In algebraic geometry and commutative algebra, a ring homomorphism 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 , any A-algebra homomorphism may be lifted to an A-algebra map . If moreover any such lifting is unique, then f is said to be formally étale.[1][2]
Formally smooth maps were defined by Alexander Grothendieck in Éléments de géométrie algébrique IV.
For finitely presented morphisms, formal smoothness is equivalent to usual notion of smoothness.
^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.
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