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%From: James McClure
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%To: Reinhard Schultz
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%Dear Reinhard,
%All that Stefan and Bob and I have written up at present is a bulletin
%announcement, which I sent to you a couple of days ago. We are hoping to have a
%full writeup by December. Here is a preliminary version of my paper with
%Ross it is a LaTeX file.
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%Yours,
%Jim
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\documentstyle[leqno,12pt]{article}
\headheight 12pt
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{conjecture}[theorem]{Conjecture}
\title{On the topological Hochschild homology of $bu$. I.}
\author{J.E. McClure and R.E. Staffeldt\\
Department of Mathematics\\
University of Kentucky\\
Lexington, KY 40506\\
USA}
\date{Preliminary Version, August 10, 1989}
\begin{document}
\maketitle
\section{Introduction}
The purpose of this paper and its sequel is to determine the
homotopy groups of the spectrum $THH(\ell)$. Here $p$ is an odd prime, $\ell$
is the Adams summand of $p$local connective $K$theory (see for example
\cite{aa}) and $THH$ is the topological Hochschild homology
construction introduced by B\"{o}kstedt in \cite{Marcel1}. In
the present paper we will determine the mod $p$ homotopy groups of
$THH(\ell)$ and also the integral homotopy groups of
$THH(L)$ (where $L$ denotes the periodic Adams summand). In the sequel we
will investigate the integral homotopy groups of $THH(\ell)$ using our
present results as a starting point.
The $THH$ construction appears to be of basic importance in algebraic
$K$theory because it combines two useful properties: it can be used to
construct good approximations to the algebraic $K$theory functor, and
it is very accessible to calculation.
We shall review what is known about the first property in a moment;
the second property was demonstrated by
B\"{o}kstedt's calculation, in his
paper \cite{Marcel2}, of the
homotopy groups of $THH(H{\bf Z}/p)$ and $THH(H{\bf Z})$
(here $H{\bf Z}/p$ and $H{\bf Z}$ denote the evident EilenbergMac Lane
spectra).
It is natural to ask about $THH(R)$ for other popular ring spectra $R$, and our
work is a first step in this direction. We pay special attention to the
connective case because this is the case which is likely to be relevant in
applications (see Subsection 1.4 below).
The calculation which we present in this paper is a homotopytheoretic one
which uses the Adams spectral sequence (hereafter abbreviated
$ASS$). This calculation has several interesting
features; in particular it is a pleasing example
of an $ASS$ calculation in which, although there are
infinitely many differentials, it is still possible to get the complete
answer.
Here is a summary of the contents of the paper.
In Section 2 we review the facts we need to know about ordinary
Hochschild homology. In Section 3 we do the same for topological
Hochschild homology. In Section 4 we calculate the mod $p$ homology of
$THH(\ell)$ and use it to find the $E_2$ term of the $ASS$ converging to
$\pi_* (THH(\ell); {\bf Z}/p)$. This section also contains a quick
calculation, which was pointed out to us by Larry Smith and Andy Baker, of the
homotopy groups of $THH(BP)$, where $BP$ is the
BrownPeterson summand of complex cobordism. In Section 5 we calculate the
mod $p$ $K$theory of $THH(\ell)$ and use it to determine the
``$v_1$inverted'' homotopy of $THH(\ell)$. In Section 6 we work backwards
from this result to determine
the behavior of the $v_1$inverted $ASS$ for $THH(\ell)$. In Section 7 we
show that the behavior of the $v_1$inverted $ASS$ completely determines that
of the $ASS$ itself, thereby completing the calculation of $\pi_* (THH(\ell);
{\bf Z}/p)$. In Section 8, which depends only on Sections 2, 3, and 5, we
calculate $\pi_* THH(L)$. In Section 9 we
confess that our definition of the spectrum $\ell$ is not the usual one; on the
other hand we show that it agrees with the usual one up to $p$adic completion.
Our definition has the advantage that it provides an $E_{\infty}$
structure for $\ell$;
this implies that $\ell$ has an $A_{\infty}$ structure,
which is necessary in order for $THH(\ell)$ to be defined, and it also provides
extra structure for $THH(\ell)$ which will be used in the
sequel to determine differentials and extensions in the $ASS$ converging to
$\pi_* THH(\ell)$.
\vspace{.2in}
{\bf Acknowledgements:} We would like to thank everyone who has discussed this
subject with us, especially Andy Baker, Marcel B\"{o}kstedt, Nick Kuhn, and
Friedhelm Waldhausen.
\vspace{.2in}
In the remainder of the introduction we shall give a short summary of some
things which are known or suspected
about algebraic $K$theory; these provide motivation for the $THH$
construction, but none of what follows will actually be used in our work.
\subsection{The Dennis Trace Map}
The simplest way in which Hochschild homology is related to algebraic
$K$theory is via the Dennis trace map,
which is a natural transformation
\[
\tau: K_*S \rightarrow {\bf HH}_* S;
\]
here $S$ is a discrete ring and ${\bf HH}_* S$ denotes ordinary
Hochschild homology.
(See \cite[pages 106114]{II} or \cite[Section II.1]{gg} for a discussion of
$\tau$). Unfortunately, the map $\tau$ usually loses too much
information to be useful for purposes of calculation. It is, however, possible
to improve $\tau$ by factoring it through one of the variants of cyclic
homology: that is, there is a commutative diagram
\[
\begin{picture}(90,70)
\put(0,0){\makebox(0,0){$K_* S$}}
\put(0,65){\makebox(0,0){${\bf HC}^{}_* S$}}
\put(102,0){\makebox(0,0){${\bf HH}_* S$}}
\put(18,0){\vector(1,0){64}}
\put(50,2){\makebox(0,0)[b]{$\tau$}}
\put(18,60){\vector(3,2){73}}
\put(55,36){\makebox(0,0)[bl]{$\pi$}}
\put(0,8){\vector(0,1){49}}
\put(2,32){\makebox(0,0)[r]{$\alpha$}}
\end{picture}
\]
(see \cite[page 364]{gg} for the definitions of ${\bf HC}^{}$ and $\pi$ and
\cite[Section 2.3]{gg} for the definition of $\alpha$).
The following basic theorem, due to Goodwillie \cite[Theorem II.3.4]{gg},
says that the map $\alpha$ can be used to calculate
{\em rationalized} relative algebraic $K$theory in certain situations.
\begin{theorem}
If $S_1 \rightarrow S_2$ is a surjection with nilpotent
kernel then
\[
\alpha \otimes {\bf Q}:
K_*(S_1 \rightarrow S_2) \otimes {\bf Q}
\rightarrow
{\bf HC}^{}_* (S_1 \rightarrow S_2) \otimes {\bf Q}
\]
is an isomorphism.
\end{theorem}
See \cite[pages 365 and 373]{gg} for the definitions of ${\bf HC}^{}$ and
$\alpha$ in the relative situation.
The most important application of Theorem 1.1 is to Waldhausen's functor
$A(X)$. For this, one needs to generalize Theorem 1.1 to apply to
simplicial rings $S$. This can be done (see \cite{gg}),
and in this generality the hypothesis of Theorem 1.1 is replaced by the much
less stringent hypothesis that the map
\[
\pi_0 S_1 \rightarrow \pi_0 S_2
\]
be a surjection with nilpotent kernel (see \cite{gg}).
Now given a space $X$, it is easy to construct a simplicial ring whose
$K$theory agrees rationally with $A(X)$, and thus
Theorem 1.1 can be applied to calculate $A(X \rightarrow Y) \otimes {\bf Q}$
whenever $X \rightarrow Y$ is a 2connected map (see \cite[pages
348349]{gg}).
\subsection{Algebraic $K$theory of Ring Spectra}
The reason for introducing topological Hochschild homology is to try to
formulate and prove an analog of Theorem 1.1 which holds {\em integrally} and
not just rationally.
One can get a hint as to how
to do this by recalling that one of the basic principles of
Waldhausen's work on algebraic
$K$theory is that
the $K$functor should be applied not just to rings but to
ring {\em spectra} (also called ``brave new rings'').
Waldhausen gave a sketch of how to do this in \cite{WW},
and a precise construction was given by May in \cite{MM}
(also see \cite{SV}). For technical reasons one must
restrict to $A_{\infty}$ ring spectra, but in practice this is not an
inconvenience. We shall refer to this functor as Waldhausen $K$theory and
denote it by $K^W$; when $R$ is an $A_{\infty}$ ring spectrum,
$K^W(R)$ is a spectrum whose homotopy groups will be denoted by $K^W_*(R)$.
The functor $K^W_*$ generalizes both $K_*$ and $A(X)$, for
when $R$ is the
EilenbergMac Lane spectrum $HS$ associated to a discrete ring $S$
one has the equation
\begin{equation}
K^W_* HS =
K_* S;
\end{equation}
and when $R$ is the sphere spectrum $S^0$, or
more generally the suspension spectrum $\Sigma^{\infty} (\Omega X)_+$,
one has
\[
K^W (S^0) = A(*)
\]
and
\[
K^W (\Sigma^{\infty} (\Omega X)_+)
= A(X)
\]
(here $(\Omega X)_+$ denotes the space obtained by adding a disjoint basepoint
to the loop space of $X$).
\subsection{Topological Hochschild Homology}
In view of what has been said so far, it is natural to try to approximate
$ K^W_* R$ by means of a Hochschild
homology construction which can be applied to $A_{\infty}$ ring spectra $R$.
This is what topological Hochschild homology $THH(R)$ is. It is
clear enough in principle how one should construct $THH(R)$ (see Section 3),
although the technical details are quite complicated (see \cite{Marcel1} and
\cite{Elmendorf}).
$THH(R)$ is a spectrum and we shall denote its homotopy $\pi_* THH(R)$ by
${\bf THH}_* (R)$.
There is a natural transformation
\[
\tau': K^W_* R \rightarrow {\bf THH}_* R
\]
which is analogous to the Dennis trace map. (See \cite[Section 2]{Marcel1} for
the construction of $\tau'$).
In the special case $R=HS$ it is
important to note that the analog of equation (1) does {\em not} hold for
${\bf THH}_*$; that is, it is not true that ${\bf THH_*} (HS)$ agrees with
${\bf HH}_* (S)$ for a discrete ring $S$. Instead, there is a
commutative diagram which shows that $\tau'$ gives a second way of lifting
the Dennis trace map:
\[
\begin{picture}(90,70)
\put(0,65){\makebox(0,0){$K_* S$}}
\put(18,65){\vector(1,0){64}}
\put(50,67){\makebox(0,0)[b]{$\tau$}}
\put(102,65){\makebox(0,0){${\bf HH}_* S$}}
%
\put(0,57){\vector(0,1){49}}
\put(2,32){\makebox(0,0)[r]{$\tau'$}}
\put(0,0){\makebox(0,0){${\bf THH}_* HS$}}
%
\put(18,8){\vector(3,2){73}}
\put(56,31){\makebox(0,0)[tl]{$\phi$}}
\end{picture}
\]
(See Remark 3.5 for a hint about the construction of
the map $\phi$). In the special case $S={\bf Z}$, B\"{o}kstedt has shown that
$\tau'$ is nonzero in infinitely many dimensions;
more precisely, what he shows is that for each prime $p$ the
localization of $\tau'$ at $p$ is an epimorphism in dimension $2p1$ (see
\cite{Marcel3}).
Note that this cannot be true for $\tau$ for the trivial reason that
${\bf HH}_* {\bf Z}$ is zero in all
positive dimensions.
In the cases $R=S^0$ and $R=\Sigma^{\infty} (\Omega X)_+$ mentioned above one
can give explicit descriptions of $THH(R)$:
\[
THH(S^0) = S^0
\]
and
\[
THH(\Sigma^{\infty} (\Omega X)_+) = \Sigma^{\infty} (\Lambda X)_+,
\]
where $\Lambda$ denotes the free loop space; the first equation is obvious
from the definition in Section 3 and the second follows from that definition
and \cite[Theorem ?]{Jones}.
