  We show that the unbounded derived category of a Grothendieck category with
enough projective objects is the base category of a derivator whose category of
diagrams is the full 2-category of small categories. With this structure, we
give a description of the localization functor associated to a specialization
closed subset of the spectrum of a commutative noetherian ring. In addition,
using the derivator of modules, we prove some basic theorems of group
cohomology for complexes of representations over an arbitrary base ring.
