Ausoni-B\"okstedt duality for topological Hochschild homology
J.P.C.Greenlees
We consider the Gorenstein condition for topological Hochschild homology, and show that it holds remarkably often. More precisely, if $R$ is a commutative ring spectrum and $R\lra k$ is a map to a field of characteristic $p$ then, provided $k$ is small as an $R$-module, $THH(R;k)$ is Gorenstein in the sense of \cite{DGI1. In particular, this holds if $R$ is a (conventional) regular local ring with residue field $k$ of characteristic $p$. Using only B\"okstedt's calculation of $THH(k)$, this gives a non-calculational proof of dualities observed by B\"okstedt \cite{Bok and Ausoni \cite{Ausoni, Lindenstrauss-Madsen \cite{LM, Angeltweit-Rognes \cite{AR and others.
1406.2162.tex
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\begin{document}
\title{Ausoni-B\"okstedt duality for topological Hochschild homology}
\author{J.P.C.Greenlees}
\address{School of Mathematics and Statistics, Hicks Building, 
Sheffield S3 7RH. UK.}
\email{j.greenlees@sheffield.ac.uk}
\date{}

\begin{abstract}
We consider the Gorenstein condition for topological Hochschild
homology, and show that it holds remarkably often. More precisely,
if $R$ is a commutative ring spectrum and $R\lra k$ is a map to a 
field of characteristic $p$ then, provided $k$ is small as an
$R$-module,  $THH(R;k)$ is Gorenstein
in the sense of \cite{DGI1}. In particular, this holds if  $R$ is a
(conventional) regular local ring with residue field $k$ of
characteristic $p$. 

Using only B\"okstedt's calculation of 
$THH(k)$, this gives a non-calculational proof of dualities observed
by  B\"okstedt \cite{Bok} and Ausoni \cite{Ausoni},
Lindenstrauss-Madsen \cite{LM},   Angeltweit-Rognes \cite{AR}
and others.    
\end{abstract}

\maketitle

\tableofcontents

% Tame ramified. semisplit=module fg?!

\section{Introduction}

The present work was stimulated by calculations of topological
Hochschild homology by B\"okstedt \cite{Bok} and Ausoni
\cite{Ausoni}.  Given a map of commutative ring spectra $R\lra k$, we may view
$k$ as an $R$-bimodule and hence define $THH_*(R;k)$. Identifying
conventional rings with Eilenberg-MacLane ring spectra, we may take
$k$ to be a field, and the calculations gave striking examples so that 
$THH_*(R;k)$ is Gorenstein. These and other calculations are summarized in  Section \ref{sec:egs},
and the reader unfamiliar with them may wish to glance at them before
proceeding. 

The purpose of the present paper is to give a non-calculational
explanation of this duality. Most are instances of the following result
which covers many cases where there is currently no complete
calculation. The basic definitions of $THH$ are described in Section 
\ref{sec:HH}  and the Gorenstein apparatus is described in Section 
\ref{sec:Gor}. The following theorem appears below as Corollary
\ref{cor:gor}. 



\begin{thm} 
If $R$ is a connective commutative ring spectrum with a map $R\lra k$ where $k$ is
a field of characteristic $p>0$ then provided (i) $k$ is small as a
$R$-module and (ii) $R$ is Gorenstein of shift $a$ then $THH(R; k)$ 
is Gorenstein of shift $-a-3$ and has Noetherian homotopy groups. 
\end{thm}

This should be contrasted with the algebraic case (i.e., working under
$k$ rather than under the sphere spectrum). In this case, if $R$ is a
$k$-algebra, under the same hypotheses we expect $HH_*(R|k;k)$ to 
be Gorenstein a shift of $-a$ (see Appendix \ref{sec:NAR1}). 


It is essential to the argument that we are working in characteristic
$p$,  and the  only calculational input is B\"okstedt's result that  
$THH_*(\Fp)=\Fp [\mu_2]$ (this is Gorenstein of shift  $-3$, which
explains the $-3$  in the statement of the theorem). The two
technical ingredients are  (A) a cofibre sequence
conjectured to explain the Gorenstein calculations and proved by
Dundas and (B) an extension of the usual Gorenstein
ascent  theorem.% and (C) a spectral sequence for calculation. 

There is a strong precedent for calculations based on Gorenstein ascent.  Indeed
if $S\lra R \lra Q$ is a cofibre sequence (i.e., $Q\simeq
R\tensor_Sk$)\footnote{These are often called {\em fibre} sequences in
algebra because of the fact that cochains on topological fibre
sequences give examples, but this would lead to confusion in the
present context.},  the Gorenstein property often behaves well 
in the sense that if $S$ and $Q$ are Gorenstein then so is $R$. 
To illustrate its use we show how this, together with Morita
invariance,  lets us generate most Gorenstein rings from from
0-dimensional ones.  To start with,  exterior algebras $E$ are Poincar\'e
duality algebras (and hence 0-dimensional Gorenstein rings) and since any 
polynomial algebra $P$ is Morita equivalent to an exterior algebra
$E$, polynomial algebras are Gorenstein. Next, by Noether normalization 
any Noetherian $k$-algebra  $R$ is finitely generated as a module over a polynomial
subring $P$  so that the cofibre $Q=R\tensor_Pk$ is finite 
dimensional. Since $P$ is Gorenstein, $R$ is Gorenstein if and only 
if $Q$ is a Poincar\'e duality algebra, so Gorenstein rings are constructed from a
polynomial algebra and a Poincar\'e duality algebra. Altogether,
Gorenstein rings are constructed from  Poincar\'e duality algebras using
Morita equivalences and cofibre sequences.  

Duality phenomena are also ubiquitous in topology, starting with 
Poincar\'e duality and moving on to  coefficient rings of many equivariant
cohomology theories \cite{DGI1}. Once again it seems these all come
from a rather small collection of basic examples, namely the chains on
a group or the cochains on a manifold. Using Morita equivalences and
cofibre sequences one can generate a wide variety of further examples,
perhaps most notably $C^*(BG)$ for compact Lie
groups $G$ whose adjoint representation is orientable \cite{DGI1}. The present
paper shows that B\"okstedt's calculation provides a new source of
Gorenstein examples. \\[2ex]

The rest of the paper is organized as follows. Sections \ref{sec:HH}, \ref{sec:SS} and \ref{sec:Gor} provide
summaries of relevant background. Section \ref{sec:egs} gives
summaries of various calculations from the literature. Section
\ref{sec:Gorascent} proves the Gorenstein Ascent result we need, and
finally in Section \ref{sec:ABduality} we prove the main result, and
give a further discussion of examples. There are two appendices which 
describe similar results. In Appendix A we consider   THH of Thom spectra via the 
work of Blumberg-Cohen-Schlichtkrull \cite{BCS}, and in Appendix B
we consider algebraic Hochschild homology (i.e., under $k$  rather
than under the sphere spectrum) where a result of Dwyer-Miller gives
analogous duality statements. 



During the genesis of the paper, I have discussed my speculations with
many people, and I am grateful to V.Angeltveit, C.Ausoni, D.Benson,
B.Dundas,  W.G.Dwyer and A.Lindenstrauss for their patience and sharing their ideas. I
 am particularly grateful to B.Dundas for providing a proof of the
 critical conjectured cofibration described in Lemma
 \ref{lem:cofibresequence}, and allowing me to publish it here. 
I would like to thank the University of Lille for inviting me to give
lectures on duality in  2012, when these ideas started to make progress, and 
MSRI for providing an excellent environment for completing this
account. 

\section{Hochschild homology and cohomology}
\label{sec:HH}
We suppose given maps $S\lra R\lra k$ of ring spectra; as usual we
include the case of conventional rings through the use of
Eilenberg-MacLane spectra. 
% For most purposes it is sufficient to be given $S\lra R$ and an $R$-module $k$, but usually the module arises
% from a ring map $R\lra k$. 
We write $R^e=R^e_S=R\tensor_SR$, and we write
$P$ for an $R^e$-module, which we refer to as an $(R,R)$-bimodule over
$S$. Thus we may talk of the Hochschild homology spectrum
$$HH_{\bullet}(R|S;P)=R\tensor_{R^e} P$$
and the Hochschild cohomology spectrum 
$$HH^{\bullet}(R|S;P)=\Hom_{R^e}(R, P). $$
The Hochschild homology and cohomology groups are obtained by taking
homotopy groups
$$HH_*(R|S;P)=\pi_*HH_{\bullet}(R|S;P) \mbox{ and }
HH^*(R|S;P)=\pi_*HH^{\bullet}(R|S;P). $$
If $R$ and $S$ are conventional rings, $R$ is flat over $S$ and $P$ is
a conventional module, this agrees with the standard definitions in
algebra. 

In the examples of most concern to us here, 
$S=\bS$ is the sphere spectrum, and we have 
we have topological Hochschild homology $THH(R;P):=HH_{\bullet}(R|\bS ;P)$.

\section{Two spectral sequences}
\label{sec:SS}
We are going to be concerned with cofibre sequences of commutative
ring spectra, $S\lra R \lra Q$, in the sense that $Q\simeq R\tensor_S k$.
In the topological context one is used to having spectral sequences as
basic calculational tools. It is convenient to have them available
more generally. 

\subsection{The connective case}

This is the situation when the ring spectra $S, R$ and $Q$ are all connective.
The first example of this is analogous to what happens when we
have a short exact sequence of compact Lie groups $1\lra
N \lra G \lra G/N\lra 1$. We take $S=C_*(N), R=C_*(G)$ and $Q=C_*(G/N)$ and in
this case we have the homological Serre spectral sequence
$$E^2_{*,*}=H_*(G/N; H_*(N))\Rightarrow H_*(G)$$
 of the fibration $N\lra G \lra G/N$.


\begin{lemma}
\label{lem:connSS}
If  $S\lra R \lra Q$ is a cofibre sequence of connective
commutative algebras augmented over $k$ and $\pi_0(S)=k$, and $R$ is 
upward finite type as an $S$ module (for example
\cite[3.13]{DGI1}  if $\pi_n(R)$ is
finite dimensional for each $n$)
 then there is a multiplicative spectral sequence
$$E^2_{s,t}=\pi_{s}(Q)\tensor_k \pi_{t}(S)\Rightarrow
\pi_{s+t}(R),  $$
with differentials
$$d^r: E^r_{s,t}\lra E^r_{s-r,t+r-1}.$$
\end{lemma}



\begin{proof}
We construct a tower 
$$S\lra \cdots \lra S^{(n)}\lra
S^{(n-1)} \lra \cdots \lra S^{(1)}\lra S^{(0)}=k$$
of commutative rings by killing homotopy groups, where $S\lra S^{(n)}$ is an isomorphism of
$\pi_{\leq n}$ and $\pi_i(S^{(n)})=0 $ for $i>n$. Taking $R^{(n)}=R\tensor_SS^{(n)}$ we
obtain a corresponding multiplicative filtration of $R$, and consider the resulting
spectral sequence. In view of the cofibre sequence $\Sigma^t
\pi_tS\lra S^{(t)}\lra S^{(t-1)}$ of $S$-modules it is easy to write
down an exact couple, and we grade it so that  
$$D^{2}_{s,t}=\pi_{s+t}(R\tensor_S S^{(t)}) \mbox{ and }  E^{2}_{s,t}=\pi_{s+t}(R\tensor_S \Sigma^t\pi_t(S)).  $$
Now the action of $S$ on $\pi_tS$ factors through $S^{(0)}=k$, and
hence
$$R\tensor_S \Sigma^t\pi_t(S))\simeq \Sigma^t\pi_t(S)\tensor_k Q.   $$
The $d^2$ differential is induced by 
$$R\tensor_S \Sigma^t\pi_t(S)\lra R\tensor_S S^{(t)}\lra
R\tensor_S \Sigma^{t+2}\pi_{t+1}(S).    $$

The convergence of the spectral sequence is the statement that the
natural map 
$$\kappa: R=R\tensor_S S\simeq R\tensor_S [\holim_nS^{(n)}]\lra \holim_n
[R\tensor_S S^{(n)}]$$
is an equivalence. Since the $S^{(n)}$ are uniformly bounded below (by
$-1$) the result follows (as summarized in Lemma \ref{lem:limtensor} below).
\end{proof}

\subsection{The coconnective case}
This is the situation when the ring spectra $S, R$ and $Q$ are all coconnective.

