section[1.3em]1.3em0.7em subsection[2.0em]2.0em0.7em

#  Whitehead filtrations for computations in Topological Hochschild Homology

Logan Hyslop

(mm/dd/yyyy)

###### Abstract

We discuss spectral sequences coming from Whitehead filtrations in the
computation of topological Hochschild homology of ring spectra. Using cyclic
invariance, this makes for simple computations of $\operatorname{THH}$ of
connective rings $R$ with coefficients in discrete ring spectra.111Where
discrete means Eilenberg-MacLane. In particular, we show how to use this to
compute $\operatorname{THH}(\operatorname{tmf},\mathbb{F}_{2})$, and
$\operatorname{THH}(\operatorname{tmf},\mathbb{Z}_{(2)})$, where
$\operatorname{tmf}$ denotes the $\mathbb{E}_{\infty}$ ring spectrum of
topological modular forms. Then, we obtain a description of
$\operatorname{THH}(\ell/v_{1}^{n})$ in terms of
$\operatorname{THH}(\ell,\ell/v_{1}^{n})$, where the latter can be computed by
results of [AHL09]. We next explain how the methods of this computation
generalize to give us information about
$\operatorname{THH}(\operatorname{cofib}(x^{k}:\Sigma^{k|x|}R\to R))$ for $R$
and $\operatorname{cofib}(x^{k})$ suitably structured connective ring spectra,
$k>1$, and $x\in\pi_{*}(R)$ an arbitrary element in positive degree. Finally,
we examine the general framework to describe the topological Hochschild
homology of 2-local connective self-conjugate K-theory,
$\operatorname{ksc}_{2}$.

###### Contents

  1. 1 Introduction
  2. 2 Filtered Objects and Spectral Sequences
  3. 3 Topological Hochschild Homology of $\operatorname{tmf}$ with Coefficients
  4. 4 THH of Quotients of $\ell$
  5. 5 The General Case
  6. 6 The THH of $\operatorname{ksc}_{2}$

## §1. Introduction

Topological Hochschild homology was introduced by Bökstedt in 1985 as a
generalization of ordinary Hochschild homology to general homotopy coherent
ring spectra, which has lead to many recent advances in algebraic K-theory
[NS18], which in turn, lead to the recent disproof of Ravenel’s telescope
conjecture [Bur+23]. The aim of this paper is to discuss spectral sequences
arising from Whitehead filtrations as a means to compute topological
Hochschild homology (possibly with coefficients) over connective ring spectra.
When combined with cyclic invariance, this recovers the Brun spectral
sequence, and we get nice comparison maps for computing $\operatorname{THH}$
with coefficients.

For an $\mathbb{E}_{1}$-ring spectrum $R$, the topological Hochschild homology
of $R$ is defined as
$\operatorname{THH}(R)=R\otimes_{R\otimes_{\mathbb{S}}R^{op}}R$, and the
topological Hochschild homology with coefficients in a $R$-bimodule $M$ is
defined as $\operatorname{THH}(R,M)=R\otimes_{R\otimes_{\mathbb{S}}R^{op}}M$.
Assuming that $R$ is connective, we can apply the Whitehead filtration to $R$
to get a filtered $R\otimes_{\mathbb{S}}R^{op}$-module spectrum
$\tau_{\geq*}R$, to which we can then apply the functor
$R\otimes_{R\otimes_{\mathbb{S}}R^{op}}-:\operatorname{Fil}(R\otimes_{\mathbb{S}}R^{op}-\operatorname{Mod}_{L})\to\operatorname{Fil}(R-\operatorname{Mod}_{L})$,
giving us a filtered $R$-module spectrum. This gives rise to a spectral
sequence with signature

$E_{1}^{s,t}=\operatorname{THH}_{-s}(R,\pi_{t}(R))\implies\operatorname{THH}_{t-s}(R),$

where we consider $\pi_{t}(R)$ as a discrete
$R\otimes_{\mathbb{S}}R^{op}$-module in degree zero. We get a similar spectral
sequence for $\operatorname{THH}(R,M)$ whenever $R$ is a connective
$\mathbb{E}_{1}$-ring, and $M$ an $R$-bimodule. In section 2, we discuss some
basic results on spectral sequences of this type, to be used in the rest of
the paper.

In section 3, we will use the results of section 2 in order to compute
$\operatorname{THH}(\operatorname{tmf},\mathbb{F}_{2})$, and then combining
the Brun spectral sequence with the Bockstein spectral sequence, we compute
$\operatorname{THH}(\operatorname{tmf},\mathbb{Z}_{(2)})$, using this result.
The main theorem of section 3 is the computation:

######  Theorem 1.1.

$\operatorname{THH}_{*}(\operatorname{tmf},\mathbb{Z}_{(2)})=\begin{cases}\mathbb{Z}_{(2)}&\qquad\text{if
}*=0,9,13,22\\\ \mathbb{Z}/2^{k}\mathbb{Z}&\qquad\text{if
}*=2r^{k+3}-1,2^{k+3}r-1+9,2^{k+3}r-1+13,2^{k+3}r-1+22,\\\
0&\qquad\text{otherwise},\end{cases}$

for all $k>0$ and $r$ odd.

Section 4 uses the results of section 2, together with the spectral sequences
constructed by Lee-Levy [LL23] in order to compute the topological Hochschild
homology of quotients of $\ell$, $\operatorname{THH}(\ell/v_{1}^{n})$, where
$\ell$ is the mod $p$ Adams summand for some fixed odd prime $p$, and
$v_{1}\in\pi_{2}(\ell)$ generates $\pi_{*}(\ell)$ as a polynomial algebra over
$\mathbb{Z}_{(p)}$. The methods used in this section extend to prove:

######  Theorem 1.2.

Suppose that $R$ is a connective $\mathbb{E}_{m}$-ring spectrum for some
$m\geq 3$, and $x\in\pi_{*}(R)$ is a positive degree class such that, for some
fixed $k>1$, $R/x^{k}:=\operatorname{cofib}(x^{k}:R\to R)$ admits an
$\mathbb{E}_{3}$-algebra structure. Then, we have an equivalence of
$\tau_{\leq 0}R$-modules
$\operatorname{THH}(R/x^{k},\pi_{0}(R))\simeq\operatorname{THH}(R,\pi_{0}(R))\otimes_{\pi_{0}(R)}(\pi_{0}(R)\otimes_{\pi_{0}(R)\otimes_{R}R/x^{k}}\pi_{0}(R))$.

Under these same hypotheses, this result allows us to construct a spectral
sequence with signature

$E_{1}^{*,*}=\operatorname{THH}_{-*}(R,\pi_{*}(R/x^{k}))\otimes_{\pi_{0}(R)}(\bigoplus_{i\geq
0}\pi_{0}(R)\cdot a_{i})\implies\operatorname{THH}_{*}(R/x^{k}),$

where $a_{i}$ is a class in bidegree $(-i(2k|x|+2),0)$. The differentials on
the classes in $\operatorname{THH}_{-*}(R,\pi_{*}(R/x^{k}))$ are determined by
the spectral sequence
$\operatorname{THH}_{-*}(R,\pi_{*}(R/x^{k}))\implies\operatorname{THH}_{*}(R,R/x^{k})$,
so in order to understand this spectral sequence, one must only understand
what the differentials do to the classes $a_{i}$.

As another sample application, we compute the topological Hochschild homology
of 2-local self-conjugate $K$-theory $\operatorname{ksc}_{2}$ in section §6.

Notation/Conventions

  * •

We will say “category” to refer to $\infty$-categories.

  * •

Our filtered objects and spectral sequences will be as in [LL23], in
particular, the $d_{r}$ differential will go from $E_{r}^{s,t}$ to
$E_{r}^{s+r+1,t+r}$.

  * •

$p$ will denote a fixed odd prime.

  * •

$\ell$ will denote the mod $p$ Adams summand, and $\operatorname{tmf}$ will
denote the spectrum of topological modular forms.

  * •

When dealing with divided power algebras, we will use the notation $x^{(k)}$
for the class “$\frac{x^{k}}{k!}$.”

Acknowledgments. I am grateful to Alicia Lima for mentoring me during this
project, Peter May for hosting the REU at which this work was done, and Mike
Hill for useful conversations including his suggestion to examine
$\operatorname{THH}_{*}(\operatorname{ksc}_{2})$. I am especially grateful to
Ishan Levy for recommending that I do the computations that eventually grew
into this paper, and for several helpful conversations.

## §2. Filtered Objects and Spectral Sequences

We recall the definitions of filtered objects, following the description in
[LL23]. For a stable category $\mathcal{C}$, we can associate the stable
categories $\operatorname{Fil}(\mathcal{C}):=$
$\operatorname{Fun}(\mathbb{Z}_{\leq}^{op}$, $\mathcal{C})$ and
$\mathcal{C}^{gr}:=$$\operatorname{Fun}(\mathbb{Z}^{ds,op}$, $\mathcal{C})$,
of filtered objects and graded objects in $\mathcal{C}$, respectively. Here
$\mathbb{Z}_{\leq}$ denotes the poset category on $\mathbb{Z}$ with the $\leq$
order, and $\mathbb{Z}^{ds}$ denotes the discrete category with objects the
integers. For $c\in\mathcal{C}$, we have the object
$c^{n,m}\in\operatorname{Fil}(\mathcal{C})$, where
$c^{n,m}=\Sigma^{n}c^{0,n+m}$, and $c^{0,i}$ is defined as the image of $c$
under the left adjoint to
$ev_{i}:\operatorname{Fil}(\mathcal{C})\to\mathcal{C}$ (i.e.,
$(c^{0,i})_{j}=c$ for $j\leq i$, with the identity map as transition maps, and
is 0 for $j>i$). If $\mathcal{C}$ is symmetric monoidal,
$\operatorname{Fil}(\mathcal{C})$ and $\mathcal{C}^{gr}$ inherit symmetric
monoidal structures by Day convolution, with unit $1=1^{0,0}$, where $1$ is
the unit in $\mathcal{C}$. In this case, define bigradings on the homotopy
groups of a filtered object by
$\pi_{n,m}(A)=\pi_{0}\operatorname{Hom}_{\operatorname{Fil}(\mathcal{C})}(1^{n,m},A)$.
There is an element $\tau:1^{0,-1}\to 1^{0,0}$, which when tensored with any
filtered object $A$, gives a morphism of filtered objects, which is the
filtration map $A_{i+1}\to A_{i}$ in degree $i$. Modules over the cofiber of
$\tau$ are identified with graded objects, and for any filtered object $A$,
the cofiber sequence $A\otimes 1^{0,-1}\xrightarrow{\tau}A\to A^{gr}:=A/\tau$
gives rise to an exact couple. Out of this arises a spectral sequence with
signature (using the grading conventions from [LL23]):

$E_{1}^{s,t}=\pi_{t-s,s}A^{gr}\implies\pi_{t-s}(A[\tau^{-1}]),$

where $A[\tau^{-1}]$ is the "underlying" object of $A$ obtained by inverting
$\tau$.

We now introduce the main spectral sequences that we will be using in this
paper. Consider a connective $\mathbb{E}_{1}$-ring $R$, and an $R$-bimodule
$M$, i.e., a left $R\otimes_{\mathbb{S}}R^{op}$-module. Working in the
category
$\operatorname{Fil}(\operatorname{LMod}_{R\otimes_{\mathbb{S}}R^{op}})$, we
can take $\tau_{\geq*}M$ to be the Whitehead filtration on $M$, i.e., the
filtered spectrum with $m$th graded piece $\tau_{\geq m}M$, with maps
$\tau_{\geq m}M\to\tau_{\geq m-1}M$ the obvious ones. Applying the functor
$\operatorname{THH}(R;-):\operatorname{Fil}(\operatorname{LMod}_{R\otimes_{\mathbb{S}}R^{op}})\to\operatorname{Fil}(\operatorname{LMod}_{R})$,
we can then form the associated spectral sequence with signature
$E_{1}^{s,t}=\operatorname{THH}_{-s}(R,\pi_{t}(R))\Rightarrow\operatorname{THH}_{t-s}(R)$,
where we are treating $\pi_{t}(R)$ as a discrete
$R\otimes_{\mathbb{S}}R^{op}$-module concentrated in degree 0. We find that:

######  Lemma 2.1.

The filtration on $\operatorname{THH}(R;M)$ induced by the above construction
is complete.

