On the K-theory of coordinate axes in affine space

TL;DR

This paper computes the relative K-groups of coordinate axes in affine space over perfect fields of positive characteristic and extends results to ind-smooth algebras over the rationals, linking to Witt vectors and de Rham forms.

Contribution

It provides explicit calculations of K_q(A_d, I_d) in terms of Witt vectors and de Rham forms, generalizing previous results for specific cases and characteristics.

Findings

K_q(A_d, I_d) expressed via p-typical Witt vectors for characteristic p>0

Explicit computation of K_q(A_d, I_d) over ind-smooth algebras in characteristic zero

Extension of known results to higher dimensions and more general base fields

Abstract

Let k be a perfect field of characteristic p>0, let A_d be the coordinate ring of the coordinate axes in affine d-space over k, and let I_d be the ideal defining the origin. We evaluate the relative K-groups K_q(A_d,I_d) in terms of p-typical Witt vectors of k. When d=2 the result is due to Hesselholt, and for K_2 it is due to Dennis and Krusemeyer. We also compute the groups K_q(A_d,I_d) in the case where k is an ind-smooth algebra over the rationals, the result being expressed in terms of algebraic de Rham forms. When k is a field of characteristic zero this calculation is due to Geller, Reid and Weibel.

Equations

$$K_{q}(A_{d},I_{d})\cong\left\{\begin{array}[]{ll}\prod_{\begin{subarray}{c}m^{\prime}\in J_{p}\\ \text{even}\end{subarray}}\prod_{\begin{subarray}{c}s^{\prime}\mid m^{\prime}\\ \text{even}\end{subarray}}\prod_{u\leq t_{ev}}W_{t_{ev}-u}(k)^{\oplus\operatorname{cyc}_{d}(p^{u}s^{\prime})}&\quad q=2r\\ \prod_{\begin{subarray}{c}m^{\prime}\in J_{p}\\ \text{odd}\end{subarray}}\prod_{s^{\prime}\mid m^{\prime}}\prod_{u\leq t_{od}}W_{t_{od}-u}(k)^{\oplus\operatorname{cyc}_{d}(p^{u}s^{\prime})}&\quad q=2r+1\end{array}\right.$$

$$K_{q}(A_{d},I_{d})\cong\left\{\begin{array}[]{ll}\prod_{\begin{subarray}{c}m^{\prime}\geq 1\\ \text{odd}\end{subarray}}\prod_{s^{\prime}\mid m^{\prime}}\prod_{1\leq u\leq t_{ev}}W_{t_{ev}-u}(k)^{\oplus\operatorname{cyc}_{d}(2^{u}s^{\prime})}&\quad q=2r\\ \prod_{\begin{subarray}{c}m^{\prime}\geq 1\\ \text{odd}\end{subarray}}\prod_{s^{\prime}\mid m^{\prime}}\prod_{0\leq v\leq t_{ev}}k^{\oplus\operatorname{cyc}_{d}(s^{\prime})}&\quad q=2r+1\end{array}\right.$$

$$B_d=k[x_1]\times \cdots \times k[x_d].$$

$$A_d$$

$$K(A_d)$$

$$K(A_d,B_d,I_d)\text{leavevmode}\to \text{leavevmode}K(A_d,I_d)$$

$$K_q(A_d)\cong K_q(k)\oplus K_q(A_d,I_d).$$

$$K_q(A,B,I,\mathbb{Z}/p^{\nu })\stackrel{\sim }{\text{leavevmode}\to \text{leavevmode}}{\mathrm{TC}}_q(A,B,I;p,\mathbb{Z}/p^{\nu })$$

$$K(A,B,I)\to \mathrm{TC}(A,B,I,p;{\mathbb{Z}}_p)$$

$$\displaystyle\operatorname{TC}_{q}(A_{d},B_{d},I_{d})\cong\left\{\begin{array}[]{ll}\prod_{\begin{subarray}{c}m^{\prime}\in J_{p}\\ \text{even}\end{subarray}}\prod_{\begin{subarray}{c}s^{\prime}\mid m^{\prime}\\ \text{even}\end{subarray}}\prod_{u\leq t_{ev}}W_{t_{ev}-u}(k)^{\oplus\operatorname{cyc}_{d}(p^{u}s^{\prime})}&\quad q=2r\\ \prod_{\begin{subarray}{c}m^{\prime}\in J_{p}\\ \text{odd}\end{subarray}}\prod_{s^{\prime}\mid m^{\prime}}\prod_{u\leq t_{od}}W_{t_{od}-u}(k)^{\oplus\operatorname{cyc}_{d}(p^{u}s^{\prime})}&\quad q=2r{+}1\end{array}\right.$$

$$\displaystyle\operatorname{TC}_{q}(A_{d},B_{d},I_{d})\cong\left\{\begin{array}[]{ll}\prod_{\begin{subarray}{c}m^{\prime}\geq 1\\ \text{odd}\end{subarray}}\prod_{s^{\prime}\mid m^{\prime}}\prod_{1\leq u\leq t_{ev}}W_{t_{ev}-u}(k)^{\oplus\operatorname{cyc}_{d}(2^{u}s^{\prime})}&\quad q=2r\\ \prod_{\begin{subarray}{c}m^{\prime}\geq 1\\ \text{odd}\end{subarray}}\prod_{s^{\prime}\mid m^{\prime}}\prod_{0\leq v\leq t_{ev}}k^{\oplus\operatorname{cyc}_{d}(s^{\prime})}&\quad q=2r{+}1.\end{array}\right.$$

$$\mathrm{THH}(A)$$

$${\mathrm{B}}^{\mathrm{cy}}(\Pi )[k]={\Pi }^{\wedge (k+1)}$$

$$d_i({\pi }_0\wedge \cdots \wedge {\pi }_m)$$

$$s_i({\pi }_0\wedge \cdots \wedge {\pi }_m)$$

$$t_m({\pi }_0\wedge \cdots \wedge {\pi }_m)$$

$${\psi }_p:{\mathrm{B}}^{\mathrm{cy}}(\Pi )\to {\mathrm{B}}^{\mathrm{cy}}(\Pi {)}^{C_p}$$

$$d_p(z)=\frac{1}{p}z\oplus \cdots \oplus \frac{1}{p}z.$$

$$D_p:|({\mathrm{sd}}_{p}X)[-]|\to |X[-]|$$

$$\widetilde{{\Delta }_p}:{\mathrm{B}}^{\mathrm{cy}}(\Pi )[-]\to {\mathrm{sd}}_p{\mathrm{B}}^{\mathrm{cy}}(\Pi )[-]$$

$$\overrightarrow{\pi }={\pi }_0\wedge \cdots \wedge {\pi }_m⟼\overrightarrow{\pi }\wedge \cdots \wedge \overrightarrow{\pi }\text{(p copies)}$$

$${\psi }_p:{\mathrm{B}}^{\mathrm{cy}}(\Pi )\to {\mathrm{B}}^{\mathrm{cy}}(\Pi {)}^{C_p}$$

$${\mathrm{B}}^{\mathrm{cy}}(\Pi )=|{\mathrm{B}}^{\mathrm{cy}}(\Pi )[-]|$$

$$\mathrm{THH}(\mathbb{S}(\Pi ))={\Sigma }^{\infty }{\mathrm{B}}^{\mathrm{cy}}(\Pi )$$

$$\mathrm{THH}(k)\otimes {\mathrm{B}}^{\mathrm{cy}}(\Pi )\stackrel{\sim }{\to }\mathrm{THH}(k(\Pi )).$$

$$\mathrm{THH}(k(\Pi ))=\mathrm{THH}(k\otimes \mathbb{S}(\Pi ))\simeq \mathrm{THH}(k)\otimes \mathrm{THH}(\mathbb{S}(\Pi )).$$

$$\mathrm{THH}(k(\Pi ))\simeq \mathrm{THH}(k)\otimes \mathbb{S}({\mathrm{B}}^{\mathrm{cy}}(\Pi ))$$

$${\Pi }^d$$

$$\omega :\{1,2,\dots ,m\}\to S.$$

$$\omega ({\pi }_0\wedge \cdots \wedge {\pi }_m)=\omega ({\pi }_0)\star \cdots \star \omega ({\pi }_m)$$

$${\mathrm{B}}^{\mathrm{cy}}({\Pi }^d,\overline{\omega })[m]\subseteq {\mathrm{B}}^{\mathrm{cy}}({\Pi }^d)[m]$$

Full text

On the $K$-theory of coordinate axes in affine space

Martin Speirs

1. Introduction

Let $k$ be a perfect field of characteristic $p>0$ and let $A_{d}$ denote the $k$-algebra $k[x_{1},\dots,x_{d}]/(x_{i}x_{j})_{i\neq j}$, which is the coordinate ring of the coordinate $(x_{1},\dots,x_{d})$-axes in affine $d$-space $\mathbb{A}^{d}=\operatorname{Spec}(k[x_{1},\dots,x_{d}])$ over $k$. This is an affine curve with a singularity at the origin, the singularity being defined by the ideal $I_{d}=(x_{1},\dots,x_{d})$. Our main result is the computation of the relative algebraic $K$-theory, $K(A_{d},I_{d})$ of the pair $(A_{d},I_{d})$. The relative $K$-theory is defined to be the mapping fiber of the map $K(A_{d})\to K(k)$ induced by the projection onto $k=A_{d}/I_{d}$.