One example of this arises if we start from a fibration $F\lra E
\lra B$  with $B$ simply connected and take  $S=C^*(B)$, $R=C^*(E)$
and $Q=C^*(F)$. The construction generalizes the Serre spectral 
sequence
$$E_2^{*,*}=H^*(B; H^*(F))\Rightarrow H^*(E). $$

\begin{lemma}
\label{lem:coconnSS}
If 
$S\lra R \lra Q$ is a cofibre sequence of coconnective
commutative algebras augmented over $k$ and $\pi_0(S)=k$ and $R$ is 
downward finite type as an $S$ module (for example \cite[3.14]{DGI1} if $\pi_*(S)$ is
simply connected and $\pi_n(R)$ is finite dimensional for each $n$)
 then there is a multiplicative spectral sequence
$$E_2^{s,t}=\pi_{-s}(S)\tensor_k \pi_{-t}(Q)\Rightarrow
\pi_{-s-t}(R),  $$
with differentials
$$d_r: E_r^{s,t}\lra E_r^{s-r,t+r-1}.$$
\end{lemma}

\begin{proof}
We sketch the construction. 

The filtration  arises from a tower
$$S\lra \cdots \lra S_{(n)}\lra
S_{(n-1)} \lra \cdots \lra S_{(1)}\lra S_{(0)}=k$$
of commutative rings by killing homotopy groups; since $S$ is
coconnective with $\pi_0(S)=k$, this behaves very much like the
connective case. 
% passing to quotients by successively
%larger differential ideals. 
%Thus $S_{(n)}$ is the quotient by the
%ideal  of elements of degrees $\leq -n-1$, so that 
Thus the map  $S\lra S_{(n)}$ 
is an isomorphism of $\pi_{\geq -n}$ and we have cofibre sequences of $S$-modules
$$\Sigma^{-s}\pi_{-s}S\lra S_{(s)}\lra S_{(s-1)}.$$

We then get a spectral sequence
$$D_2^{s,t}=\pi_{-s-t}(R\tensor_S S_{(s)})
\mbox{ and } E_2^{s,t}=
\pi_{-s-t}(R\tensor_S \Sigma^{-s}\pi_{-s}S)$$
The differentials then take the familiar cohomological form
$$d_r: E_r^{s,t}\lra E_r^{s+r, t-r+1}. $$


The convergence of the spectral sequence is the statement that the
natural map 
$$R=R\tensor_S S\simeq R\tensor_S \holim_nS_{(n)}\lra \holim_n
R\tensor_S S_{(n)}$$
is an equivalence, and by Lemma \ref{lem:limtensor} below this holds if $R$ is of downward finite type
over $S$. 
\end{proof}





\section{Gorenstein ring spectra}
\label{sec:Gor}
We are considering duality phenomena modelled on those in commutative
algebra of Noetherian rings, namely those associated to Gorenstein
local rings. For ring spectra there is a corresponding development, starting
by restricting the class of rings by a finiteness condition and then
the core Gorenstein condition followed by a duality statement. 
 We recall some definitions from \cite{DGI1}. 

\subsection{Finiteness conditions}


In a triangulated category if $N$ can be {\em finitely built} from $M$ using cofibre
sequences, finite sums and retracts,  we write $M\finbuilds N$; if $N$ 
can be {\em built} from $M$ using cofibre sequences and arbitrary sums we 
write $M\builds N$.

We consider a map $R\lra k$ of rings. The terminology comes from
 the special case when $R$ is a commutative local ring with residue 
field $k$.  The first requirement is a finiteness condition, which
plays the role of the Noetherian condition from classical commutative
algebra. The Auslander-Buchsbaum-Serre theorem in commutative algebra
states that if $R$ is a Noetherian local ring, $k$ is small if and only if $R$ is
regular. This is far too strong a condition for us to assume,  but
there is a much weaker and more practical one in the same
vein. Indeed, for commutative Noetherian local rings,  we can always form the Koszul
complex $K$ associated to a finite set of generators of the maximal
ideal; this has the properties (i) $K$ is small ($R\finbuilds K$) (ii) $K$ is finitely
built from $k$ ($k\finbuilds K$) and (iii) $k$ is built from $K$
($K\builds k$). In the context of more general ring objects, 
we say $R$ is {\em proxy-regular} if there is an
$R$-module $K$ so that (i), (ii) and (iii) hold, and we think of this
as a finiteness condition playing a similar role to that of being Noetherian. 


\subsection{The Gorenstein condition}
We now say that   $S\lra k$ is {\em  Gorenstein} of shift
$a$ (and write $\shift (S)=\shift (S|k)=a$) if we have an equivalence 
$$\Hom_S(k, S)\simeq \Sigma^a k $$
of $R$-modules. More generally, we say that $S\lra R$ is 
{\em relatively Gorenstein of shift $a$} (and write $\shift (S|R)=a$) if 
$$\Hom_S(R, S)\simeq \Sigma^a R. $$

Just as in the classical case, we are interested in proxy-regular
rings which satisfy the Gorenstein condition. 




\subsection{Gorenstein duality}
Although the Gorenstein condition itself is convenient to work with, the real reason for
considering it is the duality property that it implies. 

In classical local commutative algebra the Gorenstein duality property is
that all local cohomology is in a single cohomological degree, where it is the
injective hull $I(k)$  of the residue
field. To give a formula, we write $\Gamma_{\fm}M$ for the $\fm$-power torsion in an
$R$-module $M$, and $H^*_{\fm}(M)$ for the local cohomology of
$M$, recalling Grothendieck's theorem that if $R$ is Noetherian,
$H^*_{\fm}(M)=R^*\Gamma_{\fm}(M)$. 
The Gorenstein duality statement  for a local ring of Krull dimension $r$ therefore states
$$H^*_{\frak{m}}(R)=H^r_{\frak{m}}(R)=I(k). $$
If $R$ is a $k$-algebra, $I(k)=R^{\vee}=\Gamma_{\fm} \Hom_k(R,k)$.

Turning to the general case, if $R$ is a $k$-algebra, we may again 
define  $R^{\vee}=\cell_k(\Hom_k(R, k))$ and observe this has the Matlis lifting property
$$\Hom_R(T, R^{\vee})\simeq \Hom_k(T,k) $$
for any $T$ built from $k$. 

In particular, if $R$ is Gorenstein of shift $a$ we have eqivalences
of $R$-modules
$$\Hom_R(k, \cell_kR)\simeq \Hom_R(k,R)\stackrel{(g)}\simeq \Sigma^a
k\stackrel{(m)}\simeq \Hom_R(k, \Sigma^a R^{\vee}),  $$
where the equivalence (g) is the Gorenstein property and the
equivalence (m) is the Matlis lifting property. 
We would like to remove the $\Hom_R(k, \cdot)$ to deduce
$$\cell_kR\simeq \Sigma^a R^{\vee}. $$
Such an equivalence is known as {\em Gorenstein duality},  since
$\cell_k(R)$  is a covariant functor of $R$ and $R^{\vee}$ is a contravariant functor of
$R$. 

Morita theory \cite{DGI1} says that if $R$ is proxy-regular we may make this
deduction provided $R$ is orientably Gorenstein in the sense that the right
actions of $\cE=\Hom_R(k,k)$  on $\Sigma^ak$ implied by the two
equivalences (g) and (m) agree. This is automatic when the ring
spectrum is both a $k$-algebra and connected. 


\begin{prop}
Suppose $R$ is a proxy-regular, connected $k$-algebra
and $\pi_*(R)$ is Noetherian with $\pi_0(R)=k$ and 
maximal ideal $\fm$ of positive degree elements.  If $R$ is 
Gorenstein of shift $a$, then $R$ it is automatically orientable and so 
has Gorenstein duality.  Accordingly there is a local cohomology spectral sequence
$$H^*_{\fm}(R_*)\Rightarrow \Sigma^a R_*^{\vee}. $$
\end{prop}




\begin{proof} 
First we argue that if $R$ is Gorenstein, it is automatically
orientable. Indeed, we show that $\cE$ has a
unique action on $k$. Since $R$ is a $k$-algebra, the
action of $\cE$ on $k$ factors through 
$$\cE=\Hom_R(k,k) \lra \Hom_k(k,k)=k,$$
so since $k$ is an Eilenberg-MacLane spectrum,  
the action is through $\pi_0(\cE)$. Now we observe that since $R$ is connected, 
$\Ext_{R_*}^s(k,k)$ is in degrees $\leq -s$, so that the spectral
sequence for calculating $\pi_*(\Hom_R(k,k))$ shows $\cE$ is
coconnective with $\pi_0(\cE)=k$ which must act trivially on $k$. 
\end{proof}

We note that if the coefficient ring $\pi_*(R)$ is Gorenstein and $R$ is
connective then $R$ is Gorenstein. Indeed,  the spectral sequence 
$$\Ext_{R_*}^{*,*}(k,R_*)\Rightarrow \pi_*(\Hom_R(k,R)) $$ 
collapses, to show $\pi_*(\Hom_R(k,R) =\Sigma^a k$ for some $a$.  The $R$-module $k$ 
is characterised by its homotopy, so $\Hom_R(k,R)\simeq \Sigma^a
k$.  Conversely, if $R$ is Gorenstein, this shows that the ring
$\pi_*(R)$ has very special properties (even if it falls short of
being Gorenstein). 

\begin{cor} \cite{ringlct}
Suppose $R$ has Gorenstein duality of shift $a$, that $\pi_*(R)$ is
Noetherian of Krull dimension $r$ and Hilbert series $p(t)=\sum_s\dim_k(R_s)t^s$.

