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When rings of differential operators are maximal orders

Published online by Cambridge University Press:  24 October 2008

M. Chamarie
Affiliation:
Département de Mathematiques, Université Claude Bernard, Lyon, France
J. T. Stafford
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT

Abstract

Let A be a commutative domain, finitely generated as an algebra over a field k of characteristic zero and write (A) for the ring of k -linear differential operators. Then A is an Ore domain with quotient division ring, say Q. Our main result is that A is a maximal order in Q if and only if (i) A = ∩{Ap: height (p) = 1} and (ii) A is geometrically unibranched. In this case A is also a Krull domain with no reflexive ideals. We also determine some conditions under which A is simple.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

REFERENCES

[1]Bass, H.. Finitistic dimension and a homological generalisation of semiprimary rings. Trans. Amer. Math. Soc. 95 (1960), 466488.CrossRefGoogle Scholar
[2]Bernstein, J. N., Gelfand, I. M. and Gelfand, S. I.. Differential operators on the cubic cone. Russian Math. Surveys 27 (1972), 169174.CrossRefGoogle Scholar
[3]Bjork, J. E.. Rings of Differential Operators (North Holland, 1979).Google Scholar
[4]Bourbaki, N.. Commutative Algebra (Hermann, 1972).Google Scholar
[5]Chamarie, M.. Anneaux de Krull non commutatifs. J. Algebra 72 (1981), 210222.CrossRefGoogle Scholar
[6]Chamarie, M.. Modules sur les anneaux de Krull non commutatifs. In Séminaire d'Algèbre P. Dubreil et M.-P. Malliavin, Lecture Notes in Math., vol. 1029 (Springer-Verlag, 1983), 283310.CrossRefGoogle Scholar
[7]Ferrand, D.. Monomorphismes et morphismes absolument plats. Bull. Soc. Math. France 100 (1972), 97128.CrossRefGoogle Scholar
[8]Hart, R.. Differential operators on affine algebras. J. London Math. Soc. 28 (1983), 470476.CrossRefGoogle Scholar
[9]Hart, R. and Smith, S. P.. Differential operators on some singular surfaces. Bull. London Math. Soc. 19 (1987), 145148.CrossRefGoogle Scholar
[10]Humphreys, J. E.. Linear Algebraic Groups (Springer-Verlag, 1981).Google Scholar
[11]Ishibashi, Y.. Remarks on a conjecture of Nakai. J. Algebra 95 (1985), 3145.Google Scholar
[12]Levasseur, T.. Anneaux d'operateurs différentiels. In Séminaire d'Algèbre P. Dubreil et M.-P. Malliavin, Lecture Notes in Math., vol. 867 (Springer-Verlag, 1981), 157173.CrossRefGoogle Scholar
[13]Matsumura, H.. Commutative Algebra (W. A. Benjamin, 1970).Google Scholar
[14]McConnell, J. C. and Robson, J. C.. Non-commutative Noetherian Rings (J. Wiley, to appear).Google Scholar
[15]Muhasky, J. L.. The differential operator rings on an affine curve. Trans. Amer. Math. Soc. (to appear).Google Scholar
[16]Smith, S. P.. Curves, differential operators and finite dimensional algebras. In Séminaire d'Algèbre P. Dubreil et M.-P. Malliavin, Lecture Notes in Math. (Springer-Verlag, to appear).Google Scholar
[17]Smith, S. P. and Stafford, J. T.. Differential operators on an affine curve. Proc. London Math. Soc. (to appear).Google Scholar
[18]Stafford, J. T.. Modules over prime Krull rings. J. Algebra 95 (1985), 332342.CrossRefGoogle Scholar
[19]Stafford, J. T.. Module structure of Weyl algebras. J. London Math. Soc. 18 (1978), 429442.CrossRefGoogle Scholar
[20]Sweedler, M. E.. Groups of simple algebras. Publ. Math. IHES 45 (1975), 79189.Google Scholar
[21]Van Doorn, M. G. M. and Van Den Essen, A. R. P.. Dn -modules with support on a curve. To appear.Google Scholar