This entry is about smooth morphisms of schemes. There are many notions of smoothness in algebra and algebraic geometry, many under the name of (such and such) regularity (nonsingularity); even in EGA there are something like 11 notions, one of which is called the smooth morphism of schemes. In the literature there are sometimes even small variations of the latter (e.g. whether we allow globally varying dimension of the smooth morphism or not). In Lab a prominent role is played by the formal smoothness, which is weaker than smoothness.
Smooth morphism is a relativization of the notion of a smooth scheme.
it is flat
and finitely presented (cf. relativization in algebraic geometry)
Smoothness of a morphism is a higher dimensional analogue of the notion of a morphism being étale (which is a smooth morphism of relative dimension ), but stronger than the notion of formal smoothness.
For a morphism of schemes, and a point of , the following are equivalent
(i) is a smooth morphism at
(ii) is locally of finite presentation at and there is an open neighborhood of such that is formally smooth
(iii) is flat at , locally of finite presentation at and the sheaf of Kähler differentials is locally free in a neighborhood of
The relative dimension of at will equal the rank of the module of Kähler differentials.
This is (EGAIV 17.5.2 and 17.15.15)
See also formally smooth morphism.