Quasi-free algebra

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 R \to R/I, each homomorphism A \to R/I lifts to A \to R. • The cohomological dimension of A with respect to Hochschild cohomology is at most one. Let (\Omega A, d) 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 \Omega^1 A is projective as a bimodule over A. A left connection is defined in the similar way. Then A is quasi-free if and only if \Omega^1 A 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 (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. == References ==