Diagonal embedding

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

\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 embedding is a closed immersion. Also, a morphism $p:X\to S$ is an unramified morphism if and only if the diagonal embedding $\delta :X\to X\times _{S}X$ is an open immersion.

Explanation

As an example, consider an algebraic variety over an algebraically closed field k and $p:X\to \operatorname {Spec} (k)$ 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 embedding.

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 $\delta :X\to X\times X$ .

References

Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157

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Diagonal embedding

Explanation

Use in intersection theory

See also

References