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 $R\to R/I$, 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 $(\mathrm{\Omega }A,d)$ 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 ${\mathrm{\Omega }}^{1}A$ 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

${\mathrm{\nabla }}_r:E\to E{\otimes }_A{\mathrm{\Omega }}^{1}A$

that satisfies ${\mathrm{\nabla }}_r(as)=a{\mathrm{\nabla }}_r(s)$ and ${\mathrm{\nabla }}_r(sa)={\mathrm{\nabla }}_r(s)a+s\otimes da$. A left connection is defined in the similar way. Then A is quasi-free if and only if ${\mathrm{\Omega }}^{1}A$ 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

Bibliography

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• Maxim Kontsevich, Alexander Rosenberg, Noncommutative spaces, preprint MPI-2004-35
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Further reading

• https://ncatlab.org/nlab/show/quasi-free+algebra

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