###### Proof.

Since $\operatorname{THH}(R;-)$ is right t-exact, the $n$th filtered piece of
$\operatorname{THH}(R;M)$ is $n$-connective, and the result follows. ∎

We recall that static modules over $R\otimes_{\mathbb{S}}R^{op}$ in degree 0
are exactly the modules which live in the heart of
$\operatorname{LMod}_{R\otimes_{\mathbb{S}}R^{op}}$, and are thus in bijection
with
$\pi_{0}(R\otimes_{\mathbb{S}}R^{op})\simeq\pi_{0}(R)\otimes_{\mathbb{Z}}\pi_{0}(R)^{op}$-modules.
It is clear that this spectral sequence is functorial in $M$, and we also note
that:

######  Proposition 2.2.

Let $R$ and $S$ be connective $\mathbb{E}_{1}$-ring spectra, and $f:R\to S$ a
$\mathbb{E}_{1}$-ring map. If $M$ is an $R$-bimodule, then the natural map
$\operatorname{THH}(R,M)\to\operatorname{THH}(S,(S\otimes
S^{op}\otimes_{R\otimes R^{op}}M)$ induces a map on the associated spectral
sequences.

###### Proof.

It suffices to show that this holds on the level of filtered objects. Using
the fact that we have natural factorizations $\tau_{\geq n}M\to(S\otimes
S^{op})\otimes_{R\otimes R^{op}}\tau_{\geq n}M\to\tau_{\geq n}(S\otimes
S^{op})\otimes_{R\otimes R^{op}}M$, we find that we have a map of filtered
objects
$\operatorname{THH}(R,\tau_{\geq\text{\textbullet}}M)\to\operatorname{THH}(S,(S\otimes
S^{op})\otimes_{R\otimes
R^{op}}\tau_{\geq\text{\textbullet}}M)\to\operatorname{THH}(S,\tau_{\geq\text{\textbullet}}(S\otimes
S^{op})\otimes_{R\otimes R^{op}}M)$, giving the claim. ∎

To make use of the spectral sequences above, it would be useful to have a
point of comparison so we can start getting some differentials, and to know
about the multiplicative structure. On the side of multiplicative structure,
we have:

######  Lemma 2.3.

Suppose $R$ is a connective $\mathbb{E}_{n}$-ring spectrum for some $n\geq 4$.
Then the above spectral sequence is multiplicative whenever $M$ is an
$\mathbb{E}_{2}$-$R$-algebra.

###### Proof.

Suppose $R$ is a $\mathbb{E}_{n}$-algebra for some $n\geq 4$. In particular,
$R\otimes_{\mathbb{S}}R^{op}$ is an $\mathbb{E}_{n}$-algebra as well, and the
canonical multiplication map $R\otimes_{\mathbb{S}}R^{op}\to R$ is an
$\mathbb{E}_{n-1}$-algebra map. Thus, the functor
$R\otimes_{R\otimes_{\mathbb{S}}R^{op}}-:\operatorname{LMod}_{R\otimes_{\mathbb{S}}R^{op}}\to\operatorname{LMod}_{R}$
is $\mathbb{E}_{n-2}$-monoidal, and in particular, $\mathbb{E}_{2}$-monoidal.
Since the Whitehead filtration is compatible with multiplicative structure,
this gives the claim. ∎

Next, we want to be able to find other useful spectral sequences to compare
these to that will allow us to figure out some of their differentials. This is
where cyclic invariance comes in. We work here in the case where $R$ is an
$\mathbb{E}_{\infty}$-ring, but the setup should work more generally with
minimal modifications:

######  Proposition 2.4.

For $R$ an $\mathbb{E}_{1}$-ring, and $S$ a $\mathbb{E}_{1}$-$R$-algebra, we
have an equivalence
$\operatorname{THH}(R,S)\simeq\operatorname{THH}(S,S\otimes_{R}S)$. If $R$ and
$S$ are both $\mathbb{E}_{\infty}$-algebras, then this is an equivalence of
$\mathbb{E}_{\infty}$-algebras.

###### Proof.

See [LL23] where this equivalence is proven for $R$ and $S$
$\mathbb{E}_{1}$-algebras. The multiplicative identification for the
$\mathbb{E}_{\infty}$ case follows from the string of equivalences
$\operatorname{THH}(R,S)\simeq R\otimes_{R\otimes R}S\simeq R\otimes_{R\otimes
R}((S\otimes S)\otimes_{S\otimes S}S)\simeq(R\otimes_{R\otimes R}(S\otimes
S))\otimes_{S\otimes S}S\simeq(S\otimes_{R}S)\otimes_{S\otimes
S}S\simeq\operatorname{THH}(S,S\otimes_{R}S)$, where each of these
equivalences can be viewed as rewriting the same colimit in
$\mathbb{E}_{\infty}$-rings, giving the identification as
$\mathbb{E}_{\infty}$-rings. ∎

When we apply the Whitehead filtration to $S\otimes_{R}S$ in a situation as
above, the resulting spectral sequence has been termed the Brun spectral
sequence, introduced by Brun in [Bru00] and studied by Höning in [Hö20].

For $R$ an arbitrary connective $\mathbb{E}_{n}$-ring spectrum, this already
gives us a lot to compare $R$ with. Namely, it is clear that
$\operatorname{THH}(R,R\otimes_{\mathbb{S}}R^{op})\simeq R$, so applying the
Whitehead filtration to the coefficients produces a spectral sequence
$E_{1}^{s,t}=\operatorname{THH}_{-s}(R,\pi_{t}(R))\Rightarrow\pi_{t-s}(R)$.
Furthermore, the natural map $R\otimes R^{op}\to R$ provides a map on spectral
sequences which is surjective on the $\mathbb{E}_{1}$-page. Thus, if one could
determine the structure of the spectral sequence associated to
$\operatorname{THH}(R,R\otimes R^{op})$, one could compute
$\operatorname{THH}(R)$, although the structure of the former can get quite
complicated in general.

## §3. Topological Hochschild Homology of $\operatorname{tmf}$ with
Coefficients

In this section, we apply the tools developed in section 2 to study the
topological Hochschild homology of connective topological modular forms with
coefficients in $\mathbb{F}_{2}$ and $\mathbb{Z}_{(2)}$. We begin with
$\operatorname{THH}(\operatorname{tmf},\mathbb{F}_{2})$, where cyclic
invariance makes this computation easy. As a precursor, we analyze
$\operatorname{THH}(\mathbb{F}_{2},\mathbb{F}_{2}\otimes_{\mathbb{S}}\mathbb{F}_{2})\simeq\mathbb{F}_{2}$:

######  Example 3.1.

We have a spectral sequence
$E_{1}^{s,t}=\operatorname{THH}_{-s}(\mathbb{F}_{2},\pi_{t}(\mathbb{F}_{2}\otimes_{\mathbb{S}}\mathbb{F}_{2}))\Rightarrow\mathbb{F}_{2}$.
By Bökstedt’s computation of $\operatorname{THH}(\mathbb{F}_{2})$, we know
that the signature of this spectral sequence is

$E_{1}^{s,t}=\mathbb{F}_{2}[\xi_{1},\xi_{2},\xi_{3},\ldots]\otimes_{\mathbb{F}_{2}}\mathbb{F}_{2}[u]\implies\mathbb{F}_{2},$

where $\xi_{i}$ has bidegree $(0,2^{i}-1)$, and $u$ has bidegree $(-2,0)$.
Since this spectral sequence converges to $\mathbb{F}_{2}$, we find that $u$
must map to $\xi_{1}$ on the $E_{1}$-page, leaving
$\mathbb{F}_{2}[\xi_{2},\xi_{3},\ldots]\otimes_{\mathbb{F}_{2}}\mathbb{F}_{2}[u^{2}]$
on the $E_{2}$-page. Repeating inductively, we find that $u^{2^{n-1}}$ maps to
$\xi_{n}$ on the $E_{2^{n}-1}$-page for degree reasons, leaving
$\mathbb{F}_{2}[\xi_{n+1},\xi_{n+2},\ldots]\otimes_{\mathbb{F}_{2}}\mathbb{F}_{2}[u^{2^{n}}]$
on the $E_{2^{n}}$-page, and these differentials together with
multiplicativity, account for all of the nonzero differentials in the spectral
sequence.

######  Example 3.2.

In a similar fashion, we can determine the differentials in the spectral
sequence associated to
$\operatorname{THH}(\mathbb{F}_{p},\mathbb{F}_{p}\otimes_{\mathbb{S}}\mathbb{F}_{p})\simeq\mathbb{F}_{p}$,
which has signature

$E_{1}^{s,t}=\mathbb{F}_{p}[\xi_{1},\xi_{2},\ldots]\otimes\Lambda[\tau_{0},\tau_{1},\ldots]\otimes\mathbb{F}_{p}[u]\implies\mathbb{F}_{p}$

, where $|\xi_{i}|=(0,2(p^{i}-1))$, $|\tau_{i}|=(0,2p^{i}-1)$, and
$|u|=(-2,0)$. We find that the differentials are determined on multiplicative
generators (on respective pages) by $d_{1}(u)=\tau_{0}$,
$d_{2p-3}(u^{p-1}\tau_{0})=\xi_{1}$, and generally,
$d_{2p^{i}-1}(u^{p^{i}})=\tau_{i}$,
$d_{2p^{i-1}(p-1)}(u^{p^{i-1}(p-1)}\tau_{i-1})=\xi_{i}$

######  Example 3.3.

Now, we can compare with the analogous spectral sequence for
$\operatorname{tmf}$, recalling that
$\pi_{*}(\mathbb{F}_{2}\otimes_{\operatorname{tmf}}\mathbb{F}_{2})=\mathbb{F}_{2}[\xi_{1},\xi_{2},\xi_{3}]/(\xi_{1}^{8},\xi_{2}^{4},\xi_{3}^{2})$
as a quotient of the dual Steenrod algebra. The signature of this spectral
sequence is

$E_{1}^{s,t}=\mathbb{F}_{2}[u]\otimes_{\mathbb{F}_{2}}\mathbb{F}_{2}[\xi_{1},\xi_{2},\xi_{3}]/(\xi_{1}^{8},\xi_{2}^{4},\xi_{3}^{2})\implies\operatorname{THH}(\operatorname{tmf},\mathbb{F}_{2}).$

By the comparison map from
$\operatorname{THH}(\mathbb{F}_{2},\mathbb{F}_{2}\otimes_{\mathbb{S}}\mathbb{F}_{2})$,
we find that $u$ maps to $\xi_{1}$ on the $E_{1}$-page, leaving

$E_{2}^{s,t}\simeq\mathbb{F}_{2}[u^{2}]\otimes_{\mathbb{F}_{2}}\mathbb{F}_{2}[\xi_{2},\xi_{3}]/(\xi_{2}^{4},\xi_{3}^{2})\otimes_{\mathbb{F}_{2}}\Lambda[\xi_{1}^{7}u],$

$u^{2}$ maps to $\xi_{2}$ on the $E_{3}$-page, leaving

$E_{4}^{s,t}\simeq\mathbb{F}_{2}[u^{4}]\otimes_{\mathbb{F}_{2}}\mathbb{F}_{2}[\xi_{3}]/(\xi_{3}^{2})\otimes_{\mathbb{F}_{2}}\Lambda[\xi_{1}^{7}u,\xi_{2}^{3}u^{2}],$

and $u^{4}$ maps to $\xi_{3}$ on the $E_{7}$ page, leaving us with

$E_{8}^{s,t}=E_{\infty}^{s,t}=\mathbb{F}_{2}[u^{8}]\otimes_{\mathbb{F}_{2}}\Lambda(u\xi_{1}^{7},u^{2}\xi_{2}^{3},u^{4}\xi_{3}),$

on the $E_{8}$-page, where the spectral sequence must degenerate for degree
reasons since $u^{8}$ cannot hit any nonzero class. There are no
multiplicative extension problems for degree reasons, so we find that

$\pi_{*}\operatorname{THH}(\operatorname{tmf},\mathbb{F}_{2})\simeq\pi_{*}\operatorname{THH}(\mathbb{F}_{2},\mathbb{F}_{2}\otimes_{\operatorname{tmf}}\mathbb{F}_{2})\simeq\mathbb{F}_{2}[u^{8}]\otimes_{\mathbb{F}_{2}}\Lambda(u\xi_{1}^{7},u^{2}\xi_{2}^{3},u^{4}\xi_{3})$

is the tensor product of a polynomial algebra with a generator in degree 16
with an exterior algebra on classes in degrees 9, 13, and 15. For future
reference, let us write
$\pi_{*}(\operatorname{THH}(\operatorname{tmf},\mathbb{F}_{2}))=\mathbb{F}_{2}[\alpha]\otimes_{\mathbb{F}_{2}}\Lambda(\lambda_{1},\lambda_{2},\lambda_{3})$,
with $|\alpha|=16$, $|\lambda_{1}|=9$, $|\lambda_{2}|=13$, and
$|\lambda_{3}|=15$.