Probably the most important fact about $\tau'$ is that it can be identified
with the map from $K^W$ to its first Goodwillie derivative;
more precisely we mean the derivative ``at $X=S^0$\,'' of the functor
\[
X \mapsto K^W(R \wedge (\Omega X)_+)
\]
from pointed spaces to spectra (\cite{Wlecture}; also
see \cite{GoCal} for the definition of the derivative and the proof of this
fact in the special case $R=S^0$). This fact is significant in two ways: it
implies that $THH$ is a ``first order'' approximation to $K^W$ in much the same
way that stable homotopy is a first order approximation to unstable homotopy,
and it can be used to obtain a ``higher order'' approximation, as we explain in
the next subsection.
\subsection{Topological Cyclic and Epicyclic Homology}
The next step is to consider functors which combine the desirable properties
of ${\bf HC}^{}$ and $THH$. For example, one can define topological {\em
cyclic} homology $THC^{}$ by observing that the spectrum
$THH(R)$ has a natural cyclic structure and therefore has an
$S^1$ action (at least if everything works as in the category of spaces
cf.\ \cite{Jones}), and letting
\[
{\bf THC}^{}_* R = \pi_* THH(R)^{hS^1},
\]
where $hS^1$ denotes the homotopy fixedpoint spectrum (cf.\ Remark 3.5).
Unfortunately it is known that this
functor cannot satisfy an integral version of Theorem 1.1 (see \cite[Section
7]{GoLet}). On the other hand, there is considerable evidence for the
following conjecture
\begin{conjecture}
It is possible to construct a functor $THE$, related to $THH$ and $THC^{}$,
and a natural transformation
\[
\tau'' : K^W (R) \rightarrow THE(R)
\]
which induces an equivalence of derivatives.
\end{conjecture}
The notation $THE$ stands for ``topological epicyclic homology''; see
\cite[Section 6]{GoLet}.
It is possible that $THE$ can be taken to be the functor defined in \cite{BHM}.
If the conjecture is true then the calculus of functors will imply the
following integral version of Theorem 1.1:
the induced map
\[
\tau'': K^W_*(R_1 \rightarrow R_2) \rightarrow {\bf THE}_* (R_1 \rightarrow
R_2)
\]
is an isomorphism
for any map $R_1 \rightarrow R_2$ of $A_{\infty}$ ring
spectra such that
\[
\pi_i R_1 \rightarrow \pi_i R_2
\]
is an isomorphism for $i \leq 0$.
This explains the statement we made earlier that connective spectra are of
particular importance for the potential applications.
We conclude with one further remark about the potential applications of $THH$.
In \cite{Waldhausen2}, Waldhausen has proposed an interesting program for
studying the relative $K^W$ theory of the map
\[
S^0 \rightarrow H{\bf Z}
\]
by means of the intermediate spectra $K^W (L_n (S^0))$ and $K^W (L_n (S^0)_c)$,
where $L_n (S^0)$ denotes the $L_n$localization of the sphere (see
\cite{RavAmJ}) and $L_n (S^0)_c$ is the associated connective spectrum. When
$n=1$ the spectrum $L_1 (S^0)_c$ is the connective imageof$J$ spectrum $j$.
It seems likely that the results and methods of our work are a good way to
obtain information about $THH(j)$.
\section{A Brief Review of Hochschild Homology}
\label{recall}
In this section we recall the facts we need about ordinary (algebraic)
Hochschild homology. Our basic reference for this subject is
\cite[Chapters~IX and~X]{CE}.
If $S$ is a graded algebra over a ground field $k$, its Hochschild homology
${\bf HH}_* (S)$ is defined to be the homology of the {\em Hochschild complex}
\cite[page 175]{CE}
\begin{equation}
\label{HC}
\begin{array}{c}
\vdots \\
\downarrow \\
S \otimes S \otimes S \\
\downarrow \\
S \otimes S \\
\downarrow \\
S,
\end{array}
\end{equation}
in which the differential is given by the formula
\begin{eqnarray}
d (t_0 \otimes \cdots \otimes t_n)
& =
& \sum_{i=0}^{n1}
(1)^i
t_0 \otimes \cdots \otimes t_i t_{i+1} \otimes \cdots \otimes t_n \nonumber \\
& &
+ (1)^n (1)^{t_n(t_0 + \cdots + t_{n1})}
t_n t_0 \otimes t_1 \otimes \cdots \otimes t_{n1}. \nonumber
\end{eqnarray}
As one might expect, ${\bf HH}_* (S)$ can also be described in terms of
${\rm Tor}$; it is
\[
{\rm Tor}^{\, S \otimes S^{\rm op}} (S,S),
\]
where the first factor of
$S \otimes S^{\rm op}$
acts on $S$ by multiplication on the left and the second factor by
multiplication on the right \cite[page 169]{CE}.
The reader may perhaps wonder why one uses this definition
for the homology of $S$ instead of the ``obvious'' definition
${\rm Tor}^S (k,k)$.
For our purposes, the answer is that the latter is the appropriate definition
for the category of {\em augmented} algebras,
but we need to work more generally; the functor
${\bf HH}_* (S)$ is closely related to
${\rm Tor}^S (k,k)$,
but it is defined for arbitrary algebras $S$.
There is an evident natural map
\[
\iota : S \rightarrow {\bf HH}_0(S),
\]
which is an isomorphism when $S$ is commutative. There is also a
``suspension'' map
\[
\sigma :
S \rightarrow {\bf HH}_1 (S)
\]
which takes $t \in S$ to the class of $1 \otimes t$.
If $S$ is commutative there is
a product
\[
{\bf HH}_i (S)
\otimes
{\bf HH}_j (S)
\rightarrow
{\bf HH}_{i+j} (S),
\]
which gives ${\bf HH}_* (S)$ the structure of a commutative graded $S$algebra
(see \cite[page 217]{CE});
moreover $\iota$ is a ring homomorphism and $\sigma$ is a derivation:
\begin{equation}
\label{derivation}
\sigma (st)
=
s \sigma (t) +
(1)^{s t} t \sigma (s).
\end{equation}
It will not surprise the reader
to find that there are times when we actually need to
compute the ring ${\bf HH}_*(S)$.
The following result is sufficient for our purposes.
\begin{proposition}
%
\label{HH}
%
If $S$ has the form
\[
{\bf Z}/p \, [x_1, x_2, \ldots ]
\otimes
\Lambda (y_1, y_2, \ldots ),
\]
then ${\bf HH}_* (S)$ has the form
\[
S \otimes
\Lambda ( \sigma (x_1), \sigma (x_2), \ldots )
\otimes
\Gamma ( \sigma (y_1), \sigma (y_2), \ldots ),
\]
where the inclusion of the first factor is the natural map
\[
\iota :
S \stackrel{\cong}{\rightarrow}
{\bf HH}_0 (S)
\]
and $\sigma$ is the suspension map
\[
S \rightarrow
{\bf HH}_1 (S).
\]
\end{proposition}
{\bf Proof.} Let
\[
\varphi :
S \rightarrow
S \otimes S
\]
be the ring map
which takes $x_i$ and $y_j$ to
\[
x_i \otimes 1 
1 \otimes x_i
\]
and
\[
y_j \otimes 1 
1 \otimes y_j,
\]
respectively.
By \cite[Theorem X.6.1]{CE},
$\varphi$ induces an isomorphism
\[
{\rm Tor}^S (S_{\varphi}, {\bf Z}/p)
\rightarrow
{\rm Tor}^{\, S \otimes S} (S,S) =
{\bf HH}_* (S),
\]
where $S_{\varphi}$ denotes the $S$module structure on $S$ obtained by pulling
back its
$S \otimes S$module structure along $\varphi$.
In our case $S$ is commutative, so that
$S_{\varphi}$ has the trivial $S$module
structure, and we conclude that there is an isomorphism
\begin{equation}
\label{CEiso}
S \otimes
{\rm Tor}^S ({\bf Z}/p,{\bf Z}/p)
\cong
{\bf HH}_* (S).
\end{equation}
But it is well known that
\[
{\rm Tor}^S ({\bf Z}/p,{\bf Z}/p)
\cong
\Lambda (\sigma' (x_1),\sigma' (x_2), \ldots )
\otimes
\Gamma (\sigma' (y_1), \sigma' (y_2), \ldots ),
\]
where $\Gamma$ denotes a divided polynomial algebra and
$\sigma'$ is the suspension map
\[
S \rightarrow
{\rm Tor}^S_1 ({\bf Z}/p, {\bf Z}/p);
\]
and it is not hard to check that the isomorphism (\ref{CEiso}) takes
$\sigma' (x_i)$ to
$\sigma (x_i)$ and
$\sigma' (y_j)$ to
$\sigma (y_j)$. ~$\clubsuit$
\section{Introduction To Topological Hochschild Homology}
\label{recall2}
In this section we turn to the topological version of Hochschild homology. Our
references for the foundations are \cite{Marcel1} and \cite{Elmendorf},
and we refer to those sources for all technical details.
Roughly speaking, topological Hochschild homology is constructed by
replacing the
algebra $S$ in the Hochschild complex (\ref{HC}) by a ring
spectrum $R$. We will show how to carry this idea out
when the multiplication in $R$ is strictly associative
(which is the case considered in \cite[Section 1]{Marcel1}),
but in fact it can be done whenever $R$ has an $A_{\infty}$ structure
(see \cite{Elmendorf}).
First we must reformulate the definition of ${\bf HH}_* (S)$. Let
$HH_{\bullet} (S)$ be the simplicial abelian group
\[
\begin{array}{c}
\vdots \\
\downarrow \uparrow \downarrow \uparrow \downarrow \uparrow \downarrow \\
S \otimes S \otimes S \\
\downarrow \uparrow \downarrow \uparrow \downarrow \\
S \otimes S \\
\downarrow \uparrow \downarrow \\
S.
\end{array}
\]
Here the face maps $\partial_i$ and degeneracy maps $s_i$ are given by the
formulas
\[
\partial_i
(t_0 \otimes \cdots \otimes t_n)
=
\left\{
\begin{array}{ll}
t_0 \otimes \cdots \otimes t_i t_{i+1} \otimes \cdots \otimes t_n &
\mbox{if $0 \leq i < n$} \\
\\
t_n t_0 \otimes t_1 \otimes \cdots \otimes t_{n1} &
\mbox{if $i=n$,}
\end{array}
\right.
\]
and
\[
s_i (t_0 \otimes \cdots \otimes t_n)
=
t_0 \otimes \cdots \otimes t_i \otimes 1 \otimes t_{i+1}
\otimes \cdots \otimes t_n.
\]
Clearly the Hochschild complex is the chain complex associated to this
simplicial abelian group. But for any simplicial abelian group, the homology
of its associated chain complex is the same as the homotopy of its geometric
realization (see \cite[Theorem 22.1]{MaySOAT}), so in our case we conclude
\[
{\bf HH}_* (S)
=
\pi_*  HH_{\bullet} (S) .
\]
We now define the
{\em topological Hochschild homology} spectrum $THH(R)$
associated to a ring spectrum $R$
to be the geometric
realization of the simplicial spectrum
\[
\begin{array}{lc}
& \vdots \\
& \downarrow \uparrow \downarrow \uparrow \downarrow \uparrow \downarrow \\
& R \wedge R \wedge R \\
THH_{\bullet} (R)=
& \downarrow \uparrow \downarrow \uparrow \downarrow \\
& R \wedge R \\
& \downarrow \uparrow \downarrow \\
& R.
