Rouquier's cocovering theorem and well-generated triangulated categories

Abstract

We study cocoverings of triangulated categories, in the sense of Rouquier, and prove that for any regular cardinal $\alpha$ the condition of $\alpha$-compactness, in the sense of Neeman, is local with respect to such cocoverings. This was established for ordinary compactness by Rouquier. Our result yields a new technique for proving that a given triangulated category is well-generated. As an application we describe the $\alpha$-compact objects in the unbounded derived category of a quasi-compact semi-separated scheme.