Transfinite Adams representability

We consider the following problems in a well generated triangulated category T. Let α be a regular cardinal and Tα⊂T the full subcategory of α-compact objects. Is every functor H:(Tα)op→Ab that preserves products of <α objects and takes exact triangles to exact sequences of the form H≅T(−,X)|Tα for some X in T? Is every natural transformation τ:T(−,X)|Tα→T(−,Y)|Tα of the form τ=T(−,f)|Tα for some f:X→Y in T? If the answer to both questions is positive we say that T satisfies α-Adams representability. A classical result going back to Brown and Adams shows that the stable homotopy category satisfies ℵ0-Adams representability. The case α=ℵ0 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 α-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 α-Adams representability for all α≥ℵ0 and rings which do not satisfy α-Adams representability for any α≥ℵ0. Moreover, we exhibit rings for which the answer to both questions is no for all ℵω>α≥ℵ2.