Rouquier's cocovering theorem and well-generated triangulated categories

TL;DR

This paper explores Rouquier's cocoverings in triangulated categories, demonstrating that $ ext{α}$-compactness is local with respect to these cocoverings, and introduces a new method for establishing well-generatedness, with applications to derived categories of schemes.

Contribution

It extends Rouquier's results from compactness to $ ext{α}$-compactness, providing a novel technique for proving well-generatedness of triangulated categories.

Findings

$ ext{α}$-compactness is local with respect to Rouquier's cocoverings.

A new technique for proving well-generatedness of triangulated categories.

Description of $ ext{α}$-compact objects in derived categories of schemes.

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.