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 etale.

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

References

• Grothendieck, Alexandre; Dieudonné, Jean (1964). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie". Publications Mathématiques de l'IHÉS. 20: 5–259. doi:10.1007/bf02684747. MR 0173675.
• Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS. 32: 5–361. doi:10.1007/bf02732123. MR 0238860.