Algebra extensions and nonsingularity

Abstract

This paper is concerned with a notion of nonsingularity for noncommutative algebras, which arises naturally in connection with cyclic homology. Let us consider associative unital algebras over the complex numbers. We call an algebra A quasi-free, when it behaves like a free algebra with respect to nilpotent extensions in the sense that any homomorphism A -+ R/I, where I is a nilpotent ideal in R, can be lifted to a homomorphism A -+ R. If we restrict to the category of finitely generated commutative algebras, then this lifting property characterizes smooth algebras, the ones corresponding to nonsingular affine varieties. In this way quasi-free algebras appear as noncommutative analogues of smooth algebras. Stretching the analogy, we can even regard quasi-free algebras as analogues of manifolds. One of the aims of this paper is to develop the analogy further by showing that quasi-free algebras provide a natural setting for noncommutative versions of certain aspects of manifolds. To give an example, let us consider the analogue of an embedding: an extension A = R/I, where A and R are quasi-free algebras playing the role of the submanifold and ambient manifold respectively. In the manifold situation, I/I2 is the module of linear functions on the nor2 mal bundle, and the symmetric algebra SA(III ) is the algebra of polynomial functions. Now in passing from commutative to noncommutative algebras, the symmetric algebra of a module is replaced by the tensor algebra of a bimodule.