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Catenarity in a class of iterated skew polynomial rings. (English) Zbl 0872.16018
Certain iterated skew polynomial algebras of a type introduced by D. A. Jordan [e.g., J. Algebra 174, No. 1, 267-281 (1995; Zbl 0833.16025)] are studied. For a large subclass, the author establishes finiteness of the Gelfand-Kirillov dimension, Auslander-regularity, the Cohen-Macaulay property, and normal separation. It then follows from a modification of Gabber’s work by T. H. Lenagan and the reviewer [J. Pure Appl. Algebra 111, No. 1-3, 123-142 (1996; Zbl 0864.16018)] that the prime spectra of the algebras in this class are catenary, and that Tauvel’s height formula holds. Applications include the one-parameter coordinate rings \({\mathcal O}_q({\mathfrak sp} k^{2n})\) and \({\mathcal O}_q({\mathfrak o} k^n)\) of quantum symplectic and Euclidean spaces when \(q\) is not a root of unity, and the multiparameter quantized Weyl algebras \(A_n^{Q,\Gamma}(k)\) when the entries of the vector \(Q\in(k^\times)^n\) are not roots of unity. The results for \({\mathcal O}_q({\mathfrak sp} k^{2n})\) and \({\mathcal O}_q({\mathfrak o} k^n)\) are new, while that for \(A_n^{Q,\Gamma}(k)\) recovers a result of Lenagan and the reviewer [op. cit.] with less technical computations.

MSC:
16S36 Ordinary and skew polynomial rings and semigroup rings
16D25 Ideals in associative algebras
16P90 Growth rate, Gelfand-Kirillov dimension
16P40 Noetherian rings and modules (associative rings and algebras)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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