A canonical lift of Frobenius in Morava $E$-theory

Stapleton, Nathaniel

doi:10.4310/hha.2019.v21.n1.a16

Cited by 1 publication

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“…For K (1)-local E ∞ -rings, Hopkins has constructed in [21] similar looking power operations denoted θ . Generalizations of these operations to higher heights were studied by different authors including [42,47], and using them, a canonical lift of Frobenius was constructed in [46] for the Morava E-theory cohomology ring of a space. However, even for K (1)-local rings, our power operation δ turns out to be different from the operation θ constructed by Hopkins.…”

Section: Resultsmentioning

confidence: 99%

Ambidexterity in chromatic homotopy theory

2022

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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.

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Section: Resultsmentioning

confidence: 99%