The dual of Brown representability for some derived categories

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.