Failure of Brown representability in derived categories

By J. Christensen, B. Keller, A. Neeman. 2000

51 citations recorded by the source corpus.

Abstract Let T be a triangulated category with coproducts, T c ⊂ T the full subcategory of compact objects in T . If T is the homotopy category of spectra, Adams (Topology 10 (1971) 185–198), proved the following: All homological functors { T c } op → A b 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 Neeman (Topology 36 (1997) 619–645), it was 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 (Relative homological and purity in triangulated categories, 1999, preprint) 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 { T c } op → A b which are not restrictions of representables.

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Citations in the corpus

Realizability of modules over Tate cohomology (2003, background): “The question of which C-modules are realizable is precisely the question of to what extent Brown representability for homology theories holds; see the paper of Christensen, Keller, and Neeman [15] for a recent discussion.”
Cohen-Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras (2005, background): “On the other hand there exist derived equ Artin algebrasΛ andΓ such thatΛ is representation finite but Γ is not, see [31].”
Generalized Brown representability in homotopy categories (2005, unknown): “The property that every weakly continuous functor H : Ho(K) → Set is representable is called the Brown representability for cohomology, while the property that every left covering functor H : Ho(Kω) → Set is representable is called the Brown representability for homology, see [10].”
Homology and cohomology of E∞ ring spectra (2004, methodology): “A characterization of the category of homology theories on CR/B is slightly trickier because of the failure of Brown’s Representability Theorem for homology theories [ 3 ].”