Infinitesimal cohomology and the Chern character to negative cyclic homology, by Guillermo Cortiñas, Christian Haesemeyer, and Charles Weibel

There is a Chern character from K-theory to negative cyclic homology. We show that it preserves the decomposition coming from Adams operations, at least in characteristic zero. Several special cases of this result have been addressed in the literature; the case of nilpotent ideals was established by Cathelineau.

Our method is to use infinitesimal cohomology to reduce the problem to the nilpotent ideal case. The first part of the paper deals with the infinitesimal cohomology of chain complexes of sheaves, comparing the infinitesimal cohomology of the negative cyclic homology complex with that of the structure sheaf.

The second part introduces sheaf cohomology spectra for the infinitesimal topology, both as the global sections of a fibrant replacement and as a homotopy limit. Applying this construction to K-theory reduces the problem to a space-level version of Cathelineau's Theorem.

There are three appendices which provide the required space-level version of Cathelineau's Theorem. Much of this material is known to experts but does not seem to be in the literature.


Guillermo Cortiñas <gcorti@agt.uva.es>
Christian Haesemeyer <chh@math.uiuc.edu>
Charles Weibel <weibel@math.rutgers.edu>