---
abstract: 'We compute topological Hochschild homology mod $p$ and $v_1$ of the connective cover of the $K(1)$ local sphere spectrum for all primes $p\ge 3$. This is accomplished using a May-type spectral sequence in topological Hochschild homology constructed from a filtration of a commutative ring spectrum.'
author:
- 'Gabe Angelini-Knoll'
bibliography:
- 'sources.bib'
title: 'On topological Hochschild homology of the $K(1)$-local sphere'
---

Introduction
============

Algebraic K-theory of rings is known to encode deep arithmetic information; for example, algebraic K-groups of rings of integers in totally real number fields are related to special values of Dedekind zeta functions by Lichtenbaum’s conjecture. We would like to explore the arithmetic encoded in the algebraic K-theory of “brave new rings" or, more precisely, commutative ring spectra. To approach this, we use a technique, initiated by Bökstedt in [@bok2], where one approximates algebraic K-theory by topological Hochschild homology. Specifically, topological Hochschild homology is a linear approximation to algebraic K-theory in the sense of Goodwillie’s calculus of functors. The purpose of this paper is to compute topological Hochschild homology of the connective cover of the $K(1)$-local sphere mod $(p,v_1)$.

This computation uses a spectral sequence associated to a multiplicative filtration constructed by the author and A. Salch in [@thhmay]. The idea of the spectral sequence is to mimic the construction of May where he filters a Hopf algebra by powers of the augmentation ideal and constructs an associated filtration of the bar construction. In particular, the author shows in [@thhmay] that there is a model for the Whitehead tower of a connective commutative ring spectrum which produces a multiplicative filtration of that commutative ring spectrum. From this filtered ring spectrum, the author and A. Salch produce a filtration of a generalized bar construction, for example the cyclic bar construction, and construct a spectral sequence in higher topological Hochschild homology. Here we only use the version for the cyclic bar construction and this work is summarized in Section \[section 2\].

In the 1980’s, F. Waldhausen first suggested computing algebraic K-theory of connective versions of the localizations of the sphere spectrum appearing in the chromatic tower [@MR764579]. Recall that there are homology theories $E(n)$, depending on a positive integer $n$ and a prime $p$, called the Johnson-Wilson $E$-theories. Using Bousfield localization we can construct spectra $L_{E(n)}S_{(p)}$ which closely approximate the $p$-local sphere spectrum for large $n$ in the sense that the map $$S_{(p)} \lra \underset{\leftarrow}{\holim}\hspace{.05in} L_{E(n)}S_{(p)}$$ is a weak equivalence due to the chromatic convergence theorem of M. Hopkins and D. Ravenel [@rav2]. F. Waldhausen’s idea was to use his localization theorem to understand the algebraic K-theory of the fibers of the maps $$K(L_{E(n)}S_{(p)}) \lra K(L_{E(n-1)}S_{(p)})$$ and use an inductive argument to understand the whole tower. Unfortunately, Waldhausen’s argument was based on the validity of the telescope conjecture relating the finite localizations $L_{E(n)}^fS_{(p)}$ to $L_{E(n)}S_{(p)}$, which is now believed to be false for $n>1$. However, using a Thomason-Trobaugh type localization theorem in nonconnective K-theory, B. Antieau, T. Barthel, and D. Gepner provide an identification of the fiber that does not rely on the telescope conjecture [@1412.4041].

It is also natural to ask if the chromatic convergence theorem holds after applying algebraic K-theory. J. McClure and R. Staffeldt showed that after taking connective covers the tower converges; i.e. the map $$K(S_{(p)}) \lra \underset{\leftarrow}{\holim}\hspace{.05in} K(\tau_{\ge 0}L_{E(n)}S_{(p)} )$$ is a weak equivalence where we write $\tau_{\ge 0} L_{E(n)}S_{(p)}$ for the connective cover functor applied to the $E(n)$-local sphere [@MR1164148]. By convention, the first object in the tower is $K(\mathbb{Z}_{(p)})$ and the first map is the induced map in algebraic K-theory of the linearization map $\tau_{\ge 0}L_{E(1)}S_{(p)}\rightarrow \mathbb{Z}_{(p)}$. In fact, the only stage in this tower that has been completely computed, after $p$-completion inside, is mod $p$ homotopy of $K(\mathbb{Z}_p)$ due to work of S. Tsalidis [@MR1428060] and M. Bökstedt and I. Madsen [@MR1317117]. The purpose of this paper is to compute a linear approximation to the next stage in the tower after $p$-completion, that is we compute $THH((\tau_{\ge 0}L_{E(1)}S)_p)$ mod $(p,v_1)$ using the fact that $(\tau_{\ge 0}L_{E(1)}S)_p\simeq \tau_{\ge 0}L_{K(1)}S$, where $\tau_{\ge 0}L_{K(1)}S$ is notation for the connective cover of the $K(1)$-local sphere.

Algebraic K-theory of $\tau_{\ge 0}L_{K(1)}S$ is also modeled by iterated algebraic K-theory of certain finite fields, which can be explained as follows. For this explanation, let $p$ be an odd prime and let $q$ be a prime power that also topologically generates $\mathbb{Z}_p^{\times}$. Due to D. Quillen [@MR0315016], it is known that algebraic K-theory of a finite field of order $q$ can be computed, after $p$-completion, using the fiber sequence $$\xymatrix{ K(\mathbb{F}_q)_p \ar[r] & ku_p \ar[r]^{1-\psi_q} & \Sigma^{2}ku_p}$$ $\psi_q$ is the $q$-th Adams operation and $ku_p$ is the $p$-completion of connective complex K-theory. Now, E. Devinatz and M. Hopkins showed that there is an equivalence $$L_{K(1)}S \simeq KU_p^{h\mathbb{G}_1}$$ where $\mathbb{G}_1$ is the height $1$ Morava stabilizer group, which is isomorphic to $\mathbb{Z}_p^{\times}$, and $KU_p$ is the $p$ completion of periodic complex $K$-theory [@MR2030586]. Note that the Morava stabilizer group $\mathbb{G}_1=\mathbb{Z}_p^{\times}$ acts on $KU_p$ by Adams operations, so we may write $\psi_{q}$ for the Adams operation corresponding to the topological generator of $\mathbb{Z}_p^{\times}$ [@MR3328537]. The homotopy fixed points can therefore be modeled by the fiber sequence $$\xymatrix{ L_{K(1)}S \ar[r] & KU_p \ar[r]^{1- \psi_{q} } & KU_p. }$$ Under the stated conditions on $q$ and $p$, there is therefore a map of fiber sequences $$\xymatrix{
K(\mathbb{F}_q)_p \ar[r] \ar[d] & ku_p \ar[r]^{1-\psi_q} \ar[d] & \Sigma^2 ku_p \ar[d]\\
L_{K(1)}S \ar[r] & KU_p \ar[r]^{1-\psi_q} &  KU_p . }$$ By examining homotopy groups of each of the fibers, we see that the map of fibers induces an isomorphism in homotopy in degrees greater or equal to zero. Recall that the homotopy groups of $L_{K(1)}S$ are given by $$\pi_*(L_{K(1)}S) \cong \left \{ \begin{array}{ll} \mathbb{Z}_p & \text{ if } *=0,-1 \\ \mathbb{Z}_p/(q^i-1)\mathbb{Z}_p & \text{ if } *=2i-1 \text{ for } i \in \mathbb{Z} \end{array}\right.$$ and this agrees with Quillen’s computation $$K_*(\mathbb{F}_q) \cong \left \{ \begin{array}{ll} \mathbb{Z} & \text{ if } *=0  \\ \mathbb{Z}/(q^i-1)\mathbb{Z} & \text{ if } *=2i-1  \text{ for  } i \in \mathbb{N} \end{array} \right.$$ in nonnegative degrees after $p$-completion when $q$ is a prime power that topologically generates $\mathbb{Z}_p^{\times}$. Therefore, there are equivalences $K(\mathbb{F}_q)_p \simeq \tau_{\ge 0}L_{K(1)}S\simeq (\tau_{\ge 0}L_{E(1)}S)_p$, where $\tau_{\ge 0}L_{K(1)}S$ denotes the connective cover of the $K(1)$-local sphere.

We compute mod $(p,v_1)$ homotopy of $THH(\tau_{\ge 0}L_{K(1)}S)$ because this is the natural first step in approaching the chromatic red-shift conjecture of C. Ausoni and J. Rognes [@1403.4838]. Recall that there are homology theories $K(n)$ for each prime $p$ and nonnegative integer $n$ with coefficients $$K(n)_*\cong  \left \{ \begin{array}{ll} \mathbb{F}_p[v_n^{\pm 1}] & \text{ when } n>0  \\ \mathbb{Q} & \text{ when  } n=0. \end{array} \right.$$ We say that a $p$-local finite cell $S$-module $V$ has type $n$ if $K(i)_*V\cong 0$ for all $i<n$, but $K(n)_*V$ is nontrivial. Due to the thick subcategory theorem of E. Devinatz, M. Hopkins, J. Smith, the category of $p$-local finite cell $S$-modules $\mathcal{F}_{(p)}$ can be filtered into proper thick subcategories $$0 \subset \dots \cC_2 \subset \cC_1 \subset \cC_0 \subset \mathcal{F}_{(p)}$$ where $\cC_n$ contains exactly spectra of type $>n$; i.e., it consists of the $K(n)$-acyclic spectra [@MR960945]. The first examples of type $n$ spectra for small $n$ can be constructed by an iterative procedure by taking cofibers of $v_n$-self maps beginning with $v_0=p$: for example, the cofiber of the multiplication by $p$ map, denoted $S/p$, is a type one spectrum. At odd primes, we can continue this process and construct a type one spectrum $V(1)$ as the cofiber of a periodic self map $v_1: \Sigma^{2p-2}S/p\rightarrow S/p$. This iterative procedure was originally studied by L. Smith and H. Toda, however, it does not continue to produce type $n$ spectra as L. Nave showed in [@MR2630045].

To phrase the red-shift conjecture, it is necessary to have a notion of “height" for spectra that are not finite cell $S$-modules. We recall a definition due to N. Baas, B. Dundas, and J. Rognes that provides a notion of height in this context. This uses the fact that every $p$-local finite cell $S$-module $V$ admits a $v_n$-self map for some $n$, a consequence of the periodicity theorem of E. Devinatz, M. Hopkins, and J. Smith [@MR960945].

