Construction of the Cyclotomic Trace

The cyclotomic trace $\operatorname{trc}: K(A) \to TC(A)$ is the principal bridge between algebraic K-theory and topological cyclic homology. We describe its construction, from the Dennis trace to the refined cyclotomic version, and explain why it captures so much of the structure of K-theory.

From the Dennis trace to the cyclotomic trace

Definition8.9The Dennis trace

The Dennis trace $\operatorname{tr}: K(A) \to THH(A)$ is constructed as follows. Recall that $K(A) = \Omega |wS_\bullet \mathcal{P}(A)|$ (using Waldhausen's S-construction on perfect $A$ -modules) and $THH(A) = |N_\bullet^{\text{cyc}}(A)|$ .

Step 1: Matrix level. For each $n$ , there is a map:

$BGL_n(A) \to |N_\bullet^{\text{cyc}}(M_n(A))|$

sending a matrix $g \in GL_n(A)$ to the 1-simplex $g \in N_1^{\text{cyc}}(M_n(A)) = M_n(A)$ .

Step 2: Morita invariance. The trace map $\operatorname{tr}: M_n(A) \to A$ , $M \mapsto \sum M_{ii}$ induces $THH(M_n(A)) \to THH(A)$ (by Morita invariance of THH).

Step 3: Stabilization. Taking the colimit $n \to \infty$ and applying the plus construction:

$BGL(A)^+ \to THH(A)$

This is the Dennis trace on spaces. At the spectrum level, using the S-construction:

$\operatorname{tr}: \mathbf{K}(A) \to THH(A)$

is a map of spectra.

Upgrading to the cyclotomic trace

Definition8.10The cyclotomic trace

The cyclotomic trace is a refinement of the Dennis trace that factors through $TC$ :

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

Its construction uses the fact that $K(A)$ has a "cyclotomic structure" compatible with $THH(A)$ :

Step 1: Equivariant refinement. The S-construction has a natural $S^1$ -equivariant structure coming from the cyclic symmetry of traces. Specifically, for each $p^n$ -simplex in $S_\bullet$ , there is a $C_{p^n}$ -equivariant map to the cyclic bar construction $N_\bullet^{\text{cyc}}$ .

Step 2: Fixed points. The Dennis trace refines to maps:

$K(A) \to THH(A)^{C_{p^n}}$

for each $n \geq 0$ , compatible with the restriction maps $R: THH(A)^{C_{p^{n+1}}} \to THH(A)^{C_{p^n}}$ and the Frobenius maps $F: THH(A)^{C_{p^{n+1}}} \to THH(A)^{C_{p^n}}$ .

Step 3: Taking the limit. The cyclotomic trace is the induced map to the homotopy limit:

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

(the classical Bokstedt--Hsiang--Madsen definition of TC).

Step 4: Nikolaus--Scholze reformulation. Equivalently:

$\operatorname{trc}: K(A) \to TC(A) = \operatorname{fib}\left(THH(A)^{hS^1} \xrightarrow{\varphi^{hS^1} - \operatorname{can}} THH(A)^{tS^1}\right)$

where the map factors through $K(A) \to THH(A)^{hS^1}$ (the "negative cyclic trace") and the fiber condition is automatically satisfied by K-theory.

Why the cyclotomic trace works

Proof

We explain why the cyclotomic trace captures relative K-theory for nilpotent extensions.

Key idea: Goodwillie calculus. The functors $K$ and $TC$ are both "finitary" functors from ring spectra to spectra. Goodwillie's calculus of functors provides Taylor approximations:

$F(A) \to P_n F(A) \to P_{n-1} F(A) \to \cdots \to P_1 F(A) \to P_0 F(A)$

The cyclotomic trace $K \to TC$ induces maps on all polynomial approximations.

Step 1: Linear approximation. The first derivative of $K$ at $A$ is:

$D_1 K_A(M) = THH(A; M)$

for a bimodule $M$ . This is essentially the Dennis trace on the linearized level.

