# Diagonal embedding

In algebraic geometry, given a morphism of schemes ${\displaystyle p:X\to S}$, the diagonal embedding

${\displaystyle \delta :X\to X\times _{S}X}$

is a morphism determined by the universal property of the fiber product ${\displaystyle X\times _{S}X}$ of p and p applied to the identity ${\displaystyle 1_{X}:X\to X}$ and the identity ${\displaystyle 1_{X}}$.

It is a special case of a graph morphism: given a morphism ${\displaystyle f:X\to Y}$ over S, the graph morphism of it is ${\displaystyle X\to X\times _{S}Y}$ induced by ${\displaystyle f}$ and the identity ${\displaystyle 1_{X}}$. The diagonal embedding is the graph morphism of ${\displaystyle 1_{X}}$.

By definition, X is a separated scheme over S (${\displaystyle p:X\to S}$ is a separated morphism) if the diagonal embedding is a closed immersion. Also, a morphism ${\displaystyle p:X\to S}$ locally of finite presentation is an unramified morphism if and only if the diagonal embedding ${\displaystyle \delta :X\to X\times _{S}X}$ is an open immersion.

## Explanation

As an example, consider an algebraic variety over an algebraically closed field k and ${\displaystyle p:X\to \operatorname {Spec} (k)}$ the structure map. Then, identifying X with the set of its k-rational points, ${\displaystyle X\times _{k}X=\{(x,y)\in X\times X\}}$ and ${\displaystyle \delta :X\to X\times _{k}X}$ is given as ${\displaystyle x\mapsto (x,x)}$; whence the name diagonal embedding.

## Use in intersection theory

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

${\displaystyle A\cdot B=\delta ^{*}(A\times B)}$

where ${\displaystyle \delta ^{*}}$ is the pullback along the diagonal embedding ${\displaystyle \delta :X\to X\times X}$.