Cyclotomic extensions in stable homotopy theory

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.

• The paper introduces a novel framework of cyclotomic extensions in stable homotopy theory, establishing a spectral analog to classical cyclotomic Galois theory.
• It leverages semiadditive ∞-categories to distinguish between K(n)-local, cyclotomically complete, and telescopic spectra, highlighting the failure of the telescope conjecture for n > 1.
• The work demonstrates that cyclotomic redshift and higher roots of unity yield practical insights into algebraic K-theory and topological cyclic homology within chromatic localizations.

Cyclotomic Extensions in Stable Homotopy Theory

Introduction and Motivation

The paper "Cyclotomic extensions in stable homotopy theory" (2606.08166) systematically analyzes chromatic cyclotomic extensions within the context of stable homotopy theory, focusing on their structural analogy with classical Galois theory over $p$-adic fields and their role in understanding the counterexample to the telescope conjecture presented by Burklund, Hahn, Levy, and Schlank (BHLS). The paper complements earlier work on cyclotomic spectra, detailing discrete cyclotomic extensions of ring spectra and their chromatic implications.

Cyclotomic spectra, equipped with an $S^1$-action, arise in algebraic $K$-theory, topological Hochschild homology ($\mathrm{THH}$), and topological cyclic homology ($\mathrm{TC}$). The notion of higher cyclotomic extensions of commutative ring spectra parallels the classical construction of cyclotomic Galois extensions of $p$-adic fields by adjoining roots of unity, with "smooth" and "discrete" cyclotomy distinguishing the spectral and extension aspects, respectively.

Chromatic Framework and Telescope Conjecture

The chromatic perspective centers around localizations with respect to Morava $K$-theories ($K(n)$), Johnson-Wilson spectra ($E(n)$), and telescopes ($T(n)$), key objects in the Hopkins-Smith chromatic filtration. The telescope conjecture posited an equivalence of Bousfield localizations $S^1$0 and $S^1$1, but BHLS demonstrated that for $S^1$2 this fails: there exist $S^1$3-local $S^1$4-ring spectra such that $S^1$5 and $S^1$6 are not equivalent.

A central technical distinction is between finite spectra—always admitting chromatic type—and infinite spectra, exemplified by the Johnson-Wilson spectrum $S^1$7, which has $S^1$8 for $S^1$9.

Semiadditive $K$0-Categories and Ambidexterity

The analysis leverages Hopkins-Lurie semiadditivity and its generalizations by Carmeli-Schlank-Yanovski. A stable $K$1-category is $K$2-semiadditive if for every functor indexed by an $K$3-finite space, limits and colimits agree via a norm map. The $K$4-local ($K$5) and telescopic ($K$6) categories are $K$7-semiadditive, and the classification theorem confirms that semiadditivity picks out precisely those monochromatic localizations sandwiched between $K$8 and $K$9.

Cyclotomic completeness emerges as a structural interpolation between $\mathrm{THH}$0-local and $\mathrm{THH}$1-local categories, introducing the intermediate category $\mathrm{THH}$2 of height $\mathrm{THH}$3 cyclotomically complete spectra.

Cyclotomic Extensions: Galois Theory and Higher Roots of Unity

The chromatic analog of classical cyclotomic extensions is constructed via Rognes' Galois theory for commutative ring spectra. For $\mathrm{THH}$4-local spheres, the Morava $\mathrm{THH}$5-theory extension $\mathrm{THH}$6 is a faithful pro-Galois extension with Galois group $\mathrm{THH}$7. Abelianization of $\mathrm{THH}$8 aligns with the absolute Galois group of $\mathrm{THH}$9, and higher cyclotomic extensions correspond to adjoining higher roots of unity via homomorphisms $\mathrm{TC}$0, adjoint to maps $\mathrm{TC}$1.

Telescopic analogs are constructed for abelian extensions, leveraging lifts from $\mathrm{TC}$2-local to $\mathrm{TC}$3-local settings. Cyclotomic completion, $\mathrm{TC}$4, is defined as localization with respect to profinite $\mathrm{TC}$5-power cyclotomic extensions, and cyclotomically complete spectra are $\mathrm{TC}$6-local.

