Bernhard Keller

June 9, 1997

To appear in Journal of Pure and Applied Algebra

Abstract. The cyclic homology of an exact category was defined by R. McCarthy using the methods of F. Waldhausen. McCarthy's theory enjoys a number of desirable properties, the most basic being the extension property, i.e. the fact that when applied to the category of finitely generated projective modules over an algebra it specializes to the cyclic homology of the algebra.

However, we show that McCarthy's theory cannot be both, compatible with localizations and invariant under functors inducing equivalences in the derived category.

This is our motivation for introducing a new theory for which all three properties hold: extension, invariance and localization. Thanks to these properties, the new theory can be computed explicitly for a number of categories of modules and sheaves.

This work goes back to a question by P. Polo. I thank him for the interest he has continued to take in the subject. I am grateful to the referee for his thorough reading of the manuscript. His remarks and questions have been a great help and encouragment in preparing the final version of the article.

Bernhard Keller, le 9 juin, 1997.

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