If $\pi_*(R) $ is Cohen-Macaulay it is also Gorenstein, and the
Hilbert series satisfies 
$$p(1/t)=(-1)^rt^{r-a}p(t).$$

If $\pi_*(R) $ is almost Cohen-Macaulay it is also almost Gorenstein, and the
Hilbert series satisfies 
$$p(1/t)-(-1)^rt^{r-a}p(t)=(-1)^{r-1}(1+t)q(t) \mbox{ and } q(1/t)=(-1)^{r-1}t^{a-r+1}q(t).$$ 

In any case $\pi_*(R)$ is Gorenstein in codimension 0 and almost
Gorenstein in codimension 1. 
\end{cor}
 

\subsection{The relatively Gorenstein case}

We make the  elementary observation that for any ring map $\theta : S\lra R$ 
$$\Hom_R(k,\Hom_S(R,S))\simeq \Hom_R(R\tensor_S k, S)\simeq 
\Hom_S(k,S).$$
Thus we conclude that if $S\lra R$ is relatively Gorenstein then $R$ is
Gorenstein  if and only if  $S$ is Gorenstein, and in that case 
$$\shift (S|k)=\shift (R|k)+\shift (R|S).$$

\begin{example}
\label{eg:relGor}
(i) The ring map $S=ko\lra ku=R$ is relatively Gorenstein of shift
2. Indeed, the connective version of Wood's theorem states $ku\simeq ko\sm
(S^0\cup_{\eta} e^2)$, so 
$$\Hom_{ko}(ku, ko)\simeq \Sigma^{-2} ku. $$
Since $ku_*=\Z [v]$ we see that $ku$  is Gorenstein of shift $-4$ over
$\Ftwo$,  and it follows that 
$ko$ is Gorenstein of shift $-6$ over $\Ftwo$. 

(ii) Precisely similar statements hold for $tmf$. This uses results of
Hopkins-Mahowald \cite{HM}, now written up by Matthew \cite{AM}. 

At the prime 3, there is a map $tmf \lra tmf_0(2)$. Here $tmf_0(2)\simeq tmf \sm (S^0\cup_{\alpha_1} e^{4}
\cup_{\alpha_1} e^{8})$, so that 
$$\Hom_{tmf}(tmf_0(2),tmf )\simeq \Sigma^{-8} tmf_0(2). $$
Since $tmf_0(2)_*=\Z_{(3)}[c_2, c_4] $ (where $|c_i|=2i$) we see that $tmf_0(2)$ is Gorenstein of shift
$-15$.  Hence we deduce by Gorenstein descent  that $tmf\lra \mathbb{F}_3 $ is
Gorenstein of shift $-23$. 

At the prime 2, there is a map $tmf \lra tmf_1(3)$. Here $tmf_1(3)$ is
a form of $BP\langle 2\rangle$ and  $tmf_1(3)\simeq tmf \sm DA(1)$ so that 
$$\Hom_{tmf}(tmf_1(3),tmf )\simeq \Sigma^{-12} tmf_0(2). $$
Since $tmf_1(3)_*= \Z_{(2)}[\alpha_1, \alpha_3]$ (where $|\alpha_i|=2i$) we see that $tmf_1(3)$ is Gorenstein of shift
$-11$.  Hence we deduce by Gorenstein descent  that $tmf \lra \Ftwo$ is
Gorenstein of shift $-23$. 
\end{example}

In general, it can be difficult to decide if $S\lra R$ is relatively Gorenstein,
and we prefer to give conditions depending on $Q$. 
%This is called {\em Gorenstein descent} since it gives conditions
%under which the property of being Gorenstein descends from $R$ to $S$.

\subsection{Gorenstein Ascent}
In effect the Gorenstein Ascent theorem will state that under suitable
hypotheses  (see Section \ref{sec:Gorascent}) there is an equivalence
$$\Hom_R(k,R)\simeq \Hom_{Q}(k, \Hom_S(k,S)\tensor_k Q). $$
When this holds, it follows that if $S$ and $Q$ are Gorenstein, so is
$R$ and 
$$\shift (R)=\shift (S)+\shift (Q).$$

\subsection{Arithmetic of shifts}
We summarize the behaviour of Gorenstein shifts  in the ideal
situation when ascent and descent both hold. If all rings and maps are Gorenstein of the indicated
shifts
$$\stackrel{a} S \stackrel{\lambda}\lra 
\stackrel{b} R \stackrel{\mu}\lra 
\stackrel{c} Q$$
then $b=a+c, \lambda =-c$ and $\mu=a$


\section{Some examples}
\label{sec:egs}
The paper is motivated by several calculations when $S=\bbS$ is the sphere
spectrum. 

\begin{example} {\bf (The map $R=\Fp \lra \Fp=k$.)}
We consider the homotopy of  $THH(\Fp)$.  
 B\"okstedt \cite{Bok} has calculated  $THH_*(\Fp )=\Fp  [\mu_{2}]$. 
Tautologically, $\Fp$ is small over $\Fp $ and it is immediate that 
$\Fp \lra \Fp$  is Gorenstein of shift 0. The Gorenstein shift of $THH_*(\Fp)$ is 
$-3$. 
\end{example}


\begin{example} {\bf (The map $R=\Z \lra \Fp=k$.)}
We consider the  mod $p$ homotopy of $THH(\Z)$. B\"okstedt \cite{Bok}
has calculated $THH_*(\Z;\Fp)=\Fp [\mu_{2p}]\tensor \Lambda_{\Fp} (\lambda_{2p-1})$. 
It is immediate that $\Fp$ is small over $\Z$ and easy to check that $\Z\lra \Fp$ 
is Gorenstein of shift $-1$. The shift of $THH_*(\Z; \Fp)$ is 
$(-2p-1)+(2p-1)=-2$. The ring but not the shift depends on $p$.


%But if the we are using the Dwyer-Miller duality with shift  $-(2p-1)$ (which 
%is not what would come out from the given argument) the shift in
%the Gorenstein condition is $(-2p-1)-(2p-1)=-4p$. 

The calculations of Lindenstrauss and Madsen \cite[4.4]{LM} have a
similar pattern. Indeed, if $\cO$ is a ring of integers in a number field, which is either
unramified or wildly ramified, $THH_*(\cO , \cO /p)$ is polynomial tensor
exterior (over $\cO /p$) on generators differing in degree by 1. In the
tamely ramified case  the ring $THH_*(\cO, \cO /p)$ is more complicated, but it is
still Gorenstein of shift $-2$. 
\end{example}

\begin{example} {\bf (The map $R=lu \lra \Fp=k$.)}
We consider mod $v_1,p$ homotopy of $THH (lu)$ where $lu$ is the Adams
summand of $p$-local connective $K$-theory with coefficients
$lu_*=\Z_{(p)}[v_1]$.  McClure-Staffeldt \cite{MS} (see also
Ausoni-Rognes \cite{AR}) have calculated
  $THH_*(lu;\Fp)=\Fp [\mu_{2p^2}]\tensor \Lambda_{\Fp}
 (\lambda_{2p-1}, \lambda_{2p^2-1})$. 
It is easy to check that $\Fp$ is small over $lu$ and that $lu\lra \Fp$ 
is Gorenstein of shift $-(2p-2)-1-1=-2p$. The shift of $THH_*(lu; \Fp)$ is 
$(-2p^2-1)+(2p-1+2p^2-1)=2p-3$. 


%I'm not even sure what the Dwyer-Miller shift would be, but probably $-(2p-1)-(2p^2-1)$. 
%This would give a Gorenstein shift of 
%$(-2p^2-1)-(2p-1+2p^2-1)=-4p^2-2p+2$. 
\end{example}


\begin{example} {\bf (The map $R=ku \lra ku/(p,v_1)=k$.)}
For primes $p>2$, Ausoni calculates the mod $p,v_1$ homotopy of $THH(ku)$ 
and shows that 
$THH_*(ku;ku/(p,v_1))= \Lambda (\lambda_{2p-1})\tensor
\Fp[\mu_{2p^2}]\tensor Q, $ where $Q$ is Poincar\'e duality algebra of 
formal dimension $2p^2-1$ \cite[9.15]{Ausoni}. Making due allowance
for the fact that $\pi_*(ku/(p,v_1))=\Fp
[v]/(v^{p-1})$ is not a field,  we may make similar calculations. 
It is easy to see that $ku/(p,v_1)$ is small over $ku$ and $ku\lra ku/(p,v_1)$ 
is Gorenstein of shift $-2p$. The shift of $THH_*(ku; ku/(p,v_1))$
over $\Fp$ is  $(-2p^2-1)+(2p-1)+(2p^2-1)=2p-3$. 
\end{example}

\begin{example} {\bf (The map $R=ko \lra H\Ftwo=k$.)}
Angeltveit and Rognes \cite{AR} show that 
$THH_*(ko;\Ftwo )= \Lambda (\lambda_{5}, \lambda_7 )\tensor
\Ftwo [\mu_8]. $ This is Gorenstein of shift $5+7-8-1=3$. 
\end{example}





\section{Gorenstein ascent}
\label{sec:Gorascent}

We have begun to see the value of understanding the behaviour of the
Gorenstein condition in cofibre sequences, and we turn to a more
systematic discussion.

 We suppose that $S\lra R \lra Q$ is a cofibre sequence of commutative
algebras with a map to $k$, and we now consider the Gorenstein ascent
question.  When does the fact that $S$ is Gorenstein imply that $R$ is
Gorenstein? It is natural to assume that $Q$ is Gorenstein, but it is
known this is not generally sufficient. We identify a number of
circumstances in which it is sufficient, and  in characteristic $p$ we
give a useful general result. Before we do this, we look at the finiteness conditions. 

\subsection{Proxy-regularity}
We provide a tool for proxy-regular ascent. It seems that some
hypothesis is necessary and the form given here applies in cases of
interest here. 

First, we should introduce notation for the standard Koszul complex
associated to a sequence $r_1, r_2, \ldots , r_n$ of elements of
elements of  $\pi_*R$. For $x \in \pi_*(R)$ we define $K(R;x)$ by 
the cofibre sequence
$$R\stackrel{x} \lra R \lra K(R;x), $$
and now we take
$$K(R;r_1, \ldots , r_n)=K(R;r_1)\tensor_R \cdots \tensor_R K(R;r_n). $$
In the usual way, a concrete realization requires the choice of
specific cocycle representatives, but the homotopy type does not
depend on these choices. If $R$ is a classical ring, this gives the
standard construction  $K(R;x)=[R\stackrel{x}\lra R]$ with the copies
of $R$ in degrees 0 and 1. 

 
\begin{lemma}
\label{lem:proxyascent}
Suppose that $S$ is proxy-regular with Koszul complex $K_S$ and that $Q$ has a Koszul complex of
the special form $K_Q=K(Q; q_1, \ldots , q_n)$ where $q_i\in \pi_*(Q)$ lifts to $r_i
\in \pi_*(R)$ for $i=1, \ldots , n$. Then $R$ is proxy-regular with Koszul complex
$K_R:=K_S\tensor_S K(R;r_1, \ldots, r_n)$. 
\end{lemma}
\begin{proof} There are three things to prove.

