separated scheme
SeparatedScheme
2002-07-14 08:17:41
2004-04-15 03:33:38
Definition
separated
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\newcommand{\p}{{\mathfrak{p}}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\renewcommand{\H}{\mathcal{H}}
\newcommand{\A}{\mathbb{A}}
\renewcommand{\c}{\mathcal{C}}
\renewcommand{\O}{\mathcal{O}}
\newcommand{\D}{\mathcal{D}}
\newcommand{\st}{\mid}
\newcommand{\lra}{\longrightarrow}
\newcommand{\res}{\operatorname{res}}
\newcommand{\id}{\operatorname{id}}
\newcommand{\diff}{\operatorname{diff}}
\newcommand{\incl}{\operatorname{incl}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Spec}{\operatorname{Spec}}
A scheme $X$ is defined to be a {\em 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 \emph{separated} over $Y$ if the diagonal embedding
$$
d: X \to X \times_{Y} X
$$
is a closed immersion.