  Let $k$ be a field of characteristic zero. By using Hironaka's
desingularisation theorem, we prove an extension criterion for a functor
defined on nonsingular k-schemes and taking values on a category of complexes.
Roughly speaking, the criterion shows that if such a functor satisfies the
standard exact sequence of a blowing-up, then the functor can be extended to
all separated k-schemes of finite type. The result is applied to the
Grothendieck's theory of motives, to the Hodge-De Rham filtered complex of an
analytic space, and to the rational homotopy of k-schemes in algebraic De Rham
theory.
