Zusammenfassung
We consider the following problems in a well generated triangulated category T. Let alpha be a regular cardinal and T-alpha subset of T. the fall subcategory of alpha-compact objects. Is every functor H: (T-alpha)(oP) -> Ab that preserves products of < alpha objects and takes exact triangles to exact sequences of the form H congruent to T(-, X)(vertical bar T alpha) for some X in T? Is every ...
Zusammenfassung
We consider the following problems in a well generated triangulated category T. Let alpha be a regular cardinal and T-alpha subset of T. the fall subcategory of alpha-compact objects. Is every functor H: (T-alpha)(oP) -> Ab that preserves products of < alpha objects and takes exact triangles to exact sequences of the form H congruent to T(-, X)(vertical bar T alpha) for some X in T? Is every natural transformation tau : T(-,X)(vertical bar T alpha) -> T(-,Y)(vertical bar T alpha) of the form tau = T(-,f)(vertical bar T alpha) for some f : X -> Y in T? If the answer to both questions is positive we say that T satisfies a-Adams representability. A classical result going back to Brown and Adams shows that the stable homotopy category satisfies N-0-Adams representability. The case alpha = N-0 is well understood thanks to the work of Christensen, Keller, and Neeman. In this paper we develop an obstruction theory to decide whether T satisfies alpha-Adams representability. We derive necessary and sufficient conditions of homological nature, and we compute several examples. In particular, we show that there are rings satisfying alpha-Adams representability for all alpha >= N-0 and rings which do not satisfy alpha-Adams representability for any a >= No. Moreover, we exhibit rings for which the answer to both questions is no for all N-w > alpha >= N-2. (C) 2016 Elsevier Inc. All rights reserved.
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