## ON THE CYCLIC HOMOLOGY OF EXACT CATEGORIES

### 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.

http://www.math.jussieu.fr/~keller/publ/cyexabs.html

Bernhard Keller, le 9 juin, 1997.

keller@math.jussieu.fr

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