Dan Christensen, Bernhard Keller, Amnon Neeman

September 12, 1999


Let T be a triangulated category with coproducts, Tc the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams (1971) proved the following : All cohomological functors  from Tc to abelian groups 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. Neeman proved that Adams' theorem remains true as long as Tc 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 homological functors on D(R) which are not restrictions of representables.

Bernhard Keller, le 12 Septembre, 1999.
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