  To a big n-tilting object in a complete, cocomplete abelian category A with
an injective cogenerator we assign a big n-cotilting object in a complete,
cocomplete abelian category B with a projective generator, and vice versa. Then
we construct an equivalence between the (conventional or absolute) derived
categories of A and B. Under various assumptions on A, which cover a wide range
of examples (for instance, if A is a module category or, more generally, a
locally finitely presentable Grothendieck abelian category), we show that B is
the abelian category of contramodules over a topological ring and that the
derived equivalences are realized by a contramodule-valued variant of the usual
derived Hom-functor.