Let $X$ be a spectrum and let $\mathcal{T}_X$ be the thick subcategory of finite $p$-local spectra $V$ such that $$\label{eq red shift} V\wedge X \rightarrow v_n^{- 1} V\wedge  X$$ is induces an isomorphism in homotopy groups in sufficiently high degrees. Then if $\mathcal{T}_X=\cC_{n-1}$, we say that $X$ has telescopic complexity $n$.

To make sense of this definition, one needs to prove that $\mathcal{T}_X$ is indeed a thick subcategory, and this is proven in [@MR2079370]. One way to phrase the red-shift conjecture is, therefore, as follows.

\[conj\] If $R$ is a commutative ring spectrum with telescopic complexity $n$, then $K(R)$ is a commutative ring spectrum with telescopic complexity $n+1$.

Note that the version of the conjecture due to C. Ausoni and J. Rognes states that if $R$ is a “suitably finite" $K(n)$-local commutative ring spectrum then for some $V$ in $\mathfrak{S}_{n}$, the map (\[eq red shift\]) induces an isomorphism in sufficiently high degrees. Since this statement is conjectural, it is possible that it may need to be reformulated for it to be true. We therefore see no issue examining the validity of statement Conjecture \[conj\], which clearly applies in our case of interest and only slightly differs from the original conjecture of C. Ausoni and J.Rognes.

In the example of interest, we know that $$S/p \wedge \tau_{\ge 0}L_{K(1)}S \longrightarrow v_1^{-1} S/p \wedge \tau_{\ge 0}L_{K(1)}S$$ induces an isomorphism in sufficiently high degrees. Since every object in $\mathfrak{S}_{0}$ can be constructed as a finite cell $S/p$-module, it suffices to check telescopic complexity for $S/p$. Therefore, $\tau_{\ge 0}L_{K(1)}S$ has telescopic complexity one. The red-shift conjecture may therefore be phrased in this case as follows: For any $V$ in $\mathfrak{S}_{1}$, the map $$\label{eq case of red shift} V\wedge K(\tau_{\ge 0}L_{K(1)}S)\rightarrow v_2^{-1}V\wedge K(\tau_{\ge 0}L_{K(1)}S)$$ induces an isomorphism in homotopy in sufficiently high degrees. A conjecture of Adams states that all objects in $\mathfrak{S}_1$ can be constructed as finite cell $V(1)$-modules [@MR2707393]. Assuming the validity of Adams’ conjecture, proving the map (\[eq case of red shift\]) induces an isomorphism in homotopy for $V=V(1)$ implies that $K(\tau_{\ge 0}L_{K(1)}S)$ has telescopic complexity $2$. The author plans to address this question in future work as a natural extension of the results of this paper.

In the present paper, we compute topological Hochschild homology of the connective $K(1)$-local sphere, after smashing with the Smith-Toda complex $V(1)$, as follows.

 \[main thm\] There is an isomorphism of graded rings $$V(1)_*(THH(\tau_{\ge 0}L_{K(1)}S))\cong P(\mu_2)\otimes \Gamma(\sigma b) \otimes \mathbb{F}_p\{\alpha_1, \lambda_1', \lambda_2\alpha_1, \lambda_2\lambda_1', \lambda_2\lambda_1'\alpha_1 \}$$ where the products between the classes $$\{\alpha_1, \lambda_1', \lambda_2\alpha_1, \lambda_2\lambda_1', \lambda_2\lambda_1'\alpha_1 \}$$ are zero except for $$\alpha_1\cdot \lambda_2\lambda_1'=\lambda_1'\cdot \lambda_2\alpha_1 =\lambda_2\lambda_1'\alpha_1.$$

The paper is divided into two sections. Section \[section 2\] gives a brief summary of the construction of the THH-May spectral sequence and provides a large class of examples of decreasingly filtered commutative ring spectra; i.e., those that can be produced as a multiplicative model for the Whitehead tower of a connective commutative ring spectrum. In Section \[section 3\], we provide all the details needed to prove Theorem \[main thm\].

Notation and Conventions
------------------------

Throughout, let $\mathfrak{S}$ be the category of $S$-modules in symmetric spectra of pointed simplicial sets, where $S$ is a cofibrant replacement for the sphere spectrum. We equip $\mathfrak{S}$ with the flat model structure [@schwedebook]. Note that $\mathfrak{S}$ is a closed symmetric monoidal model category with $\wedge$ as symmetric monoidal product and $S$ as the unit. We will write $\Comm \mathcal{D}$, for the category of commutative monoids in a symmetric monoidal category $\mathcal{D}$. When a map $X\rightarrow Y$ between objects in $\mathfrak{S}$ is a cofibration, we will write $Y/X$ for the cofiber. We will fix a prime $p\ge 3$ throughout. This ensures that the mod $p$ Moore spectrum has a $v_1$-self map $v_1:\Sigma^{2p-2}S/p\lra S/p$ and the cofiber, denoted $V(1)$, exists.

As we observed in introduction, there is an equivalence $K(\mathbb{F}_q)_p\simeq \tau_{\ge 0} L_{K(1)}S$. There is also an equivalence $\tau_{\ge 0}L_{K(1)}S\simeq j_p$ where $j_p$ is the $p$ completion of the connective image of $J$ spectrum. We use the model $K(\mathbb{F}_q)_p$, which is easily seen to be a commutative ring spectrum and we write $j$ throughout for a cofibrant replacement of this ring spectrum in $\Comm \mathfrak{S}$.

Throughout, we will write $\dot{=}$ to mean equivalence up to multiplication by a unit in $\mathbb{F}_p$. We write $E(x_1,x_2, ...)$ for an exterior algebra over $\mathbb{F}_p$, $P(x_1,x_2,...)$ for a polynomial algebra over $\mathbb{F}_p$, $P_p(x_1,x_2, ...)$ for a truncated polynomial algebra over $\mathbb{F}_p$ truncated at the $p$-th power, and $\Gamma(x_1,x_2,...)$ for a divided power algebra over $\mathbb{F}_p$ on generators $x_i$ for $i\ge 1$. We recall that a divided power algebra generated by $x$ has generators $\gamma_i(x)$ for $i\ge 0$ with relations $\gamma_0(x)=1$, $\gamma_1(x)=x$ and $\gamma_i(x)\gamma_j(x)=(i,j)\gamma_{i+j}(x)$ where $$(i,j)=\left ( \begin{array}{c} i+j \\ j\end{array} \right ).$$ In particular, over the finite field $\mathbb{F}_p$, there is an isomorphism $$\Gamma(z)\cong P_{p}(z, \gamma_p(z) ,\gamma_{p^2}(z) , ...).$$ When not otherwise indicated, $\otimes$ will represent $\otimes_{\mathbb{F}_p}$ and $HH_*(R)$ will indicate $HH_*^{\mathbb{F}_p}(R)$ when $R$ is a $\mathbb{F}_p$-algebra.

Acknowledgements
----------------

It is a pleasure to thank Andrew Salch for countless conversations about material related to this paper. The author also benefited from conversations with Bjorn Dundas, Robert Bruner and Dan Isaksen. The author also acknowledges the concurrent work of Eva Höning on topological Hochschild homology of $K(\mathbb{F}_q)$ using a different approach. Personal communication with Eva Höning resolved a differential that the author originally thought to be trivial.

THH-May spectral sequence {#section 2}
=========================

The goal of this section will be to construct the THH-May spectral sequence in a minimal way. This section is a review of results from the author’s paper with A. Salch and more details about the constructions can be found in that paper [@thhmay]. To construct the spectral sequence, we first need to define what we mean by a filtered object in $\mathfrak{S}$. We first give a description that has the advantage of concreteness.

\[def:DFO1\] A decreasingly filtered commutative monoid in $\mathfrak{S}$ is a sequence of objects in $\mathfrak{S}$ $$\xymatrix{ ... \ar[r]^{f_3} & I_2 \ar[r]^{f_2} & I_1 \ar[r]^{f_1} & I_0 } ,$$ along with structure maps $$\rho_{i,j}: I_i \wedge I_j \lra I_{i+j}$$ satisfying commutativity, associativity, unitality, and compatibility [@thhmay]. We say that $I$ is a cofibrant decreasingly filtered commutative monoid in $\mathfrak{S}$ if each $I_i$ is cofibrant and the maps $f_i$ are all cofibrations.

We avoid writing out all the commutativity, associativity, unitality, and compatibility diagrams by providing an alternate definition of a decreasingly filtered commutative monoid in $\mathfrak{S}$ using properties of functors between symmetric monoidal categories.

\[def:DFO\] A *[decreasingly filtered commutative monoid]{} in $\mathfrak{S}$ is a lax symmetric monoidal functor $$I : (\mathbb{N}^{\op},+,0) \lra (\mathfrak{S},\wedge ,S).$$*

We will write $I_n$ for the functor evaluated on the object $n\in\mathbb{N}^{\op}$. We equip the category of functors $\mathfrak{S}^{\mathbb{N}^{\op}}$ with the projective model structure, where fibrations are defined to be natural transformations $I\rightarrow J$ such that $I_i\rightarrow J_i$ are fibrations for each $i$ and weak equivalences are defined to be natural transformations $I\rightarrow J$ such that $I_i\rightarrow J_i$ is a weak equivalence for each $i$. If $I$ is a cofibrant object in the projective model structure on $\mathfrak{S}^{\mathbb{N}^{\op}}$ then it is a sequence $$\xymatrix{ ... \ar[r]^{f_3} & I_2 \ar[r]^{f_2} & I_1 \ar[r]^{f_1} & I_0 } ,$$ such that each $I_i$ is cofibrant as an object in $\mathfrak{S}$ and the maps $f_i$ are cofibrations for each $i$ [@thhmay].