\end{array}
\]
The face map
\[
\partial_i :
\overbrace{ R \wedge \cdots \wedge R}^{n+1}
\rightarrow
\overbrace{ R \wedge \cdots \wedge R}^{n}
\]
is defined by the following equation (suitably interpreted):
\[
\partial_i
(r_0 \wedge \cdots \wedge r_n)
=
\left\{
\begin{array}{ll}
r_0 \wedge \cdots \wedge r_i r_{i+1} \wedge \cdots \wedge r_n &
\mbox{if $0 \leq i < n$} \\
\\
r_n r_0 \wedge r_1 \wedge \cdots \wedge r_{n1} &
\mbox{if $i=n$.}
\end{array}
\right.
\]
The degeneracy map
\[
s_i :
\overbrace{ R \wedge \cdots \wedge R}^{n+1}
\rightarrow
\overbrace{ R \wedge \cdots \wedge R}^{n+2}
\]
is defined to be the composite
\[
\overbrace{ R \wedge \cdots \wedge R}^{n+1}
\stackrel{\cong}{\rightarrow}
\overbrace{ R \wedge \cdots \wedge R}^{i+1}
\wedge \, S^0 \! \wedge
\overbrace{R \wedge \cdots \wedge R}^{ni}
\, \stackrel{1 \wedge e \wedge 1}{\longrightarrow}
\overbrace{ R \wedge \cdots \wedge R}^{n+2},
\]
where $e$ is the unit map $S^0 \rightarrow R$;
thus the $i$th degeneracy inserts a unit in the $(i+1)$st position.
(Our assumption that the multiplication in $R$ is strictly associative is
necessary in order that the maps $\partial_i$ and $s_i$ defined in this way
satisfy the simplicial identities).
We shall write
\[
\tilde{\iota} :
R \rightarrow
THH(R)
\]
for the inclusion of the 0th simplicial filtration in $THH(R)$.
If the multiplication in $R$ is sufficiently commutative
then $THH(R)$ inherits a
ringspectrum structure and $\tilde{\iota}$ is a ring map
(see \cite[Section 2]{Marcel1}).
If $R$ is an $E_{\infty}$ ring spectrum
then $THH(R)$ inherits an $E_{\infty}$ ring
structure and $\tilde{\iota}$ is an $E_{\infty}$ ring map
(see \cite{Elmendorf}). Our definition of the spectrum $\ell$, which is given
in Section \ref{Einfty}, automatically implies that
$\ell$ is an $E_{\infty}$ ring spectrum, so we conclude that $THH(\ell)$ is
also.
Now suppose that we are given a homology theory $h_*$
with a multiplication
and that we want to know
$h_* (THH(R))$. In \cite{Marcel2}, B\"{o}kstedt introduced the following
spectral sequence
for this sort of calculation.
\begin{proposition}
\label{filtSS}
If $h_*$
satisfies the strict K\"{u}nneth formula
\[
h_* ( X \wedge Y )
\cong
h_* X \otimes_{h_* S^0} h_* Y
\]
then there is a spectral sequence
\begin{equation}
\label{SS}
{\bf HH}_* (h_*(R))
\Rightarrow
h_* (THH(R)),
\end{equation}
where ${\bf HH}_*$ is defined with respect to the ground ring $h_* S^0$.
For each $x \in h_* (R)$ the element
\[
\iota_* (x) \in {\bf HH}_0 (h_* (R))
\]
survives to
\[
\tilde{\iota}_* (x) \in h_* (THH(R)).
\]
\end{proposition}
We warn the reader that there is no Hopf algebra structure in this spectral
sequence.
It is likely, although we shall not attempt to prove it,
that Proposition \ref{filtSS} holds without
the assumption that $h_*$ satisfies the strict K\"{u}nneth formula
(cf. \cite[Theorem 13.1]{RobinsonTop}).
\vspace{.25in}
{\bf Proof of Proposition \ref{filtSS}.}
For any simplicial spectrum $X_{\bullet}$, we may
apply the theory $h_*$ to the simplicial filtration
of $ X_{\bullet} $ in the usual way to obtain a spectral
sequence converging to $h_* ( X_{\bullet} )$
(cf. \cite[Theorem 11.14]{MayG}).
If $X_{\bullet}$ is ``proper'' then
the $E_2$ term of this spectral sequence is the homology of the complex
\begin{equation}
\label{complex}
\cdots
\rightarrow
h_*(X_n)
\rightarrow
\cdots
\rightarrow
h_* (X_1)
\rightarrow
h_* (X_0),
\end{equation}
with differential
\[
d=
\sum (1)^i (\partial_i)_*.
\]
Now when $X_{\bullet}$ is $THH_{\bullet}(R)$
and $h_*$ satisfies the strict K\"{u}nneth formula
this complex is just the Hochschild complex for
$h_* (R)$, and we conclude that
\[
E_2 \cong
{\bf HH}_* (h_* (R))
\]
as required. $\clubsuit$
\vspace{.25in}
At the end of
the next section we shall need to have somewhat tighter control of
the spectral sequence (\ref{SS}). The information we need
is provided by our next result.
\begin{proposition}
\label{suspension}
There is a natural transformation
\[
\tilde{\sigma} :
\Sigma R
\rightarrow THH (R)
\]
such that the element
\[
\sigma_* (x) \in {\bf HH}_1 (h_* (R))
\]
survives to
\[
\tilde{\sigma}_* (\Sigma x) \in h_* (THH(R)).
\]
for each $x \in h_* (R)$.
\end{proposition}
{\bf Proof.}
Before we can define the natural transformation
$\tilde{\sigma}$ we need some preliminary constructions.
Let $S_{\bullet} (R)$ be the simplicial spectrum
obtained by ``replacing all $\wedge$'s in $THH_{\bullet} (R)$ by $\vee$'s.''
More precisely, the $n$th simplicial degree of $S_n (R)$ is
\[
\overbrace{R \vee \ldots \vee R }^{n+1}.
\]
The $i$th face operator
\[
\partial_i :
\overbrace{R \vee \ldots \vee R }^{n+1}
\rightarrow
\overbrace{R \vee \ldots \vee R }^{n}
\]
is defined by the equation
\[
\partial_i \circ I_j
=
\left\{
\begin{array}{ll}
I_{j1} & \mbox{ if $i < j$} \\
\\
I_j & \mbox{ if $i \geq j$ and $j < n$} \\
\\
I_0 & \mbox{ if $i=j=n$;}
\end{array}
\right.
\]
here
\[
I_j :
R \rightarrow
R \vee \ldots \vee R
\]
is the inclusion of the $j$th wedge summand.
The $i$th degeneracy map $s_i$ is defined by the equation
\[
s_i \circ I_j =
\left\{
\begin{array}{ll}
I_{j+1} & \mbox{ if $i p$};
\end{array}
\]
here we have written
$\gamma_j (\sigma(\chi\tau_i))$ for the $j$th divided power of
$\sigma(\chi\tau_i)$.
The same formula therefore holds in
$\widehat{E}^{p1}(\ell)$ for $i \geq 2$, and we conclude that
\[
\widehat{E}^p (\ell)
\cong
H_*(\ell;{\bf Z}/p)
\otimes
\Lambda [ \sigma(\chi\xi_1), \sigma(\chi\xi_2) ]
\otimes
TP_p [\sigma(\chi\tau_2), \sigma(\chi\tau_3), \ldots ],
\]
where $TP_p$ denotes a truncated polynomial algebra of height $p$
(cf. \cite[page 6]{Marcel2}). Since all indecomposables in
$\widehat{E}^p (\ell)$
are in filtrations 0 and 1 we can further conclude that
\[
\widehat{E}^p (\ell)
=
\widehat{E}^{\infty} (\ell).
\]
Proposition \ref{suspension} implies that
the elements $\sigma(\chi\xi_i)$ and $\sigma(\chi\tau_i)$ in
$\widehat{E}^{\infty}$ are
represented in $H_* (THH(\ell))$ by $\tilde{\sigma}_*(\Sigma (\chi\xi_i))$ and
$\tilde{\sigma}_*(\Sigma (\chi\tau_i))$ respectively.
Next we need to determine the multiplicative extensions in
$H_* (THH(\ell); {\bf Z}/p)$. For this we use DyerLashof operations. As we have
seen in the previous section, $THH(\ell)$ is an $E_{\infty}$ ring spectrum, and so
its homology supports DyerLashof operations
\[
Q^i :
H_n (THH(\ell); {\bf Z}/p))
\rightarrow
H_{n + 2i(p1)} (THH(\ell); {\bf Z}/p)),
\]
(see \cite{Steinberger}).
If $x$ is an element of dimension $2s$
then $Q^s x = x^p$, (\cite[Theorem 1.1(4)]{Steinberger}) so in particular
we have
\[
(\tilde{\sigma}_*(\Sigma (\chi\tau_i)))^p
=
Q^{p^i} \tilde{\sigma}_*(\Sigma (\chi\tau_i)).
\]
But B\"{o}kstedt shows that the map
\[
\tilde{\sigma}_* \Sigma :
H_n (R; {\bf Z}/p)
\rightarrow
H_{n+1} (THH(R); {\bf Z}/p)
\]
commutes with DyerLashof operations (see \cite[Lemma 2.9]{Marcel2}),
and Steinberger has calculated the action of the $Q^i$ in
$H_* (\ell; {\bf Z}/p)$:
\[
Q^{p^i} \chi \tau_i
=
\chi \tau_{i+1}
\]
(see \cite[Theorem 2.3]{Steinberger}).
We conclude that
\[
(\tilde{\sigma}_* (\Sigma \chi \tau_i))^p
=
\tilde{\sigma}_* (\Sigma \chi \tau_{i+1})
\]
for all $i \geq 2$, and hence that
\[
(\tilde{\sigma}_* (\Sigma \chi \tau_2))^{p^i}
=
\tilde{\sigma}_* (\Sigma \chi \tau_{i+2})
\]
for all $i \geq 0$.
If we denote
$\tilde{\sigma}_*\Sigma (\chi\xi_1)$
by $\lambda_1$,
$\tilde{\sigma}_*\Sigma (\chi\xi_2)$
by $\lambda_2$, and
$\tilde{\sigma}_* \Sigma (\chi \tau_2)$
by $\mu$,
we have now shown that
\[
H_* (THH(\ell); {\bf Z}/p)
\cong
H_* (\ell;{\bf Z}/p)
\otimes
\Lambda (\lambda_1, \lambda_2)
\otimes
{\bf Z}/p \, [\mu]
\]
as an algebra.
To complete the proof of Proposition \ref{H_*} we need to determine
the $A_*$coaction on $\lambda_1$, $\lambda_2$, and $\mu$. We shall give the
calculation of $\nu (\lambda_2)$; the others are similar.
Since the map $\tilde{\sigma}_* \Sigma$ commutes with $\nu$,
we have
\[
\nu (\lambda_2)
=
(1 \otimes \tilde{\sigma}_* \Sigma)
\nu (\chi \xi_2).
\]
Now $\nu (\chi \xi_2)$ is determined by Milnor's calculations: it is
\[
1 \otimes \xi_2 +
\xi_1 \otimes \xi_1^p +
\xi_2 \otimes 1
\]
(see \cite[Theorem 3.1.1]{Ravenel}).
We therefore conclude that
\[
\nu (\lambda_2)
=
1 \otimes \lambda_2 +
\xi_1 \otimes \tilde{\sigma}_* \Sigma (\xi_1^p) +
\xi_2 \otimes \tilde{\sigma}_* \Sigma (1)
\]
and it remains to show that the second and third terms are zero.
But $\tilde{\sigma}_* \Sigma (\xi_1^p)$
represents the element
$\sigma (\xi_1^p)$
in the spectral sequence,
and this element is zero because $\sigma$ is a derivation
(equation (\ref{derivation}) of Section \ref{recall}).
