Prime ideals in differential operator rings. Catenarity
HTML articles powered by AMS MathViewer
- by K. A. Brown, K. R. Goodearl and T. H. Lenagan PDF
- Trans. Amer. Math. Soc. 317 (1990), 749-772 Request permission
Abstract:
Let $R$ be a commutative algebra over the commutative ring $k$, and let $\Delta = \{ {\delta _1}, \ldots ,{\delta _n}\}$ be a finite set of commuting $k$-linear derivations from $R$ to $R$. Let $T = R[{\theta _1}, \ldots ,{\theta _n};{\delta _1}, \ldots ,{\delta _n}]$ be the corresponding ring of differential operators. We define and study an isomorphism of left $R$-modules between $T$ and its associated graded ring $R[{x_1}, \ldots ,{x_n}]$, a polynomial ring over $R$. This isomorphism is used to study the prime ideals of $T$, with emphasis on the question of catenarity. We prove that $T$ is catenary when $R$ is a commutative noetherian universally catenary $k$-algebra and one of the following cases occurs: (A) $k$ is a field of characteristic zero and $\Delta$ acts locally finitely; (B) $k$ is a field of positive characteristic; (C) $k$ is the ring of integers, $R$ is affine over $k$, and $\Delta$ acts locally finitely.References
- Allen D. Bell, Localization and ideal theory in iterated differential operator rings, J. Algebra 106 (1987), no. 2, 376–402. MR 880964, DOI 10.1016/0021-8693(87)90003-2
- Allen D. Bell and Gunnar Sigurđsson, Catenarity and Gel′fand-Kirillov dimension in Ore extensions, J. Algebra 127 (1989), no. 2, 409–425. MR 1028462, DOI 10.1016/0021-8693(89)90261-5
- Walter Borho, Peter Gabriel, and Rudolf Rentschler, Primideale in Einhüllenden auflösbarer Lie-Algebren (Beschreibung durch Bahnenräume), Lecture Notes in Mathematics, Vol. 357, Springer-Verlag, Berlin-New York, 1973 (German). MR 0376790
- Paul Moritz Cohn, Algebra. Vol. 2, John Wiley & Sons, London-New York-Sydney, 1977. With errata to Vol. I. MR 0530404
- John R. Fisher, A Goldie theorem for differentiably prime rings, Pacific J. Math. 58 (1975), no. 1, 71–77. MR 374195 O. Gabber, Equidimensionalité de la variété charactéristique, Exposé de O. Gabber, rédigé par T. Levasseur, Université de Paris VI, 1982.
- Kenneth R. Goodearl and Robert B. Warfield Jr., Primitivity in differential operator rings, Math. Z. 180 (1982), no. 4, 503–523. MR 667005, DOI 10.1007/BF01214722
- Ronald S. Irving, Generic flatness and the Nullstellensatz for Ore extensions, Comm. Algebra 7 (1979), no. 3, 259–277. MR 519702, DOI 10.1080/00927877908822347
- D. A. Jordan, Noetherian Ore extensions and Jacobson rings, J. London Math. Soc. (2) 10 (1975), 281–291. MR 389988, DOI 10.1112/jlms/s2-10.3.281
- Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
- Günter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Revised edition, Graduate Studies in Mathematics, vol. 22, American Mathematical Society, Providence, RI, 2000. MR 1721834, DOI 10.1090/gsm/022
- Martin Lorenz, Chains of prime ideals in enveloping algebras of solvable Lie algebras, J. London Math. Soc. (2) 24 (1981), no. 2, 205–210. MR 631934, DOI 10.1112/jlms/s2-24.2.205
- Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
- Louis Halle Rowen, Polynomial identities in ring theory, Pure and Applied Mathematics, vol. 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 576061
- William Schelter, Non-commutative affine P.I. rings are catenary, J. Algebra 51 (1978), no. 1, 12–18. MR 485980, DOI 10.1016/0021-8693(78)90131-X
- Gunnar Sigurdsson, Differential operator rings whose prime factors have bounded Goldie dimension, Arch. Math. (Basel) 42 (1984), no. 4, 348–353. MR 753356, DOI 10.1007/BF01246126
- P. F. Smith, Localization and the AR property, Proc. London Math. Soc. (3) 22 (1971), 39–68. MR 294383, DOI 10.1112/plms/s3-22.1.39
- Sleiman Yammine, Les théorèmes de Cohen-Seidenberg en algèbre non commutative, Séminaire d’Algèbre Paul Dubreil 31ème année (Paris, 1977–1978) Lecture Notes in Math., vol. 740, Springer, Berlin, 1979, pp. 120–169 (French). MR 563499
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 317 (1990), 749-772
- MSC: Primary 16A05; Secondary 16A66
- DOI: https://doi.org/10.1090/S0002-9947-1990-0946215-3
- MathSciNet review: 946215