Rigid dualizing complexes over quantum homogeneous spaces

A quantum homogeneous space of a Hopf algebra is a right coideal subalgebra over which the Hopf algebra is faithfully flat. It is shown that the Auslander-Gorenstein property of a Hopf algebra is inherited by its quantum homogeneous spaces. If the quantum homogeneous space B of a pointed Hopf algebra H is AS-Gorenstein of dimension d, then B has a rigid dualizing complex vB[d]. The Nakayama automorphism v is given by v = ad(g) o S-2 o Xi[tau], where ad(g) is the inner automorphism associated to some group-like element g is an element of H and Xi[tau] is the algebra map determined by the left integral of B. The quantum homogeneous spaces of U-q(sl(2)) are classified and all of them are proved to be Auslander-regular, AS-regular and Cohen-Macaulay. (C) 2012 Elsevier Inc. All rights reserved.