Failure of Brown representability in derived categories

TL;DR

This paper demonstrates that the classical Brown representability theorem does not hold universally in derived categories, providing explicit counterexamples where homological functors are not representable.

Contribution

It proves that not all homological functors in derived categories are restrictions of representable functors, settling a long-standing open problem.

Findings

Counterexamples in derived categories where homological functors are not representable

Brown representability fails in certain derived categories of rings

Clarifies limitations of representability in triangulated categories

Abstract

Let T be a triangulated category with coproducts, C the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [Adams71]: All contravariant homological functors C --> Ab are the restrictions of representable functors on T, and all natural transformations are the restrictions of morphisms in T. It has been something of a mystery, to what extent this generalises to other triangulated categories. In [Neeman97], it was proved that Adams' theorem remains true as long as C is countable, but can fail in general. The failure exhibited was that there can be natural transformations not arising from maps in T. A puzzling open problem remained: Is every homological functor the restriction of a representable functor on T? In a recent paper, Beligiannis made some progress. But in this article, we settle the problem. The answer is no. There are examples of derived categories T = D(R) of rings, and contravariant homological functors C --> Ab which are not restrictions of representables.