Due to B. Day [@Day Ex. 3.2.2], the full subcategory of lax symmetric monoidal functors in $\mathfrak{S}^{\mathbb{N}^{\op}}$ is equivalent to the category of commutative monoids in $\mathfrak{S}^{\mathbb{N}^{\op}}$ with the Day convolution symmetric monoidal product. (See [@MR3427193] for a modern treatment of this in the setting of quasi-categories.) The category $\mathfrak{S}^{\mathbb{N}^{\op}}$ is closed symmetric monoidal with the Day convolution symmetric monoidal product due to Day [@Day] and it is a symmetric monoidal model category satisfying the pushout product axiom due to Propositions 2.2.15 and 2.2.16 in the Harvard thesis of Isaacson [@MR2713397]. We may therefore consider an object in $\Comm \mathfrak{S}^{\mathbb{N}^{\op}}$ as a decreasingly filtered commutative monoid and equip the category $\Comm \mathfrak{S}^{\mathbb{N}^{\op}}$ with the model structure created by the forgetful functor $$U: \Comm \mathfrak{S}^{\mathbb{N}^{\op}}\lra \mathfrak{S}^{\mathbb{N}^{\op}};$$ i.e., a map $f$ in $\Comm \mathfrak{S}^{\mathbb{N}^{\op}}$ is a fibration or, respectively, a weak equivalence if $U(f)$ is a fibration in $\mathfrak{S}^{\mathbb{N}^{\op}}$ or, respectively, $U(f)$ is a weak equivalence in $\mathfrak{S}^{\mathbb{N}^{\op}}$.

Due to S. Schwede and B. Shipley [@SS Thm 4.1], applying the forgetful functor to a cofibration with cofibrant source in $\Comm \mathfrak{S}^{\mathbb{N}^{\op}}$ produces a cofibration in $\mathfrak{S}^{\mathbb{N}^{\op}}$, so a cofibrant object in $\Comm \mathfrak{S}^{\mathbb{N}^{\op}}$ is a decreasingly filtered commutative monoid in $\mathfrak{S}$ whose underlying object in $\mathfrak{S}^{\mathbb{N}^{\op}}$ is projective cofibrant. (Note that we need to cofibrantly replace the unit object in $\mathfrak{S}^{\mathbb{N}^{\op}}$ and consider modules over the cofibrant unit to use the theorem of S. Schwede and B. Shipley [@SS].) A projective cofibrant object $I$ in $\mathfrak{S}^{\mathbb{N}^{\op}}$ has the property that $I_i$ is cofibrant for all $i$ and $f_i:I_i\rightarrow I_{i-1}$ is a cofibration for all $i$ (see [@thhmay Lem. 2.1.5]).

A decreasingly filtered commutative monoid in $\mathfrak{S}$ is a *cofibrant decreasingly flltered commutative monoid* in $\mathfrak{S}$ if it is cofibrant in the model structure specified above on $\Comm\mathfrak{S}^{\mathbb{N}^{\op}}$.

Naturally, one would like to define the associated graded commutative monoid in $\mathfrak{S}$ of a decreasingly filtered commutative monoid in $\mathfrak{S}$. The maps $\rho_{i,j}$ are the structure necessary to make sense of such an associated graded commutative monoid in $\mathfrak{S}$.

\[assoc grd\]Let $I$ be a cofibrant decreasingly filtered commutative monoid in $\mathfrak{S}$. We define $E_0^*I$, the [*associated graded commutative monoid of $I$*]{}, to be the graded commutative monoid object in $\mathfrak{S}$ defined as follows:

-   additively, as an object in $\mathfrak{S}$, $$E_0^*I \cong \coprod_{n\in\mathbb{N}} I_n/I_{n+1},$$

-   The unit map $S \rightarrow E_0^*I$ is the composite $$S\stackrel{\eta}{\longrightarrow} I_0 \rightarrow I_0/I_1\hookrightarrow E_0^*I .$$

-   The multiplication on $E_0^*I$ is given as follows. Since the smash product commutes with colimits, hence with coproducts in $\mathfrak{S}$, to specify a map $$E_0^*I \wedge E_0^*I \rightarrow E_0^*I$$ it suffices to specify a component map $$\nabla_{i,j}: I_i/I_{i+1}\wedge I_j/I_{j+1}\rightarrow E_0^*I$$ for every $i,j\in\mathbb{N}$. We define such a map $\nabla_{i,j}$ as follows: first, we have the commutative square $$\xymatrix{ I_{i+1}\wedge I_j \ar[r]^{\rho_{i+1,j}} \ar[d]^{f_{i+1}\wedge \id_{I_j}} & I_{i+j+1}\ar[d]^{f_{i+j+1}} \\
    I_i\wedge I_j \ar[r]^{\rho_{i,j}} & I_{i+j} }$$ so, using the assumption that the maps $f_i$ are cofibrations, we take vertical cofibers to get a map $$\tilde{\nabla}_{i,j}: I_i/I_{i+1}\wedge I_j\rightarrow I_{i+j}/I_{i+j+1} .$$

    Now we have the commutative diagram $$\xymatrix{ I_{i+1}\wedge I_{j+1}\ar[rd]^{\id_{I_{i+1}}\wedge f_{j+1}}\ar[dd]^{f_{i+1}\wedge \id_{I_{j+1}}}\ar[rr]^{\rho_{i+1,j+1}} & & I_{i+j+2}\ar[rd]^{f_{i+j+2}}\ar[dd]^(.7){f_{i+j+2}} & \\
     & I_{i+1}\wedge I_j \ar[dd]^(.7){f_{i+1}\wedge \id_{I_j}} \ar[rr]^(.35){\rho_{i+1,j}} & & I_{i+j+1}\ar[dd]^{f_{i+j+1}} \\
    I_i\wedge I_{j+1}\ar[rd]^{\id_{I_i}\wedge f_{j+1}} \ar[rr]^(.4){\rho_{i,j+1}} \ar[dd] & & I_{i+j+1}\ar[rd]^{f_{i+j+1}}\ar[dd] & \\
     & I_i\wedge I_j\ar[rr]^(.4){\rho_{i,j}} \ar[dd] & & I_{i+j} \ar[dd] \\
    I_i/I_{i+1}\wedge I_{j+1} \ar[rd]_{\id_{I_i/I_{i+1}}\wedge f_{j+1}} \ar[rr]^(.4){\tilde{\nabla}_{i,j+1}} & & I_{i+j+1}/I_{i+j+2}\ar[rd]^{0} & \\
     & I_i/I_{i+1}\wedge I_j \ar[rr]^{\tilde{\nabla}_{i,j}} & & I_{i+j}/I_{i+j+1} }$$ in which the columns are cofiber sequences. We therefore have a choice of factorization of the composite map $\tilde{\nabla}_{i,j}\circ \left( \id_{I_i/I_{i+1}}\wedge f_{j+1}\right)$ through the zero object. Thus, we have the commutative square $$\xymatrix{ I_i/I_{i+1}\wedge I_{j+1}\ar[d]^{\id_{I_i/I_{i+1}}\wedge f_{j+1}} \ar[r] & 0 \ar[d] \\
    I_i/I_{i+1}\wedge I_j \ar[r]^{\tilde{\nabla}_{i,j}} & I_{i+j}/I_{i+j+1} }$$ and, taking vertical cofibers, a map $$I_i/I_{i+1}\wedge I_j/I_{j+1}\rightarrow I_{i+j}/I_{i+j+1},$$ which we compose with the inclusion map $I_{i+j}/I_{i+j+1}\hookrightarrow E_0^*I $ to produce our desired map $$\nabla_{i,j}: I_i/I_{i+1}\wedge I_j/I_{j+1}\rightarrow E_0^*I .$$

Let $S$ be a finite set. We can equip $\mathbb{N}^S$ with the $L_1$ norm so that for $x\in \mathbb{N}^S$ $$|x| = \Sigma_{s\in S} x(s).$$ We then define a sub-poset of $\mathbb{N}^S$ by $$\mathcal{D}_n^S : = \{ x\in \mathbb{N}^S ; |x|\ge n \}.$$ Let $I$ be a cofibrant decreasingly filtered commutative monoid. Define a functor $$\mathcal{F}^S(I) : (\mathbb{N}^{S})^{\op} \lra \mathfrak{S}$$ on objects by $\mathcal{F}^S(I)(x)=\bigwedge_{s\in S} I_{x(s)}$ for $x\in \mathbb{N}^{S}$ and on morphisms in the apparent way. We can precompose the functor with the inclusion to produce a functor $$\xymatrix{\mathcal{F}_n^S(I):(\mathcal{D}_n^S)^{\op} \ar[r] &  (\mathbb{N}^{S})^{\op} \ar[r]^{\mathcal{F}^S(I)} & \mathfrak{S} .}$$ We then define the May filtration associated to the finite set $S$ to be the collection of objects $$\mathcal{M}_n^S(I) := \colim \mathcal{F}_n^S(I)$$ in $\mathfrak{S}$ for $n\in \mathbb{N}$ along with maps $$\mathcal{M}_n^S(I) \lra \mathcal{M}_{n-1}^S(I)$$ given by precomposing with the inclusion $(\mathcal{D}_n^S)^{\op}\lra (\mathcal{D}_{n-1}^S)^{\op}$ and taking colimits.

Let $R$ be an object in $\Comm \mathfrak{S}$. We define the tensor product $$X_{\bullet} \otimes R$$ to be the realization of the simplicial object in $\mathfrak{S}$ $$X_{\bullet} \tilde{\otimes} R$$ where $(X_{\bullet} \tilde{\otimes} R)_n=\bigwedge_{s\in X_n} R\{s\}$ with face maps $$d_i :(X_{\bullet} \tilde{\otimes} R)_n\lra (X_{\bullet} \tilde{\otimes} R)_{n-1}$$ given on each summand of the coproduct by the map $$R\{s\}\lra \bigwedge_{t\in X_{n-1}} R\{t\}$$ which includes $R\{s\}$ into the summand corresponding to $\delta_i(s)\in X_{n-1}$. Here the maps $\delta_i$ and $\sigma_i$ are the face and degeneracy maps of the simplicial finite set $X_{\bullet}$. Note that the coproduct in commutative ring spectra is the smash product and we are using that fact here. Similarly, the degeneracy map $$s_i: (X_{\bullet} \tilde{\otimes} R)_{n-1}\lra (X_{\bullet} \tilde{\otimes} R)_{n}$$ is given on each summand by the map $$R\{t\} \lra \bigwedge_{s\in X_n} R\{s\}$$ where $R\{t\}$ includes as the smash factor corresponding to $\sigma_i(t)\in X_n$.

