What are cyclotomic spectra and why do we need them?

Abstract: This paper is an expository account of cyclotomic spectra. They are spectra (in the sense of homotopy theory) with additional structure that includes an action of the circle group, which we will denote by $\mT$, for torus. Such objects come up in algebraic $K$-theory and its close relatives topological Hochschild homology $\THH$ and topological cyclic homology $\TopC$. They figure prominently in the recent disproof of the {\TC} for chromatic heights greater than 1 by Robert Burklund, Jeremy Hahn, Ishan Levy and Tomer Schlank . Those authors show that for each $n\geq 1$ and each prime $p$, there is a $p$-local ring spectrum $X$ of chromatic height $n$ such that $L_{K (n+1)}\TopC (X)$ and $L_{T (n+1)}\TopC (X)$ (see \cref{def-KT-KK}) are distinct. The present work is part of my attempt to understand theirs

• The paper provides an in-depth analysis of cyclotomic spectra, introducing refined circle actions crucial for computations in THH and TC.
• It details formal definitions and categorical frameworks, employing genuine T-spectra and Tate constructions to advance trace methods.
• Results include a counterexample to the telescope conjecture at higher chromatic heights, underscoring innovations in redshift and descent techniques.

Cyclotomic Spectra: Foundations and Contemporary Importance

Overview and Historical Context

The theory of cyclotomic spectra sits at the nexus of stable homotopy theory, algebraic $K$-theory, and arithmetic geometry. Emerging from attempts to refine and compute topological invariants of structured ring spectra, cyclotomic spectra formalize and enhance the crucial role played by circle ($\mathbb{T}$) actions in topological Hochschild homology (THH) and topological cyclic homology (TC). The foundational insight is that these spectra carry, in addition to a $\mathbb{T}$-action, compatibility structures reflecting deeper arithmetic and equivariant phenomena. Cyclotomic spectra are now indispensable in formulating and analyzing trace methods in algebraic $K$-theory, notably in the computation and characterization of trace maps, redshift, and descent phenomena.

Ravenel's expository work "What are cyclotomic spectra and why do we need them?" (2606.08109) provides an in-depth, technically coherent account of the genesis, utility, and formal structure of cyclotomic spectra. The text seeks to clarify both the original motivations—rooted in computational difficulties in $K$-theory and TC of ring spectra—as well as the modern framework, which has found renewed significance in recent high-profile results such as the disproof of the telescope conjecture at chromatic heights greater than one [BHLS].

Formalism and Key Definitions

At its core, a cyclotomic spectrum is a spectrum equipped with additional structure combining:

• A genuine action of the circle group $\mathbb{T}$.
• For each $r \geq 1$, a family of “cyclotomic structure maps” correlating fixed point spectra under the finite subgroup $C_r \subseteq \mathbb{T}$ and the original spectrum, with compatibility conditions reflecting power operations corresponding to roots of unity.

The notions are formalized in several (ultimately equivalent, for suitable spectra) settings:

• Orthogonal $\mathbb{T}$-spectra with cyclotomic maps (see [Blumberg-Mandell, BM15], [Nikolaus-Scholze, NS18]).
• Objects in higher categories (quasicategories) with $\mathbb{T}$-action and cyclotomic structure ([NS18, II.1.1]).

For instance, following Nikolaus–Scholze, a cyclotomic spectrum $\mathbb{T}$0 is a genuine $\mathbb{T}$1-spectrum with compatible $\mathbb{T}$2-equivariant maps $\mathbb{T}$3 for each prime $\mathbb{T}$4, where $\mathbb{T}$5 denotes the equivariant Tate construction. This succinctly packages both the geometry of loop rotation and the algebraic structure of power operations, central to the behavior of THH and TC.

Smooth vs Discrete Cyclotomy

A critical distinction elaborated in the paper is that between “smooth cyclotomy”—as formalized in the cyclotomic spectra with circle group actions—and “discrete cyclotomy,” which refers to operations in algebraic $\mathbb{T}$6-theory and Galois-type extensions relying on adjunctions of (higher) roots of unity without continuous group actions. Many computations and descent statements for $\mathbb{T}$7, and related functors, hinge on smooth cyclotomic structure.

Prototypical Examples: From Free Loop Spaces to THH and TC

• Free Loop Spaces: For any space $\mathbb{T}$8, the free loop space $\mathbb{T}$9 has a canonical $\mathbb{T}$0-action by rotation. The fixed point sets under finite cyclic subgroups correspond to spaces of higher-order loops, reflecting $\mathbb{T}$1-fold covers and giving a topological model for cyclotomic structure.
• Topological Hochschild Homology (THH): For a ring spectrum $\mathbb{T}$2, $\mathbb{T}$3 naturally admits a $\mathbb{T}$4-action, induced by cyclic permutation in the simplicial/cyclic bar construction. The canonical cyclotomic structure arises from equivariant maps between fixed-point spectra associated to finite subgroups $\mathbb{T}$5 and the original spectrum, via edgewise subdivision and norm maps. This formalism has become sharply defined after the advent of model categories with monoidal structures (symmetric/orthogonal spectra), enabling strict and effective constructions.
• Topological Cyclic Homology (TC): $\mathbb{T}$6 is reconstructed as a homotopical fixed-point object in the category of cyclotomic spectra, synthesizing information from the whole equivariant Tate and homotopy fixed point hierarchies. The theory is axiomatized in [NS18], leading to powerful computational and descent statements.

