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

TL;DR

This paper develops a homological framework in triangulated categories to establish criteria for subcategory complementarity and applies it to construct the Baum-Connes assembly map for specific algebraic structures.

Contribution

It introduces a new homological criterion for subcategory complementarity and applies it to quantum groups and group C*-algebras, extending the Baum-Connes conjecture.

Findings

Established a criterion for subcategory complementarity in triangulated categories.

Constructed the Baum-Connes assembly map for quantum groups.

Linked methods to the Adams spectral sequence framework.

Abstract

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.