Ambidexterity in chromatic homotopy theory

We extend the theory of ambidexterity developed by M. J. Hopkins and J. Lurie and show that the ∞-categories of T(n)-local spectra are ∞-semiadditive for all n, where T(n) is the telescope on a v_n-self map of a type n spectrum. This generalizes 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 ∞-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 π-finite spaces depends only on the nth 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 ∞-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.