In the case where $X_{\bullet}=\Delta[1]/\delta \Delta[1]=:S^1_{\bullet}$, the minimal simplicial model for the circle, $$S^1_{\bullet} \otimes R$$ is the geometric realization of the simplicial object in $\mathfrak{S}$ $$\label{}
S^1_{\bullet} \tilde{\otimes} R := \left \{  \xymatrix{ R  & \ar@<1ex>[l]\ar@<-1ex>[l] 
 R\wedge R &                         
 \ar@<2ex>[l]\ar@<-2ex>[l]  \ar[l]R\wedge R \wedge R& 
    \ar@<3ex>[l]\ar@<-3ex>[l]\ar@<1ex>[l]\ar@<-1ex>[l]  \dots   }  \right \}$$ with face and degeneracy maps given by the following formulas: the face maps are $$d_i=\left \{ \begin{array}{ll} 
 \id_{R} \smash ... \id_{R}  \smash \mu \smash \id_{R} \smash ... \smash \id_{R}  & \text{ if } i <n \\ 
(\mu \smash   \id_{R}   \smash \dots   \smash \id_{R}) \circ t_{n}  &  \text{ if } i=n\end{array} \right .$$ where the multiplication map $\mu:R\wedge R\rightarrow R$ is in the $i$-th position on the first line and $t_n: R^{\wedge n} \rightarrow R^{\wedge n}$ is the map that cyclicly permutes the factors to the right. The degeneracy maps are $$s_i= \id_R \smash ... \smash \id_R \smash \eta \smash \id_R \smash \dots \smash \id_R$$ where the unit map $\eta:S\rightarrow R$ from the sphere spectrum is in the $i$-th position.

Using the May filtration in each simplicial degree, we produce a filtration of simplicial objects in $\mathfrak{S}$ $$\xymatrix{
\vdots \ar[d] & \vdots \ar[d] & \vdots \ar[d] & \\
 \mathcal{M}^{X_0}_2(I_{\bullet}) \ar[r]\ar[d] & 
 \mathcal{M}^{X_1}_2(I_{\bullet}) \ar@<1ex>[l] \ar@<-1ex>[l] \ar@<1ex>[r] \ar@<-1ex>[r]\ar[d] &
 \mathcal{M}^{X_2}_2(I_{\bullet}) \ar@<2ex>[l] \ar@<-2ex>[l] \ar[l] \ar@<2ex>[r] \ar@<-2ex>[r] \ar[r]\ar[d] &
 \dots \ar@<-3ex>[l] \ar@<-1ex>[l] \ar@<1ex>[l] \ar@<3ex>[l] \\
 \mathcal{M}^{X_0}_1(I_{\bullet}) \ar[r]\ar[d] & 
 \mathcal{M}^{X_1}_1(I_{\bullet}) \ar@<1ex>[l] \ar@<-1ex>[l] \ar@<1ex>[r] \ar@<-1ex>[r]\ar[d] &
 \mathcal{M}^{X_2}_1(I_{\bullet}) \ar@<2ex>[l] \ar@<-2ex>[l] \ar[l] \ar@<2ex>[r] \ar@<-2ex>[r] \ar[r]\ar[d] &
 \dots \ar@<-3ex>[l] \ar@<-1ex>[l] \ar@<1ex>[l] \ar@<3ex>[l] \\
 \mathcal{M}^{X_0}_0(I_{\bullet}) \ar[r] & 
 \mathcal{M}^{X_1}_0(I_{\bullet}) \ar@<1ex>[l] \ar@<-1ex>[l] \ar@<1ex>[r] \ar@<-1ex>[r] &
 \mathcal{M}^{X_2}_0(I_{\bullet}) \ar@<2ex>[l] \ar@<-2ex>[l] \ar[l] \ar@<2ex>[r] \ar@<-2ex>[r] \ar[r] &
 \dots \ar@<-3ex>[l] \ar@<-1ex>[l] \ar@<1ex>[l] \ar@<3ex>[l] 
}$$ and when $X_{\bullet}=S^1_{\bullet}$ the bottom row is the simplicial object whose geometric realization is $THH(I_0)$.

\[def of thh-may ss\] If $I$ is a cofibrant decreasingly filtered commutative monoid in $\mathfrak{S}$, $S^1_{\bullet}$ is the minimal simplicial model for the circle, and $H_*$ is a connective generalized homology theory then the [*topological Hochschild-May spectral sequence*]{} is the spectral sequence obtained by applying $H_*$ to the tower of cofiber sequences $$\label{tower of sequences} \xymatrix{ 
\vdots \ar[d] & \\
\left| \mathcal{M}_2^{S_{\bullet}}(I)\right| \ar[r]\ar[d] & \left| \mathcal{M}_2^{S_{\bullet}}(I)\right|/\left| \mathcal{M}_3^{S^{\bullet}}(I)\right| \\
\left| \mathcal{M}_1^{S_{\bullet}}(I)\right| \ar[r]\ar[d] & \left| \mathcal{M}_1^{S_{\bullet}}(I)\right|/\left| \mathcal{M}_2^{S_{\bullet}}(I)\right| \\
\left| \mathcal{M}_0^{S_{\bullet}}(I )\right| \ar[r] & \left| \mathcal{M}_0^{S_{\bullet}}(I )\right|/\left| \mathcal{M}_1^{S_{\bullet}}(I )\right|  .}$$ That is, it is the spectral sequence of the exact couple $$\xymatrix{ 
D^1_{*,*} \cong \bigoplus_{i,j} H_i \left| \mathcal{M}_j^{S_{\bullet}}(I )\right| \ar[rr] & & \bigoplus_{i,j} H_i \left| \mathcal{M}_j^{S_{\bullet}}(I )\right| \ar[ld] \\
 & E^1_{*,*} \cong \bigoplus_{i,j} H_i \left| \mathcal{M}_j^{S_{\bullet}}(I )\right|/\left| \mathcal{M}_{j+1}^{S_{\bullet}}(I )\right| \ar[ul] & . }$$

We need to know that the map $$\left| \mathcal{M}_n^{S_{\bullet}}(I )\right| \lra \left| \mathcal{M}_{n-1}^{S_{\bullet}}(I )\right|$$ is cofibration, and this relies on a theorem of the author and A. Salch [@asReedy], which states that a map between two “good" simplicial objects in $\mathfrak{S}$ which is a flat cofibration at each simplicial level realizes to a cofibration. This is the one part of this paper that is model dependent and uses properties of the category of symmetric spectra in simplicial sets with the flat model structure. Otherwise, all the constructions are sufficiently general to work in any nice enough model category including the categories of $\Gamma$-spaces, symmetric spectra, $S$-modules, and orthogonal spectra.

Consequently, the sequence $$\dots \lra \left | \mathcal{M}_2^{S_{\bullet}}(I)\right| \lra \left| \mathcal{M}_{1}^{S_{\bullet}}(I)\right|\lra  \left| \mathcal{M}_{0}^{S_{\bullet}}(I)\right|$$ is again a cofibrant decreasingly filtered commutative monoid in $\mathfrak{S}$, which we will call $\left | \mathcal{M}^{S_{\bullet}}(I) \right |$, and therefore, we can define $E_0^*\left | \mathcal{M}^{S_{\bullet}}(I) \right | $. The input of the THH-May spectral sequence as defined is $H_*\left (E_0^*\left| \mathcal{M}^{S_{\bullet}}(I)\right | \right )$. The main theorem of [@thhmay] produces a more computable $E_2$-page in a more general setting.

Let $X_{\bullet}$ be a simplicial finite set and $I$ be a cofibrant decreasingly filtered commutative monoid in $\mathfrak{S}$, then there is a weak equivalence in the category $\Comm \mathfrak{S}$ $$E_0^*\left| \mathcal{M}^{X_{\bullet}}(I)\right | \simeq  X_{\bullet} \otimes E_0^* I.$$

See [@thhmay] for a detailed proof.

We therefore produce a spectral sequence $$\label{thh map ss} E^2_{s,t} = H_{s,t}(THH(E_0^*(I)))\Rightarrow H_s (THH(I_0))$$ with differential $$d_r : E^r_{s,t}  \lra E^r_{s-1,t+r}$$ for any connective generalized homology theory $H_*$. The spectral sequence strongly converges as long as $\pi_k(I_i)\cong 0$ for $k<i$ and the differentials satisfy a Leibniz rule [@thhmay].

\[ thh-may ss with coeff\] Given a cofibrant decreasingly filtered commutative monoid $I$ and a cofibrant $I$-module $M$ in the category $\mathfrak{S}^{\mathbb{N}^{\op}}$, we can also construct a THH-May spectral sequence with coefficients and prove an analogous fundamental theorem of the May filtration with coefficients. This produces a spectral sequence $$\label{thh may ss coeff} E^2_{*,*} = H_*(THH(E_0^*(I),E_0^*(M) ))\Rightarrow H_* (THH(I_0, M_0))$$ with the same differential convention. The spectral sequence has strong convergence when $\pi_k(I_i)\cong 0$ for $k<i$ and $\pi_k(M_i)\cong 0$ for $k<i$. When $M$ is also itself a cofibrant decreasingly filtered commutative monoid in $\mathfrak{S}$, compatible with the $I$-module structure, the differentials will also satisfy a Leibniz rule. Full details about this construction can be found in the appendix of [@thhmay].

Decreasingly filtered commutative monoids in $\mathfrak{S}$
-----------------------------------------------------------

In order to do computations with the THH-May spectral sequence, it is necessary to have examples of cofibrant decreasingly filtered commutative monoids in $\mathfrak{S}$. The author constructed a large class of examples by constructing a specific model for Whitehead tower associated to cofibrant connective commutative ring spectrum in a joint paper with A. Salch [@thhmay]. This model for the Whitehead tower of $j$ is used to compute mod $(p,v_1)$-homotopy of $THH(j)$ in the subsequent section, so we recall the statement of the theorem.

\[filt\] Let $R$ be a cofibrant connective commutative monoid in $\mathfrak{S}$, then there is an associated cofibrant decreasingly filtered commutative monoid in $\mathfrak{S}$, $$\xymatrix{ \dots \ar[r] &  \tau_{\ge 2}R  \ar[r] & \tau_{\ge 1}R  \ar[r]&  \tau_{\ge 0}R    }$$ where $\pi_k(\tau_{\ge n}R)\cong \pi_k(R)$ for $k\ge n$ and $\pi_k(\tau_{\ge n}R)\cong 0$ for $k<n$, equipped with structure maps $$\rho_{i,j}:\tau_{\ge i}R \wedge \tau_{\ge j}R  \rightarrow \tau_{\ge i+j}R$$ satisfying commutativity, associativity, unitality, and compatibility.

See the author’s joint paper with A. Salch [@thhmay] for a detailed proof.

