Bokstedt Periodicity

Bokstedt periodicity is the foundational computation of the topological Hochschild homology of the prime field $\mathbb{F}_p$ . It establishes that $THH_*(\mathbb{F}_p)$ is a polynomial algebra, a result that stands in stark contrast to classical Hochschild homology and is the starting point for all subsequent TC computations.

Statement

Theorem8.3Bökstedt Periodicity

For any prime $p$ , the topological Hochschild homology of $\mathbb{F}_p$ with mod- $p$ coefficients is:

$THH_*(\mathbb{F}_p; \mathbb{F}_p) \cong \mathbb{F}_p[\mu]$

a polynomial algebra on one generator $\mu$ of degree 2. Integrally:

$THH_n(\mathbb{F}_p) \cong \begin{cases} \mathbb{F}_p & n = 0 \\ 0 & n = 2k, k \geq 1 \\ \mathbb{Z}/p & n = 2k - 1, k \geq 1 \end{cases}$

The class $\mu \in THH_2(\mathbb{F}_p; \mathbb{F}_p)$ is called the Bokstedt element.

Comparison with classical Hochschild homology

RemarkClassical vs topological

The classical Hochschild homology gives:

$HH_n(\mathbb{F}_p/\mathbb{Z}) \cong \begin{cases} \mathbb{F}_p & n = 0 \\ 0 & n \geq 1 \end{cases}$

since $\mathbb{F}_p$ is a perfect field and $\Omega^1_{\mathbb{F}_p/\mathbb{Z}} = 0$ .

But $THH_*(\mathbb{F}_p) \neq HH_*(\mathbb{F}_p/\mathbb{Z})$ : the topological version sees torsion phenomena from "base-changing" over the sphere spectrum $\mathbb{S}$ instead of $\mathbb{Z}$ . The nontrivial groups in $THH_*(\mathbb{F}_p)$ arise from the nontrivial homotopy groups of $\mathbb{S}$ (the stable homotopy groups of spheres) interacting with $\mathbb{F}_p$ .

This is the crucial innovation of the "brave new algebra" approach: working over $\mathbb{S}$ instead of $\mathbb{Z}$ reveals arithmetic structure invisible to classical homological algebra.

Proof outline

Proof

The proof uses the spectral sequence from the filtration of $THH$ and explicit computations with the dual Steenrod algebra.

Step 1: The Bokstedt spectral sequence. There is a spectral sequence:

$E^2_{s,t} = HH_s(\pi_*(\mathbb{F}_p); \pi_*(\mathbb{F}_p)) \implies THH_{s+t}(\mathbb{F}_p; \mathbb{F}_p)$

This spectral sequence arises from filtering $THH$ by the "skeleton filtration" of the cyclic bar construction.

Step 2: Identify the $E^2$ -page. We have $\pi_*(\mathbb{F}_p) = \mathbb{F}_p$ (concentrated in degree 0), so the $E^2$ -page is:

$HH_s(\mathbb{F}_p/\mathbb{F}_p; \mathbb{F}_p) \cong \begin{cases} \mathbb{F}_p & s = 0 \\ 0 & s \geq 1 \end{cases}$

Wait -- this is over $\mathbb{F}_p$ , but the base is the sphere spectrum, not $\mathbb{F}_p$ . The correct $E^2$ -page uses the "topological" Hochschild homology of $H\mathbb{F}_p$ relative to $\mathbb{S}$ , which involves the homotopy groups of $\mathbb{F}_p \wedge_\mathbb{S} \mathbb{F}_p$ .

By the Kunneth spectral sequence: $\pi_*(H\mathbb{F}_p \wedge_\mathbb{S} H\mathbb{F}_p) = \pi_*(H\mathbb{F}_p \wedge H\mathbb{F}_p)$ , which is the dual Steenrod algebra $\mathcal{A}_*^{\vee}$ .

