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Cyclotomic extensions in stable homotopy theory

T0 review · 0 major / 4 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read Higher cyclotomic extensions of ring spectra sit between K(n)-local and T(n)-local spectra and turn the telescope conjecture false.

desk verdict Clean, fully attributed exposition of the cyclotomic half of the BHLS telescope counterexample; no new theorems, but a usable dictionary that organizes CSY/BMCSY/BCSY and the intermediate L_Cyclo functors. read the letter →

arxiv 2606.08166 v2 pith:OZ3L7UFG submitted 2026-06-06 math.AT

Douglas C. Ravenel This is my paper · ORCID
classification math.AT MSC 55P4255N2219D55
keywords telescopeconjecturecyclotomicextensionschromatichomotopysemiadditivitycoassemblymapredshiftBP⟨n⟩Adamsoperations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This expository paper explains the discrete half of the recent disproof of the telescope conjecture: higher cyclotomic extensions of commutative ring spectra, the chromatic analogues of adjoining roots of unity to p-adic fields. Between the K(n)-local and T(n)-local categories sits an intermediate category of height-n cyclotomically complete spectra obtained by adjoining higher roots of unity (maps from Eilenberg–MacLane spaces K(Z/p^j,n)). The paper shows how these extensions interact with algebraic K-theory via cyclotomic redshift, and how a locally unipotent Z-action by Adams operations on BP⟨n⟩ can be trivialized after smashing with a type-(n+2) finite complex. The resulting coassembly map for telescopic K-theory is not an equivalence, while the same map after cyclotomic completion is; therefore the two localizations differ for every height n≥1.

What carries the argument

Height-n cyclotomic completion: localization of T(n)-local spectra at the infinite chromatic cyclotomic extension S_{T(n)}[ω_p^∞^(n)], which interpolates between L_{T(n)} and L_{K(n)} and converts coassembly into a completion map.

What would settle it

Exhibit a height n≥2 at which the T(n+1)-local coassembly map for BP⟨n⟩ with the Adams Z-action is nevertheless an equivalence, or show that no such EA_2-algebra structure exists for the Adams operations.

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Extended reading notes

Core claim

For every prime p and every height n≥1, the T(n+1)-local coassembly map for BP⟨n⟩ with its Adams Z-action is not an equivalence, yet becomes an equivalence after height-(n+1) cyclotomic completion. Consequently the categories of K(n+1)-local and T(n+1)-local spectra are distinct.

Load-bearing premise

The Adams operations on BP⟨n⟩ can be realized as algebra automorphisms of a certain intermediate multiplicative structure that become fully commutative after telescopic localization and stay nearly trivial on homotopy groups after p-completion.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. This expository companion paper develops the discrete (higher) cyclotomic-extension half of the BHLS resolution of the telescope conjecture. It reviews semiadditivity of Sp_{K(n)} and Sp_{T(n)}, Rognes Galois theory, and the construction of height-n cyclotomic extensions S_{T(n)}[\omega^{(n)}_{p^j}] obtained by adjoining higher roots of unity via maps B^n C_{p^j} \to R^\times. It introduces the intermediate localization L^{Cyclo}_n (cyclotomic completion) interpolating between L^{Fin}_n and L_n, and sketches how chromatic cyclotomic redshift (Theorem 4.33) together with asymptotic constancy for locally unipotent Z-actions on BP\langle n\rangle (Theorems 5.3, 5.11) imply that the T(n+1)-local K-theory coassembly map for BP\langle n\rangle^{h(p^k Z)} is not an equivalence while the corresponding map after height-(n+1) cyclotomic completion is an equivalence, yielding Sp_{K(n+1)} \neq Sp_{T(n+1)} (Theorem 6.3).

Significance. The paper supplies a carefully attributed dictionary that makes the cyclotomic-extension half of BHLS accessible to a broader audience of chromatic homotopy theorists. Strengths include precise cross-references (BHLS Theorems A–C become Theorems 6.3, 5.11, 5.3), the clean isolation of the intermediate category Sp^{Cyclo(n)}, and two substantial appendices that render the needed \infty-categorical and operadic background self-contained. Because the manuscript introduces no original theorems and every load-bearing statement is cited, it functions as reliable exposition rather than a new research claim; its value lies in clarifying the role of discrete cyclotomy and cyclotomic completion in the disproof of the telescope conjecture.

