Quasi-free algebra

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.^[1] 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.^[2]

Definition

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:^[3][4][5]

• Given a square-zero extension ${\displaystyle R\to R/I}$, each homomorphism ${\displaystyle A\to R/I}$ lifts to ${\displaystyle A\to R}$.
• The cohomological dimension of A with respect to Hochschild cohomology is at most one.

Let ${\displaystyle (\Omega A,d)}$ denotes the differential envelope of A; i.e., the universal differential-graded algebra generated by A.^[6][7] Then A is quasi-free if and only if ${\displaystyle \Omega ^{1}A}$ is projective as a bimodule over A.^[3]

There is also a characterization in terms of a connection. Given an A-bimodule E, a right connection on E is a linear map

${\displaystyle \nabla _{r}:E\to E\otimes _{A}\Omega ^{1}A}$

that satisfies ${\displaystyle \nabla _{r}(as)=a\nabla _{r}(s)}$ and ${\displaystyle \nabla _{r}(sa)=\nabla _{r}(s)a+s\otimes da}$.^[8] A left connection is defined in the similar way. Then A is quasi-free if and only if ${\displaystyle \Omega ^{1}A}$ admits a right connection.^[9]

Properties and examples

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).^[10] 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.^[11]