FAILURE OF BROWN REPRESENTABILITY IN DERIVED CATEGORIES
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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