Kristine Bauer
Department of Mathematics
Johns Hopkins University
3400 N. Charles St.
Baltimore, MD 21218 USA
kbbauer@math.jhu.edu
Randy McCarthy
Department of Mathematics
University of Illinois
1409 W. Green St.
Urbana, IL 61801 USA
randy@math.uiuc.edu
On vanishing Tate cohomology and decompositions in Goodwillie calculus
Mathematical Subject Classification: 55P65 (55P45, 13D03)
Our main result is that if F is a functor from a pointed category
C to spectra, the Goodwillie tower of F evaluated at X splits
rationally when X is a co-H-object of C. We show that the layers
of F(X) in this case are easy to identify. The splitting of the
Goodwillie tower gives a decomposition of F(X) into a product of its
layers. We use this to recover the rational decompositions of
Hochschild and higher Hochschild homology by Pirashvili, Loday,and
Gerstenhaber-Schack. Finally, we extend the main theorem to include
dual calculus to recover the Poincar\'e-Birkhoff-Witt theorem, and
improve the theorem in the special case in which the comultiplication
map is cocommutative.