nLab cyclotomic spectrum

Idea

A cyclotomic spectrum is an $S^1$-equivariant spectrum $E$ with fixed points for all the finite cyclic groups $C_p = \mathbb{Z}/p\mathbb{Z} \hookrightarrow S^1$ inside the circle group, and equipped with $S^1$-equivariant identifications $E^{C_p} \stackrel{\simeq}{\to} E$ of the $C_P$-fixed points with the full object.

The topological Hochschild homology-spectra $E = THH(A)$ are naturally cyclotomic spectra, and this is where the concept originates: by the discussion at Hochschild cohomology $THH(A)$ is the E-infinity ring of functions on the free loop space of $Spec(A)$, and cyclotomic structure reflects the structure of free loop spaces: loops that repeat with period $p$ are equivalent to plain loops.

Cyclotomic structure is the origin of the cyclotomic trace map $THH \longrightarrow TC$ from topological Hochschild homology to topological cyclic homology.

Definition

Throughout, for $p$ a prime number write $C_p \subset S^1$ for the cyclic group $\mathbb{Z}/p\mathbb{Z}$ of order $p$, regarded as a subgroup of the circle group.

A definition says that a cyclotomic spectrum is an circle group-genuine equivariant spectrum $X$ (modeled on orthogonal spectra) equipped with equivalences to its geometric fixed point spectra $\Phi^{C_p} X$ for all the cyclic subgroups $C_p \subset S^1$.

A more abstract definition was given in Nikolaus-Scholze 17:¹

Definition

A cyclotomic spectrum is

1. a spectrum $X$

2. a circle group ∞-action on $X$, i.e. an (∞,1)-functor $B S^1 \to Spectra$ which takes the unique point of $B S^1$ to $X$;

3. for each prime number $p$ a homomorphism of spectra with such circle group action

   $$F_p \;\colon\; X \longrightarrow X^{t C_p}$$

   to the Tate spectrum (the homotopy cofiber $X^{t C_p} \coloneqq cofib( X_{C_p} \overset{norm_p}{\to} X^{C_p} )$ of the norm map),

   where the circle action on the Tate spectrum comes from the canonical identification $S^1/C_p \simeq S^1$.

(These morphisms $F_p$ are called the Frobenius morphisms of the cyclotomic structure, due to this def., this example).

(Nikolaus-Scholze 17, def. 1.3, def. II.1.1).

Proposition

For $X$ a spectrum with stable homotopy groups bounded below, then def. is equivaent to the traditional:

There is an (∞,1)-functor

$$CycSp_-^{gen} \overset{\simeq}{\longrightarrow} CycSp_-$$

from traditional (“genuine”) cyclotomic spectra bounded below to bounded below cyclotomic spectra in the sense of def. , and this is an equivalence of (∞,1)-categories.

(Nikolaus-Scholze 17, prop. II.3.4, theorem II.6.9).

Examples

Example

(topological Hochschild homology)

For every A-∞ ring $A$, the topological Hochschild homology spectrum $THH(A)$ naturally carries the structure of a cyclotomic spectrum (def. ).

(Nikolaus-Scholze 17, section II.2, def. III.2.3)

Example

(trivial cyclotomic spectra)

Every spectrum $X$ becomes a cyclotomic spectrum $X^{triv}$ in the sense of def. by equipping it

1. with the trivial circle group ∞-action

2. for each prime $p$ with the composite morphism

   $$f_p \;\colon\; \mathbb{S} \longrightarrow \mathbb{S}^{C_p} \longrightarrow \mathbb{S}^{t C_p}$$

   (the first being the $( B C_p \times (-) \dashv (-)^{C_p} )$-unit into the homotopy fixed points, the second the defining morphism into the Tate spectrum )

3. the $S^1/C_p$-equivariant structure on these morphisms given under the adjunction between trivial action and homotopy fixed points by the adjunct morphisms

   $$X \longrightarrow \left(X^{t C_p}\right)^{S^1/C_p}$$

   as the composite

   $$X \to X^{S^1} \simeq \left( X^{C_p} \right)^{S^1/C_p} \longrightarrow \left( X^{t C_p} \right)^{S^1/C_p} \,.$$

This construction constitutes a left adjoint (infinity,1)-functor to taking topological cyclic homology

$$CycSpectra \underoverset{\underset{TC}{\longrightarrow}}{\overset{(-)^{triv}}{\longleftarrow}}{\bot} Spectra \,.$$

(Nikolaus-Scholze 17, example II.1.2 (ii) and middle of p. 126)

Example

(cyclotomic sphere spectrum)

The sphere spectrum regarded as a cyclotomic spectrum via example is called the cyclotomic sphere spectrum.

As such it is equivalently its topological Hochschild homology according to example :

$$\mathbb{S}^{triv} \simeq THH(\mathbb{S}) \,.$$

(Nikolaus-Scholze 17, example II.1.2 (ii))

Properties

Monoidal structure

The tensor unit in the symmetric monoidal (infinity,1)-category of cyclotomic spectra is the cyclotomic sphere spectrum from example (Blumberg-Mandell 13, example 4.9)

Related concepts

• cyclotomic space

• cycle category, cyclic set, cyclic object

• Frobenius homomorphisms

• Tate spectrum

References

• Andrew Blumberg, Michael Mandell, The homotopy theory of cyclotomic spectra, Geom. Topol. 19 (2015) 3105-3147 [arXiv:1303.1694, doi:10.2140/gt.2015.19.3105]

• Clark Barwick, Saul Glasman, Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin (arXiv:1602.02163)

• Clark Barwick, Saul Glasman, Noncommutative syntomic realization (pdf)

• Cary Malkiewich: A visual introduction to cyclic sets and cyclotomic spectra, talk at Young Topologists Meeting Lausanne, Switzerland (2015) [pdf, pdf]

• Thomas Nikolaus, Peter Scholze, On topological cyclic homology, Acta Math. 221 2 (2018) 203-409 [arXiv:1707.01799, doi:10.4310/ACTA.2018.v221.n2.a1]

• Andrew J. Blumberg, Michael A. Mandell: The homotopy theory of cyclotomic spectra, 10 years later [arXiv:2412.15928]

• Douglas C. Ravenel: What are cyclotomic spectra and why do we need them? [arXiv:2606.08109]

• Douglas C. Ravenel: Cyclotomic extensions in stable homotopy theory [arXiv:2606.08166]

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1. Maybe this definition does not exactly agree with other definitions in the literature; see $n$Forum discussion here. ↩