Diagonal morphism (algebraic geometry)
# Diagonal morphism (algebraic geometry)

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

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

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

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

By definition, X is a separated scheme over S ($p:X\to S$ is a separated morphism) if the diagonal morphism is a closed immersion. Also, a morphism $p:X\to S$ 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 $p:X\to \operatorname {Spec} (k)$ the structure map. Then, identifying X with the set of its k-rational points, $X\times _{k}X=\{(x,y)\in X\times X\}$ and $\delta :X\to X\times _{k}X$ is given as $x\mapsto (x,x)$; whence the name diagonal morphism.

## Separated morphism

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

As a consequence, a scheme $X$ is separated when the diagonal of $X$ within the scheme product of $X$ 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 $X\rightarrow {\textrm {Spec}}(\mathbb {Z} )$ is separated.

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

    $Y{\stackrel {\Delta }{\longrightarrow }}Y\times Y,\,y\mapsto (y,y)$

is closed. In algebraic geometry, the above formulation is used because a scheme which is a Hausdorff space is necessarily empty or zero-dimensional. The difference between the topological and algebro-geometric context comes from the topological structure of the fiber product (in the category of schemes) $X\times _{{\textrm {Spec}}(\mathbb {Z} )}X$, which is different from the product of topological spaces.

Any affine scheme Spec A is separated, because the diagonal corresponds to the surjective map of rings (hence is a closed immersion of schemes):

    $A\otimes _{\mathbb {Z} }A\rightarrow A,a\otimes a'\mapsto a\cdot a'$.

Let $S$ be a scheme obtained by identifying two affine lines through the identity map except at the origins (see gluing scheme#Examples). It is not separated. Indeed, the image of the diagonal morphism $S\to S\times S$ image has two origins, while its closure contains four origins.

## Use in intersection theory

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

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

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

