TY - JOUR

T1 - Ambidexterity in chromatic homotopy theory

AU - Carmeli, Shachar

AU - Schlank, Tomer M.

AU - Yanovski, Lior

N1 - Publisher Copyright:
© 2022, The Author(s).

PY - 2022/6

Y1 - 2022/6

N2 - 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 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.

AB - 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 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.

UR - http://www.scopus.com/inward/record.url?scp=85124525122&partnerID=8YFLogxK

U2 - 10.1007/s00222-022-01099-9

DO - 10.1007/s00222-022-01099-9

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AN - SCOPUS:85124525122

SN - 0020-9910

VL - 228

SP - 1145

EP - 1254

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 3

ER -