Quasi-free algebra

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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]

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

  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

Further reading