Quasi-free algebra

Last updated March 26, 2023

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

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 $R\to R/I$ , 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.^[6]^[7] Then A is quasi-free if and only if $\Omega ^{1}A$ 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

\nabla _{r}:E\to E\otimes _{A}\Omega ^{1}A

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

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References

↑ Cuntz & Quillen 1995
↑ Cuntz 2013 , Introduction
1 2 Cuntz & Quillen 1995 , Proposition 3.3.
↑ Vale 2009 , Proposotion 7.7.
↑ Kontsevich & Rosenberg 2000 , 1.1.
↑ Cuntz & Quillen 1995 , Proposition 1.1.
↑ Kontsevich & Rosenberg 2000 , 1.1.2.
↑ Vale 2009 , Definition 8.4.
↑ Vale 2009 , Remark 7.12.
↑ Cuntz & Quillen 1995 , Proposition 5.1.
↑ Cuntz & Quillen 1995 , § 6.

Bibliography

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.
Maxim Kontsevich, Alexander Rosenberg, Noncommutative spaces, preprint MPI-2004-35
Vale, R. (2009). "notes on quasi-free algebras" (PDF).