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We use homological ideals in triangulated categories to get a sufficientcriterion for a pair of subcategories in a triangulated category to becomplementary. We apply this criterion to construct the Baum-Connes assemblymap for locally compact groups and torsion-free discrete quantum groups. Ourmethods are related to the abstract version of the Adams spectral sequence byBrinkmann and Christensen.

Homological algebra in bivariant K-theory and other triangulated categories. II