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. 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.

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:

Given a square-zero extension, each homomorphism lifts to.
The cohomological dimension of A with respect to Hochschild cohomology is at most one.

Let denotes the differential envelope of A; i.e., the universal differential-graded algebra generated by A. Then A is quasi-free if and only if is projective as a bimodule over A.
There is also a characterization in terms of a connection. Given an A-bimodule E, a right connection on E is a linear map
that satisfies and. A left connection is defined in the similar way. Then A is quasi-free if and only if admits a right connection.

Properties and examples

One of basic properties of a quasi-free algebra is that the algebra is left and right hereditary. This puts a strong restriction for algebras to be quasi-free. For example, a hereditary 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.