It follows that $\tilde{\sigma}_* \Sigma (\xi_1^p)$
is an element in filtration 0 with dimension $2p^2 2p +1$,
and an inspection of the spectral sequence shows that the only such element is
0. Similarly, $\tilde{\sigma}_* \Sigma (1)$
is an element in filtration 0 with dimension 1, and again the only
such element is 0. This completes the proof of Proposition \ref{H_*}.
$\clubsuit$
\vspace{.2in}
{\bf Remark 4.3 (Andy Baker and Larry Smith)} \ \
Let us {\em assume} that the BrownPeterson spectrum $BP$ has an $E_{\infty}$
structure. Starting from the equation
\[
H_* (BP;{\bf Z}/p)
=
{\bf Z}/p\,[\chi\xi_1,\chi\xi_2, \ldots ],
\]
it is easy to see that $\widehat{E}^2 (BP)$ has the form
\[
H_* (BP;{\bf Z}/p)
\otimes
\Lambda [ \sigma(\chi\xi_1), \sigma(\chi\xi_2), \ldots ].
\]
For dimensional reasons there cannot be any differentials, and we conclude that
\begin{equation}
\label{hp}
H_* (THH(BP);{\bf Z}/p)
=
H_* (BP;{\bf Z}/p)
\otimes
\Lambda (\lambda_1, \lambda_2, \ldots),
\end{equation}
where
\[
\lambda_i =
\tilde{\sigma}_*\Sigma (\chi\xi_i).
\]
A similar calculation in rational cohomology, starting from the equation
\[
H_* (BP; {\bf Q})
\cong
{\bf Q} [v_1, v_2, \ldots],
\]
shows that
\begin{equation}
\label{hq}
H_* (THH(BP);{\bf Q})
=
H_* (BP;{\bf Q})
\otimes
\Lambda (\lambda'_1, \lambda'_2, \ldots),
\end{equation}
where
\[
\lambda'_i =
\tilde{\sigma}_*\Sigma (v_i).
\]
Comparing equations (\ref{hp}) and (\ref{hq}) dimensionwise shows that
$H_* (THH(BP);{\bf Z}_{(p)})$ must be torsion free. Now equation (\ref{hp}) and
Theorem 1.3 of \cite{BrownPet} imply that $THH(BP)$ is a wedge of suspensions
of $BP$ and that
\[
\pi_* THH(BP)
\cong
\pi_* BP
\otimes
\Lambda (\lambda_1, \lambda_2, \ldots).
\]
\section{Localized mod$p$ homotopy of $THH(\ell)$}
\label{homotopy}
The object of this section is to prove the following result,
which we will use in later sections to determine the differentials
in the Adams spectral sequence
$E_r(THH(\ell);{\bf Z}/p)$
for the mod$p$ homotopy of
$THH(\ell)$.
\begin{theorem}
\label{vlocal homotopy}
The inclusion
\[
\tilde\iota: \; \ell \longrightarrow THH(\ell)
\]
induces an isomorphism
\[
\tilde\iota_*: \; v_1^{1}\pi_*(\ell;{\bf Z}/p)
\stackrel{\cong}{\longrightarrow}
v_1^{1}\pi_*(THH(\ell);{\bf Z}/p).
\]
\end{theorem}
The $v_1$inverted mod $p$ homotopy of
an $\ell$module $X$ with structure map
$\alpha : \ell \wedge X \longrightarrow X$
is defined as a direct limit:
\[
v_1^{1}\pi_*(X;{\bf Z}/p) = \lim_{v_1}\pi_*(X;{\bf Z}/p),
\]
where the maps $v_1$ in the direct system
\[
\pi_{n}(X;{\bf Z}/p) \longrightarrow \pi_{2(p1) + n}(X;{\bf Z}/p)
\]
send the homotopy class $[f]$ of a stable map $f:S^n \longrightarrow
X \wedge {\bf M}$ to the homotopy class of
$[\alpha \circ (v_1 \wedge f)]$,
using $v_1$ also to denote a representative map
$v_1 : S^{2(p1)} \longrightarrow \ell$
and ${\bf M}$ to denote the mod $p$ Moore spectrum.
This result is in some sense anticipated by B\"okstedt's results
on
$THH({\bf Z}_{(p)})$
in \cite{Marcel2}.
(Actually he discusses $THH({\bf Z})$ and not $THH({\bf Z}_{(p)})$,
but it is clear that the results in \cite{Marcel2} have parallels
for $THH({\bf Z}_{(p)})$.)
It is an easy consequence of his computations that the inclusion
of spectra
\[
\iota: \; {\bf Z}_{(p)} \longrightarrow THH({\bf Z}_{(p)})
\]
induces an isomorphism in homotopy tensored with
${\bf Z}[1/p]$. In other words, inverting $p$ in
homotopy kills the difference between the EilenbergMacLane
spectrum ${\bf Z}_{(p)}$ and $THH({\bf Z}_{(p)})$.
Our theorem states that something
like this persists for $THH(\ell)$ if we consider
multiplication by $v_1$ in mod$p$ homotopy instead of multiplication
by $p$ in $p$local homotopy.
This theorem is a consequence of the following proposition.
\begin{proposition}
\label{K(1)_*THH}
The inclusion
\[
\tilde\iota: \ell \longrightarrow THH(\ell)
\]
induces an isomorphism
\[
\tilde\iota_*: K(1)_{*}(\ell) \stackrel{\cong}{\longrightarrow} K(1)_{*}(THH(\ell)).
\]
\end{proposition}
The homology theory $K(1)$ is the first Morava Ktheory, with
\[
\pi_{*}(K(1)) = {\bf Z}/p\,[v_{1}, v_{1}^{1}],
\]
where the dimension
of $v_{1}$ is $2(p1)$.
For an odd prime $p$ $K(1)$ is the Adams summand of mod $p$ periodic
$K$theory,
and we therefore have an isomorphism
\[
K(1)_*(X)
\cong
v_1^{1}\pi_*(\ell \wedge X ; {\bf Z}/p)
\]
where $\ell$ is the Adams summand of complex $K$theory,
$\pi_*(\ell) = {\bf Z}_{(p)}[v_1]$.
\vspace{.25in}
{\bf Proof of Theorem~\ref{vlocal homotopy}.}
We start by choosing some notation:
Take $\ell$ with the usual $\ell$module structure
$\mu: \ell \wedge \ell \longrightarrow \ell$
and $THH(\ell)$ with
$\alpha: \ell \wedge THH(\ell) \longrightarrow THH(\ell)$
being just the restriction of the multiplication on $THH(\ell)$.
Observe that
we can make a commutative diagram
\begin{eqnarray*}
\pi_*(l ; {\bf Z}/p) & \stackrel{\tilde\iota_*}{\longrightarrow} &
\pi_*(THH(l) ; {\bf Z}/p) \\
\downarrow h \;\; & & \;\; \downarrow h \\
\pi_*(l \wedge l ; {\bf Z}/p) & \stackrel{\tilde\iota_*}{\longrightarrow} &
\pi_*(l \wedge THH(l) ; {\bf Z}/p) \\
\downarrow \mu_* & &\;\; \downarrow \alpha_* \\
\pi_*(l ; {\bf Z}/p) & \stackrel{\tilde\iota_*}{\longrightarrow} &
\pi_*(THH(l) ; {\bf Z}/p)
\end{eqnarray*}
where the $h$'s denote Hurewicz maps.
What is important here are the facts that the compositions
$\mu_* \circ h$ and $\alpha_* \circ h$
are identities, so that
$\alpha_*$ is a surjection, among other things.
We can also localize the lower square in this diagram (but not the upper
square\,!) to obtain the following diagram.
\begin{eqnarray*}
v_1^{1}\pi_*(l \wedge l ; {\bf Z}/p) &
\stackrel{\tilde\iota_*}{\longrightarrow} &
v_1^{1}\pi_*(l \wedge THH(l) ; {\bf Z}/p) \\
\downarrow \mu_* \;\; & &\;\;\;\; \downarrow \alpha_*\\
v_1^{1}\pi_*(l ; {\bf Z}/p) &
\stackrel{\tilde\iota_*}{\longrightarrow} &
v_1^{1}\pi_*(THH(l) ; {\bf Z}/p)
\end{eqnarray*}
As we have observed that
\[
K(1)_*(X) = v_1^{1}\pi_*(\ell \wedge X ; {\bf Z}/p)
\]
Proposition \ref{K(1)_*THH} states that the upper arrow here is an
isomorphism.
As the localization of an epimorphism is an epimorphism, the right
hand arrow $\alpha_*$ in the new diagram is an epimorphism,
so we conclude
that the lower $\tilde\iota_*$,
\[
\tilde\iota_* : v_1^{1}\pi_*(\ell;{\bf Z}/p)
\longrightarrow
v_1^{1}\pi_*(THH(\ell);{\bf Z}/p),
\]
is also surjective.
Now we must prove the lower $\tilde\iota_*$ is injective. Suppose that
$x \in ker(\tilde\iota_*)$. By definition of localization we can find
an integer $m$ such that $v_1^mx = x' \in \pi_*(\ell ; {\bf Z}/p)$.
By choosing $m$ larger if necessary we can arrange that
\[
\tilde\iota_* : \pi_*(\ell ; {\bf Z}/p)
\longrightarrow
\pi_*(THH(\ell) ; {\bf Z}/p)
\]
carries $x'$ to zero. By commutativity of the upper square of the first
diagram,
\[
\tilde\iota_*(h(x')) = 0,
\]
so that injectivity of the localized $\tilde\iota_*$
implies there is $m'$ such that
\[
v_1^{m'}h(x') = 0.
\]
Then
\begin{eqnarray*}
0 & = & \mu_*(v_1^{m'}h(x'))
\\
& = & v_1^{m'}\mu_*(h(x'))
\\
& = & v_1^{m'}x'
\end{eqnarray*}
in $\pi_*(\ell ; {\bf Z}/p)$ so that in $v_1^{1}\pi_*(\ell;{\bf Z}/p)$
\[
0 = v_1^{m+m'}x.
\]
We conclude that $x = 0$, so that
\[
\tilde\iota_* : v_1^{1}\pi_*(\ell;{\bf Z}/p)
\longrightarrow
v_1^{1}\pi_*(THH(\ell);{\bf Z}/p)
\]
is also injective.
$\clubsuit$
\vspace{.25in}
To prove Proposition~\ref{K(1)_*THH} we need the following lemma.
\begin{lemma}
\label{HH(K(1)_*(l))}
The Hochschild homology of $K(1)_*(\ell)$ with respect to the ground ring
$\pi_*(K(1))$ is
\[
{\bf HH}_{i}^{\pi_*(K(1))}(K(1)_*(\ell)) =
\left\{
\begin{array}{ll}
K(1)_*(\ell) & \mbox{if $i = 0$.} \\
0 & \mbox{if $i > 0$.}
\end{array}
\right.
\]
\end{lemma}
{\bf Proof of~\ref{K(1)_*THH}.}
The inclusion of simplicial spectra $l \longrightarrow
THH(l)$ induces a morphism of the spectral sequences for
$K(1)_*(\ell)$ and $K(1)_*(THH(\ell))$ arising from
the simplicial filtrations. For $\ell$ the spectral sequence is trivial,
and for $THH(\ell)$ we can evaluate the $E^{2}$term
according to Proposition~\ref{filtSS}, since we know the homology theory
$K(1)$ has a good
K\"{u}nneth theorem \cite[page 133]{Ravenel}.
We find that the $E^{2}$term of the
spectral sequence is identified with the Hochschild homology of
$K(1)_{*}(\ell)$ over $\pi_{*}(K(1))$. Then we have
\[
K(1)_{*}(\ell) \cong E^{\infty}_{0,*} = E^{2}_{0,*}
\longrightarrow
E^{2}_{i,*} \cong {\bf HH}^{\pi_{*}(K(1))}_{i}(K(1)_{*}(\ell)).
