(Separated)
A morphism \(f : X \to Y\) is separated if the diagonal \[ \Delta_{X/Y} : X \to \fiberprod{X}{Y}{X} \] is a closed immersion. It is quasi-separated if \(\Delta_{X/Y}\) is quasicompact, equivalently if for every affine open \(V \subseteq Y\) the scheme \(f^{-1}(V)\) is quasi-separated in the sense below.
(Quasicompact and quasi-separated)
A scheme \(X\) is quasicompact if every open cover of \(X\) has a finite subcover, and a morphism \(f: X \to Y\) is quasicompact if \(f^{-1}(V)\) is quasicompact for every affine open \(V \subseteq Y\). A scheme \(X\) is quasi-separated if the intersection of any two quasicompact open subsets of \(X\) is quasicompact, equivalently if the intersection of any two affine open subsets is quasicompact; this is quasi-separatedness of \(X \to \Spec \ZZ\).
A topological space \(X\) is Hausdorff if and only if the diagonal is closed in \(X\times X\) with the product topology. An irreducible variety of positive dimension is not Hausdorff in the Zariski topology, since any two nonempty open subsets meet; and the underlying space of \(\fiberprod{X}{Y}{X}\) is in general not the product space. Separatedness asks that the diagonal be closed in the fibre product of schemes: \(\AA^1_k\) is separated over \(k\), while the line \(X\) with a doubled origin is not, since the closure of the diagonal in \(X\times_kX\) contains the point \((0_1,0_2)\) for the two origins \(0_1\ne0_2\).
Every separated morphism is quasi-separated, and every morphism from a Noetherian scheme is quasi-separated. Gluing two copies of \(\Spec k[x_1,x_2,\ldots]\) along the complement of the origin gives a scheme that is not quasi-separated: the intersection of the two affine copies is not quasicompact.
Dependencies
D-AN662Schemes, and morphisms of schemesuses
Backlinks
D-8XX95Proper morphisms and the valuative criteriausesPR-MORCANCELThe cancellation theorem for properties of morphismsusesPR-QX0VLTwo morphisms agreeing on a dense open agreeuses