Brown representability often fails for homotopy categories of complexes

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.Comment: 9 pages; version 2: minor changes, references added and updated, Remark 11 on the existence of product generators adde