Transfinite Adams representability

Abstract

We consider the following problems in a well generated triangulated category T . Let α be a regular cardinal and T α ⊂ T the full subcategory of α-compact objects. Is every functor H : (T α) op → Ab that preserves products of < α objects and takes exact triangles to exact sequences of the form H ∼= T (−, X)|T α for some X in T ? Is every natural transformation τ : T (−, X)|T α → T (−, Y )|T α of the form τ = T (−, f)|T α 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.