The hierarchy:

$\mathrm{TC}$7

shows the structural position of cyclotomic localizations between chromatic and telescopic extremes; BHLS establishes the non-equivalence for $\mathrm{TC}$8.

Higher Roots of Unity and Bousfield Comparison

The action of profinite groups is analyzed categorically in terms of spaces such as $\mathrm{TC}$9 (n-fold classifying spaces), which generalize classical group ring constructions $p$0. The cyclotomic spectra arising from higher roots of unity are controlled via their Morava $p$1-theory and telescopic analogs.

Furthermore, Bousfield comparison between the cyclotomically complete sphere, telescopic sphere, and $p$2-local sphere reveals nuanced relationships, with localization functors $p$3, $p$4, and $p$5 interpolating the stable homotopy landscape.

Cyclotomic Redshift

The chromatic redshift phenomenon, critical in the BHLS breakdown of the telescope conjecture, shows that algebraic $p$6-theory (and related functors) increase chromatic height, while the Tate construction ("blueshift") lowers it. Explicitly, for a $p$7-local ring spectrum $p$8,

$p$9

demonstrates the redshift under $K$0-theory.

Cyclotomic hyperdescent is formulated via hypersheafification in the $K$1-topos of profinite group actions, enabling a precise description of descent and completion properties for cyclotomic extensions and their Galois theories.

Adams Operations, Locally Unipotent Actions, and Coassembly Non-equivalence

The main BHLS counterexample hinges on locally unipotent $K$2-actions (via Adams operations) on $K$3. When equipped with such actions, coassembly maps for algebraic $K$4-theory and $K$5 are shown not to be equivalences after telescopic localization, a phenomenon that is trivialized by smashing with suitable finite spectra.

Key technical results demonstrate that the coassembly map

$K$6

is not an equivalence for $K$7-localization, but becomes one after $K$8-localization. This phenomenon is shown to be closely connected with cyclotomic completion and redshift, and provides a categorical and computational counterexample to the telescope conjecture.

Numerical and Structural Results

• The coassembly map for cyclotomically completed algebraic $K$9-theory is an equivalence in the intermediate $K(n)$0 category, but not for $K(n)$1.
• For $K(n)$2, the $K(n)$3-structure after telescopic localization is strictly stronger than pre-localization, enabling detailed trace arguments in the proof.
• Cyclotomic redshift can be iterated along chromatic layers, defining profinite Galois extensions at each stage.
• The Bousfield lattice between $K(n)$4-local, cyclotomically complete, and telescopic categories is strictly ordered for $K(n)$5.

Implications and Speculations

The failure of the telescope conjecture at chromatic heights $K(n)$6, elucidated via cyclotomic extensions and their semiadditive properties, suggests that the structure of stable homotopy theory and its Galois-theoretic analogs is richer than previously conjectured. The categorical framework for cyclotomic completion, semiadditivity, and roots of unity provides a robust toolkit for constructing and analyzing chromatic phenomena.

Practically, the refinement of localizations, completions, and descent properties in $K(n)$7-categories of spectra allows for greater control in computational approaches to algebraic $K(n)$8-theory, trace methods, and topological cyclic homology. Theoretically, the results indicate that understanding the interaction between Galois theory, higher roots of unity, and chromatic redshift/blueshift is crucial for decoding the structure of the stable homotopy category as filtered by chromatic height.

Future directions may include:

• Further refinement of the poset of semiadditive localizations, including novel functors interpolating beyond cyclotomic completions.
• Extensions of Rognes' and chromatic Galois theories to deeper layers, possibly incorporating more non-abelian analogs.
• Computational exploitation of cyclotomic hyperdescent and hypersheaf techniques for explicit calculations in algebraic $K(n)$9-theory and $E(n)$0.

Conclusion

"Cyclotomic extensions in stable homotopy theory" (2606.08166) rigorously constructs and analyzes the landscape of chromatic cyclotomic extensions, their semiadditive categorical framework, and their role in both the resolution and breakdown of the telescope conjecture. By developing the analogy with classical Galois theory and exploiting categorical structures, the paper advances both the conceptual and computational boundaries of stable homotopy theory and its connections to algebraic $E(n)$1-theory, cyclotomic spectra, and chromatic phenomena.