Diagonal morphism (algebraic geometry)
In algebraic geometry, the diagonal morphism is a fundamental morphism associated to any scheme or algebraic space. It provides a way to encode information about the properties of the space, such as separateness and Hausdorffness in the classical setting.
Given a scheme (or algebraic space) X, the diagonal morphism is the morphism Δ : X → X × X, where the product is understood to be the fiber product X ×Spec(k) X, for some base field k. More generally, for a morphism f: X → Y, the diagonal morphism is the morphism Δ : X → X ×Y X such that the compositions pr1 ∘ Δ = idX and pr2 ∘ Δ = idX, where pr1 and pr2 are the two projection maps X ×Y X → X.
The diagonal morphism sends a point x in X to the point (x, x) in X ×Y X. It can be thought of as the inclusion of the 'diagonal' of X ×Y X.
Properties and Importance:
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Separatedness: A morphism f: X → Y is separated if the diagonal morphism Δ : X → X ×Y X is a closed immersion. This condition is analogous to the Hausdorff property in topology. A scheme X is separated if the morphism X → Spec(k) is separated.
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Universally closedness: The diagonal morphism plays a role in defining proper morphisms. A morphism f: X → Y is proper if it is separated, of finite type, and universally closed.
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Faithfully flat descent: The diagonal morphism is crucial in the theory of faithfully flat descent. It's used to check whether a module or scheme defined on an etale cover descends to the original space.
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Schematic image: The image of the diagonal morphism is the set of points of the form (x, x). The closure of this image in the Zariski topology is often studied.
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Monoidal Category Structure: The diagonal morphism, along with the projection to the base scheme, defines a monoidal category structure on the category of schemes (or algebraic spaces) over a given base.
The properties of the diagonal morphism reveal much about the geometric structure of the underlying scheme or algebraic space. The closedness, or lack thereof, of the diagonal is an important invariant used to distinguish well-behaved spaces from those with more complicated features.