Smooth morphism of rings

Summary

A smooth morphism of rings is a map $R \to A$ of commutative rings that makes $A$ a finitely presented $R$-algebra and is formally smooth. It is the algebraic counterpart of a submersion of manifolds: the fibres are nonsingular and vary nicely. A smooth algebra is flat over the base, and its module of Kähler differentials is finitely generated and projective. Over a field, smooth algebras are the coordinate rings of nonsingular varieties, in a sense that is stable under extension of the field.