\]
But, according to the lemma above,
\[
{\bf HH}^{\pi_{*}(K(1))}_{i}(K(1)_{*}(\ell))
=
\left\{
\begin{array}{ll}
K(1)_{*}(\ell), & \mbox{if $i = 0$}\\
\\
0, & \mbox{if $i > 0$.}
\end{array}
\right.
\]
Thus we have an isomorphism of spectral sequences at the $E^{2}$level
and this fact immediately implies the proposition.
$\clubsuit$
\vspace{.25in}
The structural information about the algebra
$K(1)_*(\ell)$
which we need to prove Lemma~\ref{HH(K(1)_*(l))}
is supplied by the following result.
\begin{proposition}
\label{structure}
$K(1)_0(\ell)$
is a direct limit of semisimple ${\bf Z}/p$algebras.
In fact,
\[
K(1)_0(\ell) = \lim_{n \geq 0} \prod^{p^n} {\bf Z}/p \, ,
\]
where the $n+1$st map in the direct system is the $p^n$fold
power of the diagonal embedding
\[
{\bf Z}/p \longrightarrow \prod^{p} {\bf Z}/p.
\]
\end{proposition}
With this kind of hold on the structure of the algebra
$K(1)_*(\ell)$
it is easy to prove Lemma~\ref{HH(K(1)_*(l))}.
\vspace{.25in}
{\bf Proof of Lemma~\ref{HH(K(1)_*(l))}.}
Recall the fact that Hochschild homology commutes with direct
limits,
the fact (\cite[Theorem 5.3, page 173]{CE}) that
for $k$algebras $A$ and $B$
\[
{\bf HH}_*^k(A \times B) \cong {\bf HH}_*^k(A) \times {\bf HH}_*^k(B),
\]
and the elementary computation
\[
{\bf HH}_i^k(k) =
\left\{
\begin{array}{ll}
k, & \mbox{if $i = 0$.}
\\
0, & \mbox{if $i > 0$.}
\end{array}
\right.
\]
Since
\[
K(1)_*(\ell) = \pi_*(K(1)) \otimes K(1)_0(\ell),
\]
the structural results of Proposition~\ref{structure} imply
$K(1)_*(\ell)$ is a limit of products of $\pi_*(K(1))$
with itself.
Putting these facts together yields the proof of the Lemma
immediately.
$\clubsuit$
\vspace{.25in}
We will give two proofs of Proposition~\ref{structure},
the first being an elementary computation suggested by the methods
of \cite{Adams}.
Generally speaking, the background for the first proof
is contained in \cite[Part III, Chapter 17]{Adams}, where
results are proved for the prime two and stated for odd primes.
In the years since \cite{Adams} was published
notations and conventions have crystallized in ways that
are not always compatible with the original source. The methods
we use are adapted from the calculation of
$E(1)_*(\ell)$, where $E(1)$ is the homology theory $v_1^{1}l$
with coefficients $\pi_*(E(1)) = {\bf Z}_{(p)}\,[v_1, v_1^{1}]$.
\vspace{.25in}
{\bf First Proof of Proposition~\ref{structure}.}
Note first that
\[
K(1)_{*}(X) \cong E(1)_{*}(X;{\bf Z}/p)
\]
so that a normal universal coefficients theorem gives
$K(1)_{*}(X)$
from
$E(1)_{*}(X)$.
In our case
$E(1)_{*}(\ell)$
is torsionfree \cite[Proposition 17.2, page 354]{Adams}
so that we obtain
\[
K(1)_{*}(\ell) = E(1)_{*}(\ell)/p\,E(1)_{*}(\ell).
\]
Now we can use Adams' explicit description of
$E(1)_{*}(\ell)$
as a subalgebra of
\[
{\bf Q}[v_1, v_1^{1}, w] = E(1)_{*}(\ell;{\bf Q}),
\]
where $v_1$ and $w$ both have degree $2(p1)$.
(At odd primes Adams uses $u^{p1}$ and $v^{p1}$
where we will be using $v_1$ and $w$, respectively.)
Translating
\cite[Proposition 17.6, page 358]{Adams}
and its extension to odd primes into our notation we find that if
one defines polynomials
\begin{eqnarray*}
f_{0}(v_1, w) & = & 1, \\
f_{1}(v_1, w) & = &
\frac{w  v_1}{(p+1)  1}, \\
& \vdots & \\
f_{r}(v_1, w) & = &
\frac{(wv_1)}{((rp+1)1)} \cdot
\frac{(w(p+1)v_1)}{((rp+1)(p+1))} \cdots
\frac{(w((r1)p+1)v_1)}{(rp+1)((r1)p+1)}
\\
& = & (p^{r}r!)^{1}
(wv_1)(\cdots )(w((r1)p+1)v_1)
\end{eqnarray*}
then one obtains
$E(1)_{*}(\ell)$ embedded as a subalgebra of
${\bf Q}[v_1, v_1^{1}, w]$
as the free ${\bf Z}_{(p)}[v_1, v_1^{1}]$
module on $ \{ f_{0}, f_{1}, \ldots , f_{r}, \ldots \} $.
By using techniques of the proof of this result one can figure out
the structure of the ring $K(1)_{*}(\ell)$.
First we will prove the identity
\[
f_r \cdot f_s =
\sum_{i=0}^{r}
{s \choose {ri}} {{s+i} \choose s}
v_1^{ri} f_{s+i}
\]
in $E(1)_*(\ell)$ for $r \leq s$.
Once we know the special case
\[
f_1 \cdot f_r =
{r \choose 1}v_1f_r
+
{r+1 \choose r}f_{r+1}
\]
the general case will be proved by induction on $r$ for fixed $s$.
To prove the special case, we note, following \cite{Adams},
that $f_1 \cdot f_r$ is homogeneous of total degree
$2(r+1)(p1)$, so it has an expansion
\[
f_1 \cdot f_r
=
c_0v_1^{r+1} + c_1v_1^rf_1 +
\cdots
+ c_rv_1f_r + c_{r+1}f_{r+1},
\]
where $c_i \in {\bf Z}_{(p)}$.
>From the definition of $f_r$ it is clear that
\[
f_r(1,sp+1) = {s \choose r},
\]
where the binomial coefficient is interpreted as 0 if $s < r$,
so that successively substituting $v_1 = 1, w = sp+1$
for $0 \leq s \leq r+1$,
we can determine the coefficients $c_i$. The answers are
\[
c_0 = \cdots = c_{r1} = 0,
\]
and
\begin{eqnarray*}
c_r & = & {r \choose 1} ,
\\
c_{r+1} & = & {{r+1} \choose r}.
\end{eqnarray*}
To prove the general case,
suppose that for $r < s$ we have
\[
f_r \cdot f_s =
\sum_{i=0}^{r}
{s \choose {ri}} {{s+i} \choose s}
v_1^{ri} f_{s+i}.
\]
Then
\begin{eqnarray*}
{r \choose 1} v_1f_rf_s + {{r+1} \choose r} f_{r+1}f_s &
= &
(f_1f_r)f_s
\\
&
= &
f_1(f_rf_s)
\\
&
= &
\sum_{i=0}^{r}
{s \choose {ri}} {{s+i} \choose s}
v_1^{ri} f_1 f_{s+i}
\end{eqnarray*}
Hence
\begin{eqnarray*}
{{r+1} \choose r} f_{r+1} f_s &
= &
\sum_{i=0}^{r}
{s \choose {ri}} {{s+i} \choose s}
v_1^{ri} f_1 f_{s+i}
 {r \choose 1} v_1 f_r f_s
\\
&
= &
\sum_{i=0}^{r}
{s \choose {ri}} {{s+i} \choose s}
v_1^{ri}
\left[{{s+i} \choose 1} v_1 f_{s+i} + {{s+i+1} \choose {s+i}} f_{s+i+1}\right]
\\
&
&

{r \choose 1} v_1
\left( \sum_{j=0}^{r}
{s \choose {rj}} {{s+j} \choose s}
v_1^{rj} f_{s+j} \right)
\\
&
= &
{{r+1} \choose r}
\sum_{i=0}^{r+1}
{s \choose {r+1i}} {{s+i} \choose s}
v_1^{r+1i} f_{s+i}
\end{eqnarray*}
after patient fiddling with the binomial coefficients,
and we can cancel the extra factor to get the identity we want.
Using the mod$p$ reduction of this identity we can prove that
the reduction of \[\{f_1, f_p, \ldots, f_{p^i}, \ldots \}\]
generates $K(1)_*(\ell)$.
To prove $f_s$ is a polynomial in the $f_{p^i}$ for
$p^i \leq s$
suppose that the $p$adic expansion of $s$ begins
\[
s = s_jp^j + s_{j+1}p^{j+1}
\]
where $1 \leq s_j \leq p1$ and $s_{j+1} \geq 0$.
If $s  p^j = 0$, there is nothing to do, so suppose that
$s  p^j > 0$.
By our identity
\[
f_{p^j} \cdot f_{sp^j} =
\sum_{i=0}^{p^j}
{sp^j \choose {p^ji}} {{sp^j+i} \choose sp^j}
v_1^{p^ji} f_{sp^j+i}
\]
so that
\[
{s \choose {p^j}} f_s
=
f_{p^j} \cdot f_{sp^j} 
\sum_{i=0}^{p^j1}
{sp^j \choose {p^ji}} {{sp^j+i} \choose sp^j}
v_1^{p^ji} f_{sp^j+i}.
\]
But the coefficient of $f_s$ has nonzero reduction mod$p$, since it is
a $p$adic unit by
\cite[page 115]{Kummer},
\cite[page 270]{Dickson},
\cite{Fibonacci},
or by hand,
so induction on $s$ proves the assertion.
Thus we have a surjection
\[
m:
{\bf Z}/p\,[v_1, v_1^{1}] \otimes {\bf Z}/p\,[f_1,f_p,\ldots]
\longrightarrow
K(1)_*(\ell).
\]
Now we determine the relations satisfied by the $f_{p^j}$ in
$K(1)_*(\ell)$ and get generators for the kernel of $m$ following
similar strategy.
In
${\bf Q}[v_1, v_1^{1}, w]$
we have
\begin{eqnarray*}
(f_{p^{j}})^p & = & c_0v_1^{p^{j+1}} + c_1v_1^{p^{j+1}1}f_1 +
\cdots + \\
& & c_{p^{j}}v_1^{p^{j+1}p^{j}}f_{p^{j}} + \cdots +
c_{p^{j+1}}f_{p^{j+1}}
\end{eqnarray*}
Substituting $v_1 = 1$,
and then successively
$w = (sp+1)$
for
$ 0 \leq s \leq p^j $
we get
\[
0 = c_0 = \cdots = c_{p^{j1}}, \; 1 = c_{p^{j}}.
\]
Substituting $v_1 = 1$,
and then successively
$w = (sp+1)$
for
$p^{j} + 1 \leq s \leq p^{j+1}$
we find
\[
{s\choose p^j} ^p =
{s\choose {p^j}} + c_{p^j+1}{s\choose {p^j+1}} + \cdots + c_s
\]
so that
$pc_s$ for $s$ in this range, by induction on $s$.
Thus, the relation satisfied by the residue class
$f_{p^{j}} \in K(1)_*(\ell)$
is
\[
(f_{p^{j}})^p = v_1^{p^{j+1} p^{j}} f_{p^{j}}.
\]
Introduce the degree zero Laurent polynomial
\[
g_{p^{j}} = v_1^{p^{j}} f_{p^{j}}
\]
and the relation takes the form
\[
(g_{p^{j}})^p = g_{p^{j}}.
