A Short Introduction to the Telescope and Chromatic Splitting Conjectures

Barthel, Tobias

doi:10.1007/978-981-15-1588-0_9

Cited by 4 publications

(9 citation statements)

References 45 publications

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“…Our next result concerns the classification of 1-semiadditive localizations of p-local spectra with respect to homotopy rings. 2 We show that the ∞categories Sp K(n) and Sp T(n) are precisely the minimal and maximal examples of such localizations.…”

Section: Resultsmentioning

confidence: 88%

“…Namely, we obtain an equivalence of three different notions of "height ≤ d" for a homotopy ring: (1) the "algebraic" one using Morava K -theories, (2) the "geometric" one using finite complexes, and (3) the "categorical" one using πfinite spaces. The categorical height of a spectrum (i.e.…”

Section: Resultsmentioning

confidence: 99%

“…The triangle on the right commutes by Lemma 2.2.4 (2). The composition of the top maps is F D ν q and of the bottom maps is ν q F D .…”

Section: Remark 2211mentioning

confidence: 87%

“…map in the middle triangle is an isomorphism. Thus, the middle triangle commutes by the inductive hypothesis and Lemma 2.2.12 (2). As for the rectangle, we apply x p + y p − (x + y) p p is actually a polynomial with integer coefficients in the variables x and y and does not involve division by p. In particular, this is well defined for all x, y ∈ R, even when R has p-torsion.…”

Section: Diagram Of Spacesmentioning

confidence: 99%

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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: 88%

Section: Resultsmentioning

confidence: 99%

“…The triangle on the right commutes by Lemma 2.2.4 (2). The composition of the top maps is F D ν q and of the bottom maps is ν q F D .…”

Section: Remark 2211mentioning

confidence: 87%

Section: Diagram Of Spacesmentioning

confidence: 99%

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Ambidexterity in chromatic homotopy theory

2022

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“…A number of equivalent formulations of this conjecture and the current state of knowledge about it can be found in Mahowald-Ravenel-Schick [MRS01] and [Bar19]. The smash product theorem of Hopkins and Ravenel [Rav92, Section 8] states that L n is smashing, i.e., L n as an endofunctor on Sp commutes with colimits, while the analogous fact for L f n was proven by Miller [Mil92].…”

Section: Fpmentioning

confidence: 99%

Chromatic structures in stable homotopy theory

Barthel¹,

Beaudry²

2020

Handbook of Homotopy Theory

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Stratification in tensor triangular geometry with applications to spectral Mackey functors

Barthel,

Heard,

Sanders

2021

Preprint

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We systematically develop a theory of stratification in the context of tensor triangular geometry and apply it to classify the localizing tensorideals of certain categories of spectral G-Mackey functors for all finite groups G. Our theory of stratification is based on the approach of Stevenson which uses the Balmer-Favi notion of big support for tensor-triangulated categories whose Balmer spectrum is weakly noetherian. We clarify the role of the local-toglobal principle and establish that the Balmer-Favi notion of support provides the universal approach to weakly noetherian stratification. This provides a uniform new perspective on existing classifications in the literature and clarifies the relation with the theory of Benson-Iyengar-Krause. Our systematic development of this approach to stratification, involving a reduction to local categories and the ability to pass through finite étale extensions, may be of independent interest. Moreover, we strengthen the relationship between stratification and the telescope conjecture. The starting point for our equivariant applications is the recent computation by Patchkoria-Sanders-Wimmer of the Balmer spectrum of the category of derived Mackey functors, which was found to capture precisely the height 0 and height ∞ chromatic layers of the spectrum of the equivariant stable homotopy category. We similarly study the Balmer spectrum of the category of E(n)-local spectral Mackey functors noting that it bijects onto the height ≤ n chromatic layers of the spectrum of the equivariant stable homotopy category; conjecturally the topologies coincide. Despite our incomplete knowledge of the topology of the Balmer spectrum, we are able to completely classify the localizing tensor-ideals of these categories of spectral Mackey functors.

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A Short Introduction to the Telescope and Chromatic Splitting Conjectures

Cited by 4 publications

References 45 publications

Ambidexterity in chromatic homotopy theory

Ambidexterity in chromatic homotopy theory

Chromatic structures in stable homotopy theory

Stratification in tensor triangular geometry with applications to spectral Mackey functors

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