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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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##### References:
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