To state the result we introduce some notation. We consider words in $d$ letters $x_{1},\dots,x_{d}$, i.e. a finite string $\omega=w_{1}w_{2}\dots w_{m}$ where each $w_{i}$ is one of the letters $x_{1},\dots,x_{d}$. A word $\omega=w_{1}w_{2}\dots w_{m}$ has no cyclic repetitions if $w_{i}\neq w_{i+1}$ for all $i=1,\dots,m-1$ and if $w_{m}\neq w_{1}$. For $d\geq 1$ and $s\geq 1$ let $\operatorname{cyc}_{d}(s)$ denote the number of cyclic words in $d$ letters, of length $s$, period $s$, having no cyclic repetitions. In Section 7 we give a formula for $\operatorname{cyc}_{d}(s)$. Suppose $m^{\prime}\geq 1$ and let $t_{ev}=t_{ev}(p,r,m^{\prime})$ be the unique positive integer such that $p^{t_{ev}-1}m^{\prime}\leq 2r<p^{t_{ev}}m^{\prime}$, or zero if such $t_{ev}$ does not exist. Let $t_{od}=t_{od}(p,r,m^{\prime})$ be the unique positive integer such that $p^{t_{od}-1}m^{\prime}\leq 2r+1<p^{t_{od}}m^{\prime}$, or zero if such $t_{od}$ does not exist. Let $J_{p}$ denote the set of positive integers $m^{\prime}\geq 1$ which are coprime to $p$.

Theorem 1**.**

Let $k$ be a perfect field of characteristic $p>0$. Let $A_{d}$ be the ring $k[x_{1},\dots,x_{d}]/(x_{i}x_{j})_{i\neq j}$ of coordinate axes, and let $I_{d}=(x_{1},\dots,x_{d})$. Then if $p>2$

[TABLE]

If $p=2$ then

[TABLE]

In both cases $t_{ev}=t_{ev}(p,r,m^{\prime})$ and $t_{od}=t_{od}(p,r,m^{\prime})$ are as defined above.

Note that the products appearing in the statement are all finite since for $m^{\prime}$ large enough $t_{ev}=t_{od}=0$.

The result extends the calculation by Dennis and Krusemeyer when $q=2$, [5, Theorem 4.9]. Hesselholt carried out the computation in the case $d=2$ in [14]. Our strategy follows the one in [14], except that we use the framework for $\operatorname{TC}$ set up by Nikolaus and Scholze [21] to which we refer for background on $\operatorname{TC}$ and cyclotomic spectra.

The analogous computation when working with a base field of characteristic zero was carried out by Geller, Reid and Weibel in [9, Theorem 7.1.]. In Section 6 we provide an extension of their calculations to the case of ind-smooth ${\mathbb{Q}}$-algebras.

Our proof of Theorem 1 uses the cyclotomic trace map from $K$-theory to topological cyclic homology, $\operatorname{TC}$. See also [15] for background on cyclotomic spectra and for similar calculations. We remark that Hesselholt and Nikolaus [17] have completed a calculation for $K$-theory of cuspical curves using similar methods as this paper.

1.1. Overview

In Section 2 we reduce the $K$-theory computation to a $\operatorname{TC}$ computation. Section 3 carries out the requisite $\operatorname{THH}$ computation. Then Section 4 and Section 5 assemble this to complete the proof. Section 4 contains a new method for computing $\operatorname{TP}$, which makes crucial use of the Nikolaus-Scholze framework, see also [24] for a similar calculation. In Section 6 we consider the characteristic zero situation and extend the computation of [9]. Finally in Section 7 we derived the necessary counting formula for cyclic words.

1.2. Acknowledgements

I am grateful to my advisor Lars Hesselholt for his guidance and support during the production of this paper. Special thanks are due to Fabien Pazuki for encouraging and useful conversations. I thank Benjamin Böhme, Ryo Horiuchi, Joshua Hunt, Mikala Jansen, Manuel Krannich, and Malte Leip for several useful conversations. It is a pleasure to thank Malte Leip for carefully reading a draft version of this paper and providing several corrections and suggestions. I thank the anonymous referee for a giving the paper a detailed reading, spotting several mistakes, and providing many helpful suggestions. This paper is based upon work supported by the DNRF Niels Bohr Professorship of Lars Hesselholt as well as the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California during the Spring 2019 semester. Additionally part of the work was supported by the Independent Research Fund Denmark, Grant No. 9056-00018B with mentorship from Martin Olsson.

2. Bi-relative $K$-theory and topological cyclic homology

The normalization of $A_{d}$ is just $d$ disjoint lines, whose coordinate ring is

[TABLE]

Gluing these lines together at $x_{1}=x_{2}=\dots=x_{d}=0$ one obtains $\operatorname{Spec}(A_{d})$. Algebraically this is the statement that the following square is a pullback of rings

[TABLE]

Here the horizontal maps take the variables $x_{i}$ to zero. The left-vertical map is the normalization map. It maps $x_{i}$ to $(0,\dots,x_{i}\dots,0)$ where $x_{i}$ is in the $i$’th position. The right-vertical map is the diagonal. If $K$-theory preserved pullbacks then the diagram would give a computation of $K(A_{d})$ in terms of $K(B_{d})$ and $K(k)$. Using the fundamental theorem of $K$-theory (since $k$ is regular) one would get a formula for $K(A_{d})$ purely in terms of $K(k)$. But $K$-theory does not preserve pullbacks. However, there is still something to be done. We can form the bi-relative $K$-theory, $K(A_{d},B_{d},I_{d})$ as the iterated mapping fiber of the diagram

[TABLE]

Again, if $K$-theory preserved pullbacks then $K(A_{d},B_{d},I_{d})$ would be trivial, but (as we shall see) it is not. Since $k$ is regular the fundamental theorem of algebraic $K$-theory [22, Section 6], and the fact that $K$-theory does preserve products, shows that the canonical map

[TABLE]

is an equivalence. Here $K(A_{d},I_{d})$ is the relative $K$-theory spectrum, i.e. the mapping fiber of the map $K(A_{d})\to K(A_{d}/I_{d})$ induced by the quotient. Any splitting of the quotient map $A_{d}\to A_{d}/I_{d}$ provides a splitting of $K$-groups

[TABLE]

So if we can compute $K_{q}(A_{d},I_{d})=K_{q}(A_{d},B_{d},I_{d})$ we have a computation of $K_{q}(A_{d})$. This is what we will do.

Geisser and Hesselholt [7] have shown that the cyclotomic trace induces an isomorphism

[TABLE]

between the bi-relative $K$-theory and the bi-relative topological cyclic homology, when both are considered with ${\mathbb{Z}}/p^{v}$ coefficients. The corresponding statement with rational coefficients (using Connes-Tsygan negative cyclic homology and the Chern character) was proven by Cortiñas in [3].

In fact it suffices to compute $\operatorname{TC}(A_{d},B_{d},I_{d},p;{\mathbb{Z}}_{p})$ since the trace map

[TABLE]

is an equivalence whenever $p$ is nilpotent in $A$, as shown in [8, Theorem C]. Furthermore, since $\operatorname{THH}(A_{d},B_{d},I_{d};p)$ is an $Hk$-module, it is in particular $p$-complete, and so $\operatorname{TC}(A,B,I,p;{\mathbb{Z}}_{p})\simeq\operatorname{TC}(A,B,I,p)$ (see [21, Section II.4]). Thus, to prove Theorem 1 it suffices to prove the following result.

Theorem 2**.**

Let $k$ be a perfect field of characteristic $p>0$. Let $A_{d}$ be the ring $k[x_{1},\dots,x_{d}]/(x_{i}x_{j})_{i\neq j}$ of coordinate axes, $B_{d}=k[x_{1}]\times\dots\times k[x_{d}]$ and let $I_{d}=(x_{1},\dots,x_{d})$. Then if $p>2$

[TABLE]

If $p=2$ then

[TABLE]

3. $\operatorname{THH}$ and the cyclic bar construction

The spectrum $\operatorname{TC}(A_{d},B_{d},I_{d})$ is defined using topological Hochschild homology, so that is where we start. In this section we drop the subscript $d$, so that $A=A_{d}$, $B=B_{d}$ and $I=I_{d}$. The bi-relative topological Hochschild homology is the spectrum $\operatorname{THH}(A,B,I)$ defined as the iterated mapping fiber of the following diagram.

[TABLE]

The ring $A$ is a pointed monoid ring and, by Lemma 7 below, we may compute $\operatorname{THH}(A)$ in terms of the cyclic bar construction of the defining pointed monoid.

3.1. Unstable cyclotomic structure on the cyclic bar construction

Let $\Pi$ be a pointed monoid, that is a monoid object in the symmetric monoidal category of based spaces and smash product. The cyclic bar construction of $\Pi$ is the cyclic space $\operatorname{B^{cy}}(\Pi)[-]$ with

[TABLE]

and with the usual Hochschild-type structure maps.

[TABLE]

We write $\operatorname{B^{cy}}(\Pi)$ for the geometric realization of $\operatorname{B^{cy}}(\Pi)[-]$. It is a space with $\mathbb{T}$-action where $\mathbb{T}$ is the circle group (for a proof see for example [20, Theorem 7.1.4.]). Furthermore it is an unstable cyclotomic space, i.e. there is a map

[TABLE]

which is equivariant when the codomain is given the natural $\mathbb{T}/C_{p}$-action. This map goes back to [2, Section 2]. We briefly sketch the construction. The $C_{p}$-action on $\operatorname{B^{cy}}(\Pi)$ is not simplicial, but we can make it so by using the edge-wise subdivision functor $\operatorname{sd}_{p}:\Delta\to\Delta$ which is given by the $p$-fold concatenation, $\operatorname{sd}_{p}[m-1]=[m-1]\amalg\dots\amalg[m-1]$ and $\operatorname{sd}_{p}(\theta)=\theta\amalg\dots\amalg\theta$ for morphisms $\theta:[m-1]\to[n-1]$. Given a simplicial set $X[-]$ we let $\operatorname{sd}_{p}X[-]=X[-]\circ\operatorname{sd}^{op}_{p}$. For the topological simplex $\Delta^{m-1}\subset{\mathbb{R}}^{m}$ let $d_{p}:\Delta^{m-1}\to\Delta^{pm-1}$ be the diagonal embedding

[TABLE]

This induces a (non-simplicial) map on geometric realization

[TABLE]

by $id\times d_{p}:X[pm-1]\times\Delta^{m-1}\to X[pm-1]\times\Delta^{pm-1}$. Then [2, Lemma 1.1.] shows that $D_{p}$ is a homeomorphism – one need only check on the representables $\Delta^{k}[-]$ where it follows by explicit calculations. Furthermore, when $X[-]$ is a cyclic space, then $D_{p}$ is a $\mathbb{T}$-equivariant homoemorphism [2, Lemma 1.11.]. In the case $X[-]=\operatorname{B^{cy}}(\Pi)[-]$ one has available the simplicial diagonal

[TABLE]

given by

[TABLE]

It clearly lands in the $C_{p}$-fixed points of $\operatorname{sd}_{p}\operatorname{B^{cy}}(\Pi)[-]$ and induces an isomorphism $\tilde{\Delta_{p}}:\operatorname{B^{cy}}(\Pi)[-]\to\operatorname{sd}_{p}\operatorname{B^{cy}}(\Pi)[-]^{C_{p}}$. We define

[TABLE]

to be the composite

[TABLE]

where the middle map is the canonical equivalence witnessing the fact that geometric realization commutes with finite limits. As all three maps in the definition are $\mathbb{T}$-equivariant, so is $\psi_{p}$.

