No 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.

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 vn-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 ∞-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 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 ∞-categories. This generalizes 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, that was proved by A. Mathew, N. Naumann, and J. Noel.

We construct Galois extensions of the T (n)-local sphere, lifting all finite abelian Galois extensions of the K(n)-local sphere. This is achieved by realizing them as higher semiadditive analogues of cyclotomic extensions. Combining this with a general form of Kummer theory, we lift certain elements from the K(n)-local Picard group to the T (n)-local Picard group.Color circles from the enlarged 1708 edition of the Treatise on Miniature Painting, Claude Boutet.

We introduce and study the notion of semiadditive height for higher semiadditive ∞categories, which generalizes the chromatic height. We show that the higher semiadditive structure trivializes above the height and prove a form of the redshift principle, in which categorification increases the height by one. In the stable setting, we show that a higher semiadditive ∞-category decomposes into a product according to height, and relate the notion of height to semisimplicity properties of local systems. We place the study of higher semiadditivity and stability in the general framework of smashing localizations of Pr L , which we call modes. Using this theory we introduce and study the universal stable ∞-semiadditive ∞-category of semiadditive height n, and give sufficient conditions for a stable 1-semiadditive ∞-category to be ∞-semiadditive.