Before we compute $\operatorname{THH}(\operatorname{tmf},\mathbb{Z}_{(2)})$,
let’s warm up by redoing the well-known computation for
$\operatorname{THH}(\ell,\mathbb{Z}_{(p)})$ at an odd prime $p$:

######  Example 3.4.

Note that in a fashion similar to Example 3.3, using
$\pi_{*}(\mathbb{F}_{p}\otimes_{\ell}\mathbb{F}_{p})\simeq\Lambda[\tau_{0},\tau_{1}]$,
we find that

$\operatorname{THH}_{*}(\ell,\mathbb{F}_{p})\simeq\mathbb{F}_{p}[u^{p^{2}}]\otimes_{\mathbb{F}_{p}}\Lambda[u^{p-1}\xi_{1},u^{p(p-1)}\xi_{2}].$

Since $\pi_{*}(\ell)\simeq\mathbb{Z}_{(p)}[v_{1}]$, with $|v_{1}|=2$, it is
easy to see that
$\pi_{*}(\mathbb{Z}_{(p)}\otimes_{\ell}\mathbb{Z}_{(p)})\simeq\Lambda_{\mathbb{Z}_{(p)}}[\rho]$,
with $|\rho|=2p-1$. The $E_{1}$-page of our Whitehead spectral sequence for
$\operatorname{THH}(\mathbb{Z}_{(p)},\mathbb{Z}_{(p)}\otimes_{\ell}\mathbb{Z}_{(p)})$
is given by $\operatorname{THH}_{*}(\mathbb{Z}_{(p)})\otimes\Lambda[\rho]$,
with $\rho$ in bidegree $(0,2p-1)$, and
$a\in\operatorname{THH}_{i}(\mathbb{Z}_{(p)})$ in bidegree $(-i,0)$.
Since we only have one copy of $\mathbb{F}_{p}$ in
$\pi_{2p-1}(\operatorname{THH}(\ell,\mathbb{F}_{p}))$, we find that there must
be a multiplicative extension between $\rho$ and the $\mathbb{Z}/p\mathbb{Z}$
class in $\operatorname{THH}_{2p-1}(\mathbb{Z}_{(p)})$. By examining the
locations where elements of $\operatorname{THH}(\ell,\mathbb{F}_{p})$ are
nonzero, and working inductively, we find that for all $r>1$, there is a
nonzero differential on the class
$\operatorname{THH}_{2rp-1}(\mathbb{Z}_{(p)})$, which ends up giving us that

$\operatorname{THH}_{*}(\ell,\mathbb{Z}_{(p)})=\begin{cases}\mathbb{Z}_{(p)}&\qquad\text{if
}*=0,2p-1\\\ \mathbb{Z}/p^{k}\mathbb{Z}&\qquad\text{if
}*=2rp^{k+1}-1,2rp^{k+1}-1+2p-1\text{ with }k>0,\gcd(r,k)=1\\\
0&\qquad\text{otherwise}.\end{cases}$

The case for $\operatorname{THH}(\operatorname{tmf},\mathbb{Z}_{(2)})$ is
slightly more complicated. First, we need to know
$\pi_{*}(\mathbb{Z}_{(2)}\otimes_{tmf}\mathbb{Z}_{(2)})$. To compute this, we
start by looking at
$\pi_{*}(\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\operatorname{tmf})$.

######  Proposition 3.5.

We have
$\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\operatorname{tmf}\simeq\mathbb{Z}_{(2)}[\xi_{1}^{8}]\otimes^{\mathbb{L}}_{\mathbb{Z}_{(2)}}\mathbb{Z}_{(2)}[\xi_{2}^{4}]\otimes^{\mathbb{L}}_{\mathbb{Z}_{(2)}}\bigotimes^{\mathbb{L}}_{\mathbb{Z}_{(2)},i>2}\mathbb{Z}_{(2)}[\xi_{i}^{2}]/(2\xi_{i})$
as a $\mathbb{Z}_{(2)}$-module.

###### Proof.

We know, for instance by [Mat15], that
$\pi_{*}(\mathbb{F}_{2}\otimes_{\mathbb{S}}\operatorname{tmf})\simeq\mathbb{F}_{2}[\xi_{1}^{8},\xi_{2}^{4},\xi_{3}^{2},\xi_{4},\xi_{5},\ldots]$,
identified as a subalgebra of the dual Steenrod algebra via the algebra map
$\operatorname{tmf}\to\mathbb{F}_{2}$. Next, note that
$\mathbb{F}_{2}\otimes_{\mathbb{S}}\operatorname{tmf}\simeq\operatorname{cofib}(2:\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\operatorname{tmf}\to\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\operatorname{tmf})$,
and we have a commutative diagram of ring spectra:

${\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\operatorname{tmf}}$${\mathbb{F}_{2}\otimes_{\mathbb{S}}\operatorname{tmf}}$${\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\mathbb{Z}_{(2)}}$${\mathbb{F}_{2}\otimes_{\mathbb{S}}\mathbb{F}_{2}.}$

The long exact sequence associated to the cofiber sequence tells us that we
must have torsion-free classes in
$\pi_{8}(\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\operatorname{tmf})$ and
$\pi_{12}(\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\operatorname{tmf})$ mapping to
$\xi_{1}^{8}$ and $\xi_{2}^{4}$, respectively. Since
$\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\mathbb{Z}_{(2)}\simeq\bigotimes^{\mathbb{L}}_{\mathbb{Z}_{(2)},i>0}\mathbb{Z}_{(2)}[\xi_{i}^{2}]/(2\xi_{i})$,
we find that the class in degree $8$ must map to $\xi_{1}^{8}$ under the left
vertical map, and the class in degree 12 must map to $\xi_{2}^{4}$ under the
same map, and by abuse, we will give these classes in
$\pi_{*}(\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\operatorname{tmf})$ the same
name. For $i\geq 1$, the fact $\xi_{i}^{2}$ is 2-torsion in
$\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\mathbb{Z}_{(2)}$ reflects the fact that,
under taking the cofiber of 2 on this spectrum, the class in degree
$2(2^{i}-1)+1=2^{i+1}-1$ arising from this torsion is $\xi_{i+1}$. Since we
don’t have any classes in
$\pi_{*}(\mathbb{F}_{2}\otimes_{\mathbb{S}}\operatorname{tmf})$ of the form
$\xi_{2}\xi_{1}^{k}$ or $\xi_{3}\xi_{2}^{k}$, the powers of $\xi_{1}^{8}$ and
$\xi_{2}^{4}$ form a polynomial algebra over $\mathbb{Z}_{(p)}$, giving the
first two tensor factors above. The other tensor factors agree with those in
$\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\mathbb{Z}_{(2)}$, which follows from
examining the classes in
$\pi_{*}(\mathbb{F}_{2}\otimes_{\mathbb{S}}\operatorname{tmf})$ and using
commutativity of the above diagram. Since
$\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\operatorname{tmf}$ is 2-local, and we
don’t have any more classes showing up on the cofiber of $2$, this determines
$\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\operatorname{tmf}$, as desired. ∎

From this, we can deduce:

######  Proposition 3.6.

$\pi_{*}(\mathbb{Z}_{(2)}\otimes_{\operatorname{tmf}}\mathbb{Z}_{(2)})\simeq\begin{cases}\mathbb{Z}_{(2)}&\qquad\text{if
}*=0,22,\\\ \mathbb{Z}_{(2)}\oplus\mathbb{Z}/2\mathbb{Z}&\qquad\text{if
}*=9,13,\\\ \mathbb{Z}/2\mathbb{Z}&\qquad\text{if }*=2,4,8,10,11,12,17,19,\\\
\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}&\qquad\text{if }*=6,15\\\
0&\qquad\text{otherwise}.\end{cases}$

###### Proof.

Note that $\mathbb{S}\to\mathbb{Z}_{(2)}$ factors as
$\mathbb{S}\to\operatorname{tmf}\to\mathbb{Z}_{(2)}$. Thus, we have a diagram
of pushout squares of $\mathbb{E}_{\infty}$-ring spectra:

${\mathbb{S}}$${\operatorname{tmf}}$${\mathbb{Z}_{(2)}}$${\mathbb{Z}_{(2)}}$${\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\operatorname{tmf}}$${\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\mathbb{Z}_{(2)}}$${\mathbb{Z}_{(2)}}$${\mathbb{Z}_{(2)}\otimes_{\operatorname{tmf}}\mathbb{Z}_{(2)},}$

i.e.,
$\mathbb{Z}_{(2)}\otimes_{\operatorname{tmf}}\mathbb{Z}_{(2)}\simeq\mathbb{Z}_{(2)}\otimes_{\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\operatorname{tmf}}(\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\mathbb{Z}_{(2)})$.
This square splits up further by writing
$\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\operatorname{tmf}$ and
$\mathbb{Z}_{(2)}\otimes_{\mathbb{S}}\mathbb{Z}_{(2)}$ as an infinite
(derived) tensor product of $\mathbb{Z}_{(2)}$-algebras, as in Proposition
3.5. For $i\geq 3$, since
$\mathbb{Z}_{(2)}[\xi_{i}^{2}]/(2\xi_{i}^{2})\to\mathbb{Z}_{(2)}[\xi_{i}^{2}]/(2\xi_{i}^{2})$
is an equivalence, this only contributes a $\mathbb{Z}_{(2)}$ to the final
tensor product, and can be ignored. We are left to compute

$\mathbb{Z}_{(2)}[\xi_{1}^{2}]/(2\xi_{1}^{2})\otimes_{\mathbb{Z}_{(2)}[\xi_{1}^{8}]}\mathbb{Z}_{(2)}$

and

$\mathbb{Z}_{(2)}[\xi_{2}^{2}]/(2\xi_{2}^{2})\otimes_{\mathbb{Z}_{(2)}[\xi_{2}^{4}]}\mathbb{Z}_{(2)}.$

For the first case, note that

$\mathbb{Z}_{(2)}[\xi_{1}^{2}]/(2\xi_{1}^{2})\simeq\Sigma^{2}\mathbb{Z}_{(2)}[\xi_{1}^{8}]/2\oplus\Sigma^{4}\mathbb{Z}_{(2)}[\xi_{1}^{8}]/2\oplus\Sigma^{6}\mathbb{Z}_{(2)}[\xi_{1}^{8}]/2\oplus
T$

as a $\mathbb{Z}[\xi_{1}^{8}]$-module, where
$T=\operatorname{cofib}(2\xi_{1}^{8}:\sigma^{8}\mathbb{Z}_{(2)}[\xi_{1}^{8}]\to\mathbb{Z}_{(2)}[\xi_{1}^{8}])$.
It is then easy to compute with these free resolutions that

