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, 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.

First version: preliminary version, comments are welcome! Second version: An (important) error was removed, some details were added. Comments are also welcome! Third version: a new application is added, some typos are removed. Fourth version: final version, some important corrections are made, to appear in Arkiv Math

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