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Global Homological Dimension Of Multifiltered Rings And Quantized Enveloping Algebras

The study of global homological dimension has a major tool in the study of filtered rings and associated graded rings. In fact, [11, 7.6.18 Corollary] states that the global homological dimension of a filtered ring is bounded by the global homological dimension of its associated graded ring (this result firstly appears in [14]). Most of the classical cases work nice because the associated graded ring is commutative (polynomial commutative ring), so its global homological dimension is known. There is a lot of interesting examples in which such nice filtration does not exist (at least in a natural way), but they have what we call a multifiltration. The main idea is to replace the set of natural numbers with its order by any power p with a total order compatible with the semigroup structure. So, the amount of examples covered by this new technique increases considerably: quantum matrices (uniparametric and multiparametric), classical and quantized Weyl algebras, classical enveloping algebras of finite-dimensional Lie algebras and, as the most interesting one, quantized enveloping algebras UqC in the sense of Drinfeld and Jimbo (see Example 4.3 for definitions). This idea has been recently exploited in [6] in the study of the Gelfand–Kirillov dimension. Our development parallels the exposition given in [11, Chap. 7, Sect. 6]. To do this we have solved some technical difficulties stemming from the