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.[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 [math]\displaystyle{ R \to R/I }[/math], each homomorphism [math]\displaystyle{ A \to R/I }[/math] lifts to [math]\displaystyle{ A \to R }[/math].
- The cohomological dimension of A with respect to Hochschild cohomology is at most one.
Let [math]\displaystyle{ (\Omega A, d) }[/math] 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 [math]\displaystyle{ \Omega^1 A }[/math] 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
- [math]\displaystyle{ \nabla_r : E \to E \otimes_A \Omega^1 A }[/math]
that satisfies [math]\displaystyle{ \nabla_r(as) = a \nabla_r(s) }[/math] and [math]\displaystyle{ \nabla_r(sa) = \nabla_r(s) a + s \otimes da }[/math].[8] A left connection is defined in the similar way. Then A is quasi-free if and only if [math]\displaystyle{ \Omega^1 A }[/math] 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
- ↑ Cuntz & Quillen 1995
- ↑ Cuntz 2013, Introduction
- ↑ 3.0 3.1 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" (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. https://www.jstor.org/stable/2152819.
- Kontsevich, Maxim; Rosenberg, Alexander L. (2000). "Noncommutative Smooth Spaces" (in en). The Gelfand Mathematical Seminars, 1996–1999 (Birkhäuser): 85–108. doi:10.1007/978-1-4612-1340-6_5. 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
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