Separated morphism

A morphism $f:X\to Y$ is separated when its diagonal $X\to X{\times }_{Y}X$ is a closed immersion. This is the scheme-theoretic analogue of the Hausdorff property.

For a finite-type morphism of Noetherian schemes, separatedness is equivalent to uniqueness in every lifting problem $\mathrm{Spec}K\to X$ over $\mathrm{Spec}R\to Y$, where $R$ is a valuation ring with fraction field $K$.