$\mathbb{Z}_{(2)}[\xi_{1}^{2}]/(2\xi_{1}^{2})\otimes_{\mathbb{Z}_{(2)}[\xi_{1}^{8}]}\mathbb{Z}_{(2)}\simeq\mathbb{Z}_{(2)}\oplus\Sigma^{2}\mathbb{Z}/2\mathbb{Z}\oplus\Sigma^{4}\mathbb{Z}/2\mathbb{Z}\oplus\Sigma^{6}\mathbb{Z}/2\mathbb{Z}\oplus\Sigma^{9}\mathbb{Z}_{(2)}.$

Similar considerations give us

$\mathbb{Z}_{(2)}[\xi_{2}^{2}]/(2\xi_{2}^{2})\otimes_{\mathbb{Z}_{(2)}[\xi_{2}^{4}]}\mathbb{Z}_{(2)}\simeq\mathbb{Z}_{(2)}\oplus\Sigma^{6}\mathbb{Z}/2\mathbb{Z}\oplus\Sigma^{13}\mathbb{Z}_{(2)}.$

Computing the derived tensor product over $\mathbb{Z}_{(2)}$ of these two
$\mathbb{Z}_{(2)}$-modules gives the desired result. ∎

With this in hand, we can finally compute
$\operatorname{THH}_{*}(\operatorname{tmf},\mathbb{Z}_{(2)})$:

######  Theorem 3.7 (Theorem 1.1).

$\operatorname{THH}_{*}(\operatorname{tmf},\mathbb{Z}_{(2)})=\begin{cases}\mathbb{Z}_{(2)}&\qquad\text{if
}*=0,9,13,22\\\ \mathbb{Z}/2^{k}\mathbb{Z}&\qquad\text{if
}*=2r^{k+3}-1,2^{k+3}r-1+9,2^{k+3}r-1+13,2^{k+3}r-1+22,\\\
0&\qquad\text{otherwise},\end{cases}$

for all $k>0$ and $r$ odd.

###### Proof.

First, note that, by Proposition 3.6, the Brun spectral sequence for
$\operatorname{THH}(\mathbb{Z}_{(2)},\mathbb{Z}_{(2)}\otimes_{\operatorname{tmf}}\mathbb{Z}_{(2)})$
has $E_{1}$-page with $\operatorname{THH}_{*}(\mathbb{Z}_{(2)})$ in degrees
$(-*,0),(-*,22)$,
$\operatorname{THH}_{*}(\mathbb{Z}_{(2)},\mathbb{Z}_{(2)}\oplus\mathbb{Z}/2\mathbb{Z})$
in degrees $(-*,9),(-*,13)$,
$\operatorname{THH}_{*}(\mathbb{Z}_{(2)},\mathbb{Z}/2\mathbb{Z})$ in degrees
$(-*,2),(-*,4)$, $(-*,8),(-*,10)$, $(-*,11),(-*,12)$, $(-*,17),(-*,19)$,
$\operatorname{THH}_{*}(\mathbb{Z}_{(2)},\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z})$
in degrees $(-*,6)$ and $(-*,15)$, and is zero otherwise.

This spectral sequence has a lot of terms, so would be difficult to deal with
on its own. Instead, we examine also the Bockstein spectral sequence, with
$E_{1}$-page
$\operatorname{THH}_{*}(\operatorname{tmf},\mathbb{F}_{2})[\tilde{v_{0}}]$,
with $|\tilde{v_{0}}|=(1,1)$. Since every term of the Brun spectral sequence
in negative $s$ degree is torsion, the $\mathbb{Z}_{(2)}$-classes in degrees
0,9,13, and 22 must survive to the $E_{\infty}$-page, giving the respective
classes in $\operatorname{THH}_{*}(\operatorname{tmf},\mathbb{Z}_{(2)})$. This
is reflected in our Bockstein spectral sequence by the fact that our
$\tilde{v_{0}}$-towers on the classes $1$, $\lambda_{1}$, $\lambda_{2}$ and
$\lambda_{1}\lambda_{2}$ must survive the spectral sequence. Thus, we must
have that $\lambda_{3}$ is a permanent cycle. Since the spectral sequence is
multiplicative, we either have that it has degenerated, or else
$d_{1}(\alpha)=\tilde{v_{0}}\lambda_{3}$. The Brun spectral sequence shows
that $\pi_{15}(\operatorname{THH}(\operatorname{tmf},\mathbb{Z}_{(2)}))$ is
finite, so we cannot be in the first case, and thus
$d_{1}(\alpha)=\tilde{v_{1}}\lambda_{3}$, and $\alpha\lambda_{3}$ cannot
support any differentials since anything it could hit is either a permanent
cycle, or was killed off on the first page.

Inductively, we find that $\alpha^{2^{n}}\lambda_{3}$ cannot support any
differentials, and the only multiplicative generator that can support a
differential on or past the $E_{n+2}$-page, is $\alpha^{2^{n+1}}$, which can
only possibly support the differential
$d_{n+2}(\alpha^{2^{n+1}})=\tilde{v_{0}}^{n+2}\alpha^{2^{n}}\lambda_{3}$. If
we did not have this differential, the spectral sequence would be degenerate,
but again, the Brun spectral sequence shows this cannot be the case, so we
have $d_{n+2}(\alpha^{2^{n+1}})=\tilde{v_{0}}^{n+2}\alpha^{2^{n}}\lambda_{3}$
for all $n$. By looking at the cofiber of $2$, we find that
$\operatorname{THH}_{*}(\operatorname{tmf},\mathbb{Z}_{(2)})$ is either
$\mathbb{Z}_{(2)}$ or finite cyclic in any given degree, which determines the
multiplicative extensions in the Bockstein spectral sequence. The classes in
degree $2^{k+3}r-1$ are the classes $\alpha^{r2^{k-1}}\lambda_{3}$, and the
other classes in the second item come from multiplying by $\lambda_{1}$,
$\lambda_{2}$ and $\lambda_{1}\lambda_{2}$. ∎

## §4. THH of Quotients of $\ell$

In this section, we compute the topological Hochschild homology of quotients
$\ell/v_{1}^{n}$ of $\ell$, in terms of the topological Hochschild homology of
$\ell$, which was computed by Angeltveit-Hill-Lawson in [AHL09]. We will show
that
$\pi_{*}\operatorname{THH}(\ell/v_{1}^{n})\simeq\pi_{*}\operatorname{THH}(\ell,\ell/v_{1}^{n})\otimes_{\mathbb{Z}_{(p)}}\Gamma[x]$,
the tensor product of the topological Hochschild homology of $\ell$ with
coefficients in $\ell/v_{1}^{n}$ with a divided power algebra on a generator
$x$ in degree $2n(p-1)+2$. We start by computing
$\operatorname{THH}(\ell/v_{1}^{n},\mathbb{Z}_{(p)})$.

######  Lemma 4.1.

For $n>1$, we have that
$\mathbb{Z}_{(p)}\otimes_{\ell/v_{1}^{n}}\mathbb{Z}_{(p)}\simeq(\mathbb{Z}_{(p)}\otimes_{\ell}\mathbb{Z}_{(p)})\otimes_{\mathbb{Z}_{(p)}}(\mathbb{Z}_{(p)}\otimes_{\mathbb{Z}_{(p)}\otimes_{\ell}\ell/v_{1}^{n}}\mathbb{Z}_{(p)})$
as $\mathbb{E}_{\infty}$-algebras.

###### Proof.

We have a diagram of $\mathbb{E}_{\infty}$-rings where all squares are
pushouts:

${\ell}$${\ell/v_{1}^{n}}$${\mathbb{Z}_{(p)}}$${\mathbb{Z}_{(p)}}$${\mathbb{Z}_{(p)}\otimes_{\ell}\ell/v_{1}^{n}}$${\mathbb{Z}_{(p)}\otimes_{\ell}\mathbb{Z}_{(p)}}$${\mathbb{Z}_{(p)}}$${\mathbb{Z}_{(p)}\otimes_{\ell/v_{1}^{n}}\mathbb{Z}_{(p)}.}$

The map
$\mathbb{Z}_{(p)}\otimes_{\ell}\ell/v_{1}^{n}\to\mathbb{Z}_{(p)}\otimes_{\ell}\mathbb{Z}_{(p)}$
factors over $\tau_{\leq
2(p-1)+1}\mathbb{Z}_{(p)}\otimes_{\ell}\ell/v_{1}^{n}\simeq\mathbb{Z}_{(p)}$,
so that the cospan
$\mathbb{Z}_{(p)}\leftarrow\mathbb{Z}_{(p)}\otimes_{\ell}\ell/v_{1}^{n}\rightarrow\mathbb{Z}_{(p)}\otimes_{\ell}\mathbb{Z}_{(p)}$
may be rewritten as the tensor product over $\mathbb{Z}_{(p)}$ of the cospans
$\mathbb{Z}_{(p)}\leftarrow\mathbb{Z}_{(p)}\otimes_{\ell}\ell/v_{1}^{n}\rightarrow\mathbb{Z}_{(p)}$
and
$\mathbb{Z}_{(p)}\leftarrow\mathbb{Z}_{(p)}\rightarrow\mathbb{Z}_{(p)}\otimes_{\ell}\mathbb{Z}_{(p)}$.
Since colimits commute, we find that the pushout of our original square,
$\mathbb{Z}_{(p)}\otimes_{\ell/v_{1}^{n}}\mathbb{Z}_{(p)}$ is isomorphic to
the pushout of our first cospan tensored over $\mathbb{Z}_{(p)}$ with the
pushout of our second cospan, which is precisely
$(\mathbb{Z}_{(p)}\otimes_{\mathbb{Z}_{(p)}\otimes_{\ell}\ell/v_{1}^{n}}\mathbb{Z}_{(p)})\otimes_{\mathbb{Z}_{(p)}}(\mathbb{Z}_{(p)}\otimes_{\ell/v_{1}^{n}}\mathbb{Z}_{(p)})$,
as claimed. ∎

Next, we examine
$\mathbb{Z}_{(p)}\otimes_{\mathbb{Z}_{(p)}\otimes_{\ell}\ell/v_{1}^{n}}\mathbb{Z}_{(p)}$.
By an easy computation, its homotopy groups are

$\pi_{*}(\mathbb{Z}_{(p)}\otimes_{\ell}\ell/v_{1}^{n})=\begin{cases}\mathbb{Z}_{(p)}&\text{if
}*=0,2n(p-1)+1,\\\ 0&\text{otherwise.}\end{cases}$

In order to figure out the multiplicative structure of
$\pi_{*}(\mathbb{Z}_{(p)}\otimes_{\mathbb{Z}_{(p)}\otimes_{\ell}\ell/v_{1}^{n}}\mathbb{Z}_{(p)})$,
we must figure out the structure of
$\mathbb{Z}_{(p)}\otimes_{\ell}\ell/v_{1}^{n}$ as an algebra. We may note that
this is a $\mathbb{E}_{\infty}$-$\mathbb{Z}_{(p)}$-algebra, which is a square
zero extension of $\mathbb{Z}_{(p)}$ in
$\mathbb{E}_{\infty}$-$\mathbb{Z}_{(p)}$-algebras by [Lur17] Corollary
7.4.1.27. Since $\mathbb{Z}_{(p)}$ is initial in
$\operatorname{CAlg}(\mathbb{Z}_{(p)})$,
$\mathbb{L}^{\operatorname{CAlg}(\mathbb{Z}_{(p)})}_{\mathbb{Z}_{(p)}}$
vanishes, so there is a unique such square zero extension of
$\mathbb{Z}_{(p)}$.

