# 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