In algebraic geometry, given a morphism of schemes
, the diagonal morphism is a morphism determined by the universal property of the fiber product
It is a special case of a graph morphism: given a morphism
The diagonal embedding is the graph morphism of
By definition, X is a separated scheme over S (
is a separated morphism) if the diagonal morphism is a closed immersion.
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
Then, identifying X with the set of its k-rational points,
has its diagonal as a closed subscheme — in other words, the diagonal morphism is a closed immersion.
As a consequence, a scheme
within the scheme product of
Emphasizing the relative point of view, one might equivalently define a scheme to be separated if the unique morphism
Notice that a topological space Y is Hausdorff iff the diagonal embedding 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)
, 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): Let
be a scheme obtained by identifying two affine lines through the identity map except at the origins (see gluing scheme#Examples).
[1] Indeed, the image of the diagonal morphism
A classic way to define the intersection product of algebraic cycles
on a smooth variety X is by intersecting (restricting) their cartesian product with (to) the diagonal: precisely, where
is the pullback along the diagonal embedding
This algebraic geometry–related article is a stub.