Since we have a dga model for an
$\mathbb{E}_{\infty}$-$\mathbb{Z}_{(p)}$-algebra with the prescribed homotopy
groups, we can take the tensor product
$\mathbb{Z}_{(p)}\otimes_{\mathbb{Z}_{(p)}\otimes_{\ell}\ell/v_{1}^{n}}\mathbb{Z}_{(p)}$
with respect to this dga model, and it will tell us the structure of
$\mathbb{Z}_{(p)}\otimes_{\mathbb{Z}_{(p)}\otimes_{\ell}\ell/v_{1}^{n}}\mathbb{Z}_{(p)}$
as an $\mathbb{E}_{1}$-algebra, and in particular will tell us the
multiplicative structure of the homotopy groups. But it is easy to show that
this just returns a ring with homotopy groups
$\pi_{*}(\mathbb{Z}_{(p)}\otimes_{\mathbb{Z}_{(p)}\otimes_{\ell}\ell/v_{1}^{n}}\mathbb{Z}_{(p)})=\Gamma[x]$
with $x$ in degree $2n(p-1)+2$, as desired.

######  Lemma 4.2.

$\operatorname{THH}_{*}(\ell/v_{1}^{n},\mathbb{Z}_{(p)})\simeq\operatorname{THH}_{*}(\ell,\mathbb{Z}_{(p)})\otimes_{\mathbb{Z}_{(p)}}\pi_{*}(\mathbb{Z}_{(p)}\otimes_{\mathbb{Z}_{(p)}\otimes_{\ell}\ell/v_{1}^{n}}\mathbb{Z}_{(p)})$.

###### Proof.

This follows from

$\displaystyle\operatorname{THH}(\ell/v_{1}^{n},\mathbb{Z}_{(p)})$
$\displaystyle\simeq\operatorname{THH}(\mathbb{Z}_{(p)},\mathbb{Z}_{(p)}\otimes_{\ell/v_{1}^{n}}\mathbb{Z}_{(p)})$
$\displaystyle\simeq\mathbb{Z}_{(p)}\otimes_{\mathbb{Z}_{(p)}\otimes_{\mathbb{S}}\mathbb{Z}_{(p)}}(\mathbb{Z}_{(p)}\otimes_{\ell/v_{1}^{n}}\mathbb{Z}_{(p)})$
$\displaystyle\simeq\mathbb{Z}_{(p)}\otimes_{\mathbb{Z}_{(p)}\otimes_{\mathbb{S}}\mathbb{Z}_{(p)}}((\mathbb{Z}_{(p)}\otimes_{\ell}\mathbb{Z}_{(p)})\otimes_{\mathbb{Z}_{(p)}}(\mathbb{Z}_{(p)}\otimes_{\mathbb{Z}_{(p)}\otimes_{\ell}\ell/v_{1}^{n}}\mathbb{Z}_{(p)}))$
$\displaystyle\simeq(\mathbb{Z}_{(p)}\otimes_{\mathbb{Z}_{(p)}\otimes_{\mathbb{S}}\mathbb{Z}_{(p)}}(\mathbb{Z}_{(p)}\otimes_{\ell}\mathbb{Z}_{(p)}))\otimes_{\mathbb{Z}_{(p)}}(\mathbb{Z}_{(p)}\otimes_{\mathbb{Z}_{(p)}\otimes_{\ell}\ell/v_{1}^{n}}\mathbb{Z}_{(p)})$
$\displaystyle\simeq\operatorname{THH}(\ell,\mathbb{Z}_{(p)})\otimes_{\mathbb{Z}_{(p)}}(\mathbb{Z}_{(p)}\otimes_{\mathbb{Z}_{(p)}\otimes_{\ell}\ell/v_{1}^{n}}\mathbb{Z}_{(p)}).$

∎

Now, we are finally in a position to analyze the spectral sequence associated
to $\operatorname{THH}(\ell/v_{1}^{n})$. We have that
$E_{1}^{s,t}=\mathbb{Z}_{(p)}[v_{1}]/v_{1}^{n}\otimes_{\mathbb{Z}_{(p)}}\operatorname{THH}_{*}(\ell,\mathbb{Z}_{(p)})\otimes_{\mathbb{Z}_{(p)}}\Gamma[x]$,
where $v_{1}$ has bidegree $(0,2(p-1))$, $x$ has bidegree $(-(2n(p-1)+2),0)$,
and $\operatorname{THH}_{s}(\ell,\mathbb{Z}_{(p)})$ lives in degree $(-s,0)$.
There is a comparison map $\rho$ from the spectral sequence associated to
$\operatorname{THH}(\ell,\ell/v_{1}^{n})$ to this one, which determines many
of the differentials. In fact,

######  Theorem 4.3.

All of the differentials vanish on the classes coming from $\Gamma[x]$ in the
$\operatorname{THH}(\ell/v_{1}^{n})$ spectral sequence. In particular, the
comparison map $\rho$, together with multiplicativity, determine all of the
differentials in this spectral sequence, and we have that
$\operatorname{THH}_{*}(\ell/v_{1}^{n})\simeq\operatorname{THH}_{*}(\ell,\ell/v_{1}^{n})\otimes_{\mathbb{Z}_{(p)}}\Gamma[x]$,
where $x$ is in degree $2n(p-1)+2$.

###### Proof.

First, we note that in the case $n\neq 1\mod p^{2}$, the theorem can be proven
by looking only at the spectral sequences we have already constructed. To
prove this theorem in general, we use the filtrations coming from [LL23].
Explicitly, we work over the filtered modules over the
$\mathbb{E}_{\infty}$-algebra $\tau_{\geq*}(\ell/v_{1}^{n})$ in filtered
spectra, where $\tau_{\geq*}(\ell/v_{1}^{n})$ denotes $\ell/v_{1}^{n}$ with
the Whitehead filtration. We can then apply the construction of
$\operatorname{THH}$ to get the filtered spectrum
$\operatorname{THH}(\tau_{\geq*}(\ell/v_{1}^{n}))$, with underlying spectrum
$\operatorname{THH}(\ell/v_{1}^{n})$, and associated graded
$\operatorname{THH}(\mathbb{Z}_{(p)}[\tilde{v_{1}}]/(\tilde{v_{1}}^{n}))$,
where $|\tilde{v_{1}}|=2n(p-1)+2$. To understand the $E_{1}$-page of this
spectral sequence, we use the following lemma, the proof of which is adapted
from lemma 4.1 in [LL23]:

######  Lemma 4.4.

Suppose $k$ is a discrete ring, and $R$ is a connective (possibly graded)
$\mathbb{E}_{2}$-$k$-algebra with $\pi_{*}(R)=k[x]/x^{n}$, on some class $x$
in positive even degree, and $R$ admits an $\mathbb{E}_{2}$-algebra map from a
ring $S$ with $\pi_{*}(S)=k[x]$. Then, we have an equivalence of (graded)
$\mathbb{E}_{1}$-$k$-algebras
$\operatorname{THH}(R)=\operatorname{THH}(k)\otimes_{k}\operatorname{HH}(R/k)$.

###### Proof of lemma.

We have, as in 4.1 of [LL23] $k[x]=k\otimes_{\mathbb{S}}\mathbb{S}[x]$. Now,
as an $\mathbb{E}_{1}$-algebra
$R=k[x]\otimes_{k[x^{n}]}k\simeq(k\otimes_{\mathbb{S}}\mathbb{S}[x])\otimes_{k\otimes_{\mathbb{S}}\mathbb{S}[x^{n}]}(k\otimes_{\mathbb{S}}\mathbb{S})\simeq
k\otimes_{\mathbb{S}}\mathbb{S}[x]/x^{n}$, where $\mathbb{S}[x]/x^{n}$ denotes
$\mathbb{S}[x]\otimes_{\mathbb{S}[x^{n}]}\mathbb{S}$. Since
$\operatorname{THH}$ commutes with tensor products, there are equivalences of
(graded) spectra,

$\operatorname{THH}(R)\simeq\operatorname{THH}(k)\otimes_{\mathbb{S}}\operatorname{THH}(\mathbb{S}[x]/x^{n})\simeq\operatorname{THH}(k)\otimes_{k}k\otimes_{\mathbb{S}}\operatorname{THH}(\mathbb{S}[x]/x^{n})\simeq\operatorname{THH}(k)\otimes_{k}\operatorname{HH}((k[x]/x^{n})/k).$

$k[x]/x^{n}=\tau_{\leq n|x|-1}k[x]$, so that $R$ inherits a canonical
$\mathbb{E}_{2}$-$k$-algebra structure as this truncation. We can give $x$ a
new (positive) grading 1, to make $\mathbb{S}[x]$ a nonnegatively graded
$\mathbb{E}_{2}$-ring spectrum, so an $\mathbb{E}_{2}$-algebra in
$\operatorname{Sp}^{\mathbb{Z}^{ds}_{\geq 0}}$. There is a thick
$\otimes$-ideal $\mathcal{I}$ of $\operatorname{Sp}^{\mathbb{Z}^{ds}_{\geq
0}}$ generated by elements concentrated in grading $\geq n$. Quotienting out
this $\otimes$-ideal gives a symmetric monoidal functor
$\operatorname{Sp}^{\mathbb{Z}^{ds}_{\geq
0}}\to\operatorname{Sp}^{\mathbb{Z}_{\geq 0}^{ds}}$$/\mathcal{I}$, whose right
adjoint is then lax symmetric monoidal by [Lur17] Corollary 7.3.2.7. Composing
these two functors gives a functor which sends our graded $\mathbb{S}[x]$ to a
graded $\mathbb{E}_{2}$-algebra with underlying $\mathbb{E}_{2}$-ring
$\mathbb{S}[x]/x^{n}$, as desired. Using this grading, we can get, from the
$\mathbb{E}_{2}$-map in $\operatorname{Sp}^{\mathbb{Z}^{ds}_{\geq 0}}$,
$\mathbb{S}[x]\to k[x]$. Applying the endofunctor we just described, we get a
$\mathbb{E}_{2}$-map $\mathbb{S}[x]/x^{n}\to k[x]/x^{n}$, which upgrades our
isomorphism above to an $\mathbb{E}_{2}$-algebra isomorphism, which ensures
the induced map on $\operatorname{THH}$ is an isomorphism of
$\mathbb{E}_{1}$-algebras.

∎

We wish to apply this in our case. By [LL23] lemma 2.6, the $t$-structure on
graded spectra given by saying $x_{\text{\textbullet}}$ is connective if
$x_{i}$ is $mi$-connective for some $m$ is compatible with the multiplicative
structure. In particular, choosing $m$ sufficiently large ($m>2n(p-1)+2$), we
get a $t$-structure on graded spectra such that $\tau_{\geq
0}(\pi_{*}(\ell/v_{1}^{n}))=\mathbb{Z}_{(p)}$ concentrated in degree 0, which
shows that $\pi_{*}(\ell/v_{1}^{n})$ is a graded
$\mathbb{E}_{\infty}$-$\mathbb{Z}_{(p)}$-algebra. Now, we can apply the above
theorem to get the $E_{1}$-page of the spectral sequence as

$E_{1}^{s,t}=\operatorname{THH}_{*}(\mathbb{Z}_{(p)})\otimes_{\mathbb{Z}_{(p)}}\operatorname{HH}_{*}((\mathbb{Z}_{(p)}[\tilde{v_{1}}]/\tilde{v_{1}}^{n})/\mathbb{Z}_{(p)}).$

A standard calculation shows that

$\operatorname{HH}_{*}((\mathbb{Z}_{(p)}[\tilde{v_{1}}]/\tilde{v_{1}}^{n})/\mathbb{Z}_{(p)})\simeq\Lambda_{\mathbb{Z}_{(p)}}[d\tilde{v_{1}}]\otimes\Gamma_{\mathbb{Z}_{(p)}}[\sigma^{2}\tilde{v_{1}}^{n}]\otimes\mathbb{Z}_{(p)}[\tilde{v_{1}}]/(\tilde{v_{1}}^{n}),$

where $\sigma^{2}\tilde{v_{1}}^{n}$ is a class in bidegree $(-2,2n(p-1))$, and
$dv_{1}$ is a class in bidegree $(-1,2(p-1))$.

