<aside> ❕ Abstract. We give the definitions of $\text{THH}$ and $\text{TC}$ and statement of the Dundas-Goodwillie-McCarthy theorem. We outline the computations of $\text{THH}(\mathbb F_p)$ and $\text{TC}(\mathbb F_p)$, exhibiting redshift from height $-1$ to height $0$.

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Table of Contents

Today we'll talk about $\text{THH}$ and $\text{TC}$. $\text{TC}$ is some functor out of $\mathbb{E}_1$-ring spectra, defined in terms of $\text{THH}$, that receives a natural map from $K$-theory called the cyclotomic trace. This map is important to us because of the following theorem, which says that it induces an equivalence of relative $K$-theory and $\text{TC}$ for nilpotent immersions:

<aside> ❕ Theorem (Dundas-Goodwillie-McCarthy). For $B\to A$ a morphism of connective $\mathbb{E}_1$-ring spectra such that $\pi_0(B)\to \pi_0(A)$ is surjective with kernel a nilpotent ideal, the cyclotomic trace map $K\to \text{TC}$ induces an equivalence of spectra

$\hspace{5.75em}\ker\left(K(B)\to K(A)\right)\to \ker\left(\text{TC}(B)\to \text{TC}(A)\right)$

or equivalently, the diagram

$\hspace{11.5em}\begin{matrix} K(B)& \to & \text{TC}(B) \\ \downarrow & & \downarrow \\ K(A) & \to & \text{TC}(A)\end{matrix}$

is homotopy cartesian.

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This is useful for computing $K$-theory in many scenarios. The goal for today is to understand $\text{TC}(\mathbb{F}_p)$ and how it exhibits redshift from height $-1$ to height $0$. We'll start with a lot of necessary definitions.

Topological Hochschild Homology

$\text{TC}$ is a priori defined on a category of so-called cyclotomic spectra. To get a functor, also called $\text{TC}$, out of $\mathbb{E}_1$-ring spectra, we precompose with the functor $\text{THH}$ from $\mathbb{E}_1$-ring spectra to cyclotomic spectra. Let's explain all of these terms.

$\text{THH}$ is defined as a direct analog of the ordinary Hochschild homology $\text{HH}(A) = A\otimes_{A\otimes_\mathbb Z A^\text{op}}^\mathbb LA$ (simply replace $\mathbb{Z}$ by $\mathbb{S}$):

<aside> ❕ Definition. If $R$ is an $\mathbb{E}1$-algebra in spectra, the topological Hochschild homology $\text{THH}(R)$ of $R$ is $\hspace{10.75em}\text{THH}(R) = R\mathop\otimes\limits{R\mathop\otimes\limits_\mathbb{S}R^\text{op}}R$

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Note that for right and left $R$-modules $A, B$ in general we have

$$ A\otimes_R B \simeq \left|A\otimes_\mathbb S B\substack{\leftarrow\\[-0.2em]\leftarrow}A\otimes_\mathbb SR\otimes_\mathbb S B \substack{\leftarrow\\[-0.3em]\leftarrow\\[-0.3em]\leftarrow}A\otimes_\mathbb SR\otimes_\mathbb SR\otimes_\mathbb SB \substack{\leftarrow\\[-0.4em]\leftarrow\\[-0.4em]\leftarrow\\[-0.4em]\leftarrow}\ldots\right| $$

where the $|\ldots|$s really just denotes a colimit of a simplicial diagram.

taking $A,B = R$ we get a simplicial diagram of $R,R$-bimodules and the tensor product computing $\text{THH}$ becomes

$$ R\mathop\otimes\limits_{R\otimes_\mathbb SR^\text{op}}\left|R\otimes_\mathbb S R\substack{\leftarrow\\[-0.2em]\leftarrow}R\otimes_\mathbb SR\otimes_\mathbb S R\substack{\leftarrow\\[-0.3em]\leftarrow\\[-0.3em]\leftarrow}R\otimes_\mathbb SR\otimes_\mathbb SR\otimes_\mathbb SR \substack{\leftarrow\\[-0.4em]\leftarrow\\[-0.4em]\leftarrow\\[-0.4em]\leftarrow}\ldots\right| $$