Diagonal morphism (algebraic geometry)

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

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