The terms coming from $\operatorname{THH}_{n}(\mathbb{Z}_{(p)})$ live in
bidegree $(-n,0)$, and $\tilde{v_{1}}$ lives in bidegree $(0,2(p-1))$.
Examining the above spectral sequence, we find that
$\sigma^{2}\tilde{v_{1}}^{n}$ is the only class in total degree $(-2,2n(p-1))$
and nothing lives in degree $(0,2n(p-1)+1)$. Furthermore,
$\sigma^{2}\tilde{v_{1}}^{n}$ must vanish under the differentials on every
page, so this is the same class corresponding to $x$ in our other spectral
sequence. Now, it is clear that for all classes $a$, with bigrading
$|a|=(s,t)$, we have that $t\leq-n(p-1)s+2(n-1)(p-1)$ for $s$ even, and
$t\leq-n(p-1)(s+1)+2n(p-1)$ for $s$ odd. Further, $t$ is maximized with
respect to $s$ for $s\leq 0$ even by $\tilde{v_{1}}^{n-1}x^{(-\frac{s}{2})}$,
and for $s$ odd by $d\tilde{v_{1}}\tilde{v_{1}}^{n-1}x^{(-\frac{s+1}{2})}$. In
particular, any differential off of $x^{(}k)$ on the $E_{r}$-page would have
to hit a class in bidegree $(-2k+r+1,2kn(p-1)+r)$. But, we have that, for $r$
odd, $-2k+r+1\geq-2k+2$, and then every class with
$t>2n(p-1)(k-1)+2(n-1)(p-1)$ vanishes in this $s$ degree. In particular, the
target of $d_{r}$ on $x^{(k)}$ is 0. If $r$ is even, then $-2k+r+1\geq-2k+3$,
and thus if $t>2n(p-1)(k-2)+2n(p-1)=2n(p-1)(k-1)$, the classes in degree
$(-2k+r+1,t)$ vanish, and again, the target of $d_{r}(x^{k})$ vanishes. We
have shown that the $x^{(k)}$ are all permanent cycles. Combining this with
our first spectral sequence, we get

$\operatorname{THH}_{*}(\ell/v_{1}^{n})=\operatorname{THH}_{*}(\ell,\ell/v_{1}^{n})\otimes\Gamma[\sigma^{2}v_{1}],$

where $|\sigma^{2}v_{1}|=2n(p-1)+2$.

∎

## §5. The General Case

We remark that many of the constructions in the last section admit a
generalization.

######  Theorem 5.1.

Suppose that $R$ is a connective $\mathbb{E}_{m}$-ring spectrum for some
$m\geq 3$, and $x\in\pi_{*}(R)$ is a positive degree class such that, for some
fixed $k>1$, $R/x^{k}:=\operatorname{cofib}(x^{k}:R\to R)$ admits an
$\mathbb{E}_{3}$-algebra structure. Then, we have an equivalence of
$\tau_{\leq 0}R$-modules
$\operatorname{THH}(R/x^{k},\pi_{0}(R))\simeq\operatorname{THH}(R,\pi_{0}(R))\otimes_{\pi_{0}(R)}(\pi_{0}(R)\otimes_{\pi_{0}(R)\otimes_{R}R/x^{k}}\pi_{0}(R))$.

###### Proof.

Suppose we have a connective $\mathbb{E}_{m}$-ring spectrum $R$ such that we
have a class $x$ and an integer $k$ with the desire properties. We then have
that, by Proposition 2.2,
$\operatorname{THH}(R/x^{k},\pi_{0}(R))\simeq\operatorname{THH}(\pi_{0}(R),\pi_{0}(R)\otimes_{R/x^{k}}\pi_{0}(R))$.
Now, we have equivalences

$\displaystyle\pi_{0}(R)\otimes_{R/x^{k}}\pi_{0}(R)$
$\displaystyle\simeq(\pi_{0}(R)\otimes\pi_{0}(R))\otimes_{R/x^{k}\otimes
R/x^{k}}R/x^{k}$
$\displaystyle\simeq(\pi_{0}(R)\otimes\pi_{0}(R))\otimes_{R/x^{k}\otimes
R/x^{k}}(R/x^{k}\otimes_{R}R/x^{k})\otimes_{(R/x^{k}\otimes_{R}R/x^{k})}R/x^{k}$
$\displaystyle\simeq((\pi_{0}(R)\otimes\pi_{0}(R))\otimes_{R/x^{k}\otimes
R/x^{k}}((R/x^{k}\otimes R/x^{k})\otimes_{R\otimes
R}R))\otimes_{(R/x^{k}\otimes_{R}R/x^{k})}R/x^{k}$
$\displaystyle\simeq((\pi_{0}(R)\otimes\pi_{0}(R))\otimes_{R\otimes
R}R)\otimes_{(R/x^{k}\otimes_{R}R/x^{k})}R/x^{k}$
$\displaystyle\simeq(\pi_{0}(R)\otimes_{R}\pi_{0}(R))\otimes_{(R/x^{k}\otimes_{R}R/x^{k})}R/x^{k}$
$\displaystyle\simeq(\pi_{0}(R)\otimes_{R}\pi_{0}(R))\otimes_{\pi_{0}(R)\otimes_{R}R/x^{k}}(\pi_{0}(R)\otimes_{R}R/x^{k})\otimes_{(R/x^{k}\otimes_{R}R/x^{k})}R/x^{k}$
$\displaystyle\simeq(\pi_{0}(R)\otimes_{R}\pi_{0}(R))\otimes_{\pi_{0}(R)\otimes_{R}R/x^{k}}\pi_{0}(R).$

Now, by assumption, $x^{k}$ is such that $R/x^{k}\to\pi_{0}(R)$ factors as
$R/x^{k}\to R/x^{k-1}\to\pi_{0}(R)$, so,
$\pi_{0}(R)\otimes_{R}R/x^{k}\to\pi_{0}(R)\otimes_{R}\pi_{0}(R)$ factors as
$\pi_{0}(R)\otimes_{R}R/x^{k}\to\pi_{0}(R)\otimes_{R}R/x^{k-1}\to\pi_{0}(R)\otimes_{R}\pi_{0}(R)$.
$\pi_{0}(R)\otimes_{R}R/x^{k-1}$ has homotopy groups given by

$\pi_{*}(\pi_{0}(R)\otimes_{R}R/x^{k-1})=\begin{cases}\pi_{0}(R)&\qquad\text{if
}*=0,\,(k-1)|x|+1,\\\ 0&\qquad\text{otherwise}.\end{cases}$

In particular, $\pi_{0}(R)\otimes_{R}R/x^{k}\to\pi_{0}(R)\otimes_{R}R/x^{k-1}$
factors over
$\tau_{\leq(k-1)|x|+1}(\pi_{0}(R)\otimes_{R}R/x^{k})\simeq\pi_{0}(R)$. This
implies that $\pi_{0}(R)\otimes_{R}\pi_{0}(R)$, as a right
$\pi_{0}(R)\otimes_{R}R/x^{k}$-module, is equivalent to
$(\pi_{0}(R)\otimes_{R}\pi_{0}(R))\otimes_{\pi_{0}(R)}\pi_{0}(R)$, with the
induced right module structure on $\pi_{0}(R)$. Thus,

$\displaystyle\pi_{0}(R)\otimes_{R/x^{k}}\pi_{0}(R)$
$\displaystyle\simeq(\pi_{0}(R)\otimes_{R}\pi_{0}(R))\otimes_{\pi_{0}(R)\otimes_{R}R/x^{k}}\pi_{0}(R)$
$\displaystyle\simeq((\pi_{0}(R)\otimes_{R}\pi_{0}(R))\otimes_{\pi_{0}(R)}\pi_{0}(R))\otimes_{\pi_{0}(R)\otimes_{R}R/x^{k}}\pi_{0}(R)$
$\displaystyle\simeq(\pi_{0}(R)\otimes_{R}\pi_{0}(R))\otimes_{\pi_{0}(R)}(\pi_{0}(R)\otimes_{\pi_{0}(R)\otimes_{R}R/x^{k}}\pi_{0}(R)).$

This derived tensor product in $\pi_{0}(R)$-modules can be computed as a
tensor product on underlying modules, since the second module is flat (in fact
free). Indeed, from the cofiber sequence $\Sigma^{k|x|}R\xrightarrow{x^{k}}R$,
we can tensor this with $\pi_{0}(R)$ to find that
$\pi_{0}(R)\otimes_{R}R/x^{k}\simeq\pi_{0}(R)\oplus\Sigma^{k|x|+1}\pi_{0}(R)$
as a $\pi_{0}(R)$-module. We then have a periodic resolution of $\pi_{0}(R)$
from this class in degree 2, which allows us to see that
$\pi_{0}(R)\otimes_{\pi_{0}(R)\otimes_{R}R/x^{k}}\pi_{0}(R)\simeq\bigoplus_{r\geq
0}\Sigma^{r(k|x|+2)}\pi_{0}(R)$ as a $\pi_{0}(R)$-module.

Now, we apply the Whitehead filtration to
$\pi_{0}(R)\otimes_{R/x^{k}}\pi_{0}(R)$, and examine the spectral sequence
associated to
$\operatorname{THH}(\pi_{0}(R),\tau_{\geq*}(\pi_{0}(R)\otimes_{R/x^{k}}\pi_{0}(R)))$.
Note that this spectral sequence is multiplicative, since by assumption,
$R/x^{k}$ is an $\mathbb{E}_{3}$-algebra, so that the maps
$R/x^{k}\to\pi_{0}(R)/x^{k}\simeq\pi_{0}(R)$ are $\mathbb{E}_{3}$-algebra
maps, which implies $(\pi_{0}(R)\otimes_{R/x^{k}}\pi_{0}(R))$ is an
$\mathbb{E}_{2}$-$\pi_{0}(R)$-algebra. We have a map

$\operatorname{THH}(\pi_{0}(R),\tau_{\geq*}(\pi_{0}(R)\otimes_{R}\pi_{0}(R)))\to\operatorname{THH}(\pi_{0}(R),\tau_{\geq*}(\pi_{0}(R)\otimes_{R/x^{k}}\pi_{0}(R)))$

which descends to a map on the associated spectral sequences. By what we said
above, the $E_{1}$-page of the target is multiplicatively generated by the
classes in the image of this map, together with classes generating copies of
$\pi_{0}(R)$ in degrees $(0,r(k|x|+2))$ for $r>0$. Since there are no nonzero
classes in bidegree $(s,t)$ for $s>0$, so the differentials vanish on these
classes, and there are no multiplicative extension problems between them. The
map from
$\operatorname{THH}(\pi_{0}(R),\tau_{\geq*}(\pi_{0}(R)\otimes_{R}\pi_{0}(R)))$
determine the multiplicative extension problems on the classes in its image,
and this determines all of the multiplicative extension problems, since any
nonzero class $a$ in the image of this map multiplies with any nonzero class
$b$ coming from $(\pi_{0}(R)\otimes_{\pi_{0}(R)\otimes_{R}R/x^{k}}\pi_{0}(R))$
to a nonzero class.

