Topological Cyclic Homology (TC)

Topological cyclic homology is the most powerful computable approximation to algebraic K-theory. Through the cyclotomic trace map $K \to TC$ , many important K-theory computations have been achieved, especially in the $p$ -complete setting. The Nikolaus--Scholze reformulation has made TC more accessible and computationally tractable.

The Nikolaus--Scholze formula

Definition8.8TC via the Nikolaus--Scholze formula

For a cyclotomic spectrum $(T, \varphi)$ (a spectrum $T$ with $S^1$ -action and Frobenius maps), the topological cyclic homology is:

$TC(T) = \operatorname{fib}\left(T^{hS^1} \xrightarrow{\varphi^{hS^1} - \operatorname{can}} T^{tS^1}\right)$

where:

$T^{hS^1} = \operatorname{Map}_{S^1}(ES^1_+, T)$ is the homotopy fixed points.
$T^{tS^1} = \operatorname{cofib}(T_{hS^1} \to T^{hS^1})$ is the Tate construction.
$\varphi^{hS^1}$ is induced by the cyclotomic Frobenius $\varphi: T \to T^{tC_p}$ (for a fixed prime $p$ , working $p$ -completely).
$\operatorname{can}: T^{hS^1} \to T^{tS^1}$ is the canonical map from fixed points to Tate.

For the application to K-theory: $T = THH(A)$ with its natural cyclotomic structure, giving:

$TC(A) = \operatorname{fib}\left(THH(A)^{hS^1} \xrightarrow{\varphi_p^{hS^1} - \operatorname{can}} THH(A)^{tS^1}\right)$

after $p$ -completion.

RemarkHomotopy fixed point spectral sequence

The homotopy fixed point spectral sequence for $THH(A)^{hS^1}$ :

$E_2^{s,t} = H^s(S^1; THH_t(A)) = H^s(BS^1; THH_t(A)) \implies TC^-_{t-s}(A)$

Since $H^*(BS^1; M) = M[[u]]$ for $|u| = 2$ when $M$ is a trivial module, the $E_2$ -page is:

$E_2^{2i, t} = THH_t(A), \quad E_2^{2i+1, t} = 0$

(for trivial $S^1$ -action on homotopy groups). The differentials and extensions encode the interaction between the $S^1$ -action and the homotopy of $THH$ .

Similarly, the Tate spectral sequence for $THH(A)^{tS^1}$ :

$\hat{E}_2^{s,t} = \hat{H}^s(S^1; THH_t(A)) \implies TP_{t-s}(A)$

uses Tate cohomology of $S^1$ .

Key computations

ExampleTC of 𝔽_p

The computation of $TC(\mathbb{F}_p; p)$ (Bokstedt--Hsiang--Madsen):

$TC_n(\mathbb{F}_p; p) \cong \begin{cases} \mathbb{Z}_p & n = 0, -1 \\ 0 & \text{otherwise} \end{cases}$

The computation proceeds as follows:

$THH_*(\mathbb{F}_p; \mathbb{F}_p) \cong \mathbb{F}_p[\mu] \otimes E(\epsilon)$ with $|\mu| = 2$ , $|\epsilon| = 1$ .
The $S^1$ -action on $THH$ gives $THH(\mathbb{F}_p)^{hS^1} \simeq \mathbb{Z}_p$ (the $p$ -adic integers, concentrated in degree 0) and $THH(\mathbb{F}_p)^{tS^1} \simeq \mathbb{Z}_p$ (via the Tate spectral sequence).
The map $\varphi^{hS^1} - \operatorname{can}$ is computed to be zero on $\pi_0$ and an isomorphism on higher degrees.
The fiber gives $TC_0 = \mathbb{Z}_p$ , $TC_{-1} = \mathbb{Z}_p$ .

This recovers: $K_0(\mathbb{F}_p) = \mathbb{Z}$ and confirms the cyclotomic trace is an equivalence $p$ -adically in this case.

ExampleTC of ℤ and K-groups of ℤ

Rognes computed $TC(\mathbb{Z}; p)$ for odd primes $p$ , and combined with the cyclotomic trace, obtained:

For odd primes $p$ and $n \geq 2$ , the $p$ -part of $K_n(\mathbb{Z})$ :

$K_n(\mathbb{Z}) \otimes \mathbb{Z}_p \cong TC_n(\mathbb{Z}; p) \otimes \mathbb{Z}_p \quad (\text{for } n \geq 0)$

This gives, for instance:

$K_3(\mathbb{Z}) \cong \mathbb{Z}/48$ : the 3-part is $\mathbb{Z}/3$ from $TC_3(\mathbb{Z}; 3)$ .
$K_7(\mathbb{Z}) \cong \mathbb{Z}/240$ : the $p$ -parts for $p = 2, 3, 5$ are computed via $TC$ .

The computation uses the "topological cyclic homology of $\mathbb{Z}$ " via the fiber sequence:

$TC(\mathbb{Z}; p) \to THH(\mathbb{Z})^{hS^1} \xrightarrow{\varphi_p - \operatorname{can}} THH(\mathbb{Z})^{tS^1}$

and the Bokstedt computation $THH_*(\mathbb{Z}; \mathbb{Z}/p) \cong \mathbb{F}_p[\mu_2]$ (polynomial on a degree-2 class).

The Dundas--Goodwillie--McCarthy theorem

Theorem8.1Dundas--Goodwillie--McCarthy

Let $A \to B$ be a map of connective ring spectra such that $\pi_0(A) \to \pi_0(B)$ is a surjection with nilpotent kernel $I$ (i.e., $I^N = 0$ for some $N$ ). Then the cyclotomic trace induces an equivalence on relative terms:

$K(A, I) \xrightarrow{\;\simeq\;} TC(A, I)$

where $K(A, I) = \operatorname{fib}(K(A) \to K(B))$ and $TC(A, I) = \operatorname{fib}(TC(A) \to TC(B))$ .

For ordinary rings: if $R \to R/I$ with $I$ nilpotent, then $K_n(R, I) \cong TC_n(R, I)$ for all $n$ .

RemarkSignificance and applications

The Dundas--Goodwillie--McCarthy theorem is the most important structural result relating K-theory and TC. Combined with computability of TC, it yields:

K-theory of truncated polynomial rings: $K_*(k[x]/(x^n))$ for a field $k$ and $n \geq 2$ is computed via $TC_*(k[x]/(x^n))$ . For $k = \mathbb{F}_p$ and $n = 2$ : the relative groups $K_*(k[\epsilon], (\epsilon))$ with $\epsilon^2 = 0$ are computed by Hesselholt--Madsen, giving explicit torsion groups.
K-theory of $p$ -adic integers: $K_*(\mathbb{Z}_p)$ is computed via $TC_*(\mathbb{Z}_p)$ using the fact that $\mathbb{Z}_p$ is the $p$ -completion of $\mathbb{Z}$ , and the "reduction modulo $p$ " map $\mathbb{Z}_p \to \mathbb{F}_p$ has nilpotent kernel (after completion).
Algebraic K-theory of local fields: Hesselholt--Madsen computed $K_*(\mathbb{Z}_p)$ completely, and the result matches the etale K-theory predictions from the Quillen--Lichtenbaum conjecture.
Prismatic cohomology connection: Bhatt--Morrow--Scholze showed that the fiber of $TC \to THH^{hS^1}$ is related to prismatic cohomology, providing a new bridge between K-theory and $p$ -adic Hodge theory.