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

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.

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 morphism.

A separated morphism is a morphism ${\displaystyle f}$ such that the fiber product of ${\displaystyle f}$ with itself along ${\displaystyle f}$ has its diagonal as a closed subscheme — in other words, the diagonal morphism is a closed immersion.

As a consequence, a scheme ${\displaystyle X}$ is separated when the diagonal of ${\displaystyle X}$ within the scheme product of ${\displaystyle X}$ with itself is a closed immersion. Emphasizing the relative point of view, one might equivalently define a scheme to be separated if the unique morphism ${\displaystyle X\rightarrow {\textrm {Spec}}(\mathbb {Z} )}$ is separated.

Notice that a topological space Y is Hausdorff iff the diagonal embedding