Diagonal morphism (algebraic geometry)

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

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

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 morphism 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 is an open immersion.