Quasi-free algebra

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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

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]

Let 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 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

that satisfies and . [8] A left connection is defined in the similar way. Then A is quasi-free if and only if 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

  1. Cuntz & Quillen 1995
  2. Cuntz 2013 , Introduction
  3. 1 2 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