Passing to suspension spectra now gives us a cyclotomic structure on $\operatorname{THH}(\mathbb{S}(\Pi))$ where $\mathbb{S}(\Pi)=\Sigma^{\infty}\Pi$. Indeed, there is always a $\mathbb{T}/C_{p}$-equivariant map $\operatorname{B^{cy}}(\Pi)^{C_{p}}\to\operatorname{B^{cy}}(\Pi)^{hC_{p}}$. Composing with this map gives a $\mathbb{T}\simeq\mathbb{T}/C_{p}$-equivariant map $\psi^{h}:\operatorname{B^{cy}}(\Pi)\to\operatorname{B^{cy}}(\Pi)^{hC_{p}}$. Thus, we obtain a $\mathbb{T}\simeq\mathbb{T}/C_{p}$-equivariant map

[TABLE]

where the middle map is the canonical map from the suspension spectrum of a homotopy limit, to the homotopy limit of the suspension spectrum and the last map is the canonical map from the homotopy fixed points to the Tate construction.

3.2. $\operatorname{THH}$ of monoid algebras

Both of the algebras $A$ and $B$ (and their quotients by $I$) are pointed monoid algebras, enabling us to use the following splitting result.

Lemma 7**.**

[15, Theorem 7.1.]* Let $k$ be a ring, $\Pi$ a pointed monoid, and $k(\Pi)$ the pointed monoid algebra. Let $\operatorname{B^{cy}}(\Pi)$ be the cyclic bar-construction on $\Pi$. Then there is a $\mathbb{T}$-equivariant equivalence*

[TABLE]

Under this equivalence, the Frobenius on $\operatorname{THH}(k(\Pi))$ is identified with the tensor product of the Frobenius on $\operatorname{THH}(k)$ with the one on $\operatorname{THH}(\mathbb{S}(\Pi))$, and as such is an equivalence of cyclotomic spectra.

Proof.

Since $\operatorname{THH}$ is symmetric monoidal [21, Section IV.2.] we obtain

[TABLE]

Since $\operatorname{THH}(\mathbb{S}(\Pi))=\mathbb{S}(\operatorname{B^{cy}}(\Pi))$ we obtain

[TABLE]

as claimed. ∎

Let $\Pi^{d}=\{0,1,x_{1},x_{1}^{2},\dots,x_{2},x_{2}^{2},\dots,x_{d},x_{d}^{2},\dots\}$ be the multiplicative pointed monoid with basepoint [math] and multiplication determined by $x_{i}x_{j}=0$ when $i\neq j$. Then $A_{d}\cong k(\Pi^{d})$ is the pointed monoid algebra on $\Pi^{d}$. Let $\Pi^{1}=\{0,1,t,t^{2},\dots\}$ and $\Pi^{0}=\{0,1\}$, so $B=k(\Pi^{1})\times\cdots\times k(\Pi^{1})$ (with $d$ products) and $k=k(\Pi^{0})$. The diagram of cyclotomic spectra (5) is induced by the diagram of pointed monoids

[TABLE]

The map $\rho_{j}$ takes $x_{i}$ and to [math] when $i\neq j$ and takes $x_{j}$ to $t$. The map $\varepsilon$ takes the variables $x_{1},\dots,x_{d}$ and $t$ to [math]. The map $\Delta$ is the diagonal.

The cyclic bar-construction sometimes decomposes, as a pointed $\mathbb{T}$-space, into a wedge of spaces simple enough that one can understand their $\mathbb{T}$-homotopy type. To do this for $\operatorname{B^{cy}}(\Pi^{d})$ we need the notion of cyclic words.

We consider the alphabet $S=\{x_{1},\dots,x_{d}\}$ and words

[TABLE]

Here $m$ is the length of $\omega$. The cyclic group $C_{m}$ acts on the set of words of length $m$. An orbit for this action is called a cyclic word. Such a cyclic word $\overline{\omega}$ has a period namely the cardinality of the orbit. Words may be concatenated to give new, longer words, though concatenation is not well-defined for cyclic words. The empty word $\varnothing\to S$ is the unit for concatenation. It has length [math] and period $1$.

We can associate words to non-zero elements of $\operatorname{B^{cy}}(\Pi^{d})[m]$ as follows. If $\pi\in\Pi^{d}=\operatorname{B^{cy}}(\Pi^{d})[0]$ is non-zero then it is of the form $\pi=x_{j}^{l}$ for some $1\leq j\leq d$ and $l\geq 0$. Let $\omega(\pi)$ be the unique word of length $l$ all of whose letters are $x_{j}$. For example $\omega(x_{1}^{2})=x_{1}x_{1}$ and $\omega(1)=\varnothing$. Now for a non-zero element $\pi_{0}\wedge\dots\wedge\pi_{m}\in\operatorname{B^{cy}}(\Pi^{d})[m]$ we let

[TABLE]

be the concatenation of each of the words $\omega(\pi_{j})$. Note that $\omega(1)$ is the empty word, which is the unit for concatenation. For a cyclic word $\overline{\omega}$ we define

[TABLE]

to be the subset consisting of the basepoint and all elements $\pi_{0}\wedge\dots\wedge\pi_{m}$ such that

[TABLE]

The cyclic structure maps preserve this property and so, as $m\geq 0$ varies, this defines a cyclic subset

[TABLE]

Denote by $\operatorname{B^{cy}}(\Pi^{d},\overline{\omega})$ the geometric realization of this subset. We will also often abbreviate $B(\overline{\omega})=\operatorname{B^{cy}}(\Pi^{d},\overline{\omega})$. As $\overline{\omega}$ ranges over all cyclic words every non-zero $m$-simplex $\pi_{0}\wedge\dots\wedge\pi_{m}$ appears in exactly one such cyclic subset $\operatorname{B^{cy}}(\Pi^{d},\overline{\omega})$. Thus we get a decomposition

[TABLE]

indexed on the set of all cyclic words with letters in $S=\{x_{1},\dots,x_{d}\}$.

Lemma 9**.**

There is a canonical $\mathbb{T}$-equivariant equivalence

[TABLE]

where the sum on the left-hand side is indexed over all cyclic words whose period is $\geq 2$.

Proof.

We first note that $\operatorname{THH}$ preserves product of rings, in the sense that the canonical map $\operatorname{THH}(R\times R^{\prime})\to\operatorname{THH}(R)\times\operatorname{THH}(R^{\prime})$ is a $\mathbb{T}$-equivariant equivalence [6, Proposition 4.2.4.4.]. Using these equivalences we may consider the diagram induced from (5) using Lemma 7,

[TABLE]

$\operatorname{THH}(A,B,I)$ is the iterated mapping fiber of this diagram. The mapping fiber of the left-hand vertical map consists of two parts; the part of the wedge sum indexed on cyclic words containing at least two different letters from $S$ and a part on which $\varepsilon$ is an equivalence. The map $\varepsilon$ is trivial on this part indexed on cyclic words containing at least two different letters, finishing the claim. ∎

3.3. Homotopy type of $\operatorname{B^{cy}}(\Pi^{d};\overline{\omega})$.

In this section we determine the homotopy type of the subspaces $\operatorname{B^{cy}}(\Pi^{d},\overline{\omega})\subseteq\operatorname{B^{cy}}(\Pi^{d})$.

Definition 11**.**

We say that a word $\omega=w_{1}w_{2}\dots w_{m}$ has no cyclic repetitions if $w_{i}\neq w_{i+1}$ for all $i=1,\dots,m-1$ and if $w_{m}\neq w_{1}$. If on the other hand this is not satisfied, we say $\omega$ (or $\overline{\omega}$) has cyclic repetitions.

Lemma 12**.**

Let $\overline{\omega}$ be a cyclic word of period $s\geq 2$, with letters in the alphabet $S=\{x_{1},\dots,x_{d}\}$. The homotopy type of the pointed $\mathbb{T}$-space $\operatorname{B^{cy}}(\Pi^{d},\overline{\omega})$ is given as follows.

1. (1)

If $\overline{\omega}$ has length $m=si$ and has no cyclic repetitions then a choice of representative word $\omega\in\overline{\omega}$ determines a $\mathbb{T}$-equivariant homeomorphism

[TABLE]

where ${\mathbb{R}}[C_{m}]-1$ is the reduced regular representation of $C_{m}$. 2. (2)

If $\overline{\omega}$ has cyclic repetitions, then $\operatorname{B^{cy}}(\Pi^{d},\overline{\omega})$ is $\mathbb{T}$-equivariantly contractible.

Proof.