 Since $S\finbuilds K_S$ and $R\finbuilds K(R;r_1, \ldots, r_n)$, it
follows that 
$$R=S\tensor_SR\finbuilds K_S\tensor_S K(R;r_1, \ldots, r_n)=K_R.$$ 

 Since $Q\simeq k\tensor_SR$, we find firstly 
$$k\finbuilds K_Q= k\tensor_S K(R;r_1, \ldots , r_n)\finbuilds
K_S\tensor_S K(R;r_1, \ldots , r_n)=K_R$$
and secondly 
$$K_R=K_S\tensor_SK(R; r_1, \ldots , r_n) \builds k\tensor_SK(R; r_1, \ldots
, r_n) =K_Q\builds k .$$
This completes the proof. 
\end{proof}

\subsection{Good approximation implies ascent}

The core of our results about ascent come  from \cite{DGI1}. Indeed, 
the proof of \cite[8.6]{DGI1} gives a sufficient condition for Gorenstein
ascent in the commutative context. 

\begin{lemma}
\label{lem:Gorascent}
If $S$ and $R$ are commutative and the natural map $\nu:
\Hom_S(k,S)\tensor_S R\lra  \Hom_S(k,R)$ 
is an equivalence then 
$$\Hom_R(k,R)\simeq \Hom_{Q}(k, Hom_S(k,S)\tensor_k Q). $$
In this case, if  $S$ and $Q$
are Gorenstein, so is $R$, and the shifts add up: $\shift (R)=\shift
(S)+\shift (Q)$. \qqed
\end{lemma}


Now that we have a sufficient condition for Gorenstein ascent, we want
to identify cases in which  $\nu$ is an equivalence. 
The most familiar case is when $R$ is small over $S$ (or
equivalently, when $Q$ is finitely built from $k$). 
We emphasize that the hypothesis on $\nu$ in Lemma \ref{lem:Gorascent}
only depends on $R$ as a {\em module} over $S$, and we will obtain a
useful generalization by approximating $R$ by $S$-modules for which 
$\nu$ is an equivalence. The approximation will be as an inverse
limit, and to see the approximation is accurate we need to impose
hypotheses to ensure inverse limits and tensor products commute. 

\begin{lemma}
\label{lem:limtensor}
Suppose $M$ and $N$ are $S$-modules and $N\simeq \holim_n N_n$, and
consider the natural map 
$$\kappa: M\tensor_S [\holim_n N_n]\lra \holim_n[M\tensor_S N_n].$$

The map $\kappa $ is an equivalence in either of the following
circumstances
\begin{itemize}
\item $S$ is connective, $M$ is of upward finite type and the modules
$N_n$ are uniformly bounded below. The hypothesis on $M$ holds if
$\pi_0(S)=k$, 
$\pi_*(M)$ is bounded below and $\pi_n(M)$ is finite dimensional over
$k$ for all $n$.
\item $S$ is coconnective, $M$ is of downward finite type and the modules
$N_n$ are uniformly bounded above. The hypothesis on $M$ holds if
$\pi_0(S)=k$, $\pi_{-1}(S)=0$,  $\pi_*(M)$ is bounded above and $\pi_n(M)$ is finite dimensional over
$k$ for all $n$.
\end{itemize}
\end{lemma}

\begin{proof}
In the first part the fact that $M$ is of upward finite type and the $N_n$
are uniformly bounded below is enough to see that the limit is achieved
in each degree.  It is proved as \cite[3.13]{DGI1} that the homotopy
level condition ensures $M$ is of upward finite type. 

The proof of the second part is precisely similar, with a reference to
\cite[3.14]{DGI1}. 
\end{proof}



\begin{lemma}
\label{lem:Gorascentlim}
Suppose that $\pi_*(S)$ is  Noetherian and that 
 $\pi_*(\Hom_S(k,S))$ is a finitely generated module over $\pi_*(S)$
 and that $R\simeq \ilim_n R_n$ for small $S$-modules $R_n$.
The hypothesis  of Lemma \ref{lem:Gorascent} applies in either of the following
circumstances
\begin{itemize}
\item $S$ is connected and the $R_n$ are uniformly bounded below
\item $S$ is simply coconnected and the $R_n$ are uniformly bounded above
\end{itemize}
In this case, 
$$\Hom_R(k,R)\simeq \Hom_{Q}(k, Hom_S(k,S)\tensor_k Q) $$
and Gorenstein ascent holds for the cofibre sequence $S\lra R\lra Q$.    
\end{lemma} 

\begin{proof}
First, 
$$\Hom_S(k, R) \simeq \Hom_S(k, \holim_n R_n) \simeq \holim_n
\Hom_S(k,  R_n).  $$ 
Now, since $R_n$ is a small $S$-module,  $\Hom_S(k,R_n)\simeq \Hom_S(k,S)\tensor_S R_n$.

It therefore remains to show that the natural map 
$$\kappa: M\tensor_S [\holim_n R_n] \lra \holim_n[M\tensor_S R_n] $$
is an equivalence when $M=\Hom_S(k,S)$ so the conclusion follows from 
Lemma \ref{lem:limtensor}. 
\end{proof}


\subsection{Building good approximations}
We give criteria under which  $R$ may be approximated in this
way. First, we assume that $R$ is a $k$-algebra, and  it is convenient to
introduce some further terminology.

\begin{defn}
We say that a map $R \lra Q$ is {\em \pifinite\ } if $\pi_*(Q)$ is
finitely generated as a module over  $k [x_1, \ldots , x_n]$ for some
finite set of elements $x_1, \ldots, x_n$ of $ \pi_*(R)$. A cofibration sequence $S\lra R
\lra Q$ is {\em \pifinite } if the map $R\lra Q$ is \pifinite .
\end{defn}

\begin{remark} 
 If
$\pi_*(R)$ is Noetherian, it is equivalent to ask that $\pi_*(Q)$ is
finitely generated over $\pi_*(R)$. 
%The purpose of the $\pi_*$-finiteness condition is to ensure that 
%every element of 
%$\pi_*(Q)$ has a power which is in the image of  $\pi_*(R)$. 
\end{remark}

\begin{prop}
\label{prop:GorascentNoeth}
Suppose that $\pi_*(S)$ is  Noetherian,  $\pi_*(R)$ and 
$\pi_*(\Hom_S(k,S))$ are  finitely generated $\pi_*(S)$-modules, $R$
is a $k$-algebra and the cofibration is \pifinite , and suppose that either (i) $S, R$ and
$Q$ are all  connected or (ii) that $S, R$ and $Q$ are all  coconnected and $S$ is
simply  coconnected.  Under these conditions, 
$$\Hom_R(k,R)\simeq \Hom_{Q}(k, Hom_S(k,S)\tensor_k Q) $$
and Gorenstein ascent holds for the cofibre sequence $S\lra R\lra Q$.    
\end{prop}


\begin{proof} From the \pifinite\ hypothesis, by the Noether
  normalization argument,   there is a polynomial subring $R(1)_*$ of $\pi_*(R)$ over which $\pi_*(Q)$ is finitely
  generated.
%Since the cofibration is semi-split,  suitable powers of the polynomial
%  generators lie in the image of $\pi_*(R)\lra \pi_*(Q)$ and generate
% a second polynomial ring $Q(1)_*$ over which $\pi_*Q$ is finitely 
%generated.
 Now  let $R(2)_*, R(3)_*, \ldots$ be the subrings
generated by the $2$nd, $4$th, $8$th .... powers of generators of $R(1)_*$. 


 Next we construct  (non-commutative) $k$-algebra spectra $R(n)$ with 
$\pi_*R(n)=R(n)_*$.    For a polynomial ring on a single generator of
degree $d$, we
can consider the James construction $J_k(S^d)$ on a sphere over
$k$. This is the free associative $k$-algebra spectrum on the $d$-sphere
and has homotopy $k[X_d]$.  If $R(n)_*=k[x_1, \cdots, x_s]$, we form 
$$J_k(S^{d_1})\tensor_k \cdots \tensor_k J_k(S^{d_s})$$
where $x_i$ is of degree $d_i$.  If $A$ is a commutative $k$-algebra  then we
may construct a ring map taking $X_i$ to $x_i$ as the composite
$$J_k(S^{d_1})\tensor_k \cdots \tensor_k J_k(S^{d_s})
\lra A\tensor_k\cdots\tensor_k A\lra A. $$
The first map takes $X_i$ to $x_i$ in the $i$th factor, and 
 is a map of associative rings. The second map is multiplication in
 $A$, and this is a ring map since $A$ is commutative. 

Using these ring spectra $R(n)$,  using tensor products of the single
variable case as above, we may construct maps
$$
\diagram
\ldots  \rto &R(2) \rto &R(1) \rto&R\rto &Q
\enddiagram$$
realizing the algebras we took  in homotopy. 
Now by construction $Q_n=Q\tensor_{R(n)}k$  is finitely built from $k$
since its homotopy is a finite dimensional $k$-vector space. Since the 
polynomial generators were in increasingly large degrees, 
 $Q=\holim_n Q_n$.  Similarly, if we write $R_n=R\tensor_{R(n)}k$ 
we have $R\simeq \holim_n R_n$, and  $k\tensor_SR_n\simeq 
k\tensor_S R\tensor_{R(n)} k\simeq
Q\tensor_{R(n)}k=Q_n$. Thus we have 
sequences $S\lra R_n \lra Q_n$ and $R_n$ is small as an $S$-module. 

The result now follows from Lemma \ref{lem:Gorascentlim}. 
\end{proof}

To apply this, we first note that  in characteristic $p$
 the \pifinite\  condition is automatic when $\pi_*(S)$ is in a finite
range of degrees. 

\begin{lemma}
\label{lem:charppifinite}
If $S\lra R\lra Q$ is a cofibre sequence of $k$-algebras, where $k$ is
a field of characteristic $p>0$  and $\pi_*(S)$ is Noetherian
and in a finite range of degrees, then the cofibre sequence is
\pifinite .
\end{lemma}

\begin{proof}
Using the spectral sequence of Lemma \ref{lem:connSS} or Lemma
\ref{lem:coconnSS} as appropriate we see that if $x\in \pi_*(Q)$ survives to the $r$th
page then $d_r(x^p)=0$, so that $x^p$ survives to the $(r+1)$st page. 
If $\pi_*(S)$ is in a finite range of degrees, the spectral sequence
collapses at the $N$th stage for some $N$ and the $p^{N-1}$th powers
of all elements survive, and therefore lie in the image of
$\pi_*(R)\lra \pi_*(Q)$ so that the cofibration is \pifinite . 
\end{proof} 

We may now apply  Proposition \ref{prop:GorascentNoeth} to give a
useful characteristic $p$ Gorenstein ascent theorem. 

\begin{cor}
\label{cor:GorascentNoeth}
Consider a  cofibre sequence $S\lra R\lra Q$ of $k$-algebras. 
Suppose that $\pi_*(S)$  Noetherian and either connected or simply
coconnected and $k$ is a field of of characteristic $p>0$. If $\pi_*(R)$ and 
$\pi_*(\Hom_S(k,S))$ are  finitely generated $\pi_*(S)$-modules, and 
$\pi_*(S)$ is concentrated in a finite range of degrees then 
$$\Hom_R(k,R)\simeq \Hom_{Q}(k, Hom_S(k,S)\tensor_k Q) $$
and Gorenstein ascent holds for the cofibre sequence.  
\end{cor}


\section{Ausoni-B\"okstedt duality. }
\label{sec:ABduality}
We now have the necessary ingredients to state and prove our duality
result. The idea is that if we are given  maps $\Ca\lra \Ba \lra k$ of
commutative ring spectra with the cofibre ring spectrum $A=\Ba \tensor_\Ca k$
  Gorenstein then (at least under some hypotheses on $k$ and $A$)
 if  $\TbB= THH(\Ba;k)$ is  Gorenstein then $\TbC=THH(\Ca;k)$ is also
 Gorenstein. Since we are deducing the domain $\TbC$  is Gorenstein from
  the fact that  $\TbB$  is Gorenstein,  we think of this as a {\em descent} theorem, even
  though the principal ingredient is an {\em ascent} theorem for a suitable
  cofibration. 