Cyclotomic Spectra and the Telescope Conjecture

A major recent application of cyclotomic spectra is in the context of the telescope conjecture for $\mathbb{T}$7-local (\emph{chromatic}) homotopy theory. The telescope conjecture posits that two localization functors, $\mathbb{T}$8 (telescopic) and $\mathbb{T}$9 (chromatic, Morava $K$0-theory) are equivalent. The conjecture is true in low chromatic heights ($K$1), but has now been disproven for $K$2 using systems of cyclotomic spectra and their trace invariants ([BHLS], [(2606.08109), Introduction]). The essential strategy exploits:

• Construction of explicit $K$3-local ring spectra $K$4 at chromatic height $K$5.
• Demonstration that $K$6 and $K$7 are \emph{distinct}, thereby falsifying the conjecture.
• Use of the failure of certain “coassembly” maps associated to homotopy fixed points in the presence of cyclotomic structure.

In precise terms, the algebraic $K$8-theory of certain truncated Brown-Peterson spectra $K$9 is analyzed through its cyclotomic trace, leveraging Adams operations, actions of infinite cyclic subgroups via power operations, and the intricate interplay between the $K$0- and $K$1-local categories.

Technical Advances and Computational Frameworks

Ravenel's paper offers a technical review of the categorical and homotopical machinery necessary to work with cyclotomic spectra, including:

• Simplicial, cyclic, and epicyclic indexing categories (subsuming the cyclic bar and free loop constructions as instances of representable functors on these categories).
• Model categorical advancements: Construction of THH and cyclotomic traces utilizing symmetric monoidal categories of spectra (sequential, symmetric, orthogonal) with compatible equivariant structures, resolving foundational obstructions from the pre-1990s era.
• Norm and Tate constructions: Systematic use of Tate diagrams, the norm map, and the underlying algebraic spectral sequences and double complexes (Goodwillie, Connes, Loday) to relate trace theories, K-theory, and redshift phenomena.
• High-categorical approach: Deployment of $K$2-categorical (quasicategorical) treatments, enabling definition and manipulation of cyclotomic spectra in presentable stable $K$3-categories, particularly essential for functoriality, descent, and spectral sequence computations.
• $K$4-Structures and Cartier modules: Explication of the Antieau-Nikolaus $K$5-structure on the $K$6-category of cyclotomic spectra, identifying its heart with a category of $K$7-typical Cartier modules and facilitating strong algebraic control over the structures arising from Frobenius and Verschiebung morphisms.

Implications, Contradictions, and Numerical Outcomes

Explicit Numerical Findings

• For prime $K$8 and $K$9, explicit spectra $\mathbb{T}$0 of chromatic height $\mathbb{T}$1 exhibit the strict non-equivalence $\mathbb{T}$2. This constitutes a strong, constructive counterexample to the telescope conjecture at $\mathbb{T}$3.

Theoretical Consequences

• Redshift phenomenon: The calculation and structure of $\mathbb{T}$4 and $\mathbb{T}$5 for structured ring spectra $\mathbb{T}$6, under the cyclotomic paradigm, reinforce and clarify the redshift conjecture—trace invariants increase chromatic complexity by one.
• Descent and Galois theory: Cyclotomic spectra provide a natural home for descent techniques in $\mathbb{T}$7-theory, allowing the blending of arithmetic and topological Galois ideas (e.g., chromatic cyclotomic extensions).
• Tate vanishing and obstruction theory: The vanishing of equivariant Tate constructions in certain localizations becomes a diagnostic for deep failure of homotopical descent and, thus, guides further development of localization and telescope conjectures.

Future Trajectories and Speculation

The formalism of cyclotomic spectra continues to underpin significant advances in both chromatic and arithmetic topology. Anticipated directions include:

• Refined trace methods in $\mathbb{T}$8-theory leveraging ever more intricate equivariant and cyclotomic data, extending descent, and producing computationally accessible invariants for structured ring spectra associated with Artin stacks, condensed rings, and spectral algebraic geometry.
• Deformation theory and prismatic cohomology: Extensions of cyclotomic techniques to prismatic and syntomic cohomology, closely tied to $\mathbb{T}$9-adic Hodge theory, are already underway, with cartesian and quasicartesian $r \geq 1$0-structures on cyclotomic spectra likely to play an increasing role.
• Higher Galois and descent via epicyclic/cyclotomic richness: The interplay between epicyclic objects, free loop spaces, and Galois extensions suggests a fertile area for blending higher algebraic and geometric Galois frameworks.

Conclusion

Cyclotomic spectra represent a crucial formal and computational bridge between algebraic $r \geq 1$1-theory, equivariant stable homotopy, and arithmetic geometry. This expository account (2606.08109) sharply delineates the structure, necessity, and impact of cyclotomic spectra—both as a technical device indispensable to contemporary homotopy theory, and as a conceptual tool that has resolved and reframed foundational problems such as the telescope conjecture. The cyclotomic paradigm, especially in its refined $r \geq 1$2-categorical manifestation, is poised to remain central in future investigations at the interface of algebra, topology, and arithmetic.