Auslander-regular and Cohen-macaulay Quantum Groups

Let Uq(C) be the quantum group or quantized enveloping algebra in the sense of [6, 7] associated to a Cartan matrix C. A relevant property of Uq(C) is that it can be endowed with a multi-filtration such that the associated multi-graded algebra is an easy localization of the coordinate ring of a quantum affine space [7, Proposition 10.1]. Thus, it is not surprising if we claim that Uq(C) is an Auslander-regular and Cohen-Macaulay algebra (see, e.g., [2] for these notions). However, when one tries to construct a mathematically sound argument to prove this, one realizes that there are not ready-to-use results for this in the literature. Here we use re-filtering methods (see Theorem 1) similar to that in [5] and [4] to prove, in conjunction with results from [2] and [14], that certain types of multifiltered algebras are Auslander-regular and Cohen-Macaulay (Theorem 3). This is applied to obtain that Uq(C) is Auslander-regular and Cohen-Macaulay. In this note, K denotes a commutative ring and N is the free abelian monoid with n generators ǫ1, . . . , ǫn. The elements in N n are vectors α = (α1, . . . , αn) with non-negative integer entries. An admissible order on N is a total order compatible with the sum in N and such that 0 α for every α ∈ N. In this way, N becomes a well-ordered monoid. A fundamental example of admissible order on N is the lexicographical order ≤lex with ǫ1 <lex · · · <lex ǫn. Every vector w with strictly positive entries gives an example of admissible order w by putting