##  [Formal Smoothness](/formal-smoothness) 

  ##  [Formal Smoothness](https://algebra.quantumdictionary.io/formal-smoothness-0) 

  

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**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

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.