nLab
quasi-separated morphism

Let $f:X\to Y$ be a morphism of schemes. Let ${\Delta }_f:X\to X{\times }_{Y}X$ 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 $Xo\mathrm{Spec}Z$ is quasi-separated, i.e. $\Delta :X\to X\times X$ is quasicompact. Every quasi-separated scheme is semiseparated.

• MathOverflow why-does-finitely-presented-imply-quasi-separated
• Daniel Murfet, Concentrated schemes, pdf, an expositional digest from EGA