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 $$T\!\left( n\right) $$ T n -local spectra are $$\infty $$ ∞ -semiadditive for all n, where $$T\!\left( n\right) $$ T n is the telescope on a $$v_{n}$$ v n -self map of a type n spectrum. This generalizes and provides a new proof for the analogous result of Hopkins–Lurie on $$K\!\left( n\right) $$ K n -local spectra. Moreover, we show that $$K\!\left( n\right) $$ K n -local and $$T\!\left( n\right) $$ 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 $$T\!\left( n\right) $$ T n -homology of $$\pi $$ 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 $$\infty $$ ∞ -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.

Higher semiadditive Grothendieck-Witt theory and the 𝐾(1)-local sphere

“…Hopkins and Lurie extend Theorem 1.1.1 by showing that Sp 𝐾(𝑛) is 𝑚-semiadditive for all 𝑚. This was further generalized in work by the first author, Schlank, and Yanovski [17], who proved the analogous statement for the telescopic localizations Sp 𝑇(𝑛) .…”

“…Example 2.0.9. The ∞-categories Sp 𝐾(𝑛) and Sp 𝑇(𝑛) are ∞-semiadditive (i.e., 𝑚semiadditive for every 𝑚 ≥ 0) [17,30]. We obtain higher commutative monoids This paper is dedicated to studying these higher commutative monoids in the case 𝑛 = 1 (in which case they are same because the telescope conjecture holds at height 1, see, e.g., [12,Theorem 4.1]).…”

“…However, by joint work between the second author, Schlank, and Yanovski [17], following work of Kuhn [34] (in the case of 1-semiadditivity), the ∞-category of 𝑇(𝑛)local spectra is ∞-semiadditive. It follows that one has canonical elements |𝐴| 𝕊 𝑇(𝑛) ∈ 𝜋 0 (𝕊 𝑇(𝑛) ) for all 𝜋-finite spaces 𝐴.…”

“…(1) . As an application of Theorem A, we finish the paper by computing certain well-known operations in 𝜋 0 𝕊 𝐾(1) : namely, the power operations 𝜃 (first defined by McClure [13,§IX], [31]) and 𝛿 𝑝 (appearing in the work of Schlank, Yanovski, and the first author [17]), and the 𝐾(1)-local logarithm log 𝐾(1) of Rezk [48].…”

$G$-spectra of cyclic defect

“…The n = 0 case of Kuhn's theorem is the familiar fact that the additive transfer map from orbits to fixed points is an isomorphism when working over Q. These results were further generalized and reinterpreted by the Hopkins-Lurie theory of ambidexterity [HL13], which has been generalized and developed extensively by Carmeli-Schlank-Yanovski and Barthel-Carmeli-Schlank-Yanovski [CSY22], [CSY21a], [CSY21b], [BCSY22]. For the convenience of the reader, the following proposition collects some basic features of the theory.…”