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