\]
We can finally state that
$m$ induces a surjection
\[
m:
{\bf Z}/p\,[v_{1}, v_{1}^{1}][g_{p^{j}}: \;j \geq 0]/(g_{p^{j}}^pg_{p^{j}}:
\;j \geq 0)
\longrightarrow
K(1)_*(\ell)
\]
and we claim this is an isomorphism.
It suffices to consider the restriction
\[
m_0:
{\bf Z}/p\,[g_{p^{j}}: \;j \geq 0]/(g_{p^{j}}^pg_{p^{j}}:
\;j \geq 0)
\longrightarrow
K(1)_0(\ell).
\]
We suppose $x \in ker(m_0)$.
Then there is an $n \geq 0$ such that
\[
x \in A_n =
{\bf Z}/p\,
[g_{p^{j}}: \;n \geq j \geq 0]/(g_{p^{j}}^pg_{p^{j}})
\]
which is a ${\bf Z}/p$vector space of dimension $p^{n+1}$.
The image $m_0(A_n)$ is the subspace of $K(1)_0(\ell)$ spanned
by $\{f_0 =1, \ldots, v_1^{p^{n+1}+1}f_{p^{n+1}1}\}$.
This space supports $p^{n+1}$ independent functionals
$\phi_s$ where
\[
\phi_0(f_i) = \delta_{i0}
\]
and
\[
\phi_s(f) = f(1, sp+1) \;\mbox{mod $p$}
\]
for $1 \leq s \leq p^{n+1}1$, so it is also of dimension $p^{n+1}$.
It follows that $x=0$. Hence $ker(m_0) = 0$. Since we retrieve
$m$ by tensoring with ${\bf Z}/p\,[v_{1}, v_{1}^{1}]$, $m$ is
also an isomorphism.
Since it is well known that
\[
{\bf Z}/p\,[x]/(xx^p) \cong \prod_{i=1}^{p} {\bf Z}/p,
\]
the isomorphisms $m_0$ and $m$ can be unwound to give
the rest of the proof.
$\clubsuit$
\vspace{.25in}
Our second proof of Proposition~\ref{structure}
follows a suggestion of M.J. Hopkins that a proof
could be extracted from a result of Miller and
Ravenel \cite{MillerRavenel} in stable homotopy
theory. We follow the exposition in
\cite{Ravenel} between pages 223 and 226. Obviously
the prerequisites for this proof are much more
demanding than those for the first proof, but we
can also prove more.
\vspace{.25in}
{\bf Second Proof of Proposition \ref{structure}.}
In the language of $BP$theory, we want to calculate
$K(1)_*(BP\langle 1 \rangle)$,
and it is not much more work to calculate all the rings
\[
K(n)_*(BP\langle n \rangle)
=
v_n^{1}k(n)_*(BP\langle n \rangle)
\]
for all $n \geq 1$.
The coefficient rings of the spectra here are
\[
BP\langle n \rangle_*
=
{\bf Z}_{(p)}[v_1, \ldots, v_n]
\]
where degree $v_k = 2(p^k1)$,
\[
k(n)_*
=
{\bf Z}/p\,[v_n],
\]
and
\[
K(n)_*
=
{\bf Z}/p\,[v_n, v_n^{1}].
\]
$BP\langle n \rangle$ may be constructed from the spectrum $BP$,
\[
BP_*
=
{\bf Z}_{(p)}[v_1, \ldots, v_n, \ldots],
\]
by a BaasSullivan construction
\[
BP \longrightarrow BP\langle n \rangle
\]
which realizes the quotient map
\[
{\bf Z}_{(p)}[v_1, \ldots, v_n, \ldots]
\longrightarrow
{\bf Z}_{(p)}[v_1, \ldots, v_n, \ldots]/(v_{n+1}, v_{n+2}, \ldots)
\]
on coefficients.
One has that
\[
BP_*BP
=
BP_*[t_1, t_2, \ldots]
\]
so that the dual form of one of
the universal coefficients theorems for generalized homology theories
\cite[Proposition~13.5, page 285]{Adams}
gives
\begin{eqnarray*}
k(n)_*(BP) &
\cong &
k(n)_* \otimes _{BP_*} BP_*BP
\\
&
= &
{\bf Z}/p\,[v_n][t_1, t_2, \ldots]
\end{eqnarray*}
and
\begin{eqnarray*}
k(n)_*(BP\langle n \rangle) &
\cong &
k(n)_* \otimes _{BP_*} BP_*(BP \langle n \rangle)
\\
&
= &
{\bf Z}/p\,[v_n][t_1, t_2, \ldots]/(\eta_R(v_{n+1}), \eta_R(v_{n+2}), \ldots),
\end{eqnarray*}
by definition of $BP\langle n \rangle$ and by definition of the right unit
$\eta_R$.
Now we have also for $k \geq 1$ the congruence
\[
\eta_R(v_{n+k})
\equiv
v_nt_k^{p^n}  v_n^{p^k}t_k
\; \mbox{modulo $(\eta_R(v_{n+1}), \ldots , \eta_R(v_{n+k1}))$}
\]
in $k(n)_*(BP)$
\cite[page 224]{Ravenel}.
It follows that
\[
k(n)_*(BP\langle n \rangle)
\cong
{\bf Z}/p\,[v_n][t_1, t_2, \ldots]/(v_nt_k^{p^n}  v_n^{p^k}t_k: k \geq 1)
\]
and
\[
K(n)_*(BP\langle n \rangle)
\cong
{\bf Z}/p\,[v_n,v_n^{1}]
[t_1, t_2, \ldots]/(v_nt_k^{p^n}  v_n^{p^k}t_k: k \geq 1).
\]
When $n=1$ we can introduce new variables of degree 0
\[
u_1 = v_1^{1}t_1, \ldots, u_k = v_1^{p^{k1} \cdots 1}t_k, \ldots,
\]
whereupon the presentation changes to
\[
K(1)_*(BP\langle 1 \rangle)
\cong
{\bf Z}/p\,[v_1,v_1^{1}][u_1, u_2, \ldots]/(u_k^p  u_k: k \geq 1),
\]
which is what we had before.
$\clubsuit$
\section{The localized Adams spectral sequence}
\label{localizedASS}
This section contains the last of the preliminary computations we
need before we compute
$\pi_*(THH(\ell); {\bf Z}/p)$
with the classical Adams spectral sequence, denoted
$E_r(THH(\ell);{\bf Z}/p)$
in section~\ref{E_2 term}.
>From here on this is the only spectral sequence we will be dealing
with, so for the rest of the paper we simplify the notation
to just
$E_r$.
In theorem~\ref{E_2} we computed the $E_2$term of the spectral sequence
and the result is
\begin{eqnarray*}
E_{2}^{*,*} &
= &
{\rm Ext}_{A_{*}}^{*,*}({\bf Z}/p, H{\bf Z}/p_{*}(THH(l) \wedge {\bf M})),
\\
&
\cong &
{\bf Z}/p\,[a_{1}] \otimes \Lambda (\lambda_{1}, \lambda_{2}) \otimes {\bf
Z}/p\,[\mu ],
\end{eqnarray*}
where bidegree $a_{1} = (2(p1),1)$, bidegree $\lambda_{i} = (2p^{i}1,0)$,
and bidegree $\mu = (2p^{2},0)$. As $a_{1}$ is a permanent cycle, this
is a spectral sequence of ${\bf Z}/p\,[a_{1}]$algebras, so we can
invert $a_1$ to obtain
a spectral sequence of ${\bf Z}/p\,[a_{1},a_{1}^{1}]$ algebras,
which we will denote $a_1^{1}E_r$.
Though we make no assertion concerning the convergence of the localized
spectral sequence,
the isomorphism
\[
v_{1}^{1}\pi_{*}(\ell;{\bf Z}/p)
\cong
v_{1}^{1}\pi_{*}(THH(\ell);{\bf Z}/p)
\]
established in Theorem~\ref{vlocal homotopy}
determines $a_1^{1}E_{\infty}$. Since we have inverted only
bihomogeneous elements $a_1^{1}E_r$ is still bigraded,
and this extra fact enables us to identify the pattern of
differentials which gives the required $E_{\infty}$term.
\begin{theorem}
\label{a_1inverseSS}
In the localized spectral sequence
$a_{1}^{1}E_{r}$
one has
\[
a_{1}^{1}E_{2}
\cong
{\bf Z}/p\,[a_{1}, a_{1}^{1}]
\otimes
\Lambda (\lambda_{1}, \lambda_{2}) \otimes {\bf Z}/p\,[\mu]
\]
and
\[
a_1^{1}E_{\infty} \cong {\bf Z}/p\,[a_{1}, a_{1}^{1}].
\]
The pattern of differentials may be described recursively as follows:
The $n$th nonzero differential occurs in
\[
a_1^{1}E_{r(n)}
\cong
{\bf Z}/p\,[a_{1}, a_{1}^{1}] \otimes
\Lambda (\lambda_{n}, \lambda_{n+1}) \otimes {\bf Z}/p\,[\mu^{p^{n1}}]
\]
and is given by the formulas
\[
d_{r(n)}(\lambda_{n}) = 0,\; d_{r(n)}(\lambda_{n+1}) = 0,\;\mbox{and }
d_{r(n)}(\mu^{p^{n1}}) \doteq a_{1}^{r(n)}\lambda_{n}.
\]
The $r(n)$ are given recursively by the definitions
\[
r(1) = p,\;
r(2k) = p\,r(2k1),\; \mbox{and } r(2k+1) = p\,r(2k) + p,
\]
and the $\lambda_{n}$ are defined for $n \geq 3$ by
$\lambda_{n} = \lambda_{n2}\mu^{p^{n3}(p1)}.$
\end{theorem}
We use the symbol $\doteq$ for equality up to a multiple by a
nonzero element of ${\bf Z}/p$.
For example, this says that the first few steps in the spectral
sequence are
\[
E_{p} \cong
{\bf Z}/p\,[a_{1}, a_{1}^{1}]
\otimes
\Lambda(\lambda_{1}, \lambda_{2}) \otimes {\bf Z}/p\,[\mu],
\]
when
$d_p(\mu) \doteq a_1^{p} \lambda_{1}$;
\[
E_{p^{2}} \cong
{\bf Z}/p\,[a_{1}, a_{1}^{1}]
\otimes
\Lambda(\lambda_{2}, \lambda_{1}\mu^{p1})
\otimes
{\bf Z}/p\,[\mu^{p}],
\]
when
$d_{p^2}(\mu^{p}) \doteq a_1^{p^{2}} \lambda_{2}$;
\[
E_{p^{3}+p} \cong
{\bf Z}/p\,[a_{1}, a_{1}^{1}]
\otimes
\Lambda(\lambda_{1}\mu^{p1}, \lambda_{2}\mu^{p^2  p})
\otimes
{\bf Z}/p\,[\mu^{p^2}],
\]
when
$d_{p^3+p}(\mu^{p^2}) \doteq a_1^{p^3 + p} \lambda_{1}\mu^{p1}$;
and
\[
E_{p^4+p^2} \cong
{\bf Z}/p\,[a_{1}, a_{1}^{1}]
\otimes
\Lambda(\lambda_{2}\mu^{p^2p}, \lambda_{1}\mu^{p^3  p^2 + p 1})
\otimes
{\bf Z}/p\,[\mu^{p^3}],
\]
when
$d_{p^4+p^2}(\mu^{p^3}) \doteq a_1^{p^4 + p^2} \lambda_{2}\mu^{p^2p}$.
\vspace{.25in}
{\bf Proof.}
As we have said in the opening remarks, we will show that the isomorphism
\[
v_{1}^{1}\pi_{*}(\ell;{\bf Z}/p)
\cong
v_{1}^{1}\pi_{*}(THH(\ell);{\bf Z}/p)
\]
of Theorem~\ref{vlocal homotopy}
determines the $E_{\infty}$term of the spectral sequence and that there
is exactly one way to obtain this $E_{\infty}$term from the given
$E_2$term.
