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$4^{\ast }$-$n$ category and for the derived category of quasi-coherent sheaves over a nice enough scheme, including the projective finitely dimensional space.