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 .[8] A left connection is defined in the similar way. Then A is quasi-free if and only if 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]