Transfinite Adams representability

Abstract: In a well generated triangulated category T, given a regular cardinal a, we consider the following problems: given a functor from the category of a-compact objects to abelian groups that preserves products of <a objects and takes exact triangles to exact sequences, is it the restriction of a representable functor in T? Is every natural transformation between two such restricted representable functors induced by a map between the representatives? 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 Adams representability for the first infinite cardinal. For that cardinal, Adams representability is well understood thanks to the work of Christensen, Keller and Neeman. In this paper, we develop an obstruction theory to decide when T satisfies a-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 a-Adams representability for all non-countable cardinals a and rings which do not satisfy a-Adams representability for any infinite cardinal a. Moreover, we exhibit rings for which the answer to both questions is no for infinite many cardinals. As a side result, we give an example of an infinite phantom map.