Failure of Brown representability in derived categoriesDedicated to H. Lenzing on the occasion of his sixtieth birthday

Let T be a triangulated category with coproducts, T^c⊂T the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams (Topology 10 (1971) 185-198), proved the following: All homological functors {T^c}^op→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 Neeman (Topology 36 (1997) 619-645), it was proved that Adams' theorem remains true as long as T^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 (Relative homological and purity in triangulated categories, 1999, preprint) 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 homological functors {T^c}^op→Ab which are not restrictions of representables.