The fact that we can construct an object in $\mathfrak{S}^{\mathbb{N}^{\op}}$ with these properties is well known. The main thrust of the proof in [@thhmay] is that $\tau_{\ge \bullet}R$ can be built with multiplicative structure; i.e. $\tau_{\ge \bullet}R$ can be constructed as an object in $\Comm \mathfrak{S}^{\mathbb{N}^{\op}}$.

 \[exm J\]Assume a prime $p\ge5$ is fixed and $q$ is a prime power that topologically generates $\mathbb{Z}_p^{\times}$. Recall that $j$ is a cofibrant replacement in $\Comm \mathfrak{S}$ for the connective commutative ring spectrum $\hat{K}(\mathbb{F}_q)_p$. Theorem \[filt\] produces a decreasingly filtered commutative monoid in $\mathfrak{S}$, denoted $\mathbb{J}$. The associated graded $E_0^*\mathbb{J}$ is additively equivalent to $$H\pi_0j\vee\Sigma^{2p-3}H\pi_{2p-3}j\vee \Sigma^{4p-5}H\pi_{4p-5}j\vee ...$$ or more succinctly $H\pi_*(j)$. Its homotopy groups $\pi_*(E_0^*\mathbb{J})$ are isomorphic to $\pi_*(j)$, but it is a generalized Eilenberg-Maclane spectrum. In other words, we have killed off all the Postnikov $k$-invariants. After smashing with $S/p$, there is an equivalence $$S/p \wedge E_0^*\mathbb{J}\simeq H\mathbb{F}_p \vee \bigvee_{i\ge 1} \Sigma^{(2p-2)i-1}H\mathbb{F}_p\vee \Sigma^{(2p-2)i}H\mathbb{F}_p$$ and $\pi_*(S/p \wedge E_0^*\mathbb{J})\cong P(v_1)\otimes E(\alpha_1)$. In particular, $S/p \wedge E_0^*\mathbb{J}$ is an $H\mathbb{F}_p$-algebra.

mod $(p,v_1)$-homotopy of $THH(\ell_{K(1)}S)$ {#section 3}
=============================================

In Section \[section 2\], we reviewed the construction that takes a decreasingly filtered commutative monoid $I$ in $\mathfrak{S}$ as input and produces a May-type spectral sequence $$E^2_{s,t}=E_{s,t}THH(E_0^*I)\Rightarrow E_{s}THH(I_0)$$ for any connective generalized homology theory $E$, which we we call the $E$-THH-May spectral sequence. Also, in Section \[section 2\] we produced a Whitehead-type decreasingly filtered commutative monoid in $\mathfrak{S}$, denoted $\mathbb{J}$, associated to a cofibrant commutative ring spectrum model for $p$-complete connective image of J, which we denote $j$. We therefore have a spectral sequence $$~\label{Ethhmayj} E^2_{s,t}=E_{s,t}THH(E_0^*\mathbb{J})\Rightarrow H_{s}THH(j).$$ The purpose of this section is to compute this spectral sequence in the case $E=V(1)$.

 Computing the $H\mathbb{F}_p$-THH-May spectral sequence
--------------------------------------------------------

In the case where $E=H\mathbb{F}_p$, the input of the spectral sequence is calculable, and the output is already known due the work of Angeltveit-Rognes [@MR2171809]. This computation, therefore, will allow us to see the differentials in the $V(1)$-THH-May spectral sequence that are also detected in the $H\mathbb{F}_p$-THH-May spectral sequence. To begin, let us recall the computation of Angeltveit-Rognes.

 \[HFpj\] There is an isomorphism $${H\mathbb{F}_p}_*(j)\cong P(\tilde{\xi}_1^p,\tilde{\xi}_2,\bar{\xi}_3,...)\otimes E(\tilde{\tau}_2,\bar{\tau}_3, ...)\otimes E(b) \cong (\mathcal{A}//A(1))_*\otimes E(b)$$ where all the elements in $(\mathcal{A}//A(1))_*$ besides $\tilde{\tau}_2$, $\tilde{\xi}_1^p$, and $\tilde{\xi}_2$, and $b$ have the usual $\mathcal{A}_*$-coaction and the coaction on the remaining elements $\tilde{\tau}_2$, $\tilde{\xi}_1^p$, $\tilde{\xi}_2$, and $b$ are\

  -- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
     $\psi(b)=1\otimes b$
     $\psi(\tilde{\xi}_1^p)= 1\otimes \tilde{\xi}_1^p -\tau_0\otimes b + \bar{\xi}_1^p\otimes 1 $
     $\psi(\tilde{\xi}_2)= 1\otimes \tilde{\xi}_2+\bar{\xi}_1\otimes \tilde{\xi}_1^p +\tau_1\otimes b +  \bar{\xi}_2\otimes 1$
     $\psi(\tilde{\tau}_2)= 1\otimes \tilde{\tau}_2 +\bar{\tau}_1\otimes \tilde{\xi}_1^p + \bar{\tau}_0\otimes \tilde{\xi}_2 - \tau_1 \tau_0\otimes b  + \bar{\tau}_2\otimes 1  .$
  -- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

There is also an isomorphism $${H\mathbb{F}_p}_*(THH(j))\cong {H\mathbb{F}_p}_*(j)\otimes E(\sigma \tilde{\xi}_1^p,\sigma \tilde{\xi}_2)\otimes P(\sigma \tilde{\tau}_2)\otimes \Gamma(\sigma b)$$ of $\mathcal{A}_*$-comodules and ${H\mathbb{F}_p}_*(j)$-algebras. The $\mathcal{A}_*$-coaction is given by using the formula $$\psi(\sigma x)=(1\otimes \sigma)\circ \psi(x)$$ and the previously stated coactions.

Note that Angeltveit and Rognes use a tilde over a symbol, for example $\tilde{x}$ to signify that the element has a different coaction then the coaction on $x$ or $\bar{x}$. We now want to compute the input of the spectral sequence. First, we note that as described in Example \[exm J\], $S/p\wedge E_0^*\mathbb{J}$ is an $H\mathbb{F}_p$ algebra and hence $V(1)\wedge E_0^*\mathbb{J}$ is also an $H\mathbb{F}_p$ algebra. It is known more generally that $THH(R)$ is an $R$ algebra, so $V(1)\wedge THH(E_0^*\mathbb{J})$ is a $V(1)\wedge E_0^*\mathbb{J}$-algebra and in particular an $H\mathbb{F}_p$-module. We can therefore apply the following lemma, which can be found in Ausoni-Rognes [@MR2928844 Lem. 4.1], though certainly the lemma predates their work and they refer to Whitehead as the originator. We provide our own proof.

\[prim\] Let $M$ be an $H\mathbb{F}_p$-module. Then $M$ is equivalent to a wedge of suspensions of $H\mathbb{F}_p$, and the Hurewicz map $$\pi_*(M)\lra  {H\mathbb{F}_p}_*(M)$$ induces an isomorphism between $\pi_*(M)$ and the subalgebra of $\mathcal{A}_*$-comodule primitives contained in ${H\mathbb{F}_p}_*(M)$.

We recall that in the language of Hopkins-Smith [@MR1652975] the spectrum $H\mathbb{F}_p$ is a field spectrum, so any $H\mathbb{F}_p$-module is a wedge of suspensions of $H\mathbb{F}_p$. Observe that the Adams spectral sequence $$\Ext_{\mathcal{A}_*}^{s,t}(\mathbb{F}_p;{H\mathbb{F}_p}_*(M) ) \Rightarrow \pi_{t-s}(M)$$ collapses to the $s=0$ line, and therefore the input of the spectral sequence is $$\text{Hom}_{\mathcal{A}_*}(\mathbb{F}_p; {H\mathbb{F}_p}_*(M)).$$ Since $\mathcal{A}_*$-comodule maps from $\mathbb{F}_p$ to ${H\mathbb{F}_p}_*(M)$ are equivalent to $\mathcal{A}_*$-comodule primitives in ${H\mathbb{F}_p}_*(M)$ and ${H\mathbb{F}_p}_*(M)$ is a $\mathcal{A}_*$-comodule algebra, the result follows.

Therefore, computing the algebra of comodule primitives in ${H\mathbb{F}_p}_*(V(1)\wedge THH(E_0^*\mathbb{J}))$ will suffice for computing the input of the $V(1)$-THH-May spectral sequence.

There is an isomorphism $$\pi_*(H\mathbb{F}_p\wedge E_0^*\mathbb{J}) \cong (A//E(0))_*\otimes P(v_1)\otimes E(\alpha_1) .$$

As observed in Example \[exm J\] $$S/p\wedge E_0^*\mathbb{J}\simeq H\mathbb{F}_p \vee \bigvee_{i\ge 1} \Sigma^{(2p-2)i-1}H\mathbb{F}_p\vee \Sigma^{(2p-2)i}H\mathbb{F}_p$$ and $\pi_*(S/p\wedge E_0^*\mathbb{J})\cong P(v_1)\otimes E(\alpha_1)$. By using the equivalence $H\mathbb{Z}\wedge S/p \simeq H\mathbb{F}_p$ we get $$H\mathbb{F}_p\wedge E_0^*\mathbb{J} \simeq H\mathbb{Z} \wedge S/p \wedge E_0^*\mathbb{J}$$ so additively $$H\mathbb{F}_p\wedge E_0^*\mathbb{J}\simeq H\mathbb{Z} \wedge  (H\mathbb{F}_p \vee \bigvee_{i\ge 1} \Sigma^{(2p-2)i-1}H\mathbb{F}_p\vee \Sigma^{(2p-2)i}H\mathbb{F}_p).$$ We can write this as $$(H\mathbb{Z} \wedge H\mathbb{F}_p )\wedge_{H\mathbb{F}_p} \left  (H\mathbb{F}_p \vee \bigvee_{i\ge 1} \Sigma^{(2p-2)i-1}H\mathbb{F}_p\vee \Sigma^{(2p-2)i}H\mathbb{F}_p \right )$$ and use the collapse of the Künneth spectral sequence to produce the desired isomorphism.

 \[Bok1\] There is an isomorphism $${H\mathbb{F}_p}_*(THH(E_0^*\mathbb{J}))\cong (A//E(0))_*\otimes P(v_1)\otimes E(\alpha_1)\otimes E(\sigma \bar{\xi}_1, \sigma v_1 )\otimes P(\sigma \bar{\tau}_2) \otimes \Gamma(\sigma \alpha_1)$$ where the $A_*$-coaction is the usual one, that is the coproduct in $\mathcal{A}_*$, on elements in $(A//E(0))_*$. The coaction on $\alpha_1$ and $v_1$ is given by the formulas\