This establishes the claim on the level of homotopy groups. For the full
claim, note that we have a map
$\operatorname{THH}(\pi_{0}(R),\pi_{0}(R)\otimes_{R/x^{k}}\pi_{0}(R))\to\operatorname{THH}(\pi_{0}(R),\pi_{0}(R)\otimes_{\pi_{0}(R)\otimes_{R}R/x^{k}}\pi_{0}(R))\to\pi_{0}(R)\otimes_{\pi_{0}(R)\otimes_{R}R/x^{k}}\pi_{0}(R)\simeq\bigoplus_{r\geq
0}\Sigma^{r(k|x|+2)}\pi_{0}(R)$, which admits a splitting
$\varphi:\bigoplus_{r\geq
0}\Sigma^{r(k|x|+2)}\pi_{0}(R)\to\operatorname{THH}(\pi_{0}(R),\pi_{0}(R)\otimes_{R/x^{k}}\pi_{0}(R))$.
Since $\operatorname{THH}(\pi_{0}(R),\pi_{0}(R)\otimes_{R/x^{k}}\pi_{0}(R))$
admits a
$\operatorname{THH}(\pi_{0}(R),\pi_{0}(R)\otimes_{R}\pi_{0}(R))$-module
structure coming from the natural map, $\varphi$ extends to a map
$\operatorname{THH}(\pi_{0}(R),\pi_{0}(R)\otimes_{R}\pi_{0}(R))\otimes_{\tau_{\geq
0}R}\pi_{0}(R)\otimes_{\pi_{0}(R)\otimes_{R}R/x^{k}}\pi_{0}(R)\to\operatorname{THH}(\pi_{0}(R),\pi_{0}(R)\otimes_{R/x^{k}}\pi_{0}(R))$,
which provides our desired equivalence. ∎

######  Remark 5.2.

As the proof indicates, we can replace $x^{k}$ by any class $x$ in even
positive degree such that $R/x$ is an $\mathbb{E}_{3}$-algebra, and such that
$(R/x\otimes_{R}\pi_{0}(R))\to(\pi_{0}(R)\otimes_{R}\pi_{0}(R))$ factors over
$(R/x\otimes_{R}\pi_{0}(R))\to\tau_{\leq
0}((R/x\otimes_{R}\pi_{0}(R)))\simeq\pi_{0}(R)$.

######  Corollary 5.3.

Let $R$, $x$ and $k$ be as in Theorem 5.1. Then, the $E_{1}$-page of the
spectral sequence coming from
$\operatorname{THH}(R/x^{k},\tau_{\geq*}(R/x^{k}))$ converging to
$\operatorname{THH}(R/x^{k})$ is isomorphic to the $E_{1}$-page of the
spectral sequence coming from $\operatorname{THH}(R,\tau_{\geq*}(R/x^{k}))$
tensored over $\pi_{0}(R)$ with $\bigoplus_{r\geq 0}\tau_{\geq 0}R\cdot
a_{r}$, where $a_{r}$ is a class in bidegree $(-r(k|x|+2),0)$.

###### Proof.

This follows from the proposition together with the fact that if $M$ is any
$\pi_{0}(R)$-module, then

$\displaystyle\operatorname{THH}(R/x^{k},M)$
$\displaystyle\simeq\operatorname{THH}(R/x^{k},\pi_{0}(R))\otimes_{\pi_{0}(R)}M$
$\displaystyle\simeq\operatorname{THH}(R,\pi_{0}(R))\otimes_{\pi_{0}(R)}(\pi_{0}(R)\otimes_{\pi_{0}(R)\otimes_{R}R/x^{k}}\pi_{0}(R))\otimes_{\pi_{0}(R)}M$
$\displaystyle\simeq\operatorname{THH}(R,M)\otimes_{\pi_{0}(R)}(\pi_{0}(R)\otimes_{\pi_{0}(R)\otimes_{R}R/x^{k}}\pi_{0}(R)).$

∎

######  Remark 5.4.

We don’t know whether or not this isomorphism extends to
$\operatorname{THH}_{*}(R/x^{k})\simeq\operatorname{THH}_{*}(R,R/x^{k})\otimes\pi_{*}(R/x^{k}\otimes_{R/x^{k}\otimes_{R}R/x^{k}}R/x^{k})$
in general, although this does seem to hold in many cases.

Together with the map from $\operatorname{THH}(R,\tau_{\geq*}(R/x^{k}))$, this
means that in nice cases, if we understand $\operatorname{THH}(R,R/x^{k})$, we
need only understand the differentials on the classes $a_{r}$ in order to
understand $\operatorname{THH}(R/x^{k})$.

## §6. The THH of $\operatorname{ksc}_{2}$

Using the results of section §5, we will compute
$\operatorname{THH}(\operatorname{ksc}_{2})$ and show that it fits into the
same overall framework as above. Recall that self-conjugate $K$-theory,
$\operatorname{ksc}$ is the $\mathbb{E}_{\infty}$-ring defined as the cofiber
$\operatorname{cofib}(\eta^{2}:\Sigma^{2}\operatorname{ko}\to\operatorname{ko})$,
or alternatively defined as the connective cover of the $\mathbb{Z}$-homotopy
fixed points of the periodic complex $K$-theory spectrum $\operatorname{KU}$,
where $\mathbb{Z}$ acts as $\psi^{-1}$. We have:

######  Theorem 6.1.

$\operatorname{THH}_{*}(\operatorname{ksc}_{2})\simeq\operatorname{THH}_{*}(\operatorname{ko}_{2},\operatorname{ksc}_{2})\otimes_{\mathbb{Z}_{(2)}}\Gamma[\sigma^{2}\eta^{2}]$,
where $\sigma^{2}\eta^{2}$ is a class in degree $4$.

###### Proof.

Since
$\operatorname{ksc}_{2}=\operatorname{cofib}(\eta^{2}:\Sigma^{2}\operatorname{ko}_{2}\to\operatorname{ko}_{2})$,
the results of §5 give us a spectral sequence with signature

$E_{1}^{s,t}=\operatorname{THH}_{*}(\operatorname{ko}_{2},\pi_{*}(\operatorname{ksc}_{2}))\otimes\Gamma[\sigma^{2}\eta^{2}]\implies\operatorname{THH}_{*}(\operatorname{ksc}_{2}).$

Our goal is to show that the classes coming from $\Gamma[\sigma^{2}\eta^{2}]$
are permanent cycles. By similar results to §2, we can recover the known fact
that $\operatorname{THH}_{*}(\operatorname{ko}_{2},\mathbb{Z}_{(2)})$ is
$\mathbb{Z}_{(2)}$ in degrees 0 and 5; $\mathbb{Z}/2^{k}\mathbb{Z}$ in degrees
$r2^{k+2}-1$ and $r2^{k+2}-1+5$ for $r>0$ odd; and is zero otherwise.
Similarly,

$\operatorname{THH}_{*}(\operatorname{ko}_{2},\mathbb{F}_{2})\simeq\mathbb{F}_{2}[u^{4}]\otimes\Lambda[u\xi_{1}^{3},u^{2}\xi_{2}].$

In particular, when we run the Whitehead spectral sequence for
$\operatorname{THH}(\operatorname{ksc}_{2})$, the class $\sigma^{2}\eta^{2}$
in bidegree $(-4,0)$ cannot hit anything for degree reasons, and is thus a
permanent cycle. In order to prove the theorem, it suffices to see that the
classes $(\sigma^{2}\eta^{2})^{(2^{n})}$ do not hit any $2$-torsion classes in
the spectral sequence, since then the differentials on the divided power
classes will all be trivial.

Similar to Theorem 4.3, we start with the filtration of
$\operatorname{ko}_{2}$ from [LL23] Definition 2.12. This has associated
graded given by $\mathbb{Z}_{(2)}[v_{1}^{2},\eta]/(2\eta)$, where $\eta$ is in
filtration degree 2, and $v_{1}^{2}$ is in filtration degree 4. Taking the
cofiber of $\eta^{2}$ on $\operatorname{ko}_{2}^{fil}$ gives a filtered
spectrum $\operatorname{ksc}_{2}^{fil}$ with underlying spectrum
$\operatorname{ksc}_{2}$, and associated graded with
$\pi_{*,*}(\operatorname{ksc}_{2}^{gr})=\mathbb{Z}_{(2)}[v_{1}^{2},\eta,\rho]/(2\eta,\eta^{2},\rho\eta,\rho^{2})$,
where $\rho$ is in topological degree 3 and filtration degree 4. We wish to
understand what $\operatorname{THH}(\operatorname{ksc}_{2}^{gr})$ looks like.
To accomplish this task, we start by examining
$\operatorname{THH}(\operatorname{ko}_{2}^{gr},\mathbb{Z}_{(2)})$. Recall that
$\pi_{*}(\operatorname{ko}_{2}^{gr}/2)=\mathbb{F}_{2}[v_{1},\eta]$, and
$\pi_{*}(\operatorname{ko}_{2}^{gr}/\eta)=\mathbb{Z}_{(2)}[v_{1}]$, coming
from the fact $\operatorname{ko}_{2}^{gr}/\eta$ is the associated graded for
the Whitehead filtration on $\operatorname{ku}_{2}$. It follows that

$\displaystyle\mathbb{Z}_{(2)}\otimes_{\operatorname{ko}_{2}^{gr}}\mathbb{Z}_{(2)}$
$\displaystyle\simeq\mathbb{Z}_{(2)}\otimes_{\operatorname{ko}_{2}^{gr}}\operatorname{ko}_{2}^{gr}/\eta\otimes_{\operatorname{ko}_{2}^{gr}/\eta}\mathbb{Z}_{(2)}$
$\displaystyle\simeq\Lambda_{\mathbb{Z}_{(2)}}[\sigma\eta]\otimes_{\operatorname{ko}_{2}^{gr}/\eta}\mathbb{Z}_{(2)}.$

Using that $\mathbb{Z}_{(2)}=(\operatorname{ko}_{2}^{gr}/\eta)/v_{1}$, and
noting that $v_{1}$ takes $1$ to $2\sigma\eta$, we find that

$\pi_{*}(\mathbb{Z}_{(2)}\otimes_{\operatorname{ko}_{2}^{gr}}\mathbb{Z}_{(2)})=\mathbb{Z}_{(2)}[\sigma\eta,\sigma
v_{1}^{2}]/(\sigma\eta\sigma v_{1}^{2},2\sigma\eta,(\sigma\eta)^{2},(\sigma
v_{1}^{2})^{2}).$

Running the Brun spectral sequence for
$\operatorname{THH}(\operatorname{ko}_{2}^{gr},\mathbb{Z}_{(2)})$ shows that
the only class that may appear in topological degree $\leq 4$ in
$\operatorname{THH}(\operatorname{ko}_{2}^{gr},\mathbb{Z}_{(2)})$ is a
$\mathbb{Z}/2\mathbb{Z}$ class in topological degree 3 and filtration degree 0
(which we will temporarily denote by $\alpha$ if it survives); and the class
$\sigma\eta$ from above.

In the spectral sequence associated to the Whitehead filtration on the
$\mathbb{E}_{\infty}$-$\mathbb{Z}$-algebra $\operatorname{ksc}_{2}^{gr}$, the
only possible classes in topological degree $4$ in
$\operatorname{THH}(\operatorname{ksc}_{2}^{gr})$ are $\sigma^{2}\eta^{2}$, in
filtration degree $4$, and $\alpha\eta$, in topological degree 4 and
filtration degree 2. Furthermore, for all
$x\in\operatorname{THH}(\operatorname{ksc}_{2}^{gr})$,
$\deg^{top}(x)\geq\deg^{fil}(x)-1$.

