# nLab quasi-separated morphism

Let $f:X\to Y$ be a morphism of schemes. Let ${\Delta }_{f}:X\to X{×}_{Y}X$ be the diagonal map. We say that $f$ is quasi-separated if ${\Delta }_{f}$ is a quasicompact morphism. Every separated morphism of schemes is quasi-separated; every monomorphism of schemes is separated hence also quasi-separated.

A scheme $X$ is quasi-separated if the morphism $Xo\mathrm{Spec}\phantom{\rule{thinmathspace}{0ex}}Z$ is quasi-separated, i.e. $\Delta :X\to X×X$ is quasicompact. Every quasi-separated scheme is semiseparated.