Bivariant Periodic Cyclic Homology
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Bivariant Periodic Cyclic Homology

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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 Kunneth 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.show more

Product details

  • Paperback | 120 pages
  • 152 x 230 x 10mm | 181.44g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 2003.
  • 1584880104
  • 9781584880103

Table of contents

Introduction A First Guide The List Pro-k-modules and Homological Machinery Pro-Categories Countable Pro-Modules Projective and Injective Resolutions Resolutions of Pro-Vecor Spaces Resolutions of Exact Sequences The Ext-Functors, the UCT, and the GMS Connecting Homomorphisms in Pro-Homology A Universal Coefficient Theorem The Cuntz-Quillen Theory and the Universal Coefficient Theorem Some Examples Cyclic Pro-Homology Virtues of Cyclic Pro-Homology Pro-Homological Version of Goodwillies Theorem Excision in Hochschild Pro-Homolgy Excision in Cyclic Pro-Homology Excision in the Cuntz-Quillen Theory Kunneth Type Formulas Mixed Complexes From k[u]-Comodules to Pro-k-Modules The Cotensor Product The Kunneth Isomorphism in Pro-Homology Kunneth Isomorphisms for the Periodic Theory Cyclic Homology of some Affine Algebras Calculations Additional Remarksshow more