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Separated Morphism

A separated morphism is a morphism f:X->S of schemes whose diagonal morphism Delta_(X/S):X->X×_SX is a closed immersion.

Separatedness is the algebraic-geometric analog of the T2-space condition. It is preserved by composition of morphisms and base change. Every proper morphism is separated.

See also

Base Change, Closed Immersion, Diagonal Morphism, Fiber Product, Morphism, Proper Morphism, T2-Space

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References

Grothendieck, A. and Dieudonné, J. "Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes." Publ. Math. IHES 8, 5-222, 1961. https://doi.org/10.1007/BF02699291.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. "Separated Morphism." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SeparatedMorphism.html

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