\subsection{Gorenstein descent for $THH$}

The key to method is the existence of a suitable cofibration sequence
conjectured on the basis of the examples and proved by Dundas.

\begin{lemma} (Dundas)
\label{lem:cofibresequence}
 Given a cofibre sequence 
$\Ca\lra \Ba \lra A$  of commutative ring spectra over $k$ (i.e., $\Ba$
has a map to $k$ and $A=\Ba\tensor_{\Ca} k$)
there is a cofibre sequence of commutative $k$-algebra spectra 
  $$A \lra \TbC\lra \TbB$$
where $\TbC=\Ca\tensor_{\Ca^e} k=THH(\Ca;k)$ and $\TbB=\Ba \tensor_{\Ba^e} k
=THH(\Ba; k)$.
\end{lemma}
\begin{remark}
Dundas's lemma makes $THH( \cdot ; k)$ remarkably computable. For
example, Lindenstrauss points out that if $R$ is $k$-algebra, we may apply the 
Dundas Lemma to the cofibre sequence $k\lra R \lra R$ to deduce
$$THH_*(R;k)\cong THH_*(k)\tensor \mathrm{Tor}_*^R(k,k) .$$
In particular, this allows one to deduce from B\"okstedt's calculation
that  $THH_*(k)=k[\mu_2]$ for any field $k$ of characteristic $p$. 
\end{remark}

\begin{proof}
For a $\Ca$ bimodule $M$, according to the original
definition,  the topological Hochschild
homology  $THH(\Ca;M)$ is a realization of the Hochschild simplicial
spectrum  with $n$th term $M \tensor_{\bS} \Ca^{\tensor  n}$,
where the tensor power is for $\tensor_{\bS}$;
this is natural for maps of rings and of bimodules. In particular, the
map $\Ca\lra \Ba$ of ring spectra vertically and the map 
$\Ba \tensor_{\bS} k \lra k$ of $\Ba$-bimodules horizontally give a commutative square
$$\begin{array}{ccc}
\Ba \tensor_{\bS} k \tensor_{\bS}
\Ca^{\tensor n}&\rightarrow&k\tensor_{\bS} \Ca^{\tensor n}\\
\downarrow && \downarrow\\
\Ba \tensor_{\bS} k \tensor_{\bS}
\Ba^{\tensor n}&\rightarrow&k\tensor_{\bS} \Ba^{\tensor n}
\end{array}$$
which is evidently a pushout square of commutative ring spectra. 
 Taking geometric realizations we obtain  the pushout square
$$\begin{array}{ccc}
THH (\Ca; \Ba\tensor_{\bS}k )&\rightarrow&THH(\Ca; k)\\
\downarrow && \downarrow\\
THH (\Ba; \Ba\tensor_{\bS}k )&\rightarrow&THH(\Ba; k).
\end{array}$$
Now we use the fact that $THH$ with coefficients in a  bimodule of the
form $M=X\tensor_{\bS}Y$  simplifies:
$$THH(\Ca; X\tensor_{\bS}Y)=Y\tensor_\Ca X. $$
The pushout square now gives the required result. 
%Algebra spectra augmented over $k$ are tensored over based
%spaces \cite{}. Thus we have a cofibre sequence
%$$\Ca\lra \Ba\lra A\lra \Sigma \Ca \lra \Sigma \Ba,  $$ 
%where $\Sigma$ denotes tensoring with the based space $S^1$. It
%remains to observe that $S^1$ is the pushout of $*\lla \{ 0,1\}\lra
%[0,1]$ so that $\Sigma \Ca=\Ca\tensor_{\Ca^e}k$. 
\end{proof}

We now want to take the cofibre sequence $A\lra \TbC \lra \TbB$ and deduce
that when $A$ and $\TbB$ are Gorenstein, so is $\TbC$. We need only verify 
that the hypotheses of Lemma \ref{lem:proxyascent} and
Corollary \ref{cor:GorascentNoeth} are satisfied.

\begin{thm} {\em (Gorenstein descent for THH)}
\label{thm:THHGordescent}
Suppose $\Ca\lra \Ba\lra A$ is a cofibre sequence of connective ring spectra
with maps to $k$ and that 
\begin{enumerate} 
\item $A$ and $\TbB$ are proxy-regular and  Gorenstein 
\item $\pi_*(A)$ and $\pi_*(\TbB)$ are Noetherian
\item the map $\TbC \lra \TbB$ is \pifinite\
\end{enumerate} 
then $\TbC$ is proxy-regular and  Gorenstein with 
$$\shift (\TbC)=\shift (\TbB)+\shift (A).$$
\end{thm}

\begin{proof}
We consider the cofibration $A \lra \TbC \lra \TbB$ of Lemma
\ref{lem:cofibresequence}. 

%First, we observe that  $\pi_*(\TbB)$ is a connective Noetherian
%$k$-algebra. 

By the \pifinite\ hypothesis we may choose a finite number of elements of  $\pi_*(\TbC)$ so
that $\pi_*(\TbB)$ is finitely generated over the $k$-algebra they
generate, and we may form a Koszul complex by using these
generators. This verifies the hypotheses
necessary to see that $\TbC$ is proxy-regular by Lemma
\ref{lem:proxyascent}. 

The hypotheses for Gorenstein ascent from $A$ to $\TbC$ are stated explicitly. 
\end{proof}

\begin{cor}
\label{cor:gor}
If $\Ca$ is Gorenstein of shift $a$ and augmented over a field $k$ of
characteristic $p$ and if
$\Ca$ is regular  then $THH(\Ca;k) \lra k$ is 
Gorenstein  of shift $-a-3$. 
\end{cor}

\begin{proof}
We apply the theorem to the cofibre sequence $\Ca\lra k \lra A$. 
Since $\Ca$ is regular, $\pi_*A$ is finite dimensional and hence Noetherian. 
Now note that $k$ is Gorenstein of shift 0, so by Gorenstein Ascent
for $\Ca\lra k \lra A$, the ring $A$ is Gorenstein and we have $\shift
(A)=-a$.

From B\"okstedt's calculation $\TbB = THH(k)$ has Noetherian homotopy $k[\mu_2]$, so it is
Gorenstein of shift $-3$.  Now observe that by Lemma
\ref{lem:charppifinite},  the cofibration
$A\lra \TbC\lra \TbB$ is \pifinite . 
%Indeed,  since we are in
%characteristic $p$ and the differentials are derivations, $p$th powers of generators on
%any page survive to the next page. Since $\pi_*A$ is finite dimensional, the
% spectral sequence collapses at a finite stage. Accordingly there
% exists an $n$ so that the  $p^n$th power of generators in
%$\pi_*(\TbB)$ lift to $\pi_*(\TbC)$. 
Thus the hypotheses of Theorem \ref{thm:THHGordescent} are satisfied, and we may  apply
 Gorenstein Ascent  to the cofibration $A\lra \TbC \lra \TbB$ to obtain the conclusion. 

\end{proof}


\subsection{Examples revisited}
We observe that Theorem \ref{thm:THHGordescent} gives  a
non-calculational proof of several of the dualities we
observed in Section \ref{sec:egs}  above. In each case we specify $\Ca\lra \Ba$ and $k$ and then
discuss the resulting cofibre sequence $A\lra THH(\Ca;k)\lra
THH(\Ba;k)$. 
%both B\"okstedt's
%duality for $THH_*(\Z; \Z/p)$ and Ausoni's duality for $THH_*(lu;
%\Z/p)$. 

\begin{example}
 If we take $\Ca\lra \Ba$ to be $\Z\lra \Fp$ and $k=\Fp$,  we find $A
 \sim C_*(S^1)$ (where $\sim$ means that the coefficient rings are
 isomorphic). Since $THH(\Fp)$ is Gorenstein  of shift $-3$, 
Corollary \ref{cor:gor} gives  another proof of B\"okstedt duality: $THH(\Z;
\Fp)$ is Gorenstein of shift   $1+(-3)=-2$. 

The spectral sequence \ref{lem:connSS} gives an alternative approach to the
calculational proof. The only necessary input would be to know that the differential
$d_2(\mu_2)\neq 0$. This then shows that $d_2(\mu_2^n)\neq 0$ unless
$n$ is a multiple of $p$, so that the $E_3=E_{\infty}$ term is
generated by $\mu_2^p$ (giving $\mu_{2p}$) and $\mu_2^{p-1}\tau$ 
(giving $\lambda_{2p-1}$). 

The Lindenstrauss-Madsen example in the unramified case can be treated
in the same way, since $\cO /p=k$. However, the ramified case is not
covered by our analysis since $THH_*(\cO /p)$ is not Noetherian. 
\end{example}

\begin{example}
If we take $\Ca\lra \Ba$ to be $lu\lra \Z$ and $k=\Fp$ we find $A\sim C_*(S^{2p-1})$,
and from B\"okstedt duality for $THH(\Z ; \Fp)$, Theorem
\ref{thm:THHGordescent} shows  $THH(lu;\Fp)$ is Gorenstein of
shift $(2p-1)+(-2)=2p-3$. 

The spectral sequence \ref{lem:connSS} gives an alternative approach to the
proof. The only necessary input would be  to know that the differential
$d_{2p}(\mu_{2p})\neq 0$. This then shows that $d_{2p}(\mu_{2p}^n)\neq 0$ unless
$n$ is a multiple of $p$, so that the $E_{2p+1}=E_{\infty}$ term is
generated by $\mu_{2p}^p$ (giving $\mu_{2p^2}$), $\mu_{2p}^{p-1}\tau_{2p-1}$ 
(giving $\lambda_{2p^2-1}$) and $\lambda_{2p-1}$ (which survives as it
is). 



