Ambidexterity in Chromatic Homotopy Theory

Abstract

We extend the theory of ambidexterity developed by M. J. Hopkins and J. Lurie and show that the $\infty$-categories of $T(n)$-local spectra are $\infty$-semiadditive for all $n$, where $T(n)$ is the telescope on a $v_{n}$-self map of a type $n$ spectrum. This extends and provides a new proof for the analogous result of Hopkins-Lurie on $K(n)$-local spectra. Moreover, we show that $K(n)$-local and $T(n)$-local spectra are respectively, the minimal and maximal $1$-semiadditive localizations of spectra with respect to a homotopy ring, and that all such localizations are in fact $\infty$-semiadditive. As a consequence, we deduce that several different notions of "bounded chromatic height" for homotopy rings are equivalent, and in particular, that $T(n)$-homology of $\pi$-finite spaces depends only on the $n$-th Postnikov truncation. A key ingredient in the proof of the main result is a construction of a certain power operation for commutative ring objects in stable $1$-semiadditive $\infty$-categories. This is closely related to some known constructions for Morava $E$-theory and is of independent interest. Using this power operation we also give a new proof, and a generalization, of a nilpotence conjecture of J.P. May, which was proved by A. Mathew, N. Naumann, and J. Noel.