×

Krull and global dimensions of Weyl algebras over division rings. (English) Zbl 0558.16002

The paper starts with the study of the Krull dimension of the iterated operator ring extension \(T=R[\theta_ 1,...,\theta_ u]\) of a right noetherian ring R of finite Krull dimension k; in more detail, \(T=R[\theta_ 1;\delta_ 1][\theta_ 2;\delta_ 2]...[\theta_ u;\delta_ u],\) where \(\delta_ i\) is a derivation on \(R[\theta_ 1;\delta_ 1]...[\theta_{i-1};\delta_{i-1}].\) It is shown that \(r.K.\dim T=k+u\) if and only if \(h_ u(M)=k\) and \(K.\dim (M\otimes_ RT)=u\) for some simple right R-module M. Here, the height, \(h_ u(M)\), of M is defined in terms of finite sequences of finitely generated u- clean R-modules starting with M, a u-clean R-module being, basically, a critical R-module which remains critical when tensored with T. Next, necessary and sufficient conditions to have \(K.\dim (M\otimes_ RT)=u\) are developed.
Specializing to the Weyl algebra \(A_ n(R)\) (so \(u=2n)\), where R is right noetherian, these conditions amount to the embeddability of \(A_ n({\mathbb{Z}})\) in some matrix ring over \(End_ RM\). The corresponding formula for r.K.dim \(A_ n(R)\) has various simplifications when R is a division ring for, in that case, \(h_{2n}(M)=k\) for any simple right R- module M. In particular, a well known formula of J. C. McConnell [Proc. Lond. Math. Soc., III. Ser. 28, 89-98 (1974; Zbl 0275.16005)] for r.K.dim \(A_ n(D_ t)\) is obtained, where \(D_ t\) is the quotient division ring of \(A_ t(K)\) and K is a field of characteristic zero. Further results are obtained when R is fully bounded noetherian and when R is a noetherian P.I. ring. Concerning global dimension, formulas for the global dimension of \(A_ n(D)\), D a division ring, are obtained which lead to a proof of the equality of the global dimension and the Krull dimension of \(A_ n(D)\). Moreover, if \({\mathfrak g}\) is a finite dimensional Lie algebra over a field K of characteristic zero, it is shown that \(gl.\dim A_ n(U({\mathfrak g}))=\dim_ K{\mathfrak g}+n.\) When K has positive characteristic, n must be replaced by 2n in this formula.
Reviewer: R.Gordon

MSC:

16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16E10 Homological dimension in associative algebras
16P40 Noetherian rings and modules (associative rings and algebras)
16W20 Automorphisms and endomorphisms
16D30 Infinite-dimensional simple rings (except as in 16Kxx)
16P50 Localization and associative Noetherian rings
16Kxx Division rings and semisimple Artin rings
17B35 Universal enveloping (super)algebras

Citations:

Zbl 0275.16005
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Borho, W.; Gabriel, P.; Rentschler, R., Primideale in Einhüllenden auflösbarer Lie-Algebren, (Lecture Notes in Mathematics No. 357 (1973), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0293.17005
[2] Cartan, H.; Eilenberg, S., Homological Algebra (1956), Princeton Univ. Press: Princeton Univ. Press Princeton, N.J · Zbl 0075.24305
[3] Dixmier, J., Enveloping Algebras (1977), North-Holland: North-Holland Amsterdam · Zbl 0366.17007
[4] Goodearl, K. R., Global dimension of differential operator rings, (Proc. Amer. Math. Soc., 45 (1974)), 315-322 · Zbl 0263.13003
[5] Goodearl, K. R., Global dimension of differential operator rings. II, Trans. Amer. Math. Soc., 209, 65-85 (1975) · Zbl 0306.16018
[6] Goodearl, K. R.; Lenagan, T. H., Krull dimension of differential operator rings. III-Noncommutative coefficients, Trans. Amer. Math. Soc., 275, 833-859 (1983) · Zbl 0529.16001
[7] Goodearl, K. R.; Lenagan, T. H., Krull dimension of differential operator rings. IV-Multiple derivations, (Proc. London Math. Soc. (3), 47 (1983)), 306-336 · Zbl 0498.16001
[8] Goodearl, K. R.; Warfield, R. B., Krull dimension of differential operator rings, (Proc. London Math. Soc. (3), 45 (1982)), 49-70 · Zbl 0493.16004
[9] Gordon, R.; Robson, J. C., Krull dimension, Mem. Amer. Math. Soc., 133 (1973) · Zbl 0269.16017
[10] Hart, R., A note on the tensor product of algebras, J. Algebra, 21, 422-427 (1972) · Zbl 0234.16011
[11] Hochschild, G.; Kostant, B.; Rosenberg, A., Differential forms on regular affine algebras, Trans. Amer. Math. Soc., 102, 383-408 (1962) · Zbl 0102.27701
[12] Joseph, A., A generalization of Quillen’s Lemma and its application to the Weyl algebras, Israel J. Math., 28, 177-192 (1977) · Zbl 0366.17006
[13] McConnell, J. C., A note on the Weyl algebra \(A_n\), (Proc. London Math. Soc. (3), 28 (1974)), 89-98 · Zbl 0275.16005
[14] McConnell, J. C., On the global dimension of some rings, Math. Z., 153, 253-254 (1977) · Zbl 0331.16024
[15] McConnell, J. C., On the global and Krull dimensions of Weyl algebras over affine coefficient rings, J. London Math. Soc. (2), 29, 249-253 (1984) · Zbl 0538.16011
[16] Nouazé, Y.; Gabriel, P., Idéaux premiers de l’algèbre enveloppante d’une algèbre de Lie nilpotente, J. Algebra, 6, 77-99 (1967) · Zbl 0159.04101
[17] Quillen, D., On the endomorphism ring of a simple module over an enveloping algebra, (Proc. Amer. Math. Soc., 21 (1969)), 171-172 · Zbl 0188.08901
[18] Rentschler, R.; Gabriel, P., Sur la dimension des anneaux et ensembles ordonnés, C. R. Acad. Sci. Paris Sér. A, 265, 712-715 (1967) · Zbl 0155.36201
[19] Resco, R., Transcendental division algebras and simple noetherian rings, Israel J. Math., 32, 236-256 (1979) · Zbl 0404.16012
[20] Rinehart, G. S., Note on the global dimension of a certain ring, (Proc. Amer. Math. Soc., 13 (1962)), 341-346 · Zbl 0104.26102
[21] Rinehart, G. S.; Rosenberg, A., The global dimensions of Ore extensions and Weyl algebras, (Algebra, Topology, and Category Theory (1976), Academic Press: Academic Press New York), 169-180
[22] Roos, J.-E, Détermination de la dimension homologique globale des algèbres de Weyl, C. R. Acad. Sci. Paris Sér. A, 274, 23-26 (1972) · Zbl 0227.16021
[23] Roy, A., A note on filtered rings, Arch. Math., 16, 421-427 (1965) · Zbl 0143.26701
[24] Tauvel, P., Sur les quotients premiers de l’algèbre enveloppante d’une algb̀re de Lie résoluble, Bull. Soc. Math. France, 106, 177-205 (1978) · Zbl 0399.17003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.