  We introduce the notion of a separator for a morphism of schemes f:T\to S; in
particular, it is universal among morphisms from T to separated S-schemes. A
separator is a local isomorphism; this property conveys the intuition of gluing
some affine covering more, in order to make the scheme separated. When f is
quasi-separated, its separator exists if and only if the schematic closure of
the diagonal projects on both factors by flat morphisms of finite type. In
particular, f admits a separator if T is Noetherian Dedekind and S=Spec(Z), or
if f is \'etale of finite presentation and S is normal. Any normal scheme of
finite type over a Noetherian ring admits an open subset containing all the
points of codimension 1, which has a separator. A contrario, we give several
examples of morphisms f that do not admit a separator.
  As an application, we attach to every smooth scheme T over a normal base S a
morphism to a separated \'etale S-scheme of finite presentation, which is
universal (a kind of separated alternative for "scheme of connected components
of the fibres"). This simultaneously generalizes the classical case where the
base is a field, and the case of a smooth and proper morphism (Stein
factorisation).
