nLab cyclotomic spectrum



Stable Homotopy theory

Representation theory



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

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

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


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

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

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


A cyclotomic spectrum is

  1. a spectrum XX

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

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

    F p:XX tC p F_p \;\colon\; X \longrightarrow X^{t C_p}

    to the Tate spectrum (the homotopy cofiber X tC pcofib(X C pnorm pX C p)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 pS 1S^1/C_p \simeq S^1.

(These morphisms F pF_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).


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

There is an (∞,1)-functor

CycSp genCycSp 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).



(topological Hochschild homology)

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

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


(trivial cyclotomic spectra)

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

  1. with the trivial circle group ∞-action

  2. for each prime pp with the composite morphism

    f p:𝕊𝕊 C p𝕊 tC p f_p \;\colon\; \mathbb{S} \longrightarrow \mathbb{S}^{C_p} \longrightarrow \mathbb{S}^{t C_p}

    (the first being the (BC p×()() C p)( 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 pS^1/C_p-equivariant structure on these morphisms given under the adjunction between trivial action and homotopy fixed points by the adjunct morphisms

    X(X tC p) S 1/C p X \longrightarrow \left(X^{t C_p}\right)^{S^1/C_p}

    as the composite

    XX S 1(X C p) S 1/C p(X tC p) S 1/C p. 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

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

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


(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 :

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

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


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)


  1. Maybe this definition does not exactly agree with other definitions in the literature; see nnForum discussion here.

Last revised on September 14, 2023 at 09:49:46. See the history of this page for a list of all contributions to it.