Of course we get the same conclusion by taking $\Ca\lra \Ba$ to be
$lu \lra \Fp$ and $k=\Fp$. In that case we find $A\sim C_*(S^1\times S^{2p-1})$ so
that by Corollary \ref{cor:gor} $THH(lu;\Fp)$ is Gorenstein of shift $[(2p-1)+1]-3=2p-3$. 
\end{example}


\begin{example}
If we take $\Ca\lra \Ba$ to be $ku\lra \Z$ and $k=\Fp$ we find $A\sim C_*(S^{3})$,
and from B\"okstedt duality we deduce $THH(ku;\Fp)$ is Gorenstein of
shift $3+(-2)=1$. 

The spectral sequence \ref{lem:connSS} gives a calculation. If $p$ is odd, there
can be no differentials and 
$THH(ku; \Fp)=\Fp [\mu_{2p}]\tensor \Lambda (\lambda_{2p-1},
\lambda_3)$. 

If $p=2$ the only necessary input would be  to know whether the differential
$d_{4}(\mu_{4})$ is zero or not. If it is zero then 
$THH(ku; \Ftwo)=\Ftwo [\mu_{4}]\tensor \Lambda (\lambda_{3},
\lambda_3')$. If it is non-zero then 
 $THH(ku; \Ftwo)=\Ftwo [\mu_{8}]\tensor \Lambda
 (\lambda_{3},\lambda_7)$. 

Of course we get the same conclusion by working with $\Ca\lra \Ba$ to be
$ku \lra \Fp$ and $k=\Fp$ we find $A\sim C_*(S^1\times S^{3})$ so
that $THH(lu;\Fp)$ is Gorenstein of shift $[3+1]-3=1$. 
\end{example}


\begin{example}
If we take $\Ca\lra \Ba$ to be $ku\lra ku/(p,v_1)$ and $k=ku/(p,v_1)$ we
find $A \sim C_*(S^1\times S^{2p-1}; ku/(p,v_1))$ (a Poincar\'e duality
algebra of formal dimension $4p-4$). In order to proceed we
would need to know that $THH(ku/(p,v_1))$ is Gorenstein of shift $x$. %\cite{Angeltveit}.
 We would then deduce  $THH(ku;ku/(p,v_1))$ is Gorenstein of shift $4p-4+x$. 
\end{example}


\begin{example}
We take $\Ca\lra \Ba$ to be $ko\lra ku$ and $k=\Ftwo$. In view of
Wood's theorem we have the cofibre sequence $\Sigma ko\stackrel{\eta}
\lra ko \lra ku$, showing  that $ku$ is small over $ko$ and $A\sim C_*(S^2)$. 

Since $ku$ is Gorenstein of shift 1, Theorem \ref{thm:THHGordescent}
shows $THH(ko; \Ftwo) $ is
Gorensten of shift $1+2=3$ as we saw  from the calculation 
$$THH_*(ko;\Ftwo )= \Lambda (\lambda_{5}, \lambda_7 )\tensor
\Ftwo [\mu_8]$$
of Angeltveit and Rognes \cite{AR}. 

More directly, using Example \ref{eg:relGor} we could take $\Ca \lra \Ftwo$ to be $ko\lra \Ftwo$,
and apply Corollary \ref{cor:gor} to conclude that $THH(ko;\Ftwo)$ is 
Gorenstein of shift $-(-6)-3=3$
\end{example}

\begin{example}
{\em ($tmf$ localized at 3)}
We take $\Ca\lra \Ba$ to be $tmf\lra tmf_0(3)$ at the prime 3 and
$k={\mathbb{F}}_3$.  Since $tmf_0(3)$ is $tmf$ extended by a three 
cell  complex, it is small. We have also seen $tmf_0(3)$ is Gorenstein of shift
$-15$, so that $THH(tmf_0(3); {\mathbb{F}}_3)$ is Gorenstein of shift
$12$. Since the three cell complex is self-dual of dimension 8,   $THH(tmf; {\mathbb{F}}_3))$ is Gorenstein of
shift $20$. 

More directly, using Example \ref{eg:relGor} we observe that since $tmf_0(3)$ is small over
$tmf$ and ${\mathbb{F}}_3$ is small over $tmf_0(3)$ then 
${\mathbb{F}}_3$ is small over $tmf$. Since $tmf$ is Gorenstein
of shift $-23$ we may apply Corollary
\ref{cor:gor} to deduce $THH(tmf; {\mathbb{F}}_3))$ is Gorenstein of
shift $20$. 
\end{example}

\begin{example}
{\em ($tmf$ localized at 2)}
We take $\Ca\lra \Ba$ to be $tmf\lra tmf_1(3)$ at the prime 2 and
$k={\mathbb{F}}_2$.  Since $tmf_1(3)$ is $tmf$ extended by a finite
complex, it is small. We have also seen $tmf_1(3)$ is Gorenstein of shift
$-11$, so that $THH(tmf_1(3); {\mathbb{F}}_2)$ is Gorenstein of shift
$8$. Since the complex is self-dual of dimension 12,   $THH(tmf; {\mathbb{F}}_2))$ is Gorenstein of
shift $20$. 

More directly, using Example \ref{eg:relGor}  we observe that since $tmf_1(3)$ is small over
$tmf$ and ${\mathbb{F}}_2$ is small over $tmf_1(3)$ then 
${\mathbb{F}}_2$ is small over $tmf$. Since $tmf_1(3)$ is Gorenstein
of shift $-23$ we may apply Corollary
\ref{cor:gor} to deduce $THH(tmf; {\mathbb{F}}_2))$ is Gorenstein of
shift $20$. 
\end{example}

\begin{example}
We take $\Ca \lra \Ba$ to be $e_n\lra \Fp$, where $e_n$ is the connective Lubin-Tate commutative ring spectrum with
homotopy $W({\mathbb{F}}_{p^n})[[u_1, \ldots , u_{n-1}]][u]$. From its
homotopy we see that it is Gorenstein of shift $-n-3$. From Theorem
\ref{thm:THHGordescent} we conclude $THH(e_n; \Fp)$ is Gorenstein of
shift $n$. 
\end{example}

These examples are iterable. 
\begin{example}
Veen's calculation \cite{Veen} of the 
homotopy ring of the double THH of $k$, as $k[\mu'_2, \mu''_2]\tensor
\Lambda (\lambda_3)$ gives an example whose coefficient ring is
Noetherian and of Krull dimension 2.  

If we are given map  $R\lra k$ we may take $\Ca\lra \Ba$ to be $THH(R;
k)\lra THH(k)$,  and take $A$ to be its cofibre. Now apply Dundas's
Lemma to obtain a cofibre sequence
$$A\lra THH(THH(R;k); THH(k)) \lra THH(THH(k)). $$

Unfortunately there seems to be no obvious example for which $A$ is
finite dimensional, or even Noetherian. For example, if we take $R=\Z$
it seems $\pi_*A= \Fp [\mu_2]/(\mu_2^p)\tensor \Gamma (\gamma_{2p})$. 
\end{example}




\subsection{Discussion}
The main obstacle to finding more examples is the need to ensure that the
coefficient rings should be Noetherian, which seems rather rare for
$THH$. In the known cases where this happens,  the coefficient rings are all themselves
Gorenstein. We have not yet  found an example where the ring
spectrum is proxy regular and Gorenstein except when the coefficients are already Gorenstein. 

Our analysis is based on  the  cofibre sequence $A\lra \TbC
\lra \TbB$  and requires that it is \pifinite . We have relied on the fact
that   if $A$ is finite dimensional and $k$ is of characteristic $p$, 
the cofibre sequence is \pifinite. In this case  $\TbC$ will have the
same Krull  dimension as $\TbB$. 

\appendix

\begin{center}
{\large 
Appendices }

\end{center}
We end with two appendices describing closely related
phenomena. They do not form part of the argument, and are included
for comparison. 

 \section{Thom spectra}

Given a 3-fold loop map $f: X\lra BF$, Blumberg-Cohen-Schlichtkrull \cite{BCS} prove
$$THH(Mf)\simeq Mf \sm BX_+. $$
Since $f$ is a 3-fold loop map, $X$ is a 3-fold loop space and
$BX\simeq \Omega BBX$. We expect that $THH(Mf)=C_*(BX; Mf)$ 
being Gorenstein over $Mf$ will be related to $C^*(BBX; Mf)=F(BBX_+, Mf)$ being
Gorenstein over $Mf$. 

\begin{example} {\em ($\Fp$ revisited.)} 
This example is closely related to the fact that  $THH(\Fp)$ is Gorenstein of shift $-3$.

Mahowald  \cite{Mahowald} showed that the
Eilenberg-MacLane spectrum $\Fp$ is the Thom spectrum of 
a map $\Omega^2S^3 \lra BF$. Although this is only a double loop
space map, it is shown in \cite[1.3]{BCS}  that 
$$THH(\Fp)\simeq \Fp \sm \Omega S^3_+=:C_*(\Omega S^3),  $$
but this is not an equivalence of ring spectra, since the ring
structures in homotopy groups are different. 

Since $S^3$ is a 3-manifold,  $C^*(S^3)$ is Gorenstein of shift $-3$,
and by Morita invariance of the Gorenstein condition \cite{DGI1}, we conclude that 
$C_*(\Omega S^3)$ is Gorenstein of shift $-3$. Although it is
precisely parallel, this doesn't directly imply anything about $THH(\Fp)$. 
\end{example}

\begin{example} {\em ($\Z$ revisited.)}
This is closely related to the fact that $THH(\Z; \Z/p)$ is Gorenstein of shift $-2$. 

Mahowald \cite{Mahowald} proved that the
Eilenberg-MacLane spectrum $\Z$ is the Thom spectrum of 
a map $\Omega^2S^3\langle 3\rangle \lra BF$. Although it is only a
double loop map it is shown as \cite[1.4]{BCS}  that 
$THH(\Z)\simeq \Z  \sm \Omega S^3\langle 3 \rangle_+$, and
that  
$$THH(\Z; \Z/p)\simeq \Fp \sm \Omega S^3\langle 3
\rangle_+=C_*(\Omega S^3\langle 3\rangle).$$

Applying cochains to the fibration 
$$K(\Z, 2) \lra S^3\langle 3\rangle  \lra S^3  $$
we get a cofibre sequence of ring spectra, and a standard calculation
shows this is \pifinite . By Corollary \ref{cor:GorascentNoeth}, the cochains on the total
space is Gorenstein if the cochains on the base and the cochains on
the fibre are, and that the shifts add. Since $S^3$ is a 3-manifold, $C^*(S^3)$ is Gorenstein of shift $-3$. 
On the other hand $C_*(\Omega BU(1))=C_*(U(1))$ is Gorenstein of shift 1 since
$U(1) $ is a 1-dimensional compact Lie group, and hence by Morita 
invariance of the Gorenstein condition \cite{DGI1}, we find 
$C^*(BU(1))$ is  Gorenstein of shift 1. By Gorenstein ascent, we deduce that 
$C^*(S^3\langle 3 \rangle)$ is Gorenstein of shift
$-2=-3+1$. By Morita invariance of the Gorenstein condition \cite{DGI1} we conclude that 
$C_*(\Omega S^3\langle 3 \rangle )$ is  Gorenstein of shift $-2$.
\end{example}

%The simplest possible way to continue this pattern would be if $THH(lu)\simeq lu \sm
%\Omega Y_+$,
%and then find 
%$$THH(lu, lu/(p,v_1))\simeq H\Fp \sm \Omega Y_+=C_*(\Omega Y).$$
% We would then hope that $C_*(\Omega Y)$ is Gorenstein, which we might
% show by checking $C^*(Y)$ is Gorenstein of shift $2p-3$.  
%Unfortunately $ku$ is not a Thom spectrum (and presumably $lu$ isn't 
%either), so we have to proceed more indirectly.


