Title: Tate cohomology and periodic localization of polynomial functors
Author: Nicholas J. Kuhn
AMS classification numbers: Primary 55P65; Secondary 55N22, 55P60, 55P91
Address: Department of Mathematics, University of Virginia,
Charlottesville, VA 22903
email: njk4x@virginia.edu
abstract:
In this paper, we show that Goodwillie calculus, as applied to functors
from stable homotopy to itself, interacts in striking ways with
chromatic aspects of the stable category. Localized at a fixed prime p,
let T(n) be the telescope of a v_n self map of a finite S--module of
type n. The Periodicity Theorem of Hopkins and Smith implies that the
Bousfield localization functor associated to T(n) is independent of
choices. Goodwillie's general theory says that to any homotopy functor
F from S--modules to S--modules, there is an associated tower under F,
{P_dF}, such that F --> P_dF is the universal arrow to a d--excisive
functor. Our first theorem says that P_dF --> P_{d-1}F always admits a
homotopy section after localization with respect to T(n) (and so also
after localization with respect to Morava K--theory K(n)). Thus, after
periodic localization, polynomial functors split as the product of their
homogeneous factors. This theorem follows from our second theorem which
is equivalentto the following: for any finite group G, the Tate spectrum
t_G(T(n)) is weakly contractible. This strengthens and extends previous
theorems of Greenlees--Sadofsky, Hovey--Sadofsky, and Mahowald--Shick.
The Periodicity Theorem is used in an essential way in our proof. The
connection between the two theorems is via a reformulation of a result
of McCarthy on dual calculus.