separated scheme

A scheme $X$ is defined to be a separated scheme if the morphism

$$d:X\to X{\times }_{\mathrm{Spec}\mathbb{Z}}X$$

into the fibre product $X{\times }_{\mathrm{Spec}\mathbb{Z}}X$ which is induced by the identity maps $i:X⟶X$ in each coordinate is a closed immersion.

Note the similarity to the definition of a Hausdorff topological space. In the situation of topological spaces, a space $X$ is Hausdorff if and only if the diagonal morphism $X⟶X\times X$ is a closed embedding of topological spaces. The definition of a separated scheme is very similar, except that the topological product is replaced with the scheme fibre product.

More generally, if $X$ is a scheme over a base scheme $Y$, the scheme $X$ is defined to be separated over $Y$ if the diagonal embedding

$$d:X\to X{\times }_{Y}X$$

is a closed immersion.