Abstract
We extend the theory of ambidexterity developed by M. J. Hopkins and J. Lurie and show that the infty -categories of T!left( nright) -local spectra are infty -semiadditive for all n, where T!left( nright) is the telescope on a 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( nright) -local spectra. Moreover, we show that K!left( nright) -local and T!left( nright) -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( nright) -homology of pi -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.
Highlights
1.1 Main resultsGiven an abelian group A with an action of a finite group G, summation along the orbit provides a natural map NmG : AG → AG from the co-invariants to the invariants
It seems appropriate at this point to say a few words about the results summarized in Theorem D, in light of the Telescope Conjecture, which asserts that SpK(n) = SpT(n), see [38]
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
Summary
Given an abelian group A with an action of a finite group G, summation along the orbit provides a natural map NmG : AG → AG from the co-invariants to the invariants. It seems appropriate at this point to say a few words about the results summarized in Theorem D, in light of the (still open) Telescope Conjecture, which asserts that SpK(n) = SpT(n), see [38] If this conjecture is true, the property of higher semiadditivity characterizes completely the K (n)-local ∞-categories among localizations of Sp with respect to homotopy rings (as does the existence of the Bousfield–Kuhn functor). 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. That was proved in [35] (Corollary 5.2.5)
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