minor comments (4)
  1. The twelve-page construction of the Adams operations as EA_2-automorphisms (Theorem 5.9) is deferred entirely to BHLS §5; a one-paragraph outline of the key steps (factorization homology, the map to E_n^\Psi) would improve readability without lengthening the paper substantially.
  2. Notation for the intermediate localizations (L^{Cyclo}_n versus L^{Cyclo(n)}, Sp^{Cyclo(n)} versus xSp_{T(n)}) is introduced in Definition 4.17 and (4.20) but is not always used consistently in later sections; a short notational table would help.
  3. Appendix B.5 on the Boardman–Vogt tensor product is thorough, yet the precise reason EA_2 is the strongest structure preserved by the Adams operations is only alluded to; a sentence pointing to the relevant obstruction in BHLS Remark 5.5 would close the loop.
  4. A few typographical slips appear (e.g., “semidditive”, “spactra”, “pTpnq ‘ Tpn`1qq”); a light copy-edit pass would remove them.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: pure exposition quoting BHLS theorems; self-citations are to the companion paper and standard background, none load-bearing for the telescope inequality.

full rationale

The manuscript is explicitly expository (Abstract, §1) and attributes every load-bearing statement to BHLS: Theorem 6.3 = [BHLS23, Theorem A], Theorem 5.11 = [BHLS23, Theorem B], Theorem 5.3 = [BHLS23, Theorem C], the Adams-operation construction = [BHLS23, Theorem 5.4] (quoted as Theorem 5.9), and the coassembly-as-cyclotomic-completion identification = [BHLS23, Theorem 6.25] (quoted as Theorem 6.6). The derivation chain therefore reduces to external citations, not to definitions or fits internal to this paper. Self-citations appear only for the companion [Rav26] (cyclotomic spectra) and for standard ∞-categorical/operadic background (Appendices A–B); none of them force Sp_K(n+1) ≠ Sp_T(n+1) by construction. There are no free parameters, no uniqueness theorems imported from the author, and no renaming of known empirical patterns. The score is therefore 1 solely for the presence of ordinary self-citation to the companion, which is not load-bearing.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The paper rests entirely on the standard foundations of chromatic homotopy theory, Rognes Galois theory, Hopkins-Lurie ambidexterity, and the BHLS counterexample itself. No free parameters are fitted; the only ‘invented’ language is the convenient packaging of already-published notions (higher roots of unity, cyclotomic completion).

assumptions (5)
  • domain assumption Sp_{K(n)} and Sp_{T(n)} are ∞-semiadditive (Hopkins-Lurie, CSY)
    Used throughout §3 and to guarantee that norm maps for m-finite spaces are equivalences; cited as Theorems 3.3 and 3.7.
  • domain assumption S_{K(n)} o E_n(F_p) is a faithful pro-Galois extension with group G_n (Rognes)
    Proposition 4.8; supplies the classical model that higher cyclotomic extensions lift.
  • domain assumption Every finite abelian K(n)-local Galois extension lifts to a T(n)-local extension (CSY24 Theorems A, 5.31)
    Theorem 4.10; the existence of S_{T(n)}[ω_p^j^{(n)}] rests on this.
  • domain assumption BP⟨n⟩ admits an EA_2-structure with locally unipotent Adams operations that become E_∞ after T(n)-localization (BHLS §5)
    Theorem 5.9; load-bearing for the asymptotic-constancy results that feed the coassembly comparison.
  • standard math Standard ∞-categorical notions (presentable categories, hypersheaves, Čech nerves, Boardman-Vogt tensor product)
    Collected in Appendices A and B; used without further justification.
invented entities (2)
  • height-n cyclotomic completion L_Cyclo_n / Sp_Cyclo(n) independent evidence
    purpose: Intermediate localization between L_Fin_n and L_n (and between L_T(n) and L_K(n)) obtained by localizing at the infinite cyclotomic extension R_n^Fin
    Definition 4.17 packages the colimit of the finite cyclotomic extensions already constructed by CSY; the name and the intermediate category are convenient but not new mathematical objects.
  • higher (height-n) p^j-th roots of unity independent evidence
    purpose: Maps C_{p^j} o Ω^n R^ imes that produce the chromatic cyclotomic extensions
    Definition 4.26, taken from CSY24; the analogy with classical roots of unity is expository packaging of an existing construction.

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Pith. "Pith review of Cyclotomic extensions in stable homotopy theory." pith.science (2026). https://pith.science/paper/OZ3L7UFG

@misc{pith2026260608166,
  author       = {Pith},
  title        = {Pith review of: Cyclotomic extensions in stable homotopy theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OZ3L7UFG}},
  note         = {Machine review of arXiv:2606.08166}
}
abstract

This expository paper is a companion to \cite{Rav:gjmcyc}, in which we discuss cyclotomic spectra. Both papers are intended to shed light on the recent resolution of the telescope conjecture by Robert Burklund, Jeremy Hahn, Ishan Levy and Tomer Schlank (hereafter referred to as BHLS) in \cite{BHLS}. Their proof involves both cyclotomic spectra, the subject of \cite{Rav:gjmcyc}, and cyclotomic extensions of spectra, the subject of this paper. Higher cyclotomic extensions of commutative ring spectra are analogous to Galois extensions of $p$-adic number fields (or rings of integers thereof) obtained by adjoining roots of unity.

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Works this paper leans on

4 extracted references · 1 linked inside Pith

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