affine curve
Author:
Jerry L. Muhasky
Journal:
Trans. Amer. Math. Soc. 307 (1988), 705723
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 ringtheoretic 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 .
 [1]
Allen
Altman and Steven
Kleiman, Introduction to Grothendieck duality theory, Lecture
Notes in Mathematics, Vol. 146, SpringerVerlag, BerlinNew York, 1970. MR 0274461
(43 #224)
 [2]
Frank
W. Anderson and Kent
R. Fuller, Rings and categories of modules, SpringerVerlag,
New YorkHeidelberg, 1974. Graduate Texts in Mathematics, Vol. 13. 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), 169174.
 [4]
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
(85k:13007), http://dx.doi.org/10.1216/RMJ1984142397
 [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),
no. 1, 15–20. MR 760868
(86e:13004), http://dx.doi.org/10.1112/jlms/s230.1.15
 [6]
R.
Hart, Differential operators on affine algebras, J. London
Math. Soc. (2) 28 (1983), no. 3, 470–476. MR 724716
(85b:13040), http://dx.doi.org/10.1112/jlms/s228.3.470
 [7]
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 0573083
(58 #28097)
 [8]
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 WileyInterscience Publication. MR 934572
(89j:16023)
 [9]
Hideyuki
Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture
Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading,
Mass., 1980. MR
575344 (82i:13003)
 [10]
Ian
M. Musson, Some rings of differential operators
which are Morita equivalent to the Weyl algebra 𝐴₁,
Proc. Amer. Math. Soc. 98 (1986),
no. 1, 29–30. MR 848868
(88a:16012), http://dx.doi.org/10.1090/S00029939198608488681
 [11]
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 (88a:16010), http://dx.doi.org/10.1093/qmath/37.4.475
 [12]
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 (89d:14039), http://dx.doi.org/10.1112/plms/s356.2.229
 [13]
Bo
Stenström, Rings of quotients, SpringerVerlag, New
YorkHeidelberg, 1975. Die Grundlehren der Mathematischen Wissenschaften,
Band 217; An introduction to methods of ring theory. MR 0389953
(52 #10782)
 [14]
Mark
L. Teply, On the transfer of properties to subidealizer rings,
Comm. Algebra 5 (1977), no. 7, 743–758. MR 0437602
(55 #10526)
 [1]
 A. Altman and S. Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Math., vol. 146, SpringerVerlag, 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, SpringerVerlag, 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), 169174.
 [4]
 W. C. Brown, A note on higher derivations and ordinary points of curves, Rocky Mountain J. Math. 14 (1984), 397402. 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), 1520. MR 760868 (86e:13004)
 [6]
 R. Hart, Differential operators on affine algebras, J. London Math. Soc. (2) 28 (1983), 470476. 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), 177192. MR 0573083 (58:28097)
 [8]
 J. C. McConnell and J. C. Robson, Noncommutative noetherian rings, WileyInterscience, 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 , Proc. Amer. Math. Soc. 98 (1986), 2930. 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, SpringerVerlag, 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), 743758. MR 0437602 (55:10526)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198809402235
PII:
S 00029947(1988)09402235
Article copyright:
© Copyright 1988
American Mathematical Society