This proof follows closely that of [14, Lemma 1.6]. Let $\overline{\omega}$ be a cyclic word of length $m=si$ with period $s\geq 2$. Choose $\omega\in\overline{\omega}$ and let $(\pi_{0},\dots,\pi_{m-1})$ be the unique $m$-tuple of non-zero elements in $\Pi^{d}$ such that $\omega(\pi_{0}\wedge\dots\wedge\pi_{m-1})=\omega$. The pointed cyclic set $\operatorname{B^{cy}}(\Pi^{d},\overline{\omega})[-]$ is generated by the $(m-1)$-simplex $\pi_{0}\wedge\dots\wedge\pi_{m-1}$ and so by the Yoneda lemma there is a unique surjective map of pointed cyclic sets

[TABLE]

mapping the generator of $\Lambda^{m-1}[-]$ to the generator $\pi_{0}\wedge\dots\wedge\pi_{m-1}$. Since $\omega$ has period $s$, the generator $\pi_{0}\wedge\dots\wedge\pi_{m-1}$ is fixed by the cyclic operator $t_{m}^{s}$ and so $f_{\omega}$ factors over the quotient subgroup of order $i=m/s$,

[TABLE]

From [15, Section 7.2.] we have a $\mathbb{T}$-equivariant homeomorphism

[TABLE]

(where $\mathbb{T}$ acts on the second factor) where the dual cyclic operator acts on $\Delta^{m-1}$ by the affine map that cyclicly permutes the vertices and on $\mathbb{T}$ by rotation through $2\pi/m$. Thus $f_{\omega}$ gives a continuous $\mathbb{T}$-equivariant surjection

[TABLE]

If $\overline{\omega}$ has no cyclic repetitions then all the faces of the generator

[TABLE]

are the basepoint [math], and so $f_{\omega}$ collapses the entire boundary $\partial\Delta^{m-1}$ of $\Delta^{m-1}$ to the basepoint. There are no other relations, and so we have a $\mathbb{T}$-equivariant homeomorphism

[TABLE]

Now we use the identification of the $C_{m}$-space $\Delta^{m-1}/\partial\Delta^{m-1}$ with the one-point compactification of the reduced regular representation to finish the proof of part (1).

In case $\overline{\omega}$ does have a cyclic repetition, then $\pi_{0}\wedge\dots\wedge\pi_{m-1}$ will have a non-basepoint face. So $f_{\omega}$ collapses at least one codimension $1$ face (and its $\mathbb{T}$-orbit) to the basepoint, and leaves at least one codimension $1$ face, say $F\subseteq\Delta^{m-1}$, un-collapsed. The cone on $F$ is canonically homeomorphic to $\Delta^{m-1}$ and has a canonical null-homotopy given by shrinking down to the basepoint of the cone. This null-homotopy then induces a null-homotopy on $\operatorname{B^{cy}}(\Pi^{d},\overline{\omega})$. However, it may fail to be a $\mathbb{T}$-equivariant. To get this we note that there is a $C_{i}$-equivariant homeomorphism

[TABLE]

where $C_{i}$ acts on the left by cyclically permuting the factors of the join. Again $f_{\omega}$ will collapse at least one codimension $1$ face of $\Delta^{s-1}$ and leave at least one un-collapsed, say $F\subseteq\Delta^{s-1}$. Now the null-homotopy of $cone(F)$ will induce a $\mathbb{T}$-equivariant null-homotopy

[TABLE]

This completes the proof. ∎

Let ${\mathbb{C}}(i)$ denote the $1$-dimensional complex $\mathbb{T}$-representation where $z\in\mathbb{T}\subseteq{\mathbb{C}}$ acts through multiplication with the $i$’th power $z^{i}$. For $i\geq 1$ let $\lambda_{i}={\mathbb{C}}(1)\oplus\dots\oplus{\mathbb{C}}(i)$.

Lemma 13**.**

The regular representation ${\mathbb{R}}[C_{m}]$ is isomorphic to ${\mathbb{R}}\oplus\lambda_{\frac{m-2}{2}}\oplus{\mathbb{R}}_{-}$ if $m$ is even, and ${\mathbb{R}}\oplus\lambda_{\frac{m-1}{2}}$ if $m$ is odd.

Proof.

By Maschke’s theorem ${\mathbb{R}}[C_{m}]$ is a semisimple ring. In particular it decomposes as a sum of irreducible sub-representations. Furthermore it contains a copy of every irreducible $C_{m}$-representation (since for any such $V$ and any non-zero $v\in V$, there is a surjection ${\mathbb{R}}[C_{m}]\to V$ given by $\sum r_{g}g\mapsto\sum r_{g}gv$ so, by semisimplicity, $V$ embeds into ${\mathbb{R}}[C_{m}]$). Since ${\mathbb{R}}$ and ${\mathbb{C}}(i)$, for $1\leq i\leq\lfloor\frac{m-1}{2}\rfloor$, (and ${\mathbb{R}}_{-}$ in case $m$ is even) are irreducible, pair-wise non-isomorphic, and have real dimensions summing to $\dim_{{\mathbb{R}}}{\mathbb{R}}[C_{m}]=m$ this completes the proof. ∎

Lemma 14**.**

Let $\overline{\omega}$ be a cyclic word with no cyclic repetitions, of length $m$, period $s$ and with $i=m/s$ blocks.

1. (1)

If $s$ is even, then there is a $\mathbb{T}$-equivariant homeomorphism

[TABLE] 2. (2)

If both $s$ and $i$ are odd, then there is a $\mathbb{T}$-equivariant homeomorphism

[TABLE] 3. (3)

If $s$ is odd, and $i$ is even, then there is a $\mathbb{T}$-equivariant homeomorphism

[TABLE]

where ${\mathbb{R}}P^{2}(i)$ is by definition the mapping cone of the quotient map ${\mathbb{T}/C_{i/2}}_{+}\to{\mathbb{T}/C_{i}}_{+}$.

Proof.

We use Lemma 13 in each case.

1. (1)

If $s=2k$ is even then so is $m$ and so

[TABLE]

The restriction along the inclusion $C_{i}\subseteq C_{m}$ (given by $\sigma_{i}\mapsto\sigma_{m}^{s}$ on generators) turns the sign representation ${\mathbb{R}}_{-}$ into a trivial representation. Since ${\mathbb{C}}(\frac{m}{2})={\mathbb{C}}(ki)={\mathbb{R}}\oplus{\mathbb{R}}$ as $C_{i}$-representations, we have

[TABLE]

where the last isomorphism uses that $\lambda_{m/2}$ extends to a representation of $\mathbb{T}$ and so allows the $\mathbb{T}$-equivariant untwisting map $(x,z)\mapsto(xz,zC_{i})$. 2. (2)

If both $s$ and $i$ are odd then so is $m=si$ and so

[TABLE]

Then we proceed as above, using the untwisting map. 3. (3)

If $s$ is odd and $i$ is even, then $m$ is even and so

[TABLE]

The restriction along the inclusion $C_{i}\subseteq C_{m}$ leaves the sign representation unchanged. Now repeat the argument of [15, Cor. 7.2.].

∎

3.4. Homology of $\operatorname{B^{cy}}(\Pi^{d};\overline{\omega})$.

Proposition 15**.**

Let $R$ be any commutative ring. Let $\overline{\omega}$ be a cyclic word with no cyclic repetitions, of length $m$, period $s$ and with $i=m/s$ blocks. The singular homology $\tilde{H}_{*}(B(\overline{\omega});R)$ is concentrated in degrees $m-1$ and $m$. If either $s$ is even, or both $s$ and $i$ are odd, then the $R$-modules in degree $m-1$ and $m$ are free of rank $1$. If $s$ is odd and $i$ is even, then the $R$-module in degree $m-1$ is isomorphic to $R/2R$, and the $R$-module in degree $m$ is isomorphic to $R[2]$, the $2$-torsion in $R$.

Furthermore, when $m$ and $s$ have the same parity, Connes’ operator takes a generator $y_{\overline{\omega}}$ of the $R$-module $\tilde{H}_{m-1}(B(\overline{\omega});R)$ to $i$ times a generator $z_{\overline{\omega}}$ of $\tilde{H}_{m}(B(\overline{\omega});R)$, that is $d(y_{\overline{\omega}})=iz_{\overline{\omega}}$. When $s$ is odd and $i$ is even, Connes’ operator acts trivially.

Proof.

The homology computations follow directly from Lemma 14. We also deduce the behaviour of Connes’ operator from Lemma 14. In general for a pointed $\mathbb{T}$-space $X$, Connes’ operator $d:\tilde{H}_{*}(X;k)\to\tilde{H}_{*+1}(X;k)$ is given by taking the cross-product with the fundamental class $[\mathbb{T}]$ and then applying the map induced by the action $\mu:\mathbb{T}_{+}\wedge X\to X$ on reduced homology. We consider two cases.

1. (1)

When $X={\mathbb{T}/C_{i}}_{+}$ we claim that $d:\tilde{H}_{0}(X;k)\to\tilde{H}_{1}(X;k)$ is multiplication by $i$ (up to a unit). By definition it is given as the composite of two maps. The first is the map

[TABLE]

where $\mathbb{T}\times z_{0}C_{i}$ is the $1$-cycle that factors through the map

[TABLE]

Then Eq. 16 is composed with

[TABLE]

The resulting $1$-cycle in $X$ factors through the composite map

[TABLE]

which is a degree $i$ map. 2. (2)

When $X={\mathbb{R}}P^{2}(i)$ we claim that $d:\tilde{H}_{1}(X;k)\to\tilde{H}_{2}(X;k)$ is trivial. Consider the diagram

[TABLE]

Now $H_{1}((\mathbb{T}/C_{i})_{+})$ surjects onto $H_{1}({\mathbb{R}}P^{2}(i))$ and the above diagram commutes. Since $H_{2}((\mathbb{T}/C_{i})_{+})=0$ the claim follows.

