Diagonal embedding

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

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 embedding 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

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 embedding.

Use in intersection theory

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

References

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