Coherent functors in stable homotopy theory

Krause H (2002)
Fundamenta Mathematicae 173(1): 33-56.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
Abstract / Bemerkung
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.
Erscheinungsjahr
Zeitschriftentitel
Fundamenta Mathematicae
Band
173
Ausgabe
1
Seite(n)
33-56
ISSN
PUB-ID

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Krause H. Coherent functors in stable homotopy theory. Fundamenta Mathematicae. 2002;173(1):33-56.
Krause, H. (2002). Coherent functors in stable homotopy theory. Fundamenta Mathematicae, 173(1), 33-56. doi:10.4064/fm173-1-3
Krause, H. (2002). Coherent functors in stable homotopy theory. Fundamenta Mathematicae 173, 33-56.
Krause, H., 2002. Coherent functors in stable homotopy theory. Fundamenta Mathematicae, 173(1), p 33-56.
H. Krause, “Coherent functors in stable homotopy theory”, Fundamenta Mathematicae, vol. 173, 2002, pp. 33-56.
Krause, H.: Coherent functors in stable homotopy theory. Fundamenta Mathematicae. 173, 33-56 (2002).
Krause, Henning. “Coherent functors in stable homotopy theory”. Fundamenta Mathematicae 173.1 (2002): 33-56.