∎

Remark 17*.*

From this proposition we see that if the characteristic of $k$ is different than $2$ then $\tilde{H}_{*}(B(\overline{\omega};k)$ is trivial when $s$ is odd and $i$ is even. On the other hand, if $k$ has characteristic $2$ then $k/2k=k[2]=k$. This explains why the combinatorics of Geller, Reid, and Weibel in [9], avoids cyclic words whose period is not congruent (mod $2$) to the length [9, Remark 3.9.1.] since they work over characteristic zero fields.

3.5. $\operatorname{THH}$ of the coordinate axes

We put together the various results of the previous sections to describe $\operatorname{THH}(A,B,I)$ as a cyclotomic spectrum. Again, let $A=A_{d},B=B_{d}$ and $I=I_{d}$.

We will use the Segal conjecture for $C_{p}$. This is the statement that the map

[TABLE]

identifies the codomain as the $p$-completion of the domain. For a proof of this see [21, Theorem III.1.7], though the result was originally proved by Lin [19] (for $p=2$) and Gunawardena [11] (for $p$ odd) in 1980.

Lemma 18**.**

Let $T$ be a bounded below spectrum with $C_{p}$-action and $X$ a finite pointed $C_{p}$-CW-complex. Then the lax symmetric monoidal structure map

[TABLE]

is an equivalence.

Proof.

Since both $T^{tC_{p}}\otimes(-)^{tC_{p}}$ and $(T\otimes-)^{tC_{p}}$ are exact functors we may reduce to checking the statement for the $C_{p}$-spectra $\mathbb{S}$ and $\mathbb{S}\otimes{C_{p}}_{+}$. This is because $\Sigma^{\infty}X$ may be constructed from $\mathbb{S}$ and $\mathbb{S}\otimes{C_{p}}_{+}$ using finitely many cofiber sequences (since $X$ is built by attaching finitely many $C_{p}$-cells). Replacing $\Sigma^{\infty}X$ by $\mathbb{S}$ the map in question reduces to

[TABLE]

where we use the Segal conjecture to identify $\mathbb{S}^{tC_{p}}\simeq\mathbb{S}^{\hat{}}_{p}$. Since $T$ is bounded below it follows from [21, Lemma I.2.9]) that $T^{tC_{p}}$ is $p$-complete and so the map is an equivalence. Replacing $\Sigma^{\infty}X$ by $\mathbb{S}\otimes{C_{p}}_{+}$ instead we see that both the domain and codomain of the map are zero, since $(-)^{tC_{p}}$ kills induced spectra as well as spectra of the form $T\otimes Z$ where $Z$ is induced (cf. [21, Lemma I.3.8. (i) and (ii)]). ∎

We will describe the Frobenius map on

[TABLE]

in terms of the splitting of $\operatorname{B^{cy}}(\Pi^{d})$ into the $\mathbb{T}$-equivariant subspaces $B(\overline{\omega})$. The unstable Frobenius $\psi_{p}:\operatorname{B^{cy}}(\Pi^{d})\to\operatorname{B^{cy}}(\Pi^{d})^{C_{p}}$ (defined in Section 3.1) restricts to a homeomorphism

[TABLE]

landing in the subspace $B(\overline{\omega^{\star p}})$ corresponding to the cyclic word $\overline{\omega^{\star p}}$ which has length $pm$ and period $s$ (if $\overline{\omega}$ has length $m$ and period $s$).

Proposition 19**.**

There is a $\mathbb{T}$-equivariant equivalence of spectra

[TABLE]

where the sum is indexed over cyclic words of length $m\geq 2$, having no cyclic repetitions. Under this equivalence the Frobenius map restricts to the map

[TABLE]

where the second map is the lax symmetric monoidal structure of the Tate-$C_{p}$-construction. This second map is an equivalence, while the restricted Frobenius $\tilde{{\varphi}_{p}}:\Sigma^{\infty}B(\overline{\omega})\to(\Sigma^{\infty}B(\overline{\omega^{\star p}}))^{tC_{p}}$ is a $p$-adic equivalence.

Proof.

Applying Lemma 18 with $T=\operatorname{THH}(k)$ and $X=B(\overline{\omega^{\star p}})$ we get that the map

[TABLE]

is an equivalence. It remains to show that

[TABLE]

is a $p$-adic equivalence. To do this we factor it as follows. To ease notation, let $Y=|B(\overline{\omega^{\star p}})[-]|$. By definition $\tilde{\varphi}_{p}$ factors as

[TABLE]

where $\tilde{\Delta_{p}}$ (the space-level diagonal) and $D_{p}$ are homeomorphisms as remarked in Section 3.1. The final map (denoted by $\tilde{\varphi^{\prime}_{p}}$) is the composition

[TABLE]

This map fits into the following commutative diagram

{\mathbb{S}\otimes Y^{C_{p}}}$${(\mathbb{S}\otimes Y)^{tC_{p}}}  ${(\mathbb{S}\otimes Y^{C_{p}})^{hC_{p}}}$  {\mathbb{S}^{tC_{p}}\otimes Y^{C_{p}}}$${(\mathbb{S}\otimes Y^{C_{p}})^{tC_{p}}}  \scriptstyle{\Delta_{p}\otimes id}$$\scriptstyle{\tilde{\varphi^{\prime}_{p}}}$$\scriptstyle{\operatorname{can}}$$\scriptstyle{\operatorname{can}\circ\ i}$$\scriptstyle{(1.)}$$\scriptstyle{(2.)}

Here $\Delta_{p}:\mathbb{S}\to\mathbb{S}^{tC_{p}}$ is a $p$-adic equivalence by the Segal conjecture. The map labelled $(1.)$ is the equivalence witnessing that $(-)^{tC_{p}}$ is an exact functor, and $B(\overline{\omega^{\star p}})^{C_{p}}$ is finite and has trivial $C_{p}$-action. The left-most square commutes by construction of the map $(1.)$. Indeed by exactness in the variable $Y^{C_{p}}$ (a spectrum with trivial $C_{p}$-action) one reduces to the case $Y^{C_{p}}=\mathbb{S}$ where the square becomes

{\mathbb{S}}$${\mathbb{S}^{hC_{p}}}$${\mathbb{S}^{tC_{p}}}$${\mathbb{S}^{tC_{p}}}$$\scriptstyle{\Delta_{p}}$$\scriptstyle{\operatorname{can}}$$\scriptstyle{\mathrm{id}}

The map labelled $(2.)$ is the equivalence witnessing that the inclusion of the $C_{p}$-singular set $B^{C_{p}}\subseteq B$ induces an equivalence

[TABLE]

for any $C_{p}$-spectrum $T$ and finite $C_{p}$-CW-complex $B$, cf. [15, Lemma 9.1.]. The right-most triangle (with $(2.)$ as a side) commutes since $\operatorname{can}$ is natural.

Finally, in the top triangle, the map $\mathbb{S}\otimes Y^{C_{p}}\to(\mathbb{S}\otimes Y^{C_{p}})^{hC_{p}}$ arises since $\mathbb{S}\otimes Y^{C_{p}}$ has trivial $C_{p}$-action. The homotopy $C_{p}$-fixed point functor is right adjoint to the functor equipping a spectrum with a trivial $C_{p}$ action. In particular, the map $(\mathbb{S}\otimes Y)^{hC_{p}}\to\mathbb{S}\otimes Y$ is the universal map from a spectrum with trivial $C_{p}$-action to the target. It follows that the following diagram commutes.

{\mathbb{S}\otimes Y^{C_{p}}}$${\mathbb{S}\otimes Y^{hC_{p}}}$${(\mathbb{S}\otimes Y)^{hC_{p}}}  ${(\mathbb{S}\otimes Y^{C_{p}})^{hC_{p}}}$

This completes the proof. ∎

I thank Malte Leip for abundant help with the above argument.

Corollary 20**.**

Let $\overline{\omega}$ be a cyclic word having no cyclic repetitions and length $m$. The map

[TABLE]

induces an isomorphism on homotopy groups in degrees greater than or equal to $m$

Proof.

The Frobenius map $\varphi_{p}:\operatorname{THH}(k)\to\operatorname{THH}(k)^{tC_{p}}$ induces an isomorphism on non-negative homotopy groups. This is shown in [21, Prop. IV. 4.13] for $k={\mathbb{F}}_{p}$, and in [15, Section 5.5] for $k$ a perfect field of characteristic $p$. From Proposition 19 we know that

[TABLE]

factors as the composite of an equivalence and the tensor product of the Frobenius on $\operatorname{THH}(k)$, $\varphi_{p}:\operatorname{THH}(k)\to\operatorname{THH}(k)^{tC_{p}}$ with the map $\tilde{{\varphi}_{p}}:\Sigma^{\infty}B(\overline{\omega})\to(\Sigma^{\infty}B(\overline{\omega^{\star p}}))^{tC_{p}}$ which is a $p$-adic equivalence. The result now follows from Proposition 15 as can be shown for example with the Atiyah-Hirzebruch spectral sequence. Indeed, given a map of spectra $\varphi:T\to T^{\prime}$ which is an equivalence in high degrees, and a space $B$ with homology bounded above, then $\varphi\otimes B:T\otimes B\to T^{\prime}\otimes B$ is an equivalence in high degrees. To see this, take the cofiber $\mathrm{cof}(\varphi\otimes B)=\mathrm{cof}(\varphi)\otimes B$, we must show it is highly coconnective. The Atiyah-Hirzebruch spectral sequence for $\pi_{*}(\mathrm{cof}(\varphi)\otimes B)$ has $E^{2}$-page

[TABLE]

When $i$ is large $E_{i,j}=0$ and when $j$ is large $E_{i,j}=0$ so there is some diagonal beyond which all groups are zero. In the case at hand this diagonal is determined by Proposition 15 and is given by $i+j=m$. ∎

By Proposition 15 the homology of $B(\overline{\omega})$ depends only on the length and period of $\overline{\omega}$. For $\overline{\omega}$ of length $m$ and period $s$ we therefore introduce the notation $B(m,s)=B(\overline{\omega})$. The function $\operatorname{cyc}_{d}(s)$ counts how many cyclic words with no cyclic repetitions, of length $s$, and period $s$, there are. For a fixed $m$ and $s$ a divisor of $m$, there are $\operatorname{cyc}_{d}(s)$ many cyclic words with no cyclic repetitions, of length $m$, and period $s$. Since the homology of $B(m,s)$ depends only on the parity of $m$ and $s$ (by Proposition 15) we have the following equivalences; if $p>2$ then

[TABLE]

If $p=2$ then we add the similar double sum indexed over $m$ even and $s$ odd, i.e.