Step 3: The dual Steenrod algebra. For $p = 2$ : $\mathcal{A}_*^{\vee} = \mathbb{F}_2[\xi_1, \xi_2, \ldots]$ with $|\xi_i| = 2^i - 1$ .

For $p$ odd: $\mathcal{A}_*^{\vee} = \mathbb{F}_p[\xi_1, \xi_2, \ldots] \otimes E(\tau_0, \tau_1, \ldots)$ with $|\xi_i| = 2p^i - 2$ and $|\tau_i| = 2p^i - 1$ .

The Bokstedt spectral sequence becomes:

$E^2 = HH_*(\mathcal{A}_*^{\vee}; \mathbb{F}_p) \implies THH_*(\mathbb{F}_p; \mathbb{F}_p).$

Step 4: Hochschild homology of the dual Steenrod algebra. Using the HKR theorem for smooth algebras (the polynomial part) and the standard computation for exterior algebras:

For $p$ odd: $HH_*(\mathcal{A}_*^{\vee}) \cong \mathcal{A}_*^{\vee} \otimes E(\sigma\xi_1, \sigma\xi_2, \ldots) \otimes \Gamma(\sigma\tau_0, \sigma\tau_1, \ldots)$

where $\sigma$ denotes the "suspension" operation and $\Gamma$ is the divided power algebra.

Step 5: Differentials and collapse. Bokstedt showed that the spectral sequence has differentials that kill most of the $E^2$ -page, leaving:

$E^\infty \cong \mathbb{F}_p[\mu]$

where $\mu$ is a permanent cycle of degree 2, arising from the class $\sigma\tau_0 \in E^2_{1, 2p-2}$ (for $p$ odd) after a specific differential pattern.

The key differential is $d^{2p-1}(\sigma\tau_0^{[p]}) = \sigma\xi_1 \cdot \mu$ (or similar), which triggers a cascade of differentials collapsing the spectral sequence to the polynomial algebra on $\mu$ .

Step 6: Extension problems. The multiplicative structure of $THH_*(\mathbb{F}_p; \mathbb{F}_p)$ is determined by the algebra structure of $E^\infty$ together with the fact that there are no non-trivial extensions (by degree considerations). This gives $THH_*(\mathbb{F}_p; \mathbb{F}_p) \cong \mathbb{F}_p[\mu]$ . $\square$

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Consequences

ExampleTHH of ℤ from Bökstedt periodicity

Using the cofiber sequence $\mathbb{S} \xrightarrow{p} \mathbb{S} \to \mathbb{S}/p \simeq H\mathbb{F}_p$ and the resulting exact triangle:

$THH(\mathbb{Z}) \xrightarrow{p} THH(\mathbb{Z}) \to THH(\mathbb{F}_p)$

one deduces $THH_*(\mathbb{Z})$ from Bokstedt periodicity. The long exact sequence:

$\cdots \to THH_{n+1}(\mathbb{F}_p) \to THH_n(\mathbb{Z}) \xrightarrow{p} THH_n(\mathbb{Z}) \to THH_n(\mathbb{F}_p) \to \cdots$

gives $THH_{2k-1}(\mathbb{Z}) \cong \mathbb{Z}/k$ (for $k \geq 1$ ), recovering Bokstedt's computation of $THH(\mathbb{Z})$ .

RemarkPrismatic and motivic filtrations

Recent work of Bhatt--Morrow--Scholze and Antieau--Mathew--Morrow--Nikolaus establishes that the Bokstedt element $\mu$ is intimately connected to the Breuil--Kisin twist in prismatic cohomology. The motivic filtration on $THH(\mathbb{F}_p)$ has graded pieces:

$\operatorname{gr}^i THH(\mathbb{F}_p) \simeq \Sigma^{2i} H\mathbb{F}_p$

and the polynomial generator $\mu$ corresponds to the fundamental class in $\operatorname{gr}^1$ . This connects Bokstedt periodicity to the crystalline/prismatic cohomology of $\mathbb{F}_p$ and provides a new perspective on why the polynomial structure arises.