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 homologys. 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 can be thought of as signifying smoothness on a noncommutative space.

Definition

Let R be an associative algebra over the complex numbers. Then R 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 R with respect to Hochschild cohomology is at most one.

References

• Cuntz, Joachim (June 2013). "Quillen's work on the foundations of cyclic cohomology". Journal of K-Theory. 11 (3): 559–574. doi:10.1017/is012011006jkt201. ISSN 1865-2433.
• Cuntz, Joachim; Quillen, Daniel (1995). "Algebra Extensions and Nonsingularity". Journal of the American Mathematical Society. 8 (2): 251–289. doi:10.2307/2152819. ISSN 0894-0347.
• Kontsevich, Maxim; Rosenberg, Alexander L. (2000). "Noncommutative Smooth Spaces". The Gelfand Mathematical Seminars, 1996–1999. Birkhäuser: 85–108. doi:10.1007/978-1-4612-1340-6_5.
• Vale, R. (2009). "notes on quasi-free algebras" (PDF).

Further reading

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