Quasi-free algebra

Short description: Associative algebra with lifting property

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.[1] 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.[2]

## Contents

* 1 Definition
* 2 Properties and examples
* 3 References
* 3.1 Bibliography
* 4 Further reading

## 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:[3][4][5]

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

Let (ΩA,d) denotes the differential envelope of A; i.e., the universal differential-graded algebra generated by A.[6][7] Then A is quasi-free if and only if Ω1A is projective as a bimodule over A.[3]

There is also a characterization in terms of a connection. Given an A-bimodule E, a right connection on E is a linear map

∇r:E->E⊗AΩ1A

that satisfies ∇r(as)=a∇r(s) and ∇r(sa)=∇r(s)a+s⊗da.[8] A left connection is defined in the similar way. Then A is quasi-free if and only if Ω1A 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]

## References

1. ↑ Cuntz & Quillen 1995
2. ↑ Cuntz 2013, Introduction
3. ↑ 3.0 3.1 Cuntz & Quillen 1995, Proposition 3.3.
4. ↑ Vale 2009, Proposotion 7.7.
5. ↑ Kontsevich & Rosenberg 2000, 1.1.
6. ↑ Cuntz & Quillen 1995, Proposition 1.1.
7. ↑ Kontsevich & Rosenberg 2000, 1.1.2.
8. ↑ Vale 2009, Definition 8.4.
9. ↑ Vale 2009, Remark 7.12.
10. ↑ Cuntz & Quillen 1995, Proposition 5.1.
11. ↑ Cuntz & Quillen 1995, § 6.

### Bibliography

* Cuntz, Joachim (June 2013). "Quillen's work on the foundations of cyclic cohomology" (in en). Journal of K-Theory 11 (3): 559–574. doi:10.1017/is012011006jkt201. ISSN 1865-2433. https://www.cambridge.org/core/journals/journal-of-k-theory/article/abs/quillens-work-on-the-foundations-of-cyclic-cohomology/2377B657C264A17AED11DCF85C5B40A7.
* 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" (in en). The Gelfand Mathematical Seminars, 1996–1999. Birkhäuser. pp. 85–108. doi:10.1007/978-1-4612-1340-6_5. ISBN 978-1-4612-7102-4. https://link.springer.com/chapter/10.1007/978-1-4612-1340-6_5.
* Maxim Kontsevich, Alexander Rosenberg, Noncommutative spaces, preprint MPI-2004-35
* Vale, R. (2009). "notes on quasi-free algebras". https://pi.math.cornell.edu/~rvale/ada.pdf.

## Further reading

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

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