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Catenarity in quantum algebras. (English) Zbl 0864.16018

In a celebrated but never-published work [Equidimensionalité de la variété caractéristique, Exposé de O. Gabber rédigé par T. Levasseur, Université de Paris VI (1982)] O. Gabber proved that the enveloping algebra \(U\) of a finite dimensional complex solvable Lie algebra is catenary, meaning that all saturated chains of prime ideals between any two fixed primes \(P<Q\) of \(U\) have the same length. (Gabber’s proof is included in [T. Levasseur and J. T. Stafford, Rings of differential operators on classical rings of invariants (Mem. Am. Math. Soc. 412, 1989; Zbl 0691.16019)].) The idea behind Gabber’s theorem is that catenarity is a consequence of good homological and growth properties of the algebra. In the present paper the authors first state and prove an “abstract” version of Gabber’s result, making precise the rough statement of the previous sentence; simultaneously, they derive an “abstract” version of Tauvel’s height formula [P. Tauvel, Bull. Soc. Math. Fr. 106, 177-205 (1978; Zbl 0399.17003)]. In the second part of the paper they show that the hypotheses required for these “abstract” results are satisfied by various classes of algebras associated with quantum groups.

MSC:

16P40 Noetherian rings and modules (associative rings and algebras)
16D25 Ideals in associative algebras
16S30 Universal enveloping algebras of Lie algebras
16P90 Growth rate, Gelfand-Kirillov dimension
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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