Failure of Brown Representability in Derived Categories
Dan Christensen, Bernhard Keller and Amnon Neeman
jdc@math.jhu.edu, keller@math.jussieu.fr, neeman@wintermute.anu.edu.au
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.