An associative unital -algebra is quasi-free (or formally smooth) if one of the following equivalent conditions is satisfied
Given an extension of algebras where the ideal is nilpotent and an algebra map. Then there exists a homomorphism such that .
has cohomological dimension with respect to Hochschild cohomology;
is a projective -bimodule;
the universal Hochschild 2-cocycle , is a coboundary, i.e. for some satisfying the cocycle condition ;
there exists a “right connection” i.e. a -linear map satisfying and where and .
This is due to (CuntzQuillen).
For an associative algebra, the object is formally smooth with respect to the standard infinitesimal cohesive structure over non-commutative algebras (see there for details) precisely if it is quasi-free.
Notice that the characterization via nilpotent extensions is similar to the definition of commutative formally smooth algebras as in EGAIV4 17.1.1. However most commutative formally smooth algebras are not formally smooth in the associative noncommutative sense.