Diagonal embedding
In algebraic geometry, given a morphism of schemes [math]\displaystyle{ p: X \to S }[/math], the diagonal embedding
- [math]\displaystyle{ \delta: X \to X \times_S X }[/math]
is a morphism determined by the universal property of the fiber product [math]\displaystyle{ X \times_S X }[/math] of p and p applied to the identity [math]\displaystyle{ 1_X : X \to X }[/math] and the identity [math]\displaystyle{ 1_X }[/math].
It is a special case of a graph morphism: given a morphism [math]\displaystyle{ f: X \to Y }[/math] over S, the graph morphism of it is [math]\displaystyle{ X \to X \times_S Y }[/math] induced by [math]\displaystyle{ f }[/math] and the identity [math]\displaystyle{ 1_X }[/math]. The diagonal embedding is the graph morphism of [math]\displaystyle{ 1_X }[/math].
By definition, X is a separated scheme over S ([math]\displaystyle{ p: X \to S }[/math] is a separated morphism) if the diagonal embedding is a closed immersion. Also, a morphism [math]\displaystyle{ p: X \to S }[/math] 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 [math]\displaystyle{ p: X \to \operatorname{Spec}(k) }[/math] the structure map. Then, identifying X with the set of its k-rational points, [math]\displaystyle{ X \times_k X = \{ (x, y) \in X \times X \} }[/math] and [math]\displaystyle{ \delta: X \to X \times_k X }[/math] is given as [math]\displaystyle{ x \mapsto (x, x) }[/math]; whence the name diagonal embedding.
Use in intersection theory
A classic way to define the intersection product of algebraic cycles [math]\displaystyle{ A, B }[/math] on a smooth variety X is by intersecting (restricting) their cartesian product with (to) the diagonal: precisely,
- [math]\displaystyle{ A \cdot B = \delta^*(A \times B) }[/math]
where [math]\displaystyle{ \delta^* }[/math] is the pullback along the diagonal embedding [math]\displaystyle{ \delta: X \to X \times X }[/math].
See also
References
- Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9
 |
