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FAILURE OF BROWN REPRESENTABILITY IN DERIVED CATEGORIES

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September 12, 1999

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Submitted

Let T be a triangulated category with coproducts, T^{c} 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
T^{c} 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 T^{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 homological
functors on D(R) which are not restrictions of representables.

http://www.math.jussieu.fr/~keller/publ/fbrabs.html

Bernhard Keller, le 12 Septembre, 1999.

keller@math.jussieu.fr

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