[TABLE]

Note that we do not need to know the homotopy-type of $B(m,s)$ for the above equivalences, since the homotopy type of $\operatorname{THH}(k)\otimes B(m,s)$ is determined from the homology of $B(m,s)$. Indeed the Atiyah-Hirzebruch spectral sequence for the $\operatorname{THH}(k)$-homology of $B(m,s)$ degenerates at the $E_{2}$-page with trivial extension problems, and thus computes $\pi_{*}(\operatorname{THH}(k)\otimes B(m,s))$. It is the simplicity of the homology of $B(m,s)$ that makes it possible for us to compute this Atiyah-Hirzebruch spectral sequence.

4. Negative - and periodic topological cyclic homology

The results of the previous section give us a good understanding of $\operatorname{THH}(A,B,I)$ as a cyclotomic spectrum, i.e. as a spectrum with $\mathbb{T}$-action and with the $\mathbb{T}$-equivariant Frobenius map $\varphi_{p}:THH(A,B,I)\to\operatorname{THH}(A,B,I)^{tC_{p}}$. In this section we take the next step towards computing $\operatorname{TC}(A,B,I)$ and hence towards Theorem 2. To do this we will use the results of [21] which give a formula for $\operatorname{TC}$ in terms of the cyclotomic structure on $\operatorname{THH}$. Let us briefly recall this formula. Given a ring $R$ we view $\operatorname{THH}(R)$ as a cyclotomic spectrum and form the topological negative cyclic homology,

[TABLE]

and topological periodic cyclic homology

[TABLE]

There is a canonical map from the former to the latter

[TABLE]

Assuming $R$ is characteristic $p>0$ it follows from an application of the Tate orbit lemma [21, Lemma I.2.1] that $\varphi_{p}$ induces a map, which, by abuse of notation, we still call $\varphi_{p}$,

[TABLE]

Finally, by [21, Section II.1] there is a cofiber sequence

[TABLE]

All of this readily works in the birelative case so that we also have a cofiber sequence for $\operatorname{TC}(A,B,I)$. The goal of this section is thus to compute $\operatorname{TC}^{-}(A,B,I)$ and $\operatorname{TP}(A,B,I)$.

We will need the following general lemma about the Tate $\mathbb{T}$-construction.

Lemma 24**.**

Let $\{X_{i}\}_{i\geq 0}$ be a sequence of spectra with a $\mathbb{T}$-action, such that the connectivity of (the underlying spectrum) $X_{i}$ is unbounded, as $i$ grows. Then the canonical map $(\bigoplus_{i}X_{i})^{t\mathbb{T}}\to\prod(X_{i})^{t\mathbb{T}}$ is an equivalence.

Proof.

Because of the connectivity assumption the canonical map $\bigoplus X_{i}\to\prod X_{i}$ is an equivalence. Consider the norm cofiber sequence

[TABLE]

The first term commutes with colimits, the second with limits, so we get the cofiber sequence

[TABLE]

Now $(-)_{hG}$ preserves connectivity for any group $G$. To see this one may use the homotopy orbit spectral sequence whose $E^{2}$-page consists of ordinary group homology. Thus the cofiber sequence becomes

[TABLE]

and we see that $(\bigoplus_{i}X_{i})^{t\mathbb{T}}\to\prod(X_{i})^{t\mathbb{T}}$ is an equivalence. ∎

4.1. The Tate spectral sequence

Let $X$ be a spectrum with $\mathbb{T}$-action. The Tate construction $\operatorname{TP}(X)=X^{t\mathbb{T}}$ is the target of the Tate spectral sequence for the circle group $\mathbb{T}$, see [16, end of section 4.4], and also [1, section 3]. This spectral sequence has the form

[TABLE]

Here $t$ has bi-degree $(-2,0)$ and is a generator of $H^{2}({\mathbb{C}}P^{\infty};{\mathbb{Z}})$. This spectral sequence is conditionally convergent. When $X=\operatorname{THH}(R)$ is an $E_{\infty}$ algebra the spectral sequence is multiplicative, and if $R$ is a $k$-algebra then $E^{r}(\mathbb{T},\operatorname{THH}(R))$ is a module over $E^{r}(\mathbb{T},\operatorname{THH}(k))$. By Bökstedt periodicity, when $k$ is a perfect field of characteristic $p>0$ there is an isomorphism

[TABLE]

with $x$ in degree $2$. In particular, in this case $\operatorname{TP}_{*}(R)$ is periodic. The following is well-known.

Proposition 25**.**

Let $k$ be a perfect field of characteristic $p>0$, then there is an isomorphism

[TABLE]

Proof.

By Bökstedt periodicity the $E^{2}$-term of the spectral sequence takes the form $k[t^{\pm 1}]\otimes k[x]$, and since all classes are in even total degree there are no non-trivial differentials on any page. Thus $E^{2}=E^{\infty}$ and it remains to determine the extensions. Again by periodicity it suffices to show that $\operatorname{TP}_{0}(k)=W(k)$. For $k={\mathbb{F}}_{p}$ this is done in [21, Cor. IV. 4.8.]. To conclude the result for $k$ we use functoriality along the map ${\mathbb{F}}_{p}\to k$. More precisely we argue as follows. From the Tate spectral sequence we have a complete descending multiplicative filtration

[TABLE]

with associated graded $\mathrm{gr}^{i}(k)\simeq\operatorname{THH}_{2i}(k)\simeq k$. By the universal property of the $p$-typical Witt vectors $W(k)$ [23, Chapter II, paragraph 5] we get a unique multiplicative continous map $W(k)\to\operatorname{TP}_{0}(k)$. By functoriality we have a commuting diagram

[TABLE]

of ring homomorphisms. In particular the map on associated graded induced by $W(k)\to\operatorname{TP}_{0}(k)$ maps $1$ to $1$ and so must be an isomorphism. It follows that $W(k)\to\operatorname{TP}_{0}(k)$ is itself an isomorphism. ∎

Lemma 26**.**

Let $k$ be a perfect field of characteristic $p>0$. The elements $x$ and $t^{\pm}$ from the $E^{2}$-page of the Tate spectral sequence are infinite cycles.

Proof.

This is true for degree reasons, by Bökstedt periodicity. ∎

Lemma 27**.**

Let $X$ be a spectrum with $\mathbb{T}$-action such that the underlying spectrum is an $H{\mathbb{Z}}$-module . The $d^{2}$ differential of the $\mathbb{T}$-Tate spectral sequence is given by $d^{2}(\alpha)=td(\alpha)$ where $d$ is Connes’ operator.

Proof.

See [12, Lemma 1.4.2] ∎

Following Proposition 15 we now choose some generators for the homology of the spaces $B(m,s)$. When $m$ and $s$ have the same parity let $z_{(m,s)}$ be a generator for $\tilde{H}_{m}(B(m,s);k)$ and let $y_{(m,s)}$ a generator for $\tilde{H}_{m-1}(B(m,s);k)$. When $m$ is even and $s$ is odd let $z_{(m,s)}$ be a generator of $\tilde{H}_{m-1}(B(m,s);k)$. Finally, when $p=2$, so $\tilde{H}_{m}(B(m,s);k)$ is free of rank $1$ over $k$, let $w_{(m,s)}$ be a generator in degree $m$.

Lemma 28**.**

As an element of the $E^{2}$-page of the Tate spectral sequence, $z_{(m,s)}$ is an infinite cycle, i.e. it is annihilated by all differentials.

Proof.

By [15, Theorem B] for any perfect field $k$ of positive characteristic, there is an equivalence $\tau_{\geq 0}\operatorname{TC}(k)\simeq{\mathbb{Z}}_{p}$. As a result we obtain a map

[TABLE]

of spectra. The map $\operatorname{TC}(k)\to\operatorname{TC}^{-}(k)$ is the first map of Eq. 23. The second map is the canonical map $(-)^{h\mathbb{T}}\to(-)$. If we equip the all terms except $\operatorname{THH}(k)$ with trivial $\mathbb{T}$-action then each map is $\mathbb{T}$-equivariant, and so the composition yields a $\mathbb{T}$-equivariant map

[TABLE]

Since it is $\mathbb{T}$-equivariant, this map induces a map on Tate spectral sequences, which when $p$ is odd takes the following form,

[TABLE]

For degree reasons $z_{(m,s)}$ is an infinite cycle in the top spectral sequence. It follows that $z_{(m,s)}$, viewed as a class in the bottom spectral sequence, is an infinite cycle. When $m$ is even and $s$ is odd, and $p=2$, the spectral sequences in question have a slightly different form, with generators $z_{(m,s)}$ and $w_{(m,s)}$ (for the $\operatorname{THH}(k)$-homology, cf. notation introduced in the paragraph before the lemma statement). However, the argument still works, i.e.  for degree reasons $z_{(m,s)}$ must be an infinite cycle in the top spectral sequence, which forces the same conclusion for the target spectral sequence. ∎

4.2. $\operatorname{TC}^{-}$ and $\operatorname{TP}$ of coordinate axes

In this section we compute $\operatorname{TC}^{-}$ and $\operatorname{TP}$ using the Tate spectral sequence. First assume $p>2$. By Lemma 24 and Eq. 21 we have

[TABLE]

and likewise for negative topological cyclic homology, we have

[TABLE]

When $p=2$ there is an extra double product indexed over $m\geq 2$ even and $s\mid m$ odd (cf. Proposition 15 and Remark 17). We handle this case in Section 4.3 below.

