In algebraic geometry, given a

morphism of schemes In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes. A morphism of algebraic stacks generalizes a ...

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 In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is often w ...

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 In algebraic geometry, a closed immersion of schemes is a morphism of schemes f: Z \to X that identifies ''Z'' as a closed subset of ''X'' such that locally, regular functions on ''Z'' can be extended to ''X''. The latter condition can be formaliz ...

. Also, a morphism

p: X \to S

locally of finite presentation is an

unramified morphism In algebraic geometry, an unramified morphism is a morphism f: X \to Y of schemes such that (a) it is locally of finite presentation and (b) for each x \in X and y = f(x), we have that # The residue field k(x) is a separable algebraic extension of ...

if and only if the diagonal embedding is an open immersion.

Explanation

As an example, consider an

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...

over an

algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...

''k'' and

p: X \to \operatorname(k)

the structure map. Then, identifying ''X'' with the set of its ''k''-rational points,

X \times_k 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

f

with itself along

f

has its

diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...

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 (\mathbb)

is separated. Notice that a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

''Y'' is Hausdorff iff the diagonal embedding :

Y \stackrel 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_ 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_ 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 In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theore ...

algebraic cycles In mathematics, an algebraic cycle on an algebraic variety ''V'' is a formal linear combination of subvarieties of ''V''. These are the part of the algebraic topology of ''V'' that is directly accessible by algebraic methods. Understanding the alg ...

A, B

on a

smooth variety In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smo ...

''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

* {{algebraic-geometry-stub Algebraic geometry

Explanation

Separated morphism

Use in intersection theory

See also

References