Formally smooth map

In algebraic geometry and commutative algebra, a ring homomorphism $f:A\to B$ is called formally smooth (from French: Formellement lisse) if it satisfies the following infinitesimal lifting property:

Suppose B is given the structure of an A-algebra via the map f. Given a commutative A-algebra, C, and a nilpotent ideal $N\subseteq C$, any A-algebra homomorphism $B\to C/N$ may be lifted to an A-algebra map $B\to C$. If moreover any such lifting is unique, then f is said to be formally étale.

Formally smooth maps were defined by Alexander Grothendieck in Éléments de géométrie algébrique IV. Among other things, Grothendieck proved that any such map is flat.

For finitely presented morphisms, formal smoothness is equivalent to usual notion of smoothness.