Since $\operatorname{THH}(k)$ is $p$-complete, it follows from [21, Lemma II. 4.2.] that we may identify the homotopy $\mathbb{T}$-fixed points of the Frobenius morphism for $\operatorname{THH}(A,B,I)$ with the map induced by the product of the maps

[TABLE]

Since the homotopy fixed point functor preserves co-connectivity, it follows from Corollary 20 that $\pi_{*}\varphi_{p}(m,s)$ is an isomorphism when $*\geq m$. Indeed, the fiber of

[TABLE]

has no homotopy groups above degree $m-1$, hence the same is true of the $\mathbb{T}$-homotopy fixed point spectrum.

Proposition 29**.**

Let $k$ be a perfect field of characteristic $p>0$. Let $m\geq 2$ and $s\mid m$. Write $m=p^{v}m^{\prime}$ and $s=p^{u}s^{\prime}$ with $m^{\prime}$ and $s^{\prime}$ coprime to $p$.

1. (1)

If both $m$ and $s$ are even then $\pi_{*}((\operatorname{THH}(k)\otimes B(m,s))^{t\mathbb{T}})$ as well as $\pi_{*}((\operatorname{THH}(k)\otimes B(m,s))^{h\mathbb{T}})$ are concentrated in even degrees and given by

[TABLE]

and

[TABLE] 2. (2)

If both $m$ and $s$ are odd, then $\pi_{*}((\operatorname{THH}(k)\otimes B(m,s))^{t\mathbb{T}})$ as well as $\pi_{*}((\operatorname{THH}(k)\otimes B(m,s))^{h\mathbb{T}})$ are concentrated in odd degrees and given by

[TABLE]

and

[TABLE]

Proof.

Suppose first that $m$ and $s$ are even. We proceed by induction on $v\geq 0$. Suppose $v=0$, so $m=m^{\prime}$ and $s=s^{\prime}$. Consider the Tate spectral sequence (cf. Section 4.1 )

[TABLE]

By Lemma 28 and Lemma 26 the differential structure is determined by the differentials on $y_{(m^{\prime},s^{\prime})}$. Furthermore

[TABLE]

by Lemma 27 and Proposition 15, where $i=m^{\prime}/s^{\prime}$. Since $i$ is a unit in $k$, $d^{2}$ is an isomorphism. The figure below displays the $E^{2}$-page (where we have dropped the indices for clarity). All the arrows indicate isomorphisms.

Thus $E^{3}$, hence $E^{\infty}$, is trivial as claimed. To determine the $\mathbb{T}$-homotopy fixed points, we truncate the Tate spectral sequence, removing the first quadrant. Thus the class $z_{(m^{\prime},s^{\prime})}$ and its multiples by $x^{n}$, are no longer hit by differentials and so

[TABLE]

where $z_{(m^{\prime},s^{\prime})}$ has degree $m$. This proves the claim for $v=0$. The same argument works for $v>0$ and $u=v$, since in this case $i$ is again coprime to $p$.

Suppose the claim is known for all integers less than or equal to $v$ and all $u\leq v$. As we saw above, the homotopy fixed points of the Frobenius map

[TABLE]

is an isomorphism when $*\geq p^{v}m^{\prime}$. The induction hypothesis implies that the domain is isomorphic to $W_{v-u+1}(k)$ when $*=2r\geq p^{v}m^{\prime}$. Since the Tate construction admits the structure of a lax symmetric monoidal functor [21, Corollary I.4.3.] it follows that $(\operatorname{THH}(k)\otimes B(p^{v+1}m^{\prime},p^{u}s^{\prime}))^{t\mathbb{T}}$ inherits a module structure over $\operatorname{THH}(k)^{t\mathbb{T}}$. In particular it follows from the calculation

[TABLE]

(Proposition 25) that $\pi_{*}(\operatorname{THH}(k)\otimes B(p^{v+1}m^{\prime},p^{u}s^{\prime}))^{t\mathbb{T}}$ is periodic as well. Using this periodicity we conclude that

[TABLE]

for any $r\in{\mathbb{Z}}$. Considering again the Tate spectral sequence we see that we must have

[TABLE]

(see the figure below, which is page $E^{8}$ when $v=2$ and $u=0$) and so we conclude that $E^{2(v+1-u)+3}=E^{\infty}$.

To solve the extension problem we employ a strategy exactly analogous to the one used in [24, Prop. 12]. The class $z_{(m,s)}$ in the homology of $B(m,s)$ gives a map of chain complexes

[TABLE]

which may be promoted to a map of $\mathbb{T}$-equivariant complexes, i.e. a map in the $\infty$-category $\mathcal{D}({\mathbb{Z}})^{B\mathbb{T}}$. This category is equivalent to the $\infty$-category of mixed complexes, and so to check that $f_{z_{(m,s)}}$ is $\mathbb{T}$-equivariant it suffices to check that the Connes operator on the taget ${\mathbb{Z}}\otimes B(m,s)$ is trivial. This follows from Lemma 14. Now using the multiplicative $\mathbb{T}$-equivariant map ${\mathbb{Z}}\to\operatorname{THH}(k)$ constructed in the proof of Lemma 28, we get a $\mathbb{T}$-equivariant map

[TABLE]

The induced map on Tate spectral sequences now shows us that the $W(k)$-module $\pi_{2r}(\operatorname{THH}(k)\otimes B(m,s))^{t\mathbb{T}}$ is cyclic and hence determined by its length (cf. [24, Prop. 12] for the details). The length is precisely what we have established from the $E^{\infty}$-page calculation.

Truncating the spectral sequence we now see that

[TABLE]

This completes the proof of (1).

The arguments in case (2), where $m$ and $s$ are both odd, are very similar. ∎

Remark 30*.*

In particular this shows that $\operatorname{TP}(A,B,I)$ is non-trivial. This may be contrasted with the Cuntz-Quillen result [4], that $\operatorname{HP}((A,B,I)/{\mathbb{Q}})$ is trivial, i.e. that rational periodic cyclic homology satisfies excision. Similarly it is shown in [18] that $\operatorname{TP}$ does not satisfy nil-invariance. Here again it is a result of Goodwillie [10] that rational periodic cyclic homology does satisfy nil-invariance.

Suppose $m^{\prime}$ is even and let $t_{ev}=t_{ev}(p,r,m^{\prime})$ be the unique positive integer such that $p^{t_{ev}-1}m^{\prime}\leq 2r<p^{t_{ev}}m^{\prime}$ (or zero if such $t_{ev}$ does not exist, i.e. if $m^{\prime}$ is too big). Then we may restate the $\operatorname{TC}^{-}$ calculations as

[TABLE]

Similarly if $m^{\prime}$ and $s^{\prime}$ are both odd, let $t_{od}=t_{od}(p,r,m^{\prime})$ be the unique positive integer such that $p^{t_{od}-1}m^{\prime}\leq 2r+1<p^{t_{od}}m^{\prime}$ (or zero if such $t_{od}$ does not exist, i.e. if $m^{\prime}$ is too big). Then

[TABLE]

4.3. The case $p=2$

We now deal with the case when $k$ has characteristic two. From Section 3.5 we see that the case $m$ even and $s$ odd, is missing from Proposition 29.

Proposition 31**.**

Assume $p=2$, $m$ even and $s$ odd. Write $m=2^{v}m^{\prime}$ with $m^{\prime}$ odd, and let $s^{\prime}\mid m^{\prime}$. Then the homotopy groups of $(\operatorname{THH}(k)\otimes B(m,s))^{t\mathbb{T}}$ and $(\operatorname{THH}(k)\otimes B(m,s))^{h\mathbb{T}}$ are concentrated in odd degrees where they are isomorphic to $k$.

Proof.

When $v=0$ we are in the case (2) of Proposition 29 so

[TABLE]

and

[TABLE]

From the high co-connectivity of the Frobenius we conclude that

[TABLE]

when $*$ is odd, and zero otherwise. Now the Tate spectral sequence

[TABLE]

must collapse on the $E^{2}$-page, from which we conclude that $d^{2}(w)\overset{.}{=}(tx)z$. Truncating the spectral sequence we see the homotopy fixed point spectral sequence has $E^{3}=E^{\infty}=(k[t,x]/(tx))\{z\}$ so

[TABLE]

in every odd degree. Now an induction argument shows that this pattern continues for all $v>1$. ∎

5. Topological cyclic homology of coordinate axes

In this section we complete the proof of Theorem 2, and thus of Theorem 1. We start with the case where $p>2$. For $1\leq m^{\prime}$ and $s^{\prime}\mid m^{\prime}$ let

[TABLE]

and

[TABLE]

Thus, $\operatorname{TC}(A,B,I)$ is identified with the product of the equalizer of the maps $\varphi_{p},can:\operatorname{TC}^{-}(m^{\prime},s^{\prime})\rightarrow\operatorname{TP}(m^{\prime},s^{\prime})$ as $m^{\prime}$ and $s^{\prime}$ vary accordingly. Let us denote by $\operatorname{TC}(m^{\prime},s^{\prime})$ the equalizer

[TABLE]

A priori the homotopy groups of $\operatorname{TC}(m^{\prime},s^{\prime})$ sit in a long exact sequence with those of $\operatorname{TC}^{-}(m^{\prime},s^{\prime})$ and $\operatorname{TP}(m^{\prime},s^{\prime})$. However this sequence splits into short exact sequences since $\operatorname{TC}^{-}(m^{\prime},s^{\prime})$ and $\operatorname{TP}(m^{\prime},s^{\prime})$ are concentrated in either even or odd degrees, depending on the parity of $m^{\prime}$ and $s^{\prime}$, cf.  Proposition 29.

