Quasi-free algebra

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]

## Definition

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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 {\displaystyle R\to R/I} , each homomorphism  A -> R / I {\displaystyle A\to R/I} lifts to  A -> R {\displaystyle A\to R} .
* The cohomological dimension of A with respect to Hochschild cohomology is at most one.

Let  ( Ω A , d ) {\displaystyle (\Omega A,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  Ω 1 A {\displaystyle \Omega ^{1}A} 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 ⊗ A Ω 1 A {\displaystyle \nabla _{r}:E\to E\otimes _{A}\Omega ^{1}A}

that satisfies  ∇ r ( a s ) = a ∇ r ( s ) {\displaystyle \nabla _{r}(as)=a\nabla _{r}(s)} and  ∇ r ( s a ) = ∇ r ( s ) a + s ⊗ d a {\displaystyle \nabla _{r}(sa)=\nabla _{r}(s)a+s\otimes da} .[8] A left connection is defined in the similar way. Then A is quasi-free if and only if  Ω 1 A {\displaystyle \Omega ^{1}A} admits a right connection.[9]

## Properties and examples

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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

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1. ^ Cuntz & Quillen 1995
2. ^ Cuntz 2013, Introduction
3. ^ a b 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

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* Cuntz, Joachim (June 2013). "Quillen's work on the foundations of cyclic cohomology". Journal of K-Theory. 11 (3): 559–574\. arXiv:1202.5958. doi:10.1017/is012011006jkt201. ISSN 1865-2433.559-574&rft.date=2013-06&rft_id=info:arxiv/1202.5958&rft.issn=1865-2433&rft_id=info:doi/10.1017/is012011006jkt201&rft.aulast=Cuntz&rft.aufirst=Joachim&rft_id=https://www.cambridge.org/core/journals/journal-of-k-theory/article/abs/quillens-work-on-the-foundations-of-cyclic-cohomology/2377B657C264A17AED11DCF85C5B40A7&rfr_id=info:sid/en.wikipedia.org:Quasi-free+algebra" class="Z3988">
* Cuntz, Joachim; Quillen, Daniel (1995). "Algebra Extensions and Nonsingularity". Journal of the American Mathematical Society. 8 (2): 251–289\. doi:10.2307/2152819. ISSN 0894-0347.251-289&rft.date=1995&rft_id=info:doi/10.2307/2152819&rft.issn=0894-0347&rft.aulast=Cuntz&rft.aufirst=Joachim&rft.au=Quillen,+Daniel&rft_id=https://www.jstor.org/stable/2152819&rfr_id=info:sid/en.wikipedia.org:Quasi-free+algebra" class="Z3988">
* Kontsevich, Maxim; Rosenberg, Alexander L. (2000). "Noncommutative Smooth Spaces". The Gelfand Mathematical Seminars, 1996–1999. Birkhäuser: 85–108\. arXiv:math/9812158. doi:10.1007/978-1-4612-1340-6_5.85-108&rft.date=2000&rft_id=info:arxiv/math/9812158&rft_id=info:doi/10.1007/978-1-4612-1340-6_5&rft.aulast=Kontsevich&rft.aufirst=Maxim&rft.au=Rosenberg,+Alexander+L.&rft_id=https://link.springer.com/chapter/10.1007/978-1-4612-1340-6_5&rfr_id=info:sid/en.wikipedia.org:Quasi-free+algebra" class="Z3988">
* Maxim Kontsevich, Alexander Rosenberg, Noncommutative spaces, preprint MPI-2004-35
* Vale, R. (2009). "notes on quasi-free algebras" (PDF).

## Further reading

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

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