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Zbl 1069.55007
Kuhn, Nicholas J.
Tate cohomology and periodic localization of polynomial functors. (English)
[J] Invent. Math. 157, No. 2, 345-370 (2004). ISSN 0020-9910; ISSN 1432-1297/e

Goodwillie calculus [{\it T. G. Goodwillie}, K-theory 4, 1--27 (1990; Zbl 0741.57021)] associates to any homotopy functor $F$ a tower consisting of maps $p_d: P_d F(X) \to P_{d-1} F(X)$ for each $d>0$, with maps $e_d: F \to P_d F$ which are universal maps to $d$-excisive functors. \par The first main theorem of this paper shows that for any homotopy functor $F$, the maps $p_d$ admit a natural homotopy section after localization with respect to the telescope $T(n)$ of a $v_n$ self map of a finite $S$-module of type $n$. Thus polynomial functors split as products of their homogeneous factors, at least after such localization. This result is deduced from the second main theorem which asserts that the Tate spectrum of $T(n)$, with respect to any finite group $G$, is weakly contractible.
[Martin D. Crossley (Swansea)]

MSC 2000:: *55P65 Homotopy functors
55N22 Bordism and cobordism theories, etc.
55P60 Localization and completion
55P91 Equivariant homotopy theory

Keywords: Tate cohomology; Goodwillie calculus; periodic localization

Citations: Zbl 0741.57021

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