  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.
