On the cyclic homology of exact categories Original Research Article

The cyclic homology of an exact category was defined by McCarthy (1994) using the methods of Waldhausen (1985). McCarthyʹs theory enjoys a number of desirable properties, the most basic being the agreement 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 McCarthyʹs 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.