$\psi(\alpha_1)=1\otimes \alpha_1$\
$\psi(v_1)= \bar{\tau}_0\otimes \alpha_1 + 1\otimes v_1 $\

and the coproduct on the rest of the classes uses the formula $$\psi(\sigma x)= (1\otimes \sigma )\circ \psi(x) .$$

We already know that $\pi_*(H\mathbb{F}_p\wedge E_0^*\mathbb{J})\cong (\mathcal{A}//E(0))_*\otimes P(v_1)\otimes E(\alpha_1)$. We can use the Bökstedt spectral sequence, $$E_2^{s,t}=HH_*({H\mathbb{F}_p}_*E_0^*\mathbb{J}))\Rightarrow {H\mathbb{F}_p}_*(THH(E_0^*\mathbb{J}))$$ to compute ${H\mathbb{F}_p}_*(THH(E_0^*\mathbb{J}))$. The input is $$HH_*((\mathcal{A}//E(0))_*\otimes P(v_1)\otimes E(\alpha_1))\cong (\mathcal{A}//E(0))_*\otimes P(v_1)\otimes E(\alpha_1)\otimes E(\sigma\bar{\xi}_i | i\ge 1)\otimes \Gamma(\sigma \bar{\tau}_i |i\ge 1)\otimes E(\sigma v_1)\otimes \Gamma(\sigma \alpha_1)$$ Using the well known facts that $\Tor^{E(x)}(k;k)\cong \Gamma(\sigma x)$ and $\Tor^{P(y)}(k,k)\cong E(\sigma y)$ for $|x|=2i-1$ and $|y|=2j$ for any $i,j\in \mathbb{N}$, one can show using classical methods that there are isomorphisms $HH_*(E(x))\cong E(x)\otimes \Gamma(\sigma x)$ when $|x|$ is odd, and $HH_*(P(y))\cong P(y)\otimes E(\sigma y)$ when $|y|$ is even (see [@MR2183525], for example, for a detailed calculation). Using these calculations and the fact that the functor $HH_*(-)$ is strong symmetric monoidal, that is, when $A$ and $B$ are graded $\mathbb{F}_p$-algebras there is an isomorphism $$HH_*(A\otimes B)\cong HH_*(A)\otimes HH_*(B),$$ we can compute the input of the Bökstedt spectral sequence. We observe that by the definition of $E_0^*\mathbb{J}$ there is a map of commutative ring spectra $H\mathbb{Z}\lra E_0^*\mathbb{J}$, and therefore a map of Bökstedt spectral sequences, $$\xymatrix{ 
HH_*({H\mathbb{F}_p}_*H\mathbb{Z}))\ar[d] \ar@{=>}[r] &  {H\mathbb{F}_p}_*(THH(H\mathbb{Z})) \ar[d] \\
HH_*({H\mathbb{F}_p}_*E_0^*\mathbb{J})\ar@{=>}[r] & {H\mathbb{F}_p}_*(THH(E_0^*\mathbb{J})) }$$ where the input of the top spectral sequence is $$HH_*({H\mathbb{F}_p}_*H\mathbb{Z})\cong (\mathcal{A}//E(0))_*\otimes E(\sigma \bar{\xi}_i | i \ge 1) \otimes \Gamma(\sigma \bar{\tau}_i | i\ge 1).$$ Recall that due to Bökstedt [@bok] (see also Ausoni [@MR2183525]), there are differentials $$d_{p-1}(\gamma_{p+k}(\sigma \bar{\tau}_i ) )\dot{=} \sigma \bar{\xi}_{i+1} \gamma_{k}(\sigma \bar{\tau}_i)$$ for $k\ge 0$, $i\ge 1$. Since these classes map to classes of the same names in the Bökstedt spectral sequence for $E_0^*\mathbb{J}$, the same differentials occur in the Bökstedt spectral sequence for $E_0^*\mathbb{J}$.

The only remaining possible differentials, for bidegree reasons, are possible differentials on the classes $\gamma_{p^k}(\sigma \alpha_1)$. We claim that these differentials do not occur and we will prove this by contradiction. Suppose $$\label{eq diff} d_r(\gamma_{p^k}(\sigma \alpha_1)\not=0$$ for some $r$ and some $k>0$. Then we observe that in degree $2p^k(p^2-p)$, the dimension of $(H\mathbb{F}_p)_*(THH(E_0^*\mathbb{J}))$ as an $\mathbb{F}_p$ vector space is strictly less than the dimension of $(H\mathbb{F}_p)_*(THH(j))$ in the same degree. Since the $H\mathbb{F}_p$-THH-May spectral sequence with input $(H\mathbb{F}_p)_*(THH(E_0^*\mathbb{J}))$ computes $(H\mathbb{F}_p)_*(THH(j))$, this leads to a contradiction. Therefore, the presence of any differential of the form (\[eq diff\]) contradicts the known computation of $(H\mathbb{F}_p)_*(THH(j))$ due to Angeltveit-Rognes [@MR2171809]. Thus, no differentials of the form (\[eq diff\]) occur.

There is no further room for differentials for bidegree reasons so the $E_{\infty}$-page for $E_0^*\mathbb{J}$ is $$E_{\infty}^{*,*}\cong (\mathcal{A}//E(0))_*\otimes P(v_1)\otimes E(\alpha_1)\otimes E(\sigma \bar{\xi}_1)\otimes P_p(\sigma \bar{\tau}_i | i\ge 1) \otimes E(\sigma v_1) \otimes \Gamma(\sigma \alpha_1)$$ We therefore just need to resolve hidden extensions. Due to Bökstedt [@bok], we know that $\sigma$ commutes with the Dyer-Lashof operations $$Q^{p^i}(\bar{\tau}_i)=\bar{\tau}_{i+1}$$ computed by Steinberger (see [@MR836132] for example), so we have relations $$(\sigma \bar{\tau}_i)^p =Q^{p^i} (\sigma \bar{\tau}_i) = \sigma(Q^{p^i} (\bar{\tau}_i)) = \sigma \bar{\tau}_{i+1}$$ for all $k\ge 0$. These produce hidden multiplicative extensions $$\sigma \bar{\tau}_{i+1}=(\sigma \bar{\tau}_{i})^p$$ and therefore, the multiplicative structure on the output is $${H\mathbb{F}_p}_*(THH(E_0^*\mathbb{J}))\cong (\mathcal{A}//E(0))_*\otimes P(v_1)\otimes E(\alpha_1)\otimes E(\sigma \bar{\xi}_1)\otimes P(\sigma \bar{\tau}_1) \otimes E(\sigma v_1) \otimes \Gamma(\sigma \alpha_1).$$ For the coaction, recall that the class $v_1$ arose as the Bockstein on $\alpha_1$ in $S/p\wedge E_0^*\mathbb{J}$, and therefore the coaction on $v_1$ is $$\psi(v_1)= \bar{\tau}_0\otimes \alpha_1 + 1 \otimes v_1.$$ The class dual to $\alpha_1$ cannot be the Bockstein of a class because $\beta^2=0$. Since the class dual to $\alpha_1$ is in degree $2p-3$ and the lowest class in $\mathcal{A}$ besides $\beta$ is $P^1$ in degree $2p-2$, the class $\alpha_1$ must be a comodule primitive with coaction $$\psi(\alpha_1)=1\otimes \alpha_1.$$ The rest of the coactions are determined by the coproduct in $\mathcal{A}_*$ and the formula $$\psi( \sigma x)= (1\otimes \sigma )\circ \psi(x).$$

We now use the $H\mathbb{F}_p$-THH-May spectral sequence in a case where the output is known due to Angeltveit-Rognes [@MR2171809] in order to detect differentials in the $V(1)$-THH-May spectral sequence.

\[prop d\_1\] The only differentials in the $H\mathbb{F}_p$-THH-May spectral sequence $$(H\mathbb{F}_p)_{s,t}(THH(E_0^*\mathbb{J}))\Rightarrow (H\mathbb{F}_p)_s(THH(j))$$ for $j$ are as follows: $$d_1(\bar{\xi}_1)\dot{=}\alpha_1$$ $$d_1(\sigma \bar{\xi} )\dot{=}\sigma \alpha_1$$ $$d_1(\tilde{\tau}_1)\dot{=}v_1$$ $$d_1(\sigma \bar{\tau}_1 )\dot{=} \sigma v_1.$$ The surviving classes $\bar{\xi}_1^{p-1}\alpha_1$, $\sigma \bar{\xi} \gamma_{p-1}\sigma \alpha_1$, $\gamma_p(\sigma \alpha_1)$, $(\sigma \bar{\tau}_1)^p$, and $(\sigma \bar{\tau}_1)^{p-1}\sigma v_1$ map to classes $b$, $\sigma \tilde{\xi}_1^p$, $\sigma b$, $\sigma \tilde{\tau}_2$, and $\sigma \tilde{\xi}_2$ in ${H\mathbb{F}_p}_*THH(j)$ and all other surviving classes map to classes of the same name.

The output of the spectral sequence is trivial in the range $0< s<2p^2-2p-1$, due to the computation of Angeltveit-Rognes [@MR2171809], and each of the classes which are the source or target of one of the claimed nonzero differentials lie in this range. There are no other possible differentials besides $d_1$ differentials in this range, so this forces the specified $d_1$ differentials. The resulting $E_2=E_{\infty}$-page is isomorphic to ${H\mathbb{F}_p}_*THH(j)$ with the specified correspondence in the proposition.

The behavior of the differentials above leads us to conjecture that $d_1$ commutes with the operation $\sigma$ in the $H\mathbb{F}_p$-THH-May spectral sequence.

\[computation\]There is an isomorphism $$V(1)_*(THH(E_0^*\mathbb{J}))\cong  E(\alpha_1, \lambda_1, \epsilon_1, \sigma \tilde{v_1})\otimes P(\mu_1, \tilde{v}_1)\otimes \Gamma(\sigma\alpha_1)$$ where $|\epsilon_1|=|\lambda_1|=|\sigma \tilde{v}_1|=2p-1$, $|\alpha_1|=2p-3$, $|\mu_1|=2p$, $|\tilde{v}_1|=2p-2$, and $|\sigma \alpha_1|=2p-2$.

