On the Cyclic Homology of Exact Categories

The cyclic homology of an exact category was defined by R. McCarthy [17] using the methods of F. Waldhausen [26]. 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.