According to Theorem~\ref{vlocal homotopy},
\[
\tilde{\iota} \; : \ell \longrightarrow THH(\ell)
\]
induces an isomorphism
\[
\tilde{\iota_*} :
v_{1}^{1}\pi_{*}(\ell;{\bf Z}/p)
\stackrel{\cong}{\longrightarrow}
v_{1}^{1}\pi_{*}(THH(\ell);{\bf Z}/p).
\]
Therefore we have a commutative diagram
\begin{eqnarray*}
\pi_{*}(l;{\bf Z}/p) &
\longrightarrow &
\pi_{*}(THH(l);{\bf Z}/p)
\\
\downarrow & & \downarrow
\\
v_{1}^{1}\pi_{*}(l;{\bf Z}/p) &
\stackrel{\cong}{\longrightarrow} &
v_{1}^{1}\pi_{*}(THH(l);{\bf Z}/p),
\end{eqnarray*}
where the vertical arrows are the localization maps.
As the righthand localization map is injective, we conclude that
\[
\tilde{\iota_*} :
\pi_{*}(\ell;{\bf Z}/p)
\longrightarrow
\pi_{*}(THH(\ell);{\bf Z}/p)
\]
is injective.
In section \ref{E_2 term} we showed that
\[
\tilde{\iota} \; : \ell \longrightarrow THH(\ell)
\]
induces a map of Adams spectral sequences, which may be identified with
the canonical algebra inclusion
\[
{\bf Z}/p\,[a_{1}]
\longrightarrow
{\bf Z}/p\,[a_{1}]
\otimes
\Lambda (\lambda_{1}, \lambda_{2}) \otimes {\bf Z}/p\,[\mu]
\]
on $E_2$terms.
The domain spectral sequence has no differentials, so we have
\[
{\bf Z}/p\,[a_{1}]
\longrightarrow
E_{\infty}^{*,*},
\]
and this map is injective, since we have just observed that the map
on homotopy is. Now we use the surjectivity of the induced map
on localized homotopy.
This
implies that if $x \in \pi_{*}(THH(\ell);{\bf Z}/p)$, then there is a natural
number $N$ such that $v_{1}^{N}x$ lies in the image of
$\pi_{*}(\ell;{\bf Z}/p)$ in
$\pi_{*}(THH(\ell);{\bf Z}/p)$.
Multiplication by $v_{1}$ in homotopy is represented by multiplication
by $a_{1}$ in the Adams spectral sequence \cite[Lemma 17.11(ii), page 361]
{Adams},
so we can draw conclusions about the spectral sequence $E_r$.
Take an infinite cycle $\xi \in E_{\infty}^{n,s}$ ,
not already in ${\bf Z}/p\,[a_{1}]$,
(Thus $2(p1) < n$.), representing $x \in \pi_{n}(THH(\ell);{\bf Z}/p)$,
which is itself not in the image of $\pi_n(\ell;{\bf Z}/p)$.
If $v_{1}^{N}x$ lies in the image of $\pi_{*}(l;{\bf Z}/p)$, then
$a_{1}^{N}\xi = 0$ in the spectral sequence.
This is not saying that the homotopy element represented by $\xi$
becomes zero when multiplied by $v_{1}^{N}$,
but rather that a jump in the Adams filtration
degrees has occurred as we multiplied by higher and higher powers of $v_{1}$.
All we can say is that $v_{1}^{N}x$ is now represented by an infinite cycle
in $E_{\infty}^{n',s'}$, where now $2(p1)s' = n'$.
We believe that the reader will find it helpful to sketch the spectral
sequence at this point.
These remarks are then most conveniently expressed in terms of the
localized spectral sequence
$a_1^{1}E_r$
as the assertion
\[
a_1^{1}E_{\infty}^{*,*}
\cong
{\bf Z}/p\,[a_{1},a_{1}^{1}].
\]
Now we claim that the pattern of differentials which delivers this
$E_{\infty}$term can only be that pattern described in the statement
of the theorem. Before we argue this point, recall that we have already
mentioned the fact that the localized spectral sequence retains its
bigrading, since the multiplicative set
\[
S = \{ a_1^i : i \geq 0 \}
\]
of elements we have inverted consists of bihomogeneous
elements. One checks easily that defining the bidegree of the
fraction $m/a_{1}^{i}$ to be $(n2i(p1), si)$, if the
bidegree of $m$ is $(n,s)$, leads to a well defined notion
of bidegree in the localized spectral sequence. Most importantly,
we still have
\[
d_{r}: a_{1}^{1}E_{r}^{n,s} \longrightarrow a_{1}^{1}E_{r}^{n1,s+r}.
\]
Start with the localized $E_{2}$term
\[
a_{1}^{1}E_{2}
\cong
{\bf Z}/p\,[a_{1}, a_{1}^{1}] \otimes
\Lambda (\lambda_{1}, \lambda_{2}) \otimes
{\bf Z}/p\,[\mu],
\]
where the bidegrees of $a_{1}$, $\lambda_{i}$, and $\mu$ are
$(2(p1),1)$, $(2p^{i}1, 0)$ and $(2p^{2}, 0)$, respectively.
Since the bidegree of $d_{r}$ is $(1, r)$, $\lambda_{1}$
is an infinite cycle. Considering the bigrading again,
we see that $d_{r}(\lambda_{2}) = 0$, for all $r$,
except possibly when $r = p+1$. But if $d_{p+1}(\lambda_{2})$
is not zero, we have $d_{p+1}(\lambda_{2}) \doteq a_{1}^{p+1}$,
where $\doteq$ means equal up to a nonzero scalar factor,
which is not consistent with the expression for the $E_{\infty}$term.
Therefore $\lambda_{2}$ is also an infinite cycle.
Since we have a spectral sequence of algebras, we can conclude
that $\mu$ is not an infinite cycle, and that the first nonzero
differential in the spectral sequence is determined by what it
does on $\mu$. By bigrading considerations the only possible
nonzero differential on $\mu$ is
\[
d_{p}(\mu) \doteq a_{1}^{p}\lambda_{1}.
\]
Then one easily calculates
\[
a_{1}^{1}E_{p+1}
=
{\bf Z}/p\,[a_{1}, a_{1}^{1}] \otimes
\Lambda (\lambda_{2}, \lambda_{3}) \otimes
{\bf Z}/p\,[\mu^{p}],
\]
where $\lambda_{3} = \lambda_{1}\mu^{p1}$, as in the statement of the
theorem.
Notice that the bidegrees of the new
${\bf Z}/p\,[a_{1}, a_{1}^{1}] $algebra
generators are
$(2p^{2}1,0),\; (2p^{3}  2p^{2} + 2p  1, 0)$ ,
and $(2p^{3}, 0)$.
We have already argued that $\lambda_{2}$ is an infinite cycle, and a
calculation of bidegrees shows there is one possible differential
which could be nonzero on $\lambda_{3}$, namely,
$d_{p^{2}+1}(\lambda_{3}) \doteq a_{1}^{p^{2}+1}.$
But this is also ruled out, since it would contradict what we
know about the $E_{\infty}$term.
Thus $\lambda_{3}$ is an infinite cycle, and, therefore,
$\mu^{p}$ cannot be an infinite cycle, and we need to determine
which differential takes $\mu^{p}$ out of the spectral sequence.
Since $r \geq p+1$, $d_{r}(\mu^{p}) = a_{1}^{r}\lambda_{3}$
is impossible, and the only possibility is that for some $r$
\[
d_{r}(\mu^{p})
\doteq
a_{1}^{r}\lambda_{2}.
\]
Calculating bidegrees,
we find that this can only happen, and, in fact, must happen,
when $r = p^{2}$.
Proceeding inductively, assume that the first $n1$ differentials
in the localized spectral sequence occur at the times
specified in the theorem and are given by the formulas.
Then we find that
\[
a_1^{1}E_{r(n1)+1}
\cong
{\bf Z}/p\,[a_{1}, a_{1}^{1}] \otimes
\Lambda (\lambda_{n}, \lambda_{n+1})
\otimes
{\bf Z}/p\,[\mu^{p^{n1}}],
\]
The argument that the next differential is $d_{r(n)}$ follows the
same pattern as the argument given for the second differential.
One will have to note that the bidegree of $\lambda_{m}$ is
$(2p^{m}  2p^{m1} + \cdots + 2p  1, 0)$
if $m$ is odd, and
$(2p^{m}  2p^{m1} + \cdots + 2p^{2}  1, 0)$,
if $m$ is even. Also one will need to know that
\[
r(n) = p^{n} + p^{n2} + \cdots + p,
\]
if $n$ is odd, and
\[
r(n) = p^{n} + p^{n2} + \cdots +p^{2},
\]
if $n$ is even.
By inductive hypothesis one knows that $\lambda_{n}$ is an infinite cycle.
Then one argues that the only way $\lambda_{n+1}$ could fail to be an
infinite cycle is if $d_{r(n)}(\lambda_{n+1}) \doteq a_{1}^{r(n)+1}$,
which is impossible since $E_{\infty} =
{\bf Z}/p\,[a_{1}, a_{1}^{1}]$.
These two facts imply that for some $r > r(n1)$,
$d_{r}(\mu^{p^{n1}}) \neq 0$,
and calculating bidegrees one more time one sees the only possibility
is
\[
d_{r(n)}(\mu^{p^{n1}})
\doteq
a_{1}^{r(n)}\lambda_{n}.
\]
This completes the proof of the theorem.
$\clubsuit$
\section{The mod $p$ homotopy of $THH(\ell)$}
\label{ASS}
In the following theorem we show how the differentials in the localized
Adams spectral sequence determine the differentials in the Adams spectral
sequence itself. One of the interesting things about this example is that
the spectral sequence has infinitely many nonzero differentials, and that
we can determine them all.
Notice that if one removes the explicit arithmetic from the proof below
one has a general result on the Adams spectral sequence for the mod$p$
homotopy of an $\ell$module spectrum $X$. This result is stated at the
end of the section.
\begin{theorem}
\label{maintheorem}
Let $E_{r}$ be the Adams spectral sequence convergent to the mod $p$
homotopy of $THH(\ell)$. Then the nonzero differentials in $E_{r}$ are
exactly the $d_{r(n)}$, where the function $r(n)$ is as in
Theorem \ref{a_1inverseSS}.
Moreover, the $E_{\infty}$term is a direct sum of cyclic
${\bf Z}/p\,[a_{1}]$modules with the following generators.
$1 \in E_{\infty}^{0,0}$ generates a free
${\bf Z}/p\,[a_{1}]$ module summand of $E_{\infty}$, and
all the generators of
${\bf Z}/p\,[a_{1}]$torsion summands of $E_{\infty}$ arise in
the following way.
If the $p$adic expansion of $m > 0$ starts
$m = m_{n1}p^{n1} + m_{n}p^{n}$, where $p1 > m_{n1} > 0$
and $m_{n} \geq 0$, then
\[
\lambda_{n}\mu^{p^{n1}(m_{n1} 1)}\mu^{p^{n}m_{n}}
\;\mbox{and}\;
\lambda_{n+1}\lambda_{n}\mu^{p^{n1}(m_{n1} 1)}\mu^{p^{n}m_{n}}
\]
are generators of the summands of $E_{\infty}$ isomorphic to
${\bf Z}/p\,[a_{1}]/(a_{1}^{r(n)})$. There is no torsion
present other than $a_{1}^{r(n)}$torsion for every $n$.
\end{theorem}
{\bf Proof.}
Put $r(0) = 1$, put $S_{0} = \{ \; \}$,
and take the other notation from the preceding theorem.
