The dual of Brown representability for some derived categories

TL;DR

This paper proves that for certain complete abelian categories with an injective cogenerator, their derived categories satisfy Brown representability, extending known results to new classes like quasi-coherent sheaves and algebraic stacks.

Contribution

It establishes the dual of Brown representability for derived categories of specific abelian categories, including AB4*-n categories and categories of sheaves over schemes and stacks.

Findings

Derived categories of these abelian categories satisfy Brown representability.

Includes categories of quasi-coherent sheaves over schemes.

Applies to complexes over algebraic stacks.

Abstract

Consider a complete abelian category which has an injective cogenerator. If its derived category is left--complete we show that the dual of this derived category satisfies Brown representability. In particular this is true for the derived category of an abelian AB$4^*$-$n$ category, for the derived category of quasi--coherent sheaves over a nice enough scheme (including the projective finitely dimensional space) and for the full subcategory of derived category of all sheaves over an algebraic stack consisting from complexes with quasi--coherent cohomology.