We use homological ideals in triangulated categories to get a sufficient criterion for a pair of subcategories in a triangulated category to be complementary. We apply this criterion to construct the Baum-Connes assembly map for locally compact groups and torsion-free discrete quantum groups. Our methods are related to the abstract version of the Adams spectral sequence by Brinkmann and Christensen.
Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen.
"Homological algebra in bivariant K-theory and other triangulated categories. II." Tbilisi Math. J. 1 165 - 210, 2008. https://doi.org/10.32513/tbilisi/1528768828