A classification of localizing subcategories by relative homological algebra

Abstract

In this thesis, we use the tools of relative homological algebra in triangulated categories to define a sensible notion of support for objects in the bootstrap class of a Kasparov category of C*-algebras over a finite topological space with totally ordered lattice of open subsets. This category is equivalent to a bootstrap category of filtrations of C*-algebras. As a consequence, we provide a full classification of localizing subcategories of the bootstrap category in terms of a product of lattices of noncrossing partitions of a regular polygon. In addition, we consider the 2-periodic derived category of countable modules over the ring of upper triangular matrices. Since the homological algebra is the same, the lattices of localizing subcategories in this category and the bootstrap category are isomorphic.