Quasi-separated morphism
= Quasi-separated morphism
{title2=$\Delta_{X/S}\text{ quasi-compact}$}
A <morphism of schemes> $f:X\to S$ is quasi-separated if its <diagonal morphism> $X\to X\times_SX$ is a <quasi-compact morphism>. Every <separated morphism> has this property since a <closed immersion> is a <quasi-compact morphism>. This compactness condition is needed when uniqueness in the <valuative criterion> is used to deduce <separatedness> for arbitrary <schemes>.