Brown representability often fails for homotopy categories of complexes

TL;DR

This paper demonstrates that Brown representability often fails in the homotopy category of complexes of abelian groups, providing specific counterexamples and highlighting limitations in the theory.

Contribution

It shows that both Brown representability and its dual fail for K(Ab), and presents an example of a localizing subcategory without a right adjoint.

Findings

Brown representability fails for K(Ab)

Dual Brown representability also fails

Existence of localizing subcategory without right adjoint

Abstract

We show that for the homotopy category K(Ab) of complexes of abelian groups, both Brown representability and Brown representability for the dual fail. We also provide an example of a localizing subcategory of K(Ab) for which the inclusion into K(Ab) does not have a right adjoint.