7.8 KB · ⇊ 5 parts · ⤓ Micron 7.8 KB · ⤓ HTML 42.3 KB

top

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

δ δ : X → → X × × S X {\displaystyle \delta :X\to X\times _{S}X}

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

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

By definition, X is a separated scheme over S ( p : X → → S {\displaystyle p:X\to S} is a separated morphism) if the diagonal morphism is a closed immersion. Also, a morphism p : X → → S {\displaystyle p:X\to S} locally of finite presentation is an unramified morphism if and only if the diagonal embedding is an open immersion.

• Explanation

• Separated morphism

• Use in intersection theory

• See also

• References

As an example, consider an algebraic variety over an algebraically closed field k and p : X → → Spec ⁡ ⁡ ( k ) {\displaystyle p:X\to \operatorname {Spec} (k)} the structure map. Then, identifying X with the set of its k-rational points, X × × k X = { ( x , y ) ∈ ∈ X × × X } {\displaystyle X\times _{k}X=\{(x,y)\in X\times X\}} and δ δ : X → → X × × k X {\displaystyle \delta :X\to X\times _{k}X} is given as x ↦ ↦ ( x , x ) {\displaystyle x\mapsto (x,x)} ; whence the name diagonal morphism.

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

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

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

Y ⟶ ⟶ Δ Δ Y × × Y , y ↦ ↦ ( y , y ) {\displaystyle 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 × × Spec ( Z ) X {\displaystyle 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 ⊗ ⊗ Z A → → A , a ⊗ ⊗ a ′ ↦ ↦ a ⋅ ⋅ a ′ {\displaystyle A\otimes _{\mathbb {Z} }A\rightarrow A,a\otimes a'\mapsto a\cdot a'} .

Let S {\displaystyle 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.cite-ref-1[1] Indeed, the image of the diagonal morphism S → → S × × S {\displaystyle S\to S\times S} image has two origins, while its closure contains four origins.

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

A ⋅ ⋅ B = δ δ ∗ ∗ ( A × × B ) {\displaystyle A\cdot B=\delta ^{*}(A\times B)}

where δ δ ∗ ∗ {\displaystyle \delta ^{*}} is the pullback along the diagonal embedding δ δ : X → → X × × X {\displaystyle \delta :X\to X\times X} .

• regular embedding

• Diagonal morphism

cite-note-11. ↑ Hartshorne 1977, Example 4.0.1.

• citerefhartshorne1977Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157