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Abstract

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 étale 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 étale 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).