Abstract
Given any commutative field k , denote R = O q ( M n ( k )) the coordinate ring of quantum n × n matrices over k and assume q is a nonzero element in k which is not a root of unity. Recall that R is generated by n 2 variables X i , α ((i,α)∈〚1,n〛 2 ) subject (only) to the following relations: If x y z t is any 2×2 sub-matrix of X =(X i,α ), then: (a) yx = q −1 xy , zx = q −1 xz , tz = q −1 zt , ty = q −1 yt , zy = yz ; (b) tx = xt −( q − q −1 ) yz . Denote R the k -algebra generated by the same variables X i , α subject to the same relations, except relations (b) which are replaced by: (c) tx = xt ; so that R is just the algebra of regular functions on some quantum affine space of dimension n 2 over k . The theory of “derivative elimination” defines a natural embedding ϕ : Spec (R)→ Spec ( R ) and asserts that: • The “canonical image” ϕ (Spec( R )) is a union of strata Spec w ( R ) (in the sense of [Goodearl, Letzter, in: CMS Conf. Proc., Vol. 22 (1998) 39–58]), where w describes some subset W of P (〚1,n〛 2 ) . • The sets Spec w (R):=ϕ −1 ( Spec w ( R )) ( w ∈ W ) define the Goodearl–Letzter H -stratification of Spec( R ) in the sense of [Goodearl, Letzter, Trans. Amer. Math. Soc. 352 (2000) 1381–1403]. In this paper, we give the precise description of the set W and we compute its cardinality. Using that description and the derivative elimination algorithm, we can verify (Theorems 6.3.1, 6.3.2) that H -Spec( R ) has an H -normal separation (in the sense of [Goodearl, in: Lecture Notes in Pure and Appl. Math. 210 (2000) 205–237]), so that Spec( R ) has normal separation (in the sense of [Brown, Goodearl, Trans. Amer. Math. Soc. 348 (1996) 2465–2502]). This property was conjectured by K. Brown and K. Goodearl. Since R is Auslander–Regular and Cohen–Macaulay, this implies (by [Goodearl, Lenagan, J. Pure Appl. Algebra 111 (1996) 123–142]) that R is catenary and satisfies the Tauvel's height formula.
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