Cohomological descent theory for a morphism of stacks and for equivariant derived categories

In the paper, we find necessary and sufficient conditions under which, if $X\to S$ is a morphism of algebraic varieties (or, in a more general case, of stacks), the derived category of $S$ can be recovered by using the tools of descent theory from the derived category of $X$. We show that for an action of a linearly reductive algebraic group $G$ on a scheme $X$ this result implies the equivalence of the derived category of $G$-equivariant sheaves on $X$ and the category of objects in the derived category of sheaves on $X$ with a given action of $G$ on each object.
Bibliography: 18 titles.