If $m^{\prime}$ and $s^{\prime}$ are even then the Frobenius map

[TABLE]

is an isomorphism for $0\leq v<t_{ev}$, and the canonical map

[TABLE]

is an isomorphism for $t_{ev}\leq v$. Thus we have a map of short exact sequences

[TABLE]

The top horizontal map is an isomorphism, and the bottom horizontal map is an epimorphism so, by the snake lemma, we conclude that

[TABLE]

Now suppose $m^{\prime}$ and $s^{\prime}$ are odd. Then the Frobenius map

[TABLE]

is an isomorphism for $0\leq v<t_{od}$, and the canonical map

[TABLE]

is an isomorphism for $t_{od}\leq v$. Thus we have a map of short exact sequences

[TABLE]

The top horizontal map is an isomorphism, and the bottom horizontal map is en epimorphism so, by the snake lemma, we conclude that

[TABLE]

This finishes the proof of Theorem 2 in the case $p>2$.

5.1. The case $p=2$

If $p=2$ then we must correct slightly the definition of $\operatorname{TP}(m^{\prime},s^{\prime})$ and $\operatorname{TC}^{-}(m^{\prime},s^{\prime})$. Suppose $m^{\prime}$ and $s^{\prime}$ are odd. To deal with the case $m=p^{v}m^{\prime}$ even and $s=p^{u}s^{\prime}$ even, we let

[TABLE]

and

[TABLE]

Both of these spectra are concentrated in even degrees, with their homotopy groups given by Proposition 29. As a result we see that, for $m^{\prime}\geq 1$ odd, and $s^{\prime}\mid m^{\prime}$ odd,

[TABLE]

To deal with the case where $m$ is even and $s\mid m$ is odd, let

[TABLE]

and

[TABLE]

(note that when $v=0$, we have $m=m^{\prime}$ odd, but we must include this case since the Frobenius connects it with the case $v=1$) By Proposition 31, $\operatorname{TP}(m^{\prime},s^{\prime})_{ev,od}$ and $\operatorname{TC}^{-}(m^{\prime},s^{\prime})_{ev,od}$ are concentrated in odd degrees, where they are isomorphic to $k$. Thus

[TABLE]

This completes the proof of Theorem 2 when $p=2$.

6. The characteristic zero case

In this section we compute the relative cyclic homology and the bi-relative $K$-theory of $A_{d}=k[x_{1},\dots,x_{d}]/(x_{i}x_{j})_{i\neq j}$ in the case that $k$ is an ind-smooth ${\mathbb{Q}}$-algebra. In the case where $k/{\mathbb{Q}}$ is a field extension, the results in this section were found, by different means, already in 1989 by Geller, Reid and Weibel [9, Theorem 7.1.].

We proceed as in [13, Section 3.9]. We will compute the relative groups

[TABLE]

using our understanding of the $\mathbb{T}$-homotopy type of the cyclic bar construction for $\Pi^{d}$, as found in Lemma 14. First we need the following analogue of Lemma 7

Lemma 32**.**

Let $k$ be a ring, $\Pi$ a pointed monoid, and $k(\Pi)$ the pointed monoid algebra. There is a natural equivalence of $\mathbb{T}$-spectra

[TABLE]

So the arguments from Lemma 9 still work, yielding a description of the relative (and bi-relative) Hochshild homology spectrum of $(A_{d},I_{d})$ (and $(A_{d},B_{d},I_{d})$). First we note that the relative Hochschild homology differs only slightly from the absolute version. The sole difference is that we “cut out” the part of the space $\operatorname{B^{cy}}(\Pi^{d})$ given by $\operatorname{B^{cy}}(\Pi^{d};\overline{\varnothing})$, i.e. the part corresponding to the unique cyclic word of length zero. Since the spaces $B(m,s)$ for $m$ even and $s$ odd have torsion integral homology they disappear in the rational case. So we conclude that

[TABLE]

where $B(m,s)$ is given by Lemma 14. The bottom summands, with terms $\operatorname{HH}(k)\otimes(\mathbb{T}/C_{i})_{+}$, corresponds to the spaces $\operatorname{B^{cy}}(\Pi^{d},\overline{x_{1}^{i}})$ , $\operatorname{B^{cy}}(\Pi^{d},\overline{x_{2}^{i}})$ etc. These bottom summands disappear when looking at the bi-relative theory, $\operatorname{HH}(A,B,I)$.

Theorem 33**.**

Let $k$ be any commutative unital ${\mathbb{Q}}$-algebra. There is an isomorphism

[TABLE]

and, for $q\geq 1$, an isomorphism

[TABLE]

Proof.

We must compute $\pi_{q}(X_{h\mathbb{T}})$ where $X$ ranges over the summands in the above decomposition of $\operatorname{B^{cy}}(\Pi^{d})$. If $X=\operatorname{HH}(k)\otimes S^{\lambda_{j}}\otimes(\mathbb{T}/C_{i})_{+}$ for some complex $\mathbb{T}$-representation $\lambda_{j}$ of complex dimension $j$, then since,

[TABLE]

we may regard the $C_{i}$-homology spectral sequence

[TABLE]

Since the rational group homology of $C_{i}$ is concentrated in degree [math], the edge homomorphism

[TABLE]

is an isomorphism. Furthermore, since the $C_{i}$-action on $\operatorname{HH}(k)\otimes S^{\lambda_{j}}$ extends to a $\mathbb{T}$-action, the induced action on homotopy groups is trivial. We conclude that

[TABLE]

The result follows. ∎

In the case where $k$ is an algebraic field extension of ${\mathbb{Q}}$ Theorem 33 was established by Geller, Reid and Weibel [9, Example 5.7.]. Indeed, then $k$ is ind-smooth over ${\mathbb{Q}}$ (cf. [25, 07BV]) so the Hochschild-Kostant-Rosenberg theorem calculates the Hochschild homology in terms of differential forms. Similarly, the following theorem is proved in [9, Theorem 7.1.] when $k$ is a field extension of ${\mathbb{Q}}$. We use their counting function $c_{d-1}(q)$ which we recall in Section 7, Eq. 37.

Theorem 34**.**

Suppose $k$ is an ind-smooth ${\mathbb{Q}}$-algebra. Let $d\geq 2$ and consider the ring $A_{d}=k[x_{1},\dots,x_{d}]/(x_{i}x_{j})_{i\neq j}$, and let $I_{d}=(x_{1},\dots,x_{d})$. Then

[TABLE]

Proof.

By [3, Corollary 0.2] we have $K_{n}(A_{d},I_{d})\simeq\operatorname{HC}_{n-1}(A_{d},B_{d},I_{d})$. Here we use that $K_{q}(A_{d},I_{d})$ is a rational vector space [3, Theorem 0.1], so no further rationalization is necessary. By Theorem 33 we are reduced to Hochschild homology calculations. By the Hochschild-Kostant-Rosenberg theorem [26, Theorem 9.4.7.] we have

[TABLE]

Thus,

[TABLE]

This completes the proof. ∎

Remark 35*.*

If $k$ is a field extension of ${\mathbb{Q}}$ for which we know the transcendence degree of $k$ over ${\mathbb{Q}}$ then this result completely determines the relative $K$-theory, since $\dim_{k}\Omega^{1}_{k/{\mathbb{Q}}}=tr.deg(k/{\mathbb{Q}})$. For example if $k/{\mathbb{Q}}$ is algebraic then $\Omega^{1}_{k/{\mathbb{Q}}}=0$ and so the calculation reduces to

[TABLE]

7. Appendix: counting cyclic words

In this section we use some counting techniques inspired by [9]. We first deal with all words, then with cyclic words. Let $A_{d}(m)$ denote the number of words in $d$ letters of length $m$ having no cyclic repetitions.

Lemma 36**.**

For all $d\geq 1$ and all $m\geq 1$ we have

[TABLE]

Proof.

Define the auxiliary counter, $B_{d}(m)$ counting words $\omega=w_{1}w_{2}\dots w_{m}$ with no allowed repetitions, except that we require $w_{m}=w_{1}$. Then

[TABLE]

To see this, consider a word $\omega=w_{1}w_{2}\dots w_{m}$. There are $d$ choices for $w_{1}$, and $d-1$ choices for $w_{2},w_{3},\dots$ and $w_{m-1}$. Finally for $w_{m}$ there are again $d-1$ choices since choosing any letter different from $w_{m-1}$ gives either a type $A$ ($d-2$ possibly choices) or a type $B$ word (one choice, namely $w_{m}=w_{1}$). The formula for $A_{d}(m)$ now follows by induction using the equation $B_{d}(m)=A_{d}(m-1)$ (for $m\geq 2$). This last equality is true since deleting the last letter of a type $B$ word yields a word with no cyclic repetitions. ∎

Let $\widetilde{\operatorname{cyc}}_{d}(s)$ denote the number of words in $d$ letters of length $s$, period $s$ and having no cyclic repetitions (see Definition 11) and let $\operatorname{cyc}_{d}(s)$ denote the number of cyclic words in $d$ letters of length $s$, period $s$ and having no cyclic repetitions. Then $\operatorname{cyc}_{d}(s)=\frac{1}{s}\widetilde{\operatorname{cyc}}_{d}(s)$. Also we have

[TABLE]

So using Möbius inversion we have

[TABLE]

hence we obtain a formula for $\operatorname{cyc}_{d}(s)$,

[TABLE]

Below is a table of the first few values.

[TABLE]

If we want to count the number of cyclic words without cyclic repetitions, with length $m$ but with no restrictions on period then we can just sum $\operatorname{cyc}_{d}(s)$ over all divisors $s$ of $m$. For example there are $3+2+9=14$ cyclic words in $d=3$ letter, with no repetitions having length $m=6$.

We may describe the function $c_{d-1}(m)$ that Geller, Reid, and Weibel introduce as follows

[TABLE]

That is, $c_{d-1}(m)$ is the number of cyclic words without repetitions, having length $m$ and period $s$ where $s$ and $m$ have the same parity.