Localizing subcategories in the Bootstrap category of separable C*-algebras

TL;DR

This paper classifies all localizing subcategories of the Bootstrap category of separable complex C*-algebras using the universal coefficient theorem, linking them to subsets of the Zariski spectrum of integers.

Contribution

It provides a simple classification of localizing subcategories in the Bootstrap category, connecting them to algebraic geometry via the Zariski spectrum.

Findings

Localizing subcategories correspond to subsets of the Zariski spectrum of integers

Classification aligns with that of derived categories of abelian groups

Results are contextualized with existing literature

Abstract

Using the classical universal coefficient theorem of Rosenberg-Schochet, we prove a simple classification of all localizing subcategories of the Bootstrap category of separable complex C*-algebras. Namely, they are in bijective correspondence with subsets of the Zariski spectrum of the integers -- precisely as for the localizing subcategories of the derived category of complexes of abelian groups. We provide corollaries of this fact and put it in context with similar classifications available in the literature.