About: Diagonal morphism (algebraic geometry)

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In algebraic geometry, given a morphism of schemes , the diagonal morphism is a morphism determined by the universal property of the fiber product of p and p applied to the identity and the identity . It is a special case of a graph morphism: given a morphism over S, the graph morphism of it is induced by and the identity . The diagonal embedding is the graph morphism of .

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  • In algebraic geometry, given a morphism of schemes , the diagonal morphism is a morphism determined by the universal property of the fiber product of p and p applied to the identity and the identity . It is a special case of a graph morphism: given a morphism over S, the graph morphism of it is induced by and the identity . The diagonal embedding is the graph morphism of . By definition, X is a separated scheme over S ( is a separated morphism) if the diagonal morphism is a closed immersion. Also, a morphism locally of finite presentation is an unramified morphism if and only if the diagonal embedding is an open immersion. (en)
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  • In algebraic geometry, given a morphism of schemes , the diagonal morphism is a morphism determined by the universal property of the fiber product of p and p applied to the identity and the identity . It is a special case of a graph morphism: given a morphism over S, the graph morphism of it is induced by and the identity . The diagonal embedding is the graph morphism of . (en)
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  • Diagonal morphism (algebraic geometry) (en)
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