In abstract algebra, a quasi-free algebra is an

associative algebra In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...

that satisfies the lifting property similar to that of a formally smooth algebra in

commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...

. The notion was introduced by Cuntz and Quillen for the applications to cyclic homology. A quasi-free algebra generalizes a

free algebra In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the ...

, as well as the

coordinate ring In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space. More formally, an affine algebraic set is the set of the common zeros over an algeb ...

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: *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''. Then ''A'' is quasi-free if and only if

\Omega^1 A

is projective as a

bimodule In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, i ...

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

. 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 Heredity, also called inheritance or biological inheritance, is the passing on of traits from parents to their offspring; either through asexual reproduction or sexual reproduction, the offspring cells or organisms acquire the genetic inform ...

(i.e., a

submodule In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since t ...

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 In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibilit ...

is precisely a

Dedekind domain In mathematics, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily un ...

. In particular, a

polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another 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

* * * * Maxim Kontsevich, Alexander Rosenberg
Noncommutative spaces
preprint MPI-2004-35 *

Definition

Properties and examples

References

Bibliography

Further reading