The diagonal morphism of a morphism 
of schemes is the unique
morphism 
whose compositions with both projection morphisms are the identity on 
.
The morphism 
is a separated morphism exactly when 
is a closed immersion.
Diagonal Morphism
See also
Closed Immersion, Fiber Product, Morphism, Separated MorphismExplore with Wolfram|Alpha
References
Grothendieck, A. and Dieudonné, J. "Éléments de géométrie algébrique. I. Le langage des schémas." Publ. Math. IHES 4, 5-228, 1960. https://doi.org/10.1007/BF02684778.The Stacks Project Authors. "Separation Axioms." §26.21 in The Stacks Project, Tag 01KH, 2026. https://stacks.math.columbia.edu/tag/01KH.Cite this as:
Weisstein, Eric W. "Diagonal Morphism." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DiagonalMorphism.html