We can compute ${H\mathbb{F}_p}_*(V(1)\wedge THH(j))$ where the input is ${H\mathbb{F}_p}_*(V(1)\wedge THH(E_0^*\mathbb{J}))$, using the $H\mathbb{F}_p\smash V(1)$-THH-May spectral sequence. The differentials are the same and the classes $\bar{\tau}_0$ and $\bar{\tau}_1$ map to classes of the same name in the output. This is useful because there is a map of spectral sequences from the $V(1)$-THH-May spectral sequence to the $H\mathbb{F}_p\wedge V(1)$-THH-May spectral sequence induced by the map of $S$-algebras $$\xymatrix{ S\wedge V(1) \ar[r]^{\eta \wedge \id_{V(1)}} & H\mathbb{F}_p\wedge V(1) }$$ where $\eta: S\rightarrow H\mathbb{F}_p$ is the unit map of $H\mathbb{F}_p$ as a ring spectrum. Due to Lemma \[prim\], the map $$V(1)_*(THH(E_0^* \mathbb{J} ))\lra (H\mathbb{F}_p\wedge V(1))_*(THH(E_0^* \mathbb{J}))$$ includes $V(1)_*(THH(E_0^*\mathbb{J}))$ into $(H\mathbb{F}_p\wedge V(1))_*(THH(E_0^*\mathbb{J}))$ as the $\mathcal{A}_*$-comodule primitives. By Lemma \[Bok1\], the elements $$\{ \alpha_1, v_1-\bar{\tau}_0\alpha_1 , \sigma \tilde{\tau}_1-\bar{\tau}_0\bar{\xi}_1,  \sigma \bar{\xi}_1, \sigma \alpha_1, \sigma v_1-\bar{\tau}_0\sigma\alpha_1, \tilde{\tau}_1-\bar{\tau}_1\}$$ are comodule primitives where we write $\tilde{\tau}_1$ to distinguish the class in ${H\mathbb{F}_p }_*(THH(E_0^*\mathbb{J}))$ from the class $\bar{\tau}_1\in {H\mathbb{F}_p}_*(V(1))$. These classes generate the algebra of comodule primitives. We rename these classes respectively $$\{ \alpha_1, \tilde{v}_1, \mu_1, \lambda_1, \sigma\alpha_1, \sigma \tilde{v}_1, \epsilon_1.\}$$ Thus, the result follows from Lemma \[Bok1\] and Lemma \[prim\].

We now consider the map of THH-May spectral sequences $$\xymatrix{ 
 V(1)_*(THH(E_0^*\mathbb{J}))\ar[d]^{f} \ar@{=>}[r] & V(1)_*(THH(j))\ar[d]  \\
 (H\mathbb{F}_p\wedge V(1))_*(THH(E_0^*\mathbb{J})) \ar@{=>}[r] &  (H\mathbb{F}_p\wedge V(1))_*(THH(j)) }$$ induced by the map $$\eta \wedge \id_{V(1)} : S\wedge V(1)\lra H\mathbb{F}_p\wedge V(1)$$ where $\eta:S\rightarrow H\mathbb{F}_p$ is the unit map of the ring spectrum $H\mathbb{F}_p$.

\[prop d1\]The only $d_1$ differentials are $$d_1(\lambda_1)\dot{=}\sigma \alpha_1,$$ $$d_1(\epsilon_1)\dot{=}\tilde{v}_1, \text{ and  }$$ $$d_1(\mu_1)\dot{=} \sigma \tilde{v}_1$$ in the $V(1)$-THH-May spectral sequence. The $E_2$-page of the $V(1)$-THH-May spectral sequence is therefore $$E_2^{*,*}=E(\alpha_1, \lambda_1\gamma_{p-1}(\sigma \alpha_1), (\mu_1)^{p-1}\sigma \tilde{v}_1 ) \otimes P((\mu_1)^p)\otimes \Gamma(\sigma b)$$

The classes $$\{ \tilde{v}_1, \mu_1, \lambda_1, \sigma\alpha_1,  \sigma \tilde{v}_1, \epsilon_1 \}$$ in the $V(1)$-THH-May spectral sequence map to the classes $$\{ v_1-\bar{\tau}_0\alpha_1 , \sigma \tilde{\tau}_1-\bar{\tau}_0\bar{\xi}_1,  \sigma \bar{\xi}_1, \sigma \alpha_1, \sigma v_1-\bar{\tau}_0\sigma\alpha_1, \tilde{\tau}_1-\bar{\tau}_1\}$$ in the $H\mathbb{F}_p\wedge V(1)$-THH-May spectral sequence under the map of spectral sequences $f$. There are trivial differentials $$d_1(\bar{\tau}_0)=d_1(\bar{\tau}_1)=0$$ and nontrivial differentials

  ----------------------------------- ------------------------------------------------
   $d_1(\bar{\xi}_1)\dot{=}\alpha_1$   $d_1(\sigma \bar{\xi} )\dot{=}\sigma \alpha_1$
    $d_1(\tilde{\tau}_1)\dot{=}v_1$    $d_1(\sigma \bar{\tau}_1 )\dot{=} \sigma v_1$
  ----------------------------------- ------------------------------------------------

in the $H\mathbb{F}_p\wedge V(1)$-THH-May spectral sequence by Propositions \[prop d\_1\] and \[computation\]. We will use the formula $fd_1=d_1f$ to compute the differentials. Notice that the map $f$ is injective on the $E_2$-page of the spectral sequences so it makes sense to use the formula $d_1(x)=f^{-1}d_1(f(x))$. We therefore produce differentials $$d_1(\lambda_1)= f^{-1}(d_1(\sigma \bar{\xi}_1))= f^{-1}(\sigma \alpha_1)=\sigma \alpha_1,$$ $$d_1(\epsilon_1)=f^{-1}d_1(\tilde{\tau}_1-\bar{\tau}_1)=f^{-1}(v_1)=\tilde{v}_1$$ $$d_1(\mu_1)=f^{-1}(d_1(\sigma \tilde{\tau}_1-\bar{\tau}_0\bar{\xi}_1))=f^{-1}(v_1-\bar{\tau}_0\alpha_1)=\tilde{v_1}$$ in the $V(1)$-THH-May spectral sequence as desired. There are no other possible $d_1$ differentials for bidegree reasons.

The author owes Eva Höning for pointing out that an element in $H_*(M)$ must be a comodule primitive when $M$ is an $H\mathbb{F}_p$-module. This line of thinking led to the proof of the following lemma.

There is an isomorphism $$V(1)_*(THH(j;\ell)) \cong E(\lambda_1',\lambda_2)\otimes P(\mu_2)\otimes \Gamma(\sigma b)$$

Note that there are equivalences $$V(1)\smash THH(j;\ell)\simeq THH(j;H\mathbb{F}_p)\simeq H\mathbb{F}_p\wedge_j THH(j)$$ and that $H\mathbb{F}_p\wedge_j THH(j)$ is a $H\mathbb{F}_p\wedge_j j$-module, i.e. it is a $H\mathbb{F}_p$-module. We can therefore apply Lemma \[prim\] and Theorem \[HFpj\] to compute $V(1)_*THH_*(j;\ell)$. The result is the algebra of comodule primitives in $${H\mathbb{F}_p}_*(THH(j;H\mathbb{F}_p)) \cong \mathcal{A}_*\otimes E(\sigma \tilde{\xi}_1^p, \sigma \tilde{\xi}_2)\otimes P(\sigma \tilde{\tau}_2)\otimes  \Gamma(\sigma b) ,$$ which can be seen by the collapse of the Künneth spectral sequence. The algebra of comodule primitives is isomorphic to $$E(\lambda_1', \lambda_2)\otimes P(\mu_2)\otimes  \Gamma(\sigma b)$$ where $$\lambda_1'=\sigma \tilde{\xi}_1^p -\bar{\tau}_0\sigma b,$$ $$\lambda_2= \sigma \bar{\xi}_2 - \bar{\xi}_1 \otimes \sigma \tilde{\xi}_1^p - \bar{\tau}_1\otimes \sigma b,$$ $$\mu_2 = \sigma \tau_2 - \bar{\tau}_0 \sigma \tilde{\xi}_2 - \bar{\tau}_1 \sigma \tilde{\xi}_1^p +\tau_0\tau_1 \sigma b, \text{ and }$$ $$\sigma b = \sigma b.$$

We have another approach to computing $THH_*(j; j/(p,v_1))=V(1)_*(THH(j))$, as a $V(1)_*$-module, but not as graded rings, by filtering the coefficients $j/(p,v_1)$ using the short filtration $$0 \lra \Sigma^{2p-3}H\mathbb{F}_p \lra j/(p,v_1)$$ with associated graded $j$-module $H\mathbb{F}_p\ltimes \Sigma^{2p-3}H\mathbb{F}_p$, which multiplicatively has the structure of the trivial square-zero extension of $H\mathbb{F}_p$ by $\Sigma^{2p-3}H\mathbb{F}_p$. We use the THH-May spectral sequence with filtered coefficients as follows $$THH_{s,t}(j;H\mathbb{F}_p\ltimes \Sigma^{2p-3}H\mathbb{F}_p) \rightarrow THH_s(j; j/(p,v_1)).$$ This spectral sequence reduces to the long exact sequence $$~\label{eq les} \dots \lra  \pi_{k-2p+3}(THH(j;H\mathbb{F}_p) \lra \pi_k(THH(j;j/(p,v_1)) \lra \pi_k(THH(j;H\mathbb{F}_p)) \lra \pi_{k-2p+2}(THH(j;H\mathbb{F}_p)) \lra  \dots$$ where two out of three terms are known. We claim that this exact sequence demonstrates that the $V(1)$-THH-May spectral sequence cannot collapse at $E_2$. The author owes Eva Höning for giving some evidence that there must be a longer differential in personal communication, since the author originally had an argument that said that the differential on $\lambda_2=(\mu_1)^{p-1}\sigma \tilde{v}_1$ was zero.

\[prop dp-1\] There is a differential $$d_{p-1} ((\mu_1)^{p-1}\sigma \tilde{v}_1)\dot{=}\alpha_1 \lambda_1\gamma_{p-1}(\sigma \alpha_1)$$ in the $V(1)$-THH-May spectral sequence and no remaining differentials.

