Bivariant Periodic Cyclic Homology

Recent work by Cuntz and Quillen on bivariant periodic cyclic homology has caused quite a revolution in the subject. In this self-contained exposition, the author's purpose is to understand the functorial properties of the Cuntz-Quillen theory, which motivaties his explorations of what he calls cyclic pro-homology.

Simply stated, the cyclic pro-homology of an (associative) algebra A is short for the Z/2 Z-graded inverse system of cyclic homology groups of A, considered as a pro-vector space. The author finds that this functor, taking algebras over a field k of characteristic zero into the category of pro-k-vector spaces, is remarkable. He presents a proof that it is excisive and that it satisfies a Kuenneth isomorphism for the tensor product of algebras. He explains the relation to the Cuntz-Quillen groups in a Universal Coefficient Theorem and in a Milnor lim1-sequence. This enables the lifting - to some extent- of the nice properties of cyclic pro-homology properties to the Cuntz Quillen theory itself. It is interesting to note that for the excision result, this lifting procedure goes through without constraints.

For those new to cyclic homology, Dr. Gronbaek takes care to provide an introduction to the subject, including the motivation for the theory, definitions, and fundamental results, and establishes the homological machinery needed for application to the Cuntz-Quillen theory. Mathematicians interested in cyclic homology-especially ring theorists using homological methods-will find this work original, enlightening, and thought-provoking. The author leaves the door open for deeper study into excision for the Cuntz-Quillen theory for a class of topological algebras, such as the category of m-algebras considered by Cuntz.