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.

A morphism of schemes is separated when its diagonal is a closed immersion. This is a property of the morphism, distinct from the broader morphism or diagonal constructions.

A scheme over a base $S$ is separated when its structure morphism to $S$ is a separated morphism. An absolute scheme is usually called separated when it is separated over $\mathrm{Spec}\mathbb{Z}$.

An algebraic variety that is separated over its ground field. Two affine open charts have an affine intersection: their intersection is the inverse image of the closed diagonal inside the product of the two affine charts. This ensures a finite affine cover is acyclic for quasi-coherent sheaves.

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 $K$.

This is the uniqueness part of the valuative criterion.