There is only one remaining possible differential for bidegree reasons, which is the stated differential $d_{p-1}((\mu_1)^{p-1}\sigma \tilde{v}_1)\dot{=}\alpha_1 \lambda_1\gamma_{p-1}(\sigma \alpha_1)$. Suppose the $V(1)$-THH-May spectral sequence computing $\pi_*(THH(j;j/(p,v_1))$ collapses at the $E_2$-page. Then, the long exact sequence (\[eq les\]) takes the form $$\xymatrix{  \Sigma^{2p-3} E(\lambda_1', \lambda_2)\otimes P(\mu_2)\otimes  \Gamma(\sigma b)  \ar[r] & E(\alpha_1, \lambda_1\gamma_{p-1}(\sigma \alpha_1), (\mu_1)^{p-1}\sigma \tilde{v}_1 ) \otimes P((\mu_1)^p)\otimes \Gamma(\sigma b)  \ar[d] \\ 
&E(\lambda_1', \lambda_2)\otimes P(\mu_2)\otimes  \Gamma(\sigma b)  \ar@{-->}[ul]  }$$ where the dotted arrow indicates a shift in degree by $1$. In particular, in degree $2p^2-1$ and $2p^2-2$ we have the exact sequence $$0 \lra  \mathbb{F}_p\{(\mu_1)^{p-1}\sigma \tilde{v}_1\} \lra   \mathbb{F}_p\{\lambda_2\} \lra \mathbb{F}_p\{ \lambda_1' \}\lra \mathbb{F}_p\{\alpha_1 \lambda_1\gamma_{p-1}(\sigma\alpha_1) \} \lra 0.$$ We can therefore determine if there should be a differential as stated by determining if the map $$\mathbb{F}_p\{\lambda_2 \} \lra \mathbb{F}_p\{ \lambda_1' \}$$ is nontrivial. To determine this, we note that the boundary map is exactly the map $$V(1)_*(THH(j; \ell )) \lra  V(1)_*(THH(j;\Sigma ^{2p-2}\ell ))$$ induced by the map $\ell\lra \Sigma^{2p-2}\ell$ given by $1-\psi_q$ where $q$ is the $q$-th Adams operation. This map induces multiplication by $P^1$ in homology $$\xymatrix{ H\mathbb{F}_p^*(\Sigma^{2p-2}\ell) = \Sigma^{2p-2} \mathcal{A}//E(1) \ar[rr]^{P^1} &&\mathcal{A}//E(1)=H\mathbb{F}_p^*(\ell) .}$$ In the dual, we therefore know that the map $$\xymatrix{ P(1): {H\mathbb{F}_p}_*(\ell) = ( \mathcal{A}//E(1) )_* \ar[r] &\Sigma^{2p-2}(\mathcal{A}//E(1))_*={H\mathbb{F}_p}_*(\Sigma^{2p-2}\ell) }$$ sends classes of the form $\bar{\xi}_1y$ to $y$ and the map sends all other classes to zero. The same will therefore be true for the induced map $$\xymatrix{ 
{H\mathbb{F}_p}_*(V(1)\wedge THH(j;\ell)) \ar[r] & {H\mathbb{F}_p}_*(V(1)\wedge THH(j;\Sigma^{2p-2}\ell))  \\ 
 {H\mathbb{F}_p}_*(V(1)\wedge THH(j) )\otimes_{ {H\mathbb{F}_p}_*(j) } {H\mathbb{F}_p}_*(\ell)  \ar[u]^{\cong}  \ar[r]^{ \id \otimes_{{H\mathbb{F}_p}_*(j)}P(1) }&    {H\mathbb{F}_p}_*(V(1)\wedge THH(j) )\otimes_{{H\mathbb{F}_p}_*(j)} {H\mathbb{F}_p}_*(\Sigma^{2p-2} \ell)     \ar[u]^{\cong}  ,    }$$ in particular $\bar{\xi}_1\sigma \tilde{\xi}_1^p$ maps to $\sigma\tilde{\xi}_1^p$. We therefore examine the square $$\xymatrix{
V(1)_{2p^2-1}(THH(j;\ell))\ar[r] \ar[d]^{g} & V(1)_{2p^2-2} (THH(j; \Sigma^{2p-3}\ell)) \ar[d]^{h} \\ 
(H\mathbb{F}_p\wedge V(1))_{2p^2-1}(THH(j;\ell))\ar[r] & (H\mathbb{F}_p\wedge V(1))_{2p^2-2} (THH(j; \Sigma^{2p-3}\ell)),}$$ which is isomorphic to $$\xymatrix{ 
\mathbb{F}_p\{\lambda_2\} \ar[d]^{g} \ar[r] & \mathbb{F}_p\{\lambda_1'\} \ar[d]^h \\
\mathbb{F}_p\{\sigma \tilde{\xi}_2, \bar{\xi}_1\sigma \tilde{\xi}_1^p, \bar{\tau}_1\sigma b, \bar{\xi}_1\bar{\tau}_0 \sigma b\} \ar[r] & \mathbb{F}_p\{\sigma\tilde{\xi}_1^p, \bar{\tau}_0\sigma b\} . 
}$$ As stated in the proof of Proposition \[prop d1\], the vertical maps send $\lambda_2$ and $\lambda_1'$ to classes given by the formulas

$g( \lambda_2 ) =  \sigma \tilde{\xi}_2- \bar{\xi}_1\sigma \tilde{\xi}_1^p -\bar{\tau}_1\sigma b $\
$h( \lambda_1' )= \sigma\tilde{\xi}_1^p-\bar{\tau}_0\sigma b.$\

The bottom horizontal map sends the class in the image of $\sigma \tilde{\xi}_2$ to the class $\sigma \tilde{\xi}_1^p$; i.e., $$\xymatrix{\sigma \tilde{\xi}_2- \bar{\xi}_1\sigma \tilde{\xi}_1^p -\bar{\tau}_1\sigma b \ar@{|->}[r] & \sigma \tilde{\xi}_1^p. }$$ Since the inverse image of the Hurewicz map evaluated on this element is $$h^{-1}(\sigma \tilde{\xi}_1^p)=h^{-1}(\sigma \tilde{\xi}_1^p-\bar{\tau}_0\sigma b)=\lambda_1'.$$ This proves that the top horizontal map is nontrivial and therefore, there must be a differential $$d_{p-1}((\mu_1)^{p-1}\sigma \tilde{v}_1)\dot{=} \alpha_1 \lambda_1\gamma_{p-1}(\sigma\alpha_1)$$ as stated.

Due to Oka [@MR614695 Thm. 4.4], the obstruction to a ring structure on $V(1)$ at the prime $3$ is a composite of maps including the composite map $$\xymatrix{ \beta_1: \Sigma^{11}S \ar[r] & \Sigma^{11}S/p \ar[r]^{\beta_{(1)}}  & S/p \ar[r] &  \Sigma^1 S, }$$ however we can easily compute that the induced map $\Sigma^{11}j\rightarrow \Sigma j$ is null homotopic and hence the obstruction vanishes after smashing with $j$. Thus, $V(1)\wedge j$ and hence $V(1)\wedge THH(j)$ are ring spectra, so the ring spectrum structure on $V(1)_*(THH(j))$ is also correct at the prime $3$. This type of argument is also used by Ausoni in the case of $V(1)\smash ku$ in [@MR2183525].

\[sseq:2\]

$$\begin{sseq}[entrysize=.3cm,ylabelstep=2,ylabels={0;;1;;2;;3;;4;;5;;6;;7;;8;;9;;10},xlabelstep=4]{37}{20}
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\ssdropbull \ssname{a} 

\ssmove{1}{-4}
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\ssgoto a \ssgoto b \ssstroke
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\ssdropbull
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\ssdropbull 

\ssmove{2}{4}
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\ssdropbull \ssname{c} 
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\ssmove 1 6 
\ssdropbull
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Let $p>2$ be a prime number and let $V(1)$ be the cofiber of the map $v_1 :\Sigma^{2p-2}S/p \rightarrow S/p.$ Then there is an isomorphism $$V(1)_*(THH(j))\cong P(\mu_2)\otimes \Gamma(\sigma b) \otimes \mathbb{F}_p\{\alpha_1, \lambda_1', \lambda_2\alpha_1, \lambda_2\lambda_1', \lambda_2\lambda_1'\alpha_1 \}$$ where the products between the classes $$\{\alpha_1, \lambda_1', \lambda_2\alpha_1, \lambda_2\lambda_1', \lambda_2\lambda_1'\alpha_1 \}$$ are zero except for $$\alpha_1\cdot \lambda_2\lambda_1'=\lambda_1'\cdot \lambda_2\alpha_1 =\lambda_2\lambda_1'\alpha_1.$$

This proof follows from Proposition \[prop d\_1\] and Proposition \[prop dp-1\]. There are no further possible differentials for bidegree reasons. This can be seen in Figure \[sseq:2\] since all the algebra generators are in the range specified. The only possible hidden multiplicative extension is easily ruled out by a filtration argument.

This paper mainly considers mod $(p,v_1)$ homotopy of topological Hochschild homology of the connective cover of the $K(1)$-local sphere, but one could also consider mod $(p,v_1)$ homotopy of topological Hochschild homology of the non-connective $K(1)$-local sphere. This is trivial however, since $$(S/p)_*(L_{K(1)}S)\cong P(v_1^{\pm 1})\otimes E(\alpha_1)$$ and so taking the cofiber of $v_1$ is a quotient by a unit in homotopy and $V(1)_*L_{K(1)}S\cong 0$. Since $THH(L_{K(1)}S)$ is a $L_{K(1)}S$-algebra, the spectrum $V(1)\wedge THH(L_{K(1)}S)$ is contractible. It would still be interesting to study $S/p_*THH(L_{K(1)}S)$, but this would not help us approach $S/p_*K(L_{K(1)}S)$ because the relative algebraic K-theory theorems of Dundas and McCarthy only hold for connective spectra [@DGM].

One could also consider rational topological Hochschild homology of $\ell_{E(1)}S$. Again, this will not be very interesting since $H\mathbb{Q}\wedge \ell_{E(1)}S\simeq H\mathbb{Q}$. Since rational homology has a Künneth isomorphism, there is a Bökstedt spectral sequence $$HH_*^{\mathbb{Q}}(H\mathbb{Q}_*(\ell_{E(1)}S))\Rightarrow H_*(THH(\ell_{E(1)}S);\mathbb{Q})$$ with input $HH_*^{\mathbb{Q}}(\mathbb{Q})\cong \mathbb{Q}$. Therefore, the spectral sequence collapses and the unit map $$H\mathbb{Q} \simeq H\mathbb{Q}\wedge \ell_{E(1)}S\lra H\mathbb{Q}\wedge THH(\ell_{E(1)}S)$$ is a weak equivalence.
