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.