\section{Dwyer-Miller and Kontsevich duality}
\label{sec:NAR1}

This section describes some known dualities for Hochschild homology and
cohomology with a similar flavour. The analogue to bear in mind is the case $R=k[x]$ on a
polynomial generator of even degree $d$. This is Gorenstein of shift
$-d-1$. Now we calculate $HH^*(R)=k[x,\alpha]$ where
$\alpha$ is of degree $-d-1$, and $HH_*(R)=k[x, \beta]$ with $\beta$
of  degree $d+1$, so we have $HH^*(R)=\Sigma^{-d-1}HH_*(R)$.

In this section we work under $k$ rather than under the sphere
spectrum $\bbS$. In particular, we assume that our rings are
$k$-algebras so 
the situation is very different to the one considered in the body of
the paper.  
%In particular the context does not permit us to take $S$ to be the
%sphere spectrum. 
%We also make the following significant assumption.
%satisfied in some important cases. 



\begin{assumption}
\label{ass:DwyerMiller}
We assume that $S=k$, so that $R^e=R\tensor_k R$ and that there is a ring map $R\lra R^e$ of 
$R^e$-modules. Finally, we assume that the bimodule $R$ is  induced from an
$R$-module: $R=R^e\tensor_R k$. 
\end{assumption}


\begin{prop} (Dwyer-Miller)
If Assumption \ref{ass:DwyerMiller} holds,  $k$ is small over $R$ and $R\lra k$ is Gorenstein of shift $a$ then 
$$HH^*(R;P)\cong \Sigma^a HH_*(R; P)$$ 
for all bimodules $P$. 
\end{prop}

\begin{remark}
There seems no prospect of a result like this for $THH$ since
$THH^{\bullet}(k)$ (for example) is not bounded below. 
\end{remark}

\begin{proof}
We argue as follows, where the third equivalence requires $k$ to be
small, and where $P^{ad}$ is the restriction of $P$ along the map
$R\lra R^e$. 
$$\begin{array}{rcl}
HH^*(R;P)&=&\Hom_{R^e}(R,P)\\
&=&\Hom_R(k, P^{ad})\\
&\simeq &\Hom_R(k, R)\tensor_R P^{ad}\\
&=&\Sigma^ak \tensor_R P^{ad}\\
&=&\Sigma^aR \tensor_{R^e} P\\
&=&\Sigma^aHH_*(R ;P)
\end{array}$$
\end{proof}

\begin{remark}
Take $R=C_*(\Omega X)$ for a connected orientable $d$-manifold $X$. By
the Morita invariance of the Gorenstein condition this is Gorenstein
of shift $a=-d$. Finally, note that   $HH_*(R)=H_*(\Lambda X)$, so
taking $P=R$, we see $\Sigma^d H_*(\Lambda X)=HH^*(R)$, showing that the shifted
homology of the free loop space has a ring structure; Malm \cite{Malm}
shows this 
corresponds to the Chas-Sullivan product.  
\end{remark}

\begin{cor} (Kontsevich duality)
If Assumption \ref{ass:DwyerMiller} holds, $k$ is small over $R$ and $R\lra k$ is Gorenstein with shift $a$
then $R^e\lra R$ is relatively Gorenstein with shift $a$. 
\end{cor}

\begin{proof}
Taking $P=R^e$ in the above we find
$$\Hom_{R^e}(R, R^e)=HH^*(R|S;R^e)=\Sigma^aHH_*(R|S;R^e)=\Sigma^a
R\tensor_{R^e}R^e=\Sigma^a R. $$
\end{proof}




Now consider the $(R|S)$-bimodule $P=k$, and note that if $R$ and $k$
are commutative rings, then so is 
$$HH_*(R|S;k)=R\tensor_{R^e}k.$$
Furthermore $R^e$ is an $R$-algebra, so we have an algebra map 
$$R\tensor_{R^e}k\lra R^e\tensor_{R^e} k \stackrel{\simeq}\lra k$$

\begin{cor} (Algebraic Ausoni-B\"okstedt duality)
If Assumption \ref{ass:DwyerMiller} holds, $k$ is small over $R$ and $R\lra k$ is Gorenstein with shift $a$
then 
$HH_*(R|S;k) \lra k$ is Gorenstein with shift $-a$. 
\end{cor}

\begin{proof}
Taking $P=k$ in the above we find
$$HH^{\bullet}(R|S;k)=\Sigma^a HH_{\bullet}(R|S;k). $$
Now we calculate
$$\begin{array}{rcl}
\Hom_{HH_{\bullet}(R|S;k)}(k,HH_{\bullet}(R|S;k))
 &=&\Sigma^{-a}\Hom_{HH_{\bullet}(R|S;k)} (k,HH^{\bullet}(R|S;k))\\
 &=&\Sigma^{-a}\Hom_{HH_{\bullet}(R|S;k)}(k, \Hom_{R^e}(R,k))\\
 &=&\Sigma^{-a}\Hom_{HH_{\bullet}(R|S;k)}(R\tensor_{R^e} k,k)\\
 &=&\Sigma^{-a}k
\end{array}$$
\end{proof}

% The following observation may possibly be reassuring. 

% \begin{lemma}
% If  $HH_{\bullet}(R|S)\lra R$ is regular and Gorenstein of shift $a$
% then $HH_{\bullet}(R|S;k)\lra k$  is Gorenstein of shift $a$ and 
% \end{lemma}

% \begin{proof} We apply $\tensor_R k$ to the Gorenstein hypothesis. 

% This gives 
% $$\begin{array}{rcl}
% \Sigma^a k &=&\Hom_{HH_*(R|S)}(R, HH_*(R|S))\tensor_R k\\
%  &= &
% \Hom_{HH_*(R|S)}(R, R\tensor_{R^e}R\tensor_R k) \mbox{  (if R is small over $HH_*(R|S))$}\\
% &=& \Hom_{HH_*(R|S)}(R, R\tensor_{R^e} k)\\
% &=& \Hom_{HH_*(R|S;k)}(k, R\tensor_{R^e} k)
% \end{array}$$
% (where the last is $R= (R\tensor_{R^e} R) \tensor_{R\tensor_{R^e} k} k )$, if all the actions are right). 
% \end{proof}



\begin{thebibliography}{999}
\bibitem{AHL}
V.Angeltveit, M.Hill and T.Lawson 
``Topological Hochschild homology of $\ell$ and $ko$.''
American J Math {\bf 132} (2010) 297-330
\bibitem{AR}
V.Angeltveit and J. Rognes
``Hopf algebra structure on topological Hochschild homology.''
Algebraic and Geometric Topology {\bf 5} (2005) 1223-1290
\bibitem{Ausoni}
C.Ausoni
``Topological Hochschild homology of connective complex $K$-theory.''
Amer. J. Math. 127 (2005), no. 6, 1261–1313.
\bibitem{AR}
C. Ausoni and J. Rognes,
``Algebraic K-theory of topological K-theory'', 
Acta Mathematica {\bf 188} (2002), 1-39.
\bibitem{Bass}
H. Bass ``On the ubiquity of Gorenstein rings.'' Math. Z. {\bf 82} (1963) 8–28.
\bibitem{BCS}
A. Blumberg, R.Cohen and C.Schlichtkrull
``The topological Hochschild homology of Thom spectra and the free
loop space.'' arXiv 2008
\bibitem{Bok}
M. B\"okstedt, 
``Topological Hochschild homology of $F_p$, and $\Z$'', Preprint,
Bielefeld University.
\bibitem{DGI1}
 W.G.Dwyer, J.P.C.Greenlees and S.B.Iyengar
  ``Duality in algebra and topology''
Advances in Maths. {\bf 200} (2006) 357-402
\bibitem{ringlct}
   J.P.C.Greenlees and G.Lyubeznik 
   ``Rings with a local cohomology theorem and applications to cohomology 
   rings of groups.''
   J. Pure and Applied Algebra {\bf 149} (2000) 267-285.
\bibitem{HM} M.J.Hopkins and M.E.Mahowald
``From elliptic curves to homotopy theory''
Preprint (1998)
\bibitem{LM} A.Lindenstrauss and I. Madsen ``Topological Hochschild
homology of number rings.''
Trans AMS {\bf 352} (2000) 2179-2204
\bibitem{Mahowald} M.E.Mahowald
 ``Ring spectra which are Thom complexes.'' Duke Math. J., {\bf 46}
 (1979) 549–559
\bibitem{MS} J.E.McClure and R.E.Staffeldt
``On the topological Hochschild homology of $bu$ I''
AJM {\bf 118} (1993) 1005-1012 
\bibitem{Malm} E.J.Malm
`` String topology and the based loop space''
Preprint (2011) arXiv:1103.6198 
\bibitem{AM} Akhil Matthew
``The homology of tmf'' 
Preprint (2013) arXiv:1305.6100 

\bibitem{Veen} T.Veen
``Detecting Periodic Elements in Higher Topological Hochschild Homology
Preprint (2014) arXiv:1312.5699

\end{thebibliography}
\end{document}
\part{Miscellaneous Addenda}
\section{Further thoughts}
\newcommand{\conn}{\mathrm{conn}}
\newcommand{\map}{\mathrm{map}}
\newcommand{\bmap}{\mathrm{map_*}}


One may wonder where the shifts come from and whether it has to do with $S^1$ (as in $THH(R)=R\tensor S^1$). 

    There seems to be some entertainment to be had this way. I believe that if
$R=C^*(X;k)$ then $R\tensor A =C^*(\map (A,X);k)$, under suitable
connectivity hypotheses, where $\map (A,X)$ is the unbased mapping space. 
If this isn't true, then I'll just speculate about $C^*(\map (A,X); k)$ anyway. 

     Now if $A$ is a finite complex, we have the fibration
$$\bmap (A,X)\lra \map (A,X)\lra X$$
constructing $\map (A,X)$ from $X$ and the based mapping space. For the
 based mapping space we have fibrations building $\bmap (A,X) $ from 
$\Omega^n X=\bmap (S^n,X)$. 

To make some quick progress, let's work rationally and assume that $X$
is a product of $r$ even Eilenberg-Maclane spaces. This means  that
$C^*X$ is polynomial and $C^*(\Omega^n X)$ is (rationally) either polynomial
(if $n$ is even) or exterior (if $n$ is odd)). 


\begin{lemma}
If $X$ is a product of $r$ even Eilenberg-Maclane spaces of with
homotopy in degrees $d_1, d_2, \ldots , d_r$ then $C^*(X)$ is Gorenstein of shift $a=a(X)=d-r$
where $d=d_1+\cdots +d_r$.