$P(n)$ is the following statement:
\begin{quotation}
There is a set
\[
S_{n} \subset E_{r(n)+1}^{*,0}
\]
such that
the ${\bf Z}/p\,[a_{1}]$module $E_{r(n)+1}^{*,*}$
may be decomposed as an internal direct sum
\[
E_{r(n)+1}^{*,*}
=
\langle S_n \rangle
\oplus
{\bf Z}/p\,[a_{1}] \otimes \Lambda(\lambda_{n+1}, \lambda_{n+2})
\otimes
{\bf Z}/p\,[\mu^{p^n}],
\]
where
$\langle S_n \rangle$
is the ${\bf Z}/p\,[a_{1}]$module generated by $S_n$.
For each $x \in S_n$ we have
\[
a_{1}^{r(n)}x = 0
\]
so that
\[
\langle S_n \rangle
\subset
\oplus_{0 \leq s < r(n)} E_{r(n) +1}^{*,s}.
\]
Concerning the differentials in the spectral sequence,
$E_{r(n)+1} = E_{r(n+1)}$,
and $d_{r(n+1)}$ is determined by the formulas
\begin{eqnarray*}
d_{r(n+1)}(x) & = & 0 \;
\mbox{for $x \in S_{n} \cup \{\lambda_{n+1}, \lambda_{n+2}\} $} \\
d_{r(n+1)}(\mu^{p^{n}}) & = & a_{1}^{r(n+1)}\lambda_{n+1}.
\end{eqnarray*}
\end{quotation}
As one proves the statements $P(n)$, one reexamines each
inductive step in the proof and unwinds what has happened to get
the decomposition of $E_{\infty}$ into cyclic
${\bf Z}/p\,[a_1]$modules.
\vspace{.25in}
{\bf Remark:} It will be clear from the proof of $P(n)$ that
$\langle S_n \rangle$ is also an ideal of $E_{r(n)+1}^{*,*}$, and that
the elements of
$S_n$ are all of square zero. A second glance shows that there are
many more
multiplicative relations satisfied by the elements of $S_n$
so that the $E_{r(n)+1}$term is complicated
as an algebra. The algebra structure on $E_{\infty}$ is therefore very
complicated, and we have not tried to explicitly determine it.
\vspace{.25in}
The part of the statement $P(0)$ concerning the algebraic structure
of the $E_{2}$term of the spectral sequence is the content of
theorem \ref{E_2}, so we have to
discuss the differential. As $E_{2}$ is $a_{1}$torsion free,
the natural map
\[
E_{2} \longrightarrow a_{1}^{1}E_{2}
\]
is injective and preserves the bigrading,
so the first nonzero differential in the domain is determined
by the first nonzero differential in the target.
Referring to the preceding theorem we get that the first
nonzero differential in the Adams spectral sequence is
$d_{r(1)}$ and that the following formulas are valid:
\[
d_{r(1)}(\lambda_{1}) = 0, \; d_{r(1)}(\lambda_{2}) = 0, \;
\mbox{and} \; d_{r(1)}(\mu) = a_{1}^{r(1)}\lambda_{1}.
\]
This proves $P(0)$.
To get the feel for the induction step, let us work out the proof
of $P(1)$ with some care. Recall that $r(1) = p$, so that by
$P(0)$ we may calculate
the differential on a typical element
\[
\lambda_{1}^{\epsilon_{1}}\lambda_{2}^{\epsilon_{2}}\mu^{pm_1 + m_0}
\in
E_p^{*,0},
\]
where $\epsilon_i = 0$ or 1 and $0 \leq m_0 \leq p1$, obtaining
\[
d_{p}(\lambda_{1}^{\epsilon_{1}}\lambda_{2}^{\epsilon_{2}}
\mu^{pm_1 + m_0})
\doteq
\lambda_{1}^{\epsilon_{1}}\lambda_{2}^{\epsilon_{2}}\mu^{pm_1}
(m_0\mu^{m_01}a_{1}^{p}\lambda_{1}),
\]
Then it is clear that
$\lambda_{1}^{\epsilon_{1}}\lambda_{2}^{\epsilon_{2}}
\mu^{pm_1 + m_0}$
is a cycle if
$\epsilon_{1} = 1$,
or if $m_0 = 0$.
We also see that the
${\bf Z}/p\,[a_1]$generators
of the $a_1$torsion submodule of $E_{p+1}^{*,*}$ may be taken to be
\[
S_1 =
\{
\lambda_2^{\epsilon_2}\lambda_1\mu^{pm_1},
\lambda_2^{\epsilon_2}(\lambda_1\mu)\mu^{pm_1},
\ldots,
\lambda_2^{\epsilon_2}(\lambda_1\mu^{p2})\mu^{pm_1},
\mbox{where $m_1 \geq 0$}
\}
\subset
E_{p+1}^{*,0}
\]
In fact, in $E_{p+1}$ $a_1^px = 0$ for each $x \in S_1$,
so that
\[
\langle S_{1} \rangle \subset
\oplus_{0 \leq s < p}E_{p+1}^{*,s},
\]
since $a_1$ has bidegree $(2(p1), 1)$.
Cycles which generate the
${\bf Z}/p\,[a_1]$free part of $E_{p+1}$
are of the form
\[
z =
\lambda_2^{\epsilon_2}\lambda_3^{\epsilon_3}(\mu^p)^{m_1},
\]
where
$\lambda_3 = \lambda_1\mu^{p1}$,
$\epsilon_i = 0$ or 1,
and $m_1 \geq 0$.
This gives us our ${\bf Z}/p\,[a_{1}]$module decomposition
\[
E_{p+1}^{*,*} =
\langle S_1 \rangle
\oplus
{\bf Z}/p\,[a_{1}] \otimes
\Lambda(\lambda_{2}, \lambda_{3}) \otimes
{\bf Z}/p\,[\mu^{p}]
\]
Now we can consider the next differential in the spectral sequence.
The submodule $\langle S_1 \rangle$
complementary to the subalgebra
\[
{\bf Z}/p\,[a_{1}] \otimes
\Lambda(\lambda_{2}, \lambda_{3}) \otimes
{\bf Z}/p\,[\mu^{p}]
\]
is a torsion module, and the subalgebra is torsionfree.
Subsequent differentials are the $d_r$ for $r > p$,
have bidegree $(1,r)$,
and are derivations of
${\bf Z}/p\,[a_1]$algebras.
Since
\[
\langle S_{1} \rangle \subset
\oplus_{0 \leq s < p}E_{p+1}^{*,s},
\]
so that there are no nonzero elements of
$\langle S_{1} \rangle$
of filtration degree greater than $p$,
and since algebra demands that
$d_{r}
(\langle S_{1} \rangle) \subset
\langle S_{1} \rangle$,
we find that the elements of
$\langle S_1 \rangle$
are cycles for $d_r$ for every $r > p$.
Algebraic reasons do not rule out the possibility that the
next nonvanishing differential on
an element of the complementary subalgebra
\[
{\bf Z}/p\,[a_{1}] \otimes
\Lambda(\lambda_{2}, \lambda_{3}) \otimes
{\bf Z}/p\,[\mu^{p}]
\]
has a component in
$\langle S_{1} \rangle$,
but the filtration argument does in fact rule this out.
Thus $\langle S_1 \rangle$ is a
${\bf Z}/p\,[a_1]$submodule
of
$E_{\infty}$
annihilated by $a_1^p$.
We will show that the higher differentials introduce higher
$a_1$torsion so that $S_1$ is a set of
${\bf Z}/p\,[a_1]$module generators for the
summands of $E_{\infty}$ isomorphic to
${\bf Z}/p\,[a_1]/(a_1^p)$
as claimed in the theorem.
Also, we have concluded that the next nonvanishing differential on
\[
{\bf Z}/p\,[a_{1}] \otimes
\Lambda(\lambda_{2}, \lambda_{3}) \otimes
{\bf Z}/p\,[\mu^{p}]
\]
has only a torsionfree component, so it is detected in
the localized spectral sequence.
Putting this discussion together with the preceding theorem
we obtain the rest of $P(1)$, namely,
that $E_{p+1} = E_{r(2)}$,
and that the formulas
\[
d_{r(2)}(x) = 0 \; \mbox{for $x \in S_{1} \cup
\{\lambda_{2}, \lambda_{3} \}$}
\]
and
\[
d_{r(2)}(\mu^{p}) = a_{1}^{r(2)}\lambda_2
\]
determine the next differential.
The argument for deducing $P(n)$ from $P(n1)$ is in outline
exactly the same as the argument we have just finished, so we
will just quickly repeat the steps.
According to $P(n1)$ we have the decomposition
\[
E_{r(n)}^{*,*}
=
\langle S_{n1} \rangle
\oplus
{\bf Z}/p\,[a_{1}] \otimes \Lambda(\lambda_{n}, \lambda_{n+1})
\otimes
{\bf Z}/p\,[\mu^{p^{n1}}],
\]
and we can calculate
$E_{r(n)+1}$ from $P(n1)$, first deriving the formula
\[
d_{r(n)}(\lambda_{n}^{\epsilon_{n}}\lambda_{n+1}^{\epsilon_{n+1}}
\mu^{p^nm_n + p^{n1}m_{n1}})
\doteq
\lambda_{n}^{\epsilon_{n}}\lambda_{n+1}^{\epsilon_{n+1}}\mu^{p^nm_n}
m_{n1}(\mu^{p^{n1}})^{m_{n1}1}(a_{1}^{r(n)}\lambda_{n}),
\]
where $\epsilon_i = 0$ or 1,
$0 \leq m_{n1} \leq p1$,
and $m_n \geq 0$.
One finds that the $a_1$torsion submodule of
$E_{r(n)+1}^{*,*}$
is spanned by
\begin{eqnarray*}
\lefteqn{S_{n} =
S_{n1} \cup} \\
& &
\{
\lambda_{n+1}^{\epsilon_{n+1}}\lambda_{n}\mu^{p^nm_n},
\lambda_{n+1}^{\epsilon_{n+1}}(\lambda_n\mu^{p^{n1}})\mu^{p^nm_n},
\ldots,
\lambda_{n+1}^{\epsilon_{n+1}}(\lambda_n\mu^{p^{n1}(p2)})\mu^{p^nm_n},
\mbox{where $m_n \geq 0$}
\}
\\
& &\subset E_{r(n)+1}^{*,0}
\end{eqnarray*}
and that each $x \in S_n S_{n1}$ generates a submodule isomorphic to
${\bf Z}/p\,[a_{1}]/(a_1^{r(n)})$.
This gives us
\[
\langle S_{n} \rangle \subset
\oplus_{0 \leq s < r(n)}E_{r(n)+1}^{*,s}
\]
The complementary $a_{1}$torsionfree submodule
is seen to be the algebra
\[
{\bf Z}/p\,[a_{1}] \otimes
\Lambda (\lambda_{n+1}, \lambda_{n+2}) \otimes
{\bf Z}/p\,[\mu^{p^n}],
\]
where $\lambda_{n+2} = \mu^{p^{n1}(p1)}\lambda_n$.
Then one argues with the filtration of
$\langle S_{n} \rangle$
and the bidegree of $d_{r}$ that the next differential is determined
in the localized spectral sequence.
By reference to theorem \ref{a_1inverseSS} we obtain
$E_{r(n)+1} = E_{r(n+1)}$
and the desired formulas for $d_{r(n+1)}$, completing the proof
of $P(n)$.
Thus, by the stage $E_{\infty}$ the torsionfree subalgebra
has been reduced to ${\bf Z}/p\,[a_{1}]$
and one has seen that the inductive steps
from $P(n1)$ to $P(n)$
present $E_{\infty}$
as a sum of cyclic modules over ${\bf Z}/p\,[a_{1}]$
of the required $a_1$order.
$\clubsuit$
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\end{document}