# separated scheme

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

 $d:X\to X\times_{\operatorname{Spec}\mathbb{Z}}X$

into the fibre product $X\times_{\operatorname{Spec}\mathbb{Z}}X$ which is induced by the identity maps $i:X\longrightarrow X$ 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 $X\longrightarrow X\times X$ 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.

Title separated scheme SeparatedScheme 2013-03-22 12:50:25 2013-03-22 12:50:25 djao (24) djao (24) 6 djao (24) Definition msc 14A15 separated