Diagonal morphism (algebraic geometry)

In algebraic geometry, given a morphism of schemes , the diagonal morphism

is a morphism determined by the universal property of the fiber product of p and p applied to the identity and the identity .

It is a special case of a graph morphism: given a morphism over S, the graph morphism of it is induced by and the identity . The diagonal embedding is the graph morphism of .

By definition, X is a separated scheme over S ( is a separated morphism) if the diagonal morphism is a closed immersion. Also, a morphism locally of finite presentation is an unramified morphism if and only if the diagonal embedding is an open immersion.

Explanation

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As an example, consider an algebraic variety over an algebraically closed field k and   the structure map. Then, identifying X with the set of its k-rational points,   and   is given as  ; whence the name diagonal morphism.

Separated morphism

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A separated morphism is a morphism   such that the fiber product of   with itself along   has its diagonal as a closed subscheme — in other words, the diagonal morphism is a closed immersion.

As a consequence, a scheme   is separated when the diagonal of   within the scheme product of   with itself is a closed immersion. Emphasizing the relative point of view, one might equivalently define a scheme to be separated if the unique morphism   is separated.

Notice that a topological space Y is Hausdorff iff the diagonal embedding

 

is closed. In algebraic geometry, the above formulation is used because a scheme which is a Hausdorff space is necessarily empty or zero-dimensional. The difference between the topological and algebro-geometric context comes from the topological structure of the fiber product (in the category of schemes)  , which is different from the product of topological spaces.

Any affine scheme Spec A is separated, because the diagonal corresponds to the surjective map of rings (hence is a closed immersion of schemes):

 .

Let   be a scheme obtained by identifying two affine lines through the identity map except at the origins (see gluing scheme#Examples). It is not separated.[1] Indeed, the image of the diagonal morphism   image has two origins, while its closure contains four origins.

Use in intersection theory

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A classic way to define the intersection product of algebraic cycles   on a smooth variety X is by intersecting (restricting) their cartesian product with (to) the diagonal: precisely,

 

where   is the pullback along the diagonal embedding  .

See also

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References

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  1. ^ Hartshorne 1977, Example 4.0.1.
  • Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157