These results tell us that in the spectral sequence for our filtration, with
$E_{1}$-page $\operatorname{THH}_{*}(\operatorname{ksc}_{2}^{gr})$, the class
$\sigma^{2}\eta^{2}$, which has $(s,t)$-degree $(0,4)$, cannot support any
nonzero differentials, and is a permanent cycle. To see that this
$\sigma^{2}\eta^{2}$ corresponds to the class with the same name in
$\operatorname{THH}(\operatorname{ksc})$, we have to see that there are no
multiplicative extensions. But the only possible multiplicative extension is
with the class $\alpha\eta$, should it have survived. But if $\alpha$ survived
to $\operatorname{THH}(\operatorname{ksc}_{2}^{gr},\mathbb{Z}_{(2)})$, it
would also survive to $\operatorname{THH}(\operatorname{ksc}_{2}^{gr})$ for
degree reasons, giving a class in $(s,t)$-degree $(-3,0)$ on the $E_{1}$-page.
Then there would be no classes for $\alpha$ to hit, and $\alpha$ would either
give a new $\mathbb{Z}/2\mathbb{Z}$ class in
$\operatorname{THH}_{3}(\operatorname{ksc}_{2})$, or would sit in a
multiplicative extension with $\rho$. Since the map
$\operatorname{ksc}_{2}\to\operatorname{THH}(\operatorname{ksc}_{2})$ splits,
the second case cannot happen, and the Whitehead spectral sequence for
$\operatorname{THH}(\operatorname{ksc}_{2})$ shows that the first case cannot
happen, so in fact $\alpha$ does not survive to
$\operatorname{THH}(\operatorname{ksc}_{2}^{gr})$, and there are no possible
multiplicative extensions for $\sigma^{2}\eta^{2}$, showing that this is the
desired class. Now, if all of the classes in $\Gamma[\sigma^{2}\eta^{2}]$
survive to $\operatorname{THH}(\operatorname{ksc}_{2}^{gr})$, then
$(\sigma^{2}\eta^{2})^{(2^{n})}$ sits in $(s,t)$-degree $(0,2^{n+2})$, and
thus cannot support any differentials, giving us the desired classes
$\Gamma[\sigma^{2}\eta^{2}]$ in $\operatorname{THH}(\operatorname{ksc}_{2})$.
We therefore shift our focus to proving this fact.

######  Lemma 6.2.

The classes
$\Gamma[\sigma^{2}\eta^{2}]\subseteq\operatorname{THH}_{*}(\operatorname{ksc}_{2}^{gr},\mathbb{Z}_{(2)})$
can be lifted to divided power algebra classes on $\sigma^{2}\eta^{2}$ in
$\operatorname{THH}_{*}(\operatorname{ksc}_{2}^{gr})$.

###### Proof of Lemma.

We start with the filtered spectrum $\mathbb{Z}_{(2)}^{fil}$, given in the
$2$-adic filtration, with underlying spectrum $\mathbb{Z}_{(2)}$ and
associated graded $\mathbb{F}_{2}[\tilde{v_{0}}]$. Tensoring this with the
$\mathbb{E}_{\infty}$-$\mathbb{Z}_{(2)}$-algebra $\operatorname{ksc}_{2}^{gr}$
gives us a filtered spectrum with associated graded
$\mathbb{F}_{2}[v_{1},\eta,\tilde{v_{0}}]/(\eta^{2})$, where $v_{1},\eta$ are
in graded degree $0$, and $\tilde{v_{0}}$ is in graded degree 1. This
filtration gives rise to the 2-Bockstein spectral sequence

$E_{1}^{s,t}=\operatorname{THH}_{*}(\mathbb{F}_{2}[\eta,v_{1},\tilde{v_{0}}]/(\eta^{2}))\implies\operatorname{THH}_{*}(\operatorname{ksc}_{2}^{gr}).$

By monoidality of $\operatorname{THH}$, we have

$\operatorname{THH}(\mathbb{F}_{2}[\eta,v_{1},\tilde{v_{0}}]/\eta^{2})\simeq\operatorname{THH}(\mathbb{F}_{2}[\eta]/\eta^{2})\otimes_{\operatorname{THH}(\mathbb{F}_{2})}\operatorname{THH}(\mathbb{F}_{2}[v_{1}])\otimes_{\mathbb{F}_{2}}\operatorname{THH}(\mathbb{F}_{2}[\tilde{v_{0}}]).$

Using that $\mathbb{F}_{2}[\eta]/\eta^{2}$ is a square zero extension of
$\mathbb{F}_{2}$ in $\mathbb{E}_{\infty}$-$\mathbb{F}_{2}$-algebras, we find
that

$\operatorname{THH}_{*}(\mathbb{F}_{2}[\eta]/\eta^{2})=\operatorname{THH}_{*}(\mathbb{F}_{2})\otimes\mathbb{F}_{2}[\eta]/\eta^{2}\otimes\Lambda[\sigma\eta]\otimes\Gamma[\sigma^{2}\eta^{2}].$

From this, it follows that

$\operatorname{THH}_{*}(\mathbb{F}_{2}[\eta,v_{1},\tilde{v_{0}}]/\eta^{2})=\operatorname{THH}_{*}(\mathbb{F}_{2})\otimes\mathbb{F}_{2}[\eta,v_{1},\tilde{v_{0}}]/\eta^{2}\otimes\Lambda[\sigma\eta,\sigma
v_{1},\sigma\tilde{v_{0}}]\otimes\Gamma[\sigma^{2}\eta^{2}].$

In the 2-Bockstein spectral sequence, all of the above multiplicative
generators have $t$-degree $0$ except for $\tilde{v_{0}}$, in $(s,t)$-degree
$(1,1)$, and $\sigma\tilde{v_{0}}$, in $(s,t)$-degree $(0,1)$. We begin by
examining the class $\sigma^{2}\eta^{2}$ in degree $(-4,0)$. Almost all of the
classes in the $\tilde{v_{0}}$-tower on this class must survive the spectral
sequence in order to give the $\mathbb{Z}_{(2)}\cdot\sigma^{2}\eta^{2}$ class
in degree 4 of $\operatorname{THH}_{*}(\operatorname{ksc}_{2}^{gr})$. We find
that $\sigma^{2}\eta^{2}$ cannot support any differentials, since any
nontrivial differentials would kill the entire tower. First, let’s study this
spectral sequence a little bit more:

We claim that if $a$ is a class on the $E_{k}$-page with $a\neq 0$, but
$\tilde{v_{0}}a=0$, then $\deg^{t}(a)\leq k-1$. For $k=1$, this is vacuous,
since there is no $\tilde{v_{0}}$-torsion on the $E_{1}$-page of this spectral
sequence. We proceed by induction. Suppose that $a\neq 0$ is a class on the
$E_{k}$-page with $\tilde{v_{0}}a=0$, and $\deg^{t}(a)\geq k$. The fact that
$\tilde{v_{0}}a=0$ means that at some point earlier in the spectral sequence,
say on the $E_{k-i}$-page ($i>0$), we had a class $b$ with
$d_{k-i}(b)=\tilde{v_{0}}a$. $b$ then necessarily has $t$-degree $i+1>1$. In
particular, $\tilde{v_{0}}$ must divide $b$ for degree reasons, so
$b=c\tilde{v_{0}}$, for some class $c$ (or more accurately, comes from a class
on $E_{1}$ divisible by $\tilde{v_{0}}$, and by our inductive hypothesis,
there cannot be a differential taking $c$ to a nonzero class which multiplies
with $\tilde{v_{0}}$ to zero). Now, $d(c)\neq a$, but
$d(\tilde{v_{0}}c)=d(b)=\tilde{v_{0}}a$, so that $a-d(c)\neq 0$, but
$\tilde{v_{0}}(a-d(c))=0$. Since $a-d(c)$ is a $\tilde{v_{0}}$-torsion class
on $E_{k-i}$ with $t$-degree $k\geq k-i$, it must be 0 by induction, so that
$a=d(c)$, contradicting the choice of $a$.

This claim implies that if the differential of any class in $t$-degree $0$ or
$1$ is nontrivial, then the entire $\tilde{v_{0}}$-tower on that class dies.
Let $n$ be the smallest natural such that $(\sigma^{2}\eta^{2})^{(2^{n})}$
does not live in $\operatorname{THH}_{*}(\operatorname{ksc}_{2}^{gr})$. In
particular, we must have that the entire $\tilde{v_{0}}$-tower on the
analogous class in the mod 2 Bockstein must vanish. Since
$(\sigma^{2}\eta^{2})^{(2^{n-1})}$ squares to a torsion-free class, there must
be a nonvanishing $\tilde{v_{0}}$ tower in total degree $2^{n-1}\cdot
4=2^{n+1}$ in the Bockstein spectral sequence. However, considering the map
from the spectral sequence associated to
$\operatorname{THH}(\mathbb{Z}_{(2)}^{fil})$ shows that the classes divisible
by $u$ and $\sigma\tilde{v_{0}}$ are all $\tilde{v_{0}}$-torsion. In order for
$\eta$ to be $2$-torsion, we need a differential to hit $\eta\tilde{v_{0}}$,
and this can be checked to come from $\sigma\eta$. Thus, the only classes that
can contribute to a nonvanishing $\tilde{v_{0}}$-tower are the
$(\sigma^{2}\eta^{2})^{(2^{k})}$, for $k<n$, powers of $v_{1}$, and $\sigma
v_{1}$. Since no power of $v_{1}$ divides any element of
$\Gamma[\sigma^{2}\eta^{2}]$, the only contribution can come from $\sigma
v_{1}$ and the $(\sigma^{2}\eta^{2})^{(2^{k})}$. But the total degree of
$\sigma v_{1}\cdot\prod_{k<n}(\sigma^{2}\eta^{2})^{(2^{k})}$ is $2^{n+1}$, but
the tower we need is in total degree $2^{n+2}$, and thus must come from
$(\sigma^{2}\eta^{2})^{(2^{n})}$! This shows that all of our
$(\sigma^{2}\eta^{2})^{(2^{n})}$ classes have to survive this Bockstein
spectral sequence, proving that they survive to give the divided power classes
in $\operatorname{THH}_{*}(\operatorname{ksc}_{2}^{gr})$ that we were looking
for. ∎

∎

######  Remark 6.3.

$\operatorname{THH}_{*}(\operatorname{ko}_{2},\operatorname{ksc}_{2})$ can be
computed as a graded abelian group from the work of [AHL09], noting that they
prove that $\eta^{2}$ acts as zero on
$\overline{\operatorname{THH}}_{*}(\operatorname{ko}_{2})$, which determines

$\overline{\operatorname{THH}}(\operatorname{ko}_{2},\operatorname{ksc}_{2})\simeq\operatorname{cofib}(\eta^{2}:\Sigma^{2}\overline{\operatorname{THH}}(\operatorname{ko}_{2})\to\overline{\operatorname{THH}}(\operatorname{ko}_{2}))$

up to extension problems. Since the only classes in
$\bar{\operatorname{THH}}_{*}(\operatorname{ko}_{2})$ in odd degrees are
copies of $\mathbb{Z}_{(2)}$ living in what [AHL09] call
$F^{\operatorname{ko}}$, there can only be nontrivial extension problems if
the map from

$\operatorname{THH}_{5+4n}(\operatorname{ko}_{2})\to\operatorname{THH}_{5+4n}(\operatorname{ko}_{2},\operatorname{ksc}_{2})$

is not surjective on the torsion-free parts. However, we know from the
computations in [AHL09] 7.2-7.3 that

$\operatorname{THH}_{5+4n}(\operatorname{ko}_{2})\to\operatorname{THH}_{5+4n}(\operatorname{ko}_{2},\operatorname{ku}_{2})$

induces an isomorphism on the torsion-free part, and this map factors as

$\operatorname{THH}_{5+4n}(\operatorname{ko}_{2})\to\operatorname{THH}_{5+4n}(\operatorname{ko}_{2},\operatorname{ksc}_{2})\to\operatorname{THH}_{5+4n}(\operatorname{ko}_{2},\operatorname{ku}_{2}).$

Thus, we find that, as a graded abelian group,

$\operatorname{THH}_{*}(\operatorname{ko}_{2},\operatorname{ksc}_{2})\simeq\operatorname{THH}_{*}(\operatorname{ko}_{2})\oplus\operatorname{THH}_{*-3}(\operatorname{ko}_{2}).$

Combined with the above, we have completely determined
$\operatorname{THH}_{*}(\operatorname{ksc}_{2})$.

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