Ambidexterity in chromatic homotopy theory

• 成果类型: Article
• 署名作者: Carmeli, Shachar; Schlank, Tomer M.; Yanovski, Lior
• 署名单位: Weizmann Institute of Science; Hebrew University of Jerusalem; Max Planck Society
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01099-9
• 发表日期: 2022
• 页码: 1145-1254
• 关键词: morava k-theories; tate cohomology; localization; NILPOTENCY; Respect

摘要

We extend the theory of ambidexterity developed by M. J. Hopkins and J. Lurie and show that the infinity-categories of T (n)-local spectra are infinity-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 newproof for the analogous result of Hopkins-Lurie on K(n)-local spectra. Moreover, we showthat 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 infinity-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 p-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 infinity-categories. This is closely related to some known constructions forMorava 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.