The first derivative of $TC$ at $A$ is also $THH(A; M)$ (with the appropriate cyclotomic structure). The cyclotomic trace induces the identity on first derivatives.

Step 2: Higher derivatives. More generally, for any $n$ , the $n$ -th cross-effect of $K$ and $TC$ agree, and the cyclotomic trace induces an equivalence on all multilinear layers.

This is because:

The $n$ -th derivative of $K$ involves "multi-THH" (the Hochschild homology of $n$ bimodules simultaneously).
The cyclotomic trace respects this multi-linear structure.
At each level, the Frobenius and restriction maps of the cyclotomic structure provide the necessary compatibility.

Step 3: Convergence for nilpotent extensions. If $I = \ker(A \to B)$ with $I^{N+1} = 0$ , then:

$K(A, I) \simeq P_N K_B(I) \quad \text{and} \quad TC(A, I) \simeq P_N TC_B(I)$

Since the cyclotomic trace induces equivalences on all polynomial approximations $P_n$ , we get $K(A, I) \simeq TC(A, I)$ .

Why this is special to the cyclotomic trace: The Dennis trace $K \to THH$ captures only the first derivative. The cyclotomic trace, by incorporating the $S^1$ -action and Frobenius, captures all derivatives. This is because the cyclotomic structure on $THH$ "remembers" the higher-order contributions that a single trace would miss. $\square$

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State of the art

RemarkModern developments

The cyclotomic trace has been refined and generalized in several directions:

Motivic filtration on TC (Bhatt--Morrow--Scholze): There is a filtration $\operatorname{Fil}^*_{\text{mot}} TC(A; \mathbb{Z}_p)$ whose graded pieces are: $\operatorname{gr}^n_{\text{mot}} TC(A; \mathbb{Z}_p) \simeq \mathbb{Z}_p(n)(A)[2n]$ where $\mathbb{Z}_p(n)$ is related to prismatic/syntomic cohomology. This provides a "motivic spectral sequence" for TC converging to K-theory: $E_2^{i,j} = H^{i-j}_{\text{syn}}(A, \mathbb{Z}_p(j)) \implies K_{-i-j}(A; \mathbb{Z}_p).$
Clausen--Scholze condensed approach: Using condensed mathematics, the cyclotomic trace can be extended to non-discrete rings (e.g., Banach algebras, $\ell^1$ -algebras), connecting K-theory to analytic invariants.
Even filtration (Hahn--Raksit--Wilson): A refinement of the motivic filtration using the "even filtration" on $THH$ , which provides sharper computations in low degrees.
Trace methods for ring spectra: The cyclotomic trace extends to $\mathbb{E}_\infty$ -ring spectra, and computations of $K(\ell)$ , $K(ko)$ , $K(tmf)$ (K-theory of chromatic ring spectra) use these trace methods.

ExampleSummary: the trace method pipeline

The standard approach to computing $K_n(R)$ via trace methods:

Compute $THH_*(R)$ using the Bokstedt spectral sequence.
Determine the $S^1$ -action and Frobenius on $THH(R)$ .
Compute $THH(R)^{hS^1}$ and $THH(R)^{tS^1}$ using the homotopy fixed point and Tate spectral sequences.
Compute $TC(R)$ as the fiber of $\varphi^{hS^1} - \operatorname{can}$ .
Identify $K(R, I) \cong TC(R, I)$ for a suitable nilpotent extension via Dundas--Goodwillie--McCarthy.
Use the known $K$ -groups of the quotient $R/I$ (often a field) and the long exact sequence to recover $K_*(R)$ .

This pipeline has successfully computed $K_*(\mathbb{Z}_p)$ , $K_*(k[x]/(x^n))$ , $K_*(\mathcal{O}_K)$ for local fields $K$ , and many other rings of arithmetic interest.