A short proof of telescopic Tate vanishing

TL;DR

This paper provides a concise proof of Kuhn's theorem on the vanishing of Tate constructions in telescopically localized stable homotopy theory, connecting it to the Kahn-Priddy theorem.

Contribution

It offers a simplified proof of Kuhn's Tate vanishing theorem and reveals its equivalence to the existence of a section for the transfer map after telescopic localization.

Findings

Kuhn's Tate vanishing theorem is equivalent to the transfer map admitting a section after telescopic localization.

The proof leverages the Kahn-Priddy theorem to establish the result.

The approach simplifies understanding of Tate constructions in localized stable homotopy theory.

Abstract

We give a short proof of a theorem of Kuhn that Tate constructions for finite group actions vanish in telescopically localized stable homotopy theory. In particular, we observe that Kuhn's theorem is equivalent to the statement that the transfer $BC_{p+} \to S^0$ admits a section after telescopic localization, which in turn follows from the Kahn-Priddy theorem.