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separated scheme (Definition)

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

$\displaystyle 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

$\displaystyle d: X \to X \times_{Y} X $
is a closed immersion.



"separated scheme" is owned by djao.
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Cross-references: diagonal embedding, base, product, similar, embedding, closed, diagonal, Hausdorff, topological spaces, Hausdorff topological space, similarity, closed immersion, coordinate, identity maps, induced, fibre product, morphism, scheme
There are 16 references to this entry.

This is version 3 of separated scheme, born on 2002-07-14, modified 2004-04-15.
Object id is 3166, canonical name is SeparatedScheme.
Accessed 7730 times total.

Classification:
AMS MSC: 14A15 (Algebraic geometry :: Foundations :: Schemes and morphisms)
Pending Errata and Addenda
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