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.{{harvnb|Cuntz|Quillen|1995}} 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.{{harvnb|Cuntz|2013|loc=Introduction}}

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:{{harvnb|Cuntz|Quillen|1995|loc=Proposition 3.3.}}{{harvnb|Vale|2009|loc=Proposotion 7.7.}}{{harvnb|Kontsevich|Rosenberg|2000|loc=1.1.}}

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.{{harvnb|Cuntz|Quillen|1995|loc=Proposition 1.1.}}{{harvnb|Kontsevich|Rosenberg|2000|loc=1.1.2.}} 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

: $\nabla_r : E \to E \otimes_A \Omega^1 A$

that satisfies $\nabla_r(as) = a \nabla_r(s)$ and $\nabla_r(sa) = \nabla_r(s) a + s \otimes da$ .{{harvnb|Vale|2009|loc=Definition 8.4.}} 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.{{harvnb|Vale|2009|loc=Remark 7.12.}}

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).{{harvnb|Cuntz|Quillen|1995|loc=Proposition 5.1.}} 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.{{harvnb|Cuntz|Quillen|1995|loc=§ 6.}}

References

= Bibliography =

{{cite journal |last1=Cuntz |first1=Joachim |title=Quillen's work on the foundations of cyclic cohomology |journal=Journal of K-Theory |date=June 2013 |volume=11 |issue=3 |pages=559–574 |doi=10.1017/is012011006jkt201 |url=https://www.cambridge.org/core/journals/journal-of-k-theory/article/abs/quillens-work-on-the-foundations-of-cyclic-cohomology/2377B657C264A17AED11DCF85C5B40A7 |language=en |issn=1865-2433|arxiv=1202.5958 }}
{{cite journal |last1=Cuntz |first1=Joachim |last2=Quillen |first2=Daniel |title=Algebra Extensions and Nonsingularity |journal=Journal of the American Mathematical Society |date=1995 |volume=8 |issue=2 |pages=251–289 |doi=10.2307/2152819 |url=https://www.jstor.org/stable/2152819 |issn=0894-0347|doi-access=free }}
{{cite journal |last1=Kontsevich |first1=Maxim |last2=Rosenberg |first2=Alexander L. |title=Noncommutative Smooth Spaces |journal=The Gelfand Mathematical Seminars, 1996–1999 |date=2000 |pages=85–108 |doi=10.1007/978-1-4612-1340-6_5 |url=https://link.springer.com/chapter/10.1007/978-1-4612-1340-6_5 |publisher=Birkhäuser |language=en|arxiv=math/9812158 }}
Maxim Kontsevich, Alexander Rosenberg, [https://ncatlab.org/nlab/files/KontsevichRosenbergNCSpaces.pdf Noncommutative spaces], preprint MPI-2004-35
{{cite web |title=notes on quasi-free algebras |last=Vale |first=R. |date=2009 |url=https://pi.math.cornell.edu/~rvale/ada.pdf}}

quasi-free algebra

Definition

Properties and examples

References

= Bibliography =

Further reading