In abstract algebra, a quasi-free algebra is an associative algebra that satisfies the lifting property similar to that of a formally smooth algebra in commutative algebra. The notion was introduced by Cuntz and Quillen for the applications to cyclic homology. A quasi-free algebra generalizes a free algebra, as well as the coordinate ring of a smooth affine complex curve. Because of the latter generalization, a quasi-free algebra can be thought of as signifying smoothness on a noncommutative space.
Let A be an associative algebra over the complex numbers. Then A is said to be quasi-free if the following equivalent conditions are met:
R\toR/I
A\toR/I
A\toR
Let
(\OmegaA,d)
\Omega1A
There is also a characterization in terms of a connection. Given an A-bimodule E, a right connection on E is a linear map
\nablar:E\toE ⊗ A\Omega1A
\nablar(as)=a\nablar(s)
\nablar(sa)=\nablar(s)a+s ⊗ da
\Omega1A
One of basic properties of a quasi-free algebra is that the algebra is left and right hereditary (i.e., a submodule of a projective left or right module is projective or equivalently the left or right global dimension is at most one). This puts a strong restriction for algebras to be quasi-free. For example, a hereditary (commutative) integral domain is precisely a Dedekind domain. In particular, a polynomial ring over a field is quasi-free if and only if the number of variables is at most one.
An analog of the tubular neighborhood theorem, called the formal tubular neighborhood theorem, holds for quasi-free algebras.