A smooth morphism of schemes is a locally finitely presented morphism that is also a flat
morphism and whose fibers are geometrically
regular schemes. Smooth morphisms are the algebraic-geometric analog of smooth
maps between smooth manifolds.
A algebraic variety over a field is a smooth variety exactly when its structure morphism
to the ring spectrum of the field is smooth. Open immersions are smooth morphisms, and smoothness
is preserved by composition of morphisms
and base change.
See also
Base Change,
Flat Morphism,
Geometrically Regular Scheme,
Locally Finitely Presented Morphism,
Morphism,
Open Immersion,
Smooth Variety,
Structure
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, quatrième partie." Publ. Math.
IHES 32, 5-361, 1967. https://doi.org/10.1007/BF02732123.The
Stacks Project Authors. "Smooth Morphisms." §29.35 in The Stacks
Project, Tag 01V4, 2026. https://stacks.math.columbia.edu/tag/01V4.
Cite this as:
Weisstein, Eric W. "Smooth Morphism."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SmoothMorphism.html
Subject classifications
