Dundas--Goodwillie--McCarthy Theorem

The Dundas--Goodwillie--McCarthy theorem establishes that the cyclotomic trace map from algebraic K-theory to topological cyclic homology is an equivalence on relative terms for nilpotent extensions. This is the most important result connecting K-theory to trace methods and has enabled the most precise K-theory computations achieved to date.

Statement

Theorem8.2Dundas--Goodwillie--McCarthy

Let $f: A \to B$ be a map of connective ring spectra such that $\pi_0(f): \pi_0(A) \to \pi_0(B)$ is surjective with nilpotent kernel. Then the cyclotomic trace induces an equivalence of homotopy fibers:

$\operatorname{fib}(K(A) \to K(B)) \xrightarrow{\;\simeq\;} \operatorname{fib}(TC(A) \to TC(B)).$

Equivalently, the square

$\begin{array}{ccc} K(A) & \xrightarrow{\operatorname{trc}} & TC(A) \\ \downarrow & & \downarrow \\ K(B) & \xrightarrow{\operatorname{trc}} & TC(B) \end{array}$

is homotopy Cartesian. In particular, for all $n \in \mathbb{Z}$ :

$K_n(A, I) \cong TC_n(A, I)$

where $I = \ker(\pi_0(A) \to \pi_0(B))$ and $K_n(A, I) = \pi_n(\operatorname{fib}(K(A) \to K(B)))$ .

History and proof strategy

RemarkEvolution of the theorem

The theorem was proved in stages:

Goodwillie (1986): Proved the rational version: for a nilpotent extension $R \to R/I$ : $K_n(R, I) \otimes \mathbb{Q} \cong HC_{n-1}^-(R, I) \otimes \mathbb{Q}$ relating relative K-theory to relative negative cyclic homology. This was the first indication that trace methods capture K-theory for infinitesimal extensions.
McCarthy (1997): Extended to the integral statement for split nilpotent extensions (when $I^2 = 0$ and the extension splits).
Dundas (1997): Proved the theorem for arbitrary nilpotent extensions using Goodwillie calculus.
Dundas--Goodwillie--McCarthy (2013): Published the complete proof in their monograph, incorporating the full machinery of functor calculus.

Proof outline

Proof

The proof uses Goodwillie calculus (calculus of functors) to analyze the difference between K-theory and TC as functors of ring spectra.

Step 1: Calculus of functors setup. Both $K$ and $TC$ are functors from ring spectra to spectra. Goodwillie's calculus provides a "Taylor tower" approximation for such functors:

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

where $P_n F$ is the $n$ -th polynomial approximation to $F$ . For a nilpotent extension of order $N$ ( $I^N = 0$ ), the functor depends only on its $N$ -th Taylor approximation.

Step 2: First derivative. The derivative (linearization) of K-theory at a ring spectrum $A$ is:

$D_1 K(A; M) = THH(A; M) = THH(A) \wedge_{A \wedge A^{\text{op}}} M$

for a bimodule $M$ (this is a result of Dundas--McCarthy). Similarly, the derivative of $TC$ involves the derivatives of $THH$ , $(\cdot)^{hS^1}$ , and $(\cdot)^{tS^1}$ .

Step 3: The cyclotomic trace on derivatives. The key technical result is that the cyclotomic trace induces an equivalence on first derivatives:

$D_1 K(A; M) \xrightarrow{\sim} D_1 TC(A; M)$

This is because both derivatives are computed by $THH(A; M)$ with compatible cyclotomic structure.

Step 4: Higher derivatives. For the $n$ -th derivative, both $K$ and $TC$ have the same polynomial approximations. This uses the observation that:

The $n$ -th derivative $D_n K$ is related to the $n$ -th power of $THH$ .
The cyclotomic trace respects the filtration by polynomial degree.

Step 5: Nilpotent extensions and convergence. For a nilpotent extension $A \to B$ with $I = \ker(\pi_0 A \to \pi_0 B)$ satisfying $I^N = 0$ :

The relative terms $K(A, I)$ and $TC(A, I)$ depend only on the $N$ -th polynomial approximation $P_N$ . Since the cyclotomic trace induces an equivalence on all polynomial approximations (by Step 4), it induces an equivalence:

$P_N K(A, I) \xrightarrow{\sim} P_N TC(A, I).$

Step 6: Convergence of the tower. For nilpotent $I$ , the Taylor tower converges: $F(A, I) = P_N F(A, I)$ when $I^{N+1} = 0$ . This gives the desired equivalence $K(A, I) \simeq TC(A, I)$ . $\square$

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Applications

ExampleK-theory of truncated polynomial algebras

For $A = k[x]/(x^n)$ with $k$ a perfect field of characteristic $p > 0$ , set $I = (x)$ and $B = k$ . The theorem gives:

$K_m(k[x]/(x^n), (x)) \cong TC_m(k[x]/(x^n), (x))$

Hesselholt--Madsen computed $TC_*(k[x]/(x^n))$ explicitly. For $n = 2$ ( $k = \mathbb{F}_p$ , dual numbers):

$K_m(\mathbb{F}_p[\epsilon], (\epsilon)) \cong \begin{cases} 0 & m \leq 0 \\ \mathbb{Z}/p^{v_p(j)} & m = 2j-1 \\ 0 & m \text{ even} \end{cases}$

where $v_p(j)$ is the $p$ -adic valuation. These groups detect the nilpotent structure.

Examplep-adic K-theory

For the $p$ -adic integers $\mathbb{Z}_p$ , using the surjection $\mathbb{Z}_p \to \mathbb{F}_p$ (kernel = $(p)$ , which is topologically nilpotent):

After $p$ -completion, the Dundas--Goodwillie--McCarthy theorem gives:

$\operatorname{fib}(K(\mathbb{Z}_p)_p^\wedge \to K(\mathbb{F}_p)_p^\wedge) \simeq \operatorname{fib}(TC(\mathbb{Z}_p; p) \to TC(\mathbb{F}_p; p))$

Since $K(\mathbb{F}_p)$ is known (Quillen) and $TC$ of both sides is computable, this determines $K_*(\mathbb{Z}_p)$ after $p$ -completion. Combined with the rational computation (Borel), one obtains complete information about $K_n(\mathbb{Z}_p)$ for all $n$ .

RemarkBeyond nilpotent extensions

The Dundas--Goodwillie--McCarthy theorem requires the kernel to be nilpotent. For non-nilpotent extensions, the cyclotomic trace is not an equivalence on relative terms, but it is still a very good approximation.

The Clausen--Mathew--Morrow refinement shows that for certain "henselian" situations, a modified version of the comparison holds. The emerging picture from prismatic cohomology suggests that the "correct" generalization involves the motivic filtration on TC, whose graded pieces are related to prismatic/syntomic cohomology.