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Flat Morphism

A flat morphism of schemes f:X->S is a morphism for which the induced map of local rings O_(S,f(x))->O_(X,x) makes O_(X,x) a flat module over O_(S,f(x)) for every x in X.

The composition of morphisms that are flat is flat, and every base change of a flat morphism is flat. Every smooth morphism is flat.

See also

Base Change, Flat Module, Local Ring, Morphism, Scheme, Smooth Morphism

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References

Grothendieck, A. and Dieudonné, J. "Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas, seconde partie." Publ. Math. IHES 24, 5-231, 1965. https://doi.org/10.1007/BF02684322.The Stacks Project Authors. "Flat Morphisms." §29.26 in The Stacks Project, Tag 01U2, 2026. https://stacks.math.columbia.edu/tag/01U2.

Cite this as:

Weisstein, Eric W. "Flat Morphism." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FlatMorphism.html

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