A flat morphism of schemes 
is a morphism for which
the induced map of local rings 
makes 
a flat module over 
for every 
.
Flatness is preserved by composition and base change. Every smooth morphism is flat.
Flat Morphism
See also
Base Change, Flat Module, Local Ring, Morphism, Scheme, Smooth MorphismExplore with Wolfram|Alpha
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