### Relative Chern characters for nilpotent ideals, by Guillermo Cortiñas and Chuck Weibel

[updated file received 17 Sep 2008]

If I is a nilpotent ideal in a Q-algebra A, Goodwillie defined two
isomorphisms from K_{*}(A,I) to negative cyclic homology,
HN_{*}(A,I).
One is the relative version of the absolute Chern character, and the
other is defined using rational homotopy theory. The question of
whether they agree was implicit in Goodwillie's 1986 Annals paper.

In this paper, we show that the two isomorphisms agree. Here are
three applications.

Cathelineau proved that the rational homotopy character is compatible
with the λ-filtration. It follows that the relative Chern character
is also compatible with this filtration for nilpotent ideals.
This agreement, together with Cathelineau's result, was used
by the authors and Haesemeyer to show that the absolute Chern character,
from K(A) to HN(A), is compatible with the λ-filtration
for every commutative Q-algebra. This is the main result of
Infinitesimal cohomology...
This agreement can be used to strengthen Ginot's results in
"Formules explicites pour le charactere de Chern...",
*Ann. Inst. Fourier* 54 (2004).

Guillermo Cortiñas <gcorti@dm.uba.ar>

Chuck Weibel <weibel@math.rutgers.edu>