separated scheme

A scheme $X$ is defined to be a separated scheme if the morphism $$ d: X \to X \times_{\Spec\Z} X $$ into the fibre product $X \times_{\Spec\Z} X$ which is induced by the identity maps $i: X \lra X$ 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 $X \lra X \times X$ 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.