Topological Hochschild Homology (THH)

Topological Hochschild homology (THH) is a refinement of Hochschild homology that replaces the ground ring $\mathbb{Z}$ with the sphere spectrum $\mathbb{S}$ . This "brave new algebra" approach captures torsion information invisible to ordinary Hochschild homology and serves as the foundation for topological cyclic homology (TC), the most powerful computable approximation to algebraic K-theory.

Definition

Definition8.5Topological Hochschild homology

For a ring spectrum $A$ (or more concretely, an $\mathbb{S}$ -algebra), the topological Hochschild homology $THH(A)$ is defined as the geometric realization of the cyclic bar construction:

$THH(A) = |N_\bullet^{\text{cyc}}(A)| = \left|\cdots \rightrightarrows A \wedge A \rightrightarrows A\right|$

where $N_n^{\text{cyc}}(A) = A^{\wedge (n+1)}$ (the $(n+1)$ -fold smash product) with face maps given by multiplication and the cyclic structure.

For an ordinary ring $R$ , we take $A = HR$ (the Eilenberg--MacLane spectrum) and define:

$THH(R) = THH(HR).$

The homotopy groups $THH_n(R) = \pi_n(THH(R))$ refine Hochschild homology: there is a natural map $THH_n(R) \to HH_n(R)$ which is an isomorphism for $\mathbb{Q}$ -algebras but differs integrally.

ExampleTHH of the integers

The computation of $THH(\mathbb{Z})$ by Bokstedt is foundational:

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

Compare with $HH_n(\mathbb{Z}/\mathbb{Z}) = \begin{cases} \mathbb{Z} & n = 0 \\ 0 & n > 0 \end{cases}$ (trivially, since $\mathbb{Z}$ is commutative and $\Omega^n_{\mathbb{Z}/\mathbb{Z}} = 0$ ).

The nontrivial groups in $THH_*(\mathbb{Z})$ come from the "base change" from $\mathbb{Z}$ to $\mathbb{S}$ : the sphere spectrum has nontrivial higher homotopy groups ( $\pi_n^s$ = stable homotopy groups of spheres), and these contribute to $THH$ .

At a prime $p$ : $THH_*(\mathbb{Z}; \mathbb{Z}/p) \cong \mathbb{F}_p[\mu]$ where $|\mu| = 2$ (polynomial algebra on a degree-2 class).

ExampleTHH of 𝔽_p

For the finite field $\mathbb{F}_p$ :

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

where $|\mu| = 2$ , $|\epsilon| = 1$ , and $E(\epsilon)$ is the exterior algebra. This is a polynomial algebra on a degree-2 class tensored with an exterior algebra on a degree-1 class.

The class $\mu$ is the "Bokstedt element" and plays a role analogous to the Bott element in topological K-theory. Its existence is specific to the "brave new" setting and has no classical Hochschild analogue.

The $S^1$ -action and topological cyclic homology

Definition8.6Circle action on THH

The space $THH(A)$ carries a natural action of the circle group $S^1 = U(1)$ , coming from the cyclic structure of the bar construction. This $S^1$ -action is the analogue of the cyclic operator in classical cyclic homology.

From this action, we define:

Topological negative cyclic homology: $TC^-(A) = THH(A)^{hS^1}$ (homotopy fixed points).
Topological periodic cyclic homology: $TP(A) = THH(A)^{tS^1}$ (Tate construction).
Topological cyclic homology (new definition, Nikolaus--Scholze): $TC(A) = \operatorname{fib}\left(THH(A)^{hS^1} \xrightarrow{\varphi^{hS^1} - \operatorname{can}} THH(A)^{tS^1}\right)$ where $\varphi$ is the cyclotomic Frobenius.

RemarkThe cyclotomic structure

The key feature distinguishing $THH$ from $HH$ is the cyclotomic structure: for each prime $p$ , there is a Frobenius map

$\varphi_p: THH(A) \to THH(A)^{tC_p}$

from $THH(A)$ to the $C_p$ -Tate construction. Here $C_p \subset S^1$ is the cyclic subgroup of order $p$ .

The cyclotomic Frobenius $\varphi_p$ is analogous to the Frobenius endomorphism in $p$ -adic Hodge theory and is the key ingredient in defining $TC$ . The "fixed point" structure $THH(A)^{C_{p^n}}$ for all $n$ gives a tower:

$\cdots \to THH(A)^{C_{p^2}} \xrightarrow{R} THH(A)^{C_p} \xrightarrow{R} THH(A)$

where $R$ is the restriction map. The classical definition of $TC$ uses:

$TC(A; p) = \operatorname{holim}_{R, F} THH(A)^{C_{p^n}}$

with $R$ (restriction) and $F$ (Frobenius). Nikolaus--Scholze showed this agrees with the simpler fiber definition above.

The trace map to THH

Definition8.7The cyclotomic trace

The cyclotomic trace is a map of spectra:

$\operatorname{trc}: K(A) \to TC(A)$

from algebraic K-theory to topological cyclic homology. It factors as:

$K(A) \xrightarrow{\operatorname{tr}} THH(A)^{hS^1} = TC^-(A) \to TC(A)$

where the first map is the "topological Dennis trace" and the second is the canonical map from negative cyclic to cyclic homology.

The cyclotomic trace is the single most important computational tool in algebraic K-theory. By the Dundas--Goodwillie--McCarthy theorem, the cyclotomic trace determines the relative K-theory for nilpotent extensions.

ExampleTC of p-adic integers

For $A = \mathbb{Z}_p$ (the $p$ -adic integers), Hesselholt--Madsen computed:

$TC_n(\mathbb{Z}_p; p) \cong \begin{cases} \mathbb{Z}_p & n = 0, -1 \\ 0 & n \text{ even}, n < 0 \\ \mathbb{Z}_p & n = 2k-1, k \geq 1 \\ \mathbb{Z}/p^{v_p(k)} & n = 2k, k \geq 1 \end{cases}$

This matches the known computation of $K_n(\mathbb{Z}_p; \mathbb{Z}_p)$ (p-completed K-theory), confirming the cyclotomic trace is an equivalence in this case after $p$ -completion.

Topological Hochschild Homology (THH)

Definition

The S1S^1S1-action and topological cyclic homology

The trace map to THH

The $S^1$ -action and topological cyclic homology