On compact generation of deformed schemes

TL;DR

This paper presents a new theorem that facilitates proving the compact generation of derived categories for schemes and their non-commutative deformations, extending existing results through localization and cocovering techniques.

Contribution

It introduces a theorem leveraging Rouquier's cocovering in triangulated categories to establish compact generation, applicable to both classical and non-commutative deformed schemes.

Findings

Proves compact generation for derived categories of schemes using localization coverings.

Extends Neeman's result to non-commutative deformations of schemes.

Utilizes Chevalley-Eilenberg complexes in the deformation context.

Abstract

We obtain a theorem which allows to prove compact generation of derived categories of Grothendieck categories, based upon certain coverings by localizations. This theorem follows from an application of Rouquier's cocovering theorem in the triangulated context, and it implies Neeman's result on compact generation of quasi-compact separated schemes. We prove an application of our theorem to non-commutative deformations of such schemes, based upon a change from Koszul complexes to Chevalley-Eilenberg complexes.