http://ncatlab.org/nlab/show/Dennis+trace
For trace methods related to K-theory and Hochschild homology, Dennis trace map etc, see Madsen ICM talk.
http://front.math.ucdavis.edu/1101.1883 Grothendieck’s trace map for arithmetic surfaces via residues and higher adeles from arXiv Front: math.AG by Matthew Morrow We establish the reciprocity law along a vertical curve for residues of differential forms on arithmetic surfaces, and describe Grothendieck’s trace map of the surface as a sum of residues.
nLab page on Trace