Tate cohomology and periodic localization of polynomial functors

AbstractIn 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 vn 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, {PdF}, such that F→PdF is the universal arrow to a d–excisive functor. Our first main theorem says that PdF→Pd-1F 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 main theorem which is equivalent to the following: for any finite group G, the Tate spectrum ${\mathcal{T}}_{G}(T(n))$ 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.