Formal Smoothness

- Pure Mathematics -
Algebra Dictionary
Formal smoothness (for an algebra or a morphism of algebras) is the infinitesimal lifting property: given any algebra B and a nilpotent ideal I ⊂ B, every algebra homomorphism A → B/I lifts to a homomorphism A → B. In commutative algebra this coincides with geometric formal smoothness and is related to the projectivity of the module of differentials for finitely presented algebras.