Failure of Brown representability in derived categories

Research Report MRR99-036

Failure of Brown representability in derived categories

J. Daniel Christensen, Bernhard Keller and Amnon Neeman

Abstract: Let $\ct$ be a triangulated category with coproducts, $\ct^c\subset\ct$ the full subcategory of compact objects in $\ct$. If $\ct$ is the homotopy category of spectra, Adams proved the following in [1]: All homological functors ${\{\ct^c\}}^{op}\rightarrow\ab$ are the restrictions of representable functors on $\ct$, and all natural transformations are the restrictions of morphisms in $\ct$. It has been something of a mystery, to what extent this generalises to other triangulated categories. In [35], it was proved that Adams' theorem remains true as long as $\ct^c$ is countable, but can fail in general. The failure exhibited was that there can be natural transformations not arising from maps in $\ct$. A puzzling open problem remained: Is every homological functor the restriction of a representable functor on $\ct$? In a recent paper, Beligiannis [5] made some progress. In this article, we show that the answer is no. There are rings $R$ so that when $\ct$ is the derived category of $R$, there are homological functors ${\{\ct^c\}}^{op}\rightarrow\ab$ which are not restrictions of representables.

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