PUB - Publikationen an der Universität Bielefeld

Coherent functors S --> Ab from a compactly generated triangulated category into the category of abelian groups are studied. This is inspired by Auslander's classical analysis of coherent functors from a fixed abelian category into abelian groups. We characterize coherent functors and their filtered colimits in various ways. In addition, we investigate certain subcategories of S which arise from families of coherent functors. Let S be a compactly generated triangulated category, for example the stable homotopy category of CW-spectra. We call a functor F : S --> Ab into the category of abelian groups coherent if there exists an exact sequence Hom(D,-) --> Hom(C,-) --> F --> 0 such that C and D are compact objects in S (an object X in S is compact if the representable functor Hom(X, -) preserves arbitrary coproducts). The concept of a coherent functor has been introduced explicitly for abelian categories by Auslander [2], but it is also implicit in the work [9] of Freyd on stable homotopy. In this paper we characterize coherent functors in a number of ways and use them to study a wider class of fuhctors S --> Ab which share a weak exactness property. Another purpose of this paper is to investigate certain subcategories of S which are defined in terms of coherent functors. In the category Mod Lambda of modules over an associative ring Lambda, the analogue of a compact object is a finitely presented module. This fact can be made precise (cf. the Appendix), and one has in this context the following classical result: a functor F : Mod Lambda --> Ab is coherent precisely if F preserves products and filtered colimits. There is no obvious way to formulate such a characterization for compactly generated triangulated categories because filtered colimits rarely exist in triangulated categories. Nevertheless, we are able to characterize the coherent functors as follows.