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Failure of Brown representability in derived categories | | J. Daniel Christensen
; Bernhard Keller
; Amnon Neeman
; | | Date: |
11 Dec 1999 | | Subject: | Algebraic Topology; K-Theory and Homology | math.AT math.KT | | Abstract: | Let T be a triangulated category with coproducts, C the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [Adams71]: All contravariant homological functors C --> Ab 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 [Neeman97], it was proved that Adams’ theorem remains true as long as 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 contravariant homological functors C --> Ab which are not restrictions of representables. | | Source: | arXiv, math.AT/0001056 | | Services: | Forum | Review | PDF | Favorites | |
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