Quasi-free algebra explained

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. 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.

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:

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. Then A is quasi-free if and only if \Omega ^{1} A is projective as a bimodule over A.

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 ⊗ _{A} \Omega ^{1} A that satisfies \nabla _{r(as)} = a \nabla _{r(s)} and \nabla _{r(sa)} = \nabla _{r(s)} a + s ⊗ da . 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.

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). 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.

References

Bibliography

• Cuntz . Joachim . Quillen's work on the foundations of cyclic cohomology . Journal of K-Theory . June 2013 . 11 . 3 . 559–574 . 10.1017/is012011006jkt201 . en . 1865-2433. 1202.5958 .
• Cuntz . Joachim . Quillen . Daniel . Algebra Extensions and Nonsingularity . Journal of the American Mathematical Society . 1995 . 8 . 2 . 251–289 . 10.2307/2152819 . 0894-0347. free .
• Kontsevich . Maxim . Rosenberg . Alexander L. . Noncommutative Smooth Spaces . The Gelfand Mathematical Seminars, 1996–1999 . 2000 . 85–108 . 10.1007/978-1-4612-1340-6_5 . Birkhäuser . en. math/9812158 .
• Maxim Kontsevich, Alexander Rosenberg, Noncommutative spaces, preprint MPI-2004-35
• Web site: notes on quasi-free algebras . Vale . R. . 2009 .

Further reading

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