Furthermore, provided $2n+1\leq \conn (X)$ we find
$C^*(\Omega^{2n}X)$ is Gorenstein of shift $a-2n$, and 
$C^*(\Omega^{2n+1}X)$ is Gorenstein of shift $-(a-2n)$.

\end{lemma}

\begin{proof} It suffices to  deal with the case $X=K(\Q, 2m)$, so that $C^*(X)$ is
  Gorenstein of shift $a=2m-1$, we see $C^*(\Omega X)$ is Gorenstein
  of shift $-a$ and $C^*(\Omega^2 X)$ is Gorenstein of shift $a-2$. 
\end{proof}

Now suppose that $A$ has $n(i)$ cells of dimension $i$ (where one of
the  $n(0)$ 0-cells is the basepoint).  

\begin{cor}
If $\dim (A)\leq \conn (X)$ then 
$C^*(\bmap (A,X))$ is Gorenstein of shift
$$a(\bmap (A,X))= \widetilde{\chi} (A)\cdot a(X) -[2 (n(2)-n(3))+4 (n(4)-n(5))+6 (n(6)-n(7))+\cdots].$$

\end{cor}

\begin{proof}
The shift is of degree 
$(n(0)-1)a+n(1)(-a)+n(2)(a-2)+n(3)(-(a-2))+\cdots$.
\end{proof}

\begin{cor}
If $\dim (A)\leq \conn (X)$ then 
$C^*(\map (A,X))$ is Gorenstein of shift
$$a(\map (A,X))= \chi (A)\cdot a(X) -[2 (n(2)-n(3))+4 (n(4)-n(5))+6 (n(6)-n(7))+\cdots].$$

\end{cor}

\begin{proof}
We use the fibration 
$$\bmap (A,X)\lra \map (A,X) \lra X. $$
\end{proof}    


\begin{example}
If $X$ is $c$-connected and Gorenstein of shift $a$ then for $r\leq c$
 the $r$-iterated
Hochschild homology $\map (S^1\times \cdots \times S^1, X)$ is
Gorenstein  of shift 0 if $r=1$ and of shift $-2^{r-1}$ if $r\geq 2$. 
\end{example}



\section{Eilenberg-Moore resolutions}

    Let $R=DEG_+$. [[Want to have a resolution by $G$-spectra to
    ensure the EMSS gives $W$-action. ]]
\begin{lemma}
Given any $R$-module $M$ whose homology is a finitely generated module
over the polynomial ring $R^N_*=\pi^N_*(DEG_+)=H^*(BN)$, and which is
$G$-free, there is a resolution
$$M \lla P_0 \lla P_1\lla \cdots \lla P_r , $$
of $G$-equivariant $R$-modules, where $P_i$ is a finite wedge of suspensions of $R$ and where the
associated tower converges. 
\end{lemma}

\begin{proof}
   First we construct a cofibre sequence modelled on the algebraic
complex for the resolution of $M^N_*$ by the polynomial ring $\pi^N_*(DEG_+)=H^*(BN)$. 
This takes the form
$$M \lla P_0 \lla P_1\lla \cdots \lla P_r.  $$
It remains to show that the associated tower converges. 

For this we dualize, noting that each $P_i$  is the wedge of finitely many copies of $R=DEG_+$.
Thus $P_i=DX_i$ where $X_i$ is a wedge of copies of $EG_+$. 

We thus obtain maps 
$$DM\lra DP_0\lra DP_1\lra \cdots \lra DP_r.$$
Next, note that  $DP_i=DDX_i$. Of course this won't be $X_i$, but it
is {\em nonequivariantly} $X_i$. 
Since $X_{i-1}$ is free we can complete each square
$$\diagram
   DDX_{i-1}\rto &DDX_i\\
     X_{i-1}\rto \uto &X_i\uto
\enddiagram$$
uniquely with the bottom map. The same applies to the start of the
tower, taking $M=X_{-1}$. Now it is easy to argue that the tower converges, since it is the dual of a tower
that is free and non-equivariantly contractible. 
\end{proof}

\begin{lemma}
The counit
$$DEG_+ \tensor_{DBN_+} \Sigma^d W_+ \lra G_+$$ 
of the adjunction is an equivalence. 
\end{lemma}

\begin{proof}
We note first that the domain is the $d$th suspension of
$$\Q \tensor_{C^*BN} (C^*G)^N\simeq \Q \tensor_{C^*BN} (C^*G/N)$$ 
so the Eilenberg-Moore spectral sequence shows the domain is the $d$th
suspension of\\
 $C^*G\simeq $\Sigma^{-d}C_*(G)$. This means that 
$$[DEG_+ \tensor_{DBN_+} \Sigma^{-d}W_+ ,G_+]^G=[G_+, G_+]^G=[S^0,
G_+]=\Q[W]. $$ 
A basis for the vector space on the right is given by the group
elements $w \in W$, and the element $w$ corresponds on the left to 
the map taking $1_+$ to the $w$ component of $G_+$. 
Under the adjunction the components correspond: whereever
the identity is sent, the same is true in $[W_+, W_+]^W$. By
definition the counit corresponds to the identity, so the original map
is also the identity on components and it is therefore an equivalence. 
\end{proof}

\newpage
\section{Gorenstein ascent in characteristic $p$}

Suppose $k$ is a field of characteristic $p$ and  $S\lra R\lra Q$ is a
cofibre sequence of commutative augmented, connected
$k$-algebras. 

[[Skip the following lemma to start with. It is clear from the
following proof that we just need something to let us lift suitable $p$-powers.]]
\begin{lemma}
There is a multiplicative Serre-type spectral sequence
$$H_p(S;H_q(Q))\Rightarrow H_{p+q}(R). $$ 
\end{lemma}

\begin{proof}
By killing homotopy groups there are maps $S\lra S[0, k]$ for each
$k$, and taking pushouts gives a diagram
$$\diagram
S\rto \dto &R \rto \dto & Q \dto^=\\
S[0,k]\rto &R\langle k \rangle\rto &Q
\enddiagram$$
in modules and hence a filtration of $R$. The resulting spectral sequence has
$$E^1_{p,q}=H_{p+q}(S[p]\tensor Q)$$
and the indicated $E^2$ term. The construction should also show that the spectral
sequence is multiplicative. 
\end{proof}


\begin{prop}
If $\Hom_S(k,S)$ is bounded below and $Q$ is Noetherian then 
$$\Hom_R(k, R)\simeq \Hom_S(k,\Hom_R(k,R)\tensor_k Q).$$
In particular if $S$ and $Q$ are Gorenstein, so is $R$. 
\end{prop}
\begin{proof}
From the proof in \cite{DGI1}, we see that it suffices to show that
the natural map 
$$\Hom_S(k,S)\tensor_S R\lra Hom_S(k,R)$$
is an equivalence. It is obviously an equivalence when $S=R$ and hence
it  is an equivalence whenever $R$ is small as an $S$-module. Our
strategy then  is to prove that we approximate the original cofibre sequence by
ones with small cofibre. In fact we construct a diagram
$$\diagram
S\rto \dto & R\rto \dto & Q\dto\\
S\rto \dto & R_{n+1}\rto \dto & Q_{n+1}\dto\\
S\rto  & R_{n}\rto  & Q_{n}
\enddiagram$$
with exact rows, so that 
$R\simeq \holim_n R_n$, $Q\simeq \holim_nQ_n$ and 
$R_n$ small over $S$ for all $n$. Since
$$\Hom_S(k, R)\simeq \Hom_S(k, \holim_n R_n) \simeq 
\holim_n \Hom_S(k, R_n )
\simeq 
\holim_n [\Hom_S(k, S)\tensor_S  R_n],  
$$ 
it remains to show that the natural map 
$$ H\tensor_S [\holim_n R_n] \lra \holim_n [H\tensor_S  R_n] $$
is an equivalence for  $H=\Hom_S(k,S)$.
In particular, if $H$ is bounded below and $R_n$ is uniformly bounded
below,  this holds.  

Finally, we explain how the approximations are constructed. 
Since $Q$ is Noetherian, there is a polynomial subalgebra, $Q(0)
\subseteq Q$ over which $Q$ is finite as a module. Since the Serre
spectral sequence is multiplicative, and
$k$ is of characteristic $p$, some $p$-power of each of the polynomial
generators lifts to $R$. Let $Q(1)$ be the polynomial algebra
generated by these elements, so that $Q$ is still finite over $Q(1)$
and there is a diagram 
$$\diagram 
Q(1)\dto \rto^=& Q(1)\dto \\
R\rto &Q
\enddiagram$$
Now let $Q(n)$ be the polynomial subring of $Q(1)$ generated by the
$n$th powers of the generators, and let
$Q_n=Q//Q(n), R_n=R//R(n)$ so that we have the diagram above. 
 It is easy to see that with this definition
 $Q=\holim_n Q_n$  and  $R=\holim_n R_n$; since $Q$ is finite over
 $Q(n)$, it follows that  $Q_n$ is finite dimensional,  and hence
$R_n$ is small over $S$ as required. 
\end{proof}

\begin{remark}
An essentially similar argument deals with the case of coconnective
algebras, either simply connected or nilpotent in degree $-1$. 
\end{remark}



\end{document}


\newpage
\newpage
\section{Chouinard's theorem for compact Lie groups.}
The first step to understanding the category of modules over
$C^*(BG)=C^*(BG;\Fp)$ for a compact Lie group $G$ is to show  it is 
controlled by elementary abelian subgroups. 
In effect we use a tiny part of Quillen's method for understanding the 
spectrum of $H^*(BG)$. 

Suppose $G$ is a compact Lie group, and choose an embedding in a
unitary group $U$. We write $T$ for the maximal torus of $U$ and $S$
for the elementary abelian subgroup $T[p]$. Finally $F=U/S$, and 
we note that as a $G$-space, all its isotropy groups are elementary 
abelian $p$-groups. 



\begin{lemma}
If $M$ is a  $C^*(BG)$-module with the property that
$M\tensor_{C^*(BG)}C^*(BE)\simeq 0$ for all elementary abelian
subgroups $E$ of $G$ then in fact $M\simeq 0$. 
\end{lemma}

\begin{proof}
Of course the $G$-space $F$ is built from finitely many cells $G/E$,
and hence $C^*(EG\times_GF)$ is finitely built from modules
$C^*(EG\times_G (G/E))$.  
The hypothesis states that tensoring with $C^*(EG\times_G(G/E))$ gives a
trivial module for all $E$ and hence tensoring with $C^*(EG\times_G
F)$ gives the trivial module. 

We observe that $H^*(BS)$ is free over $H^*(BT)$ and $H^*(BT)$ is free
over $H^*(BU)$. Hence $C^*(F)=C^*(U/S)\simeq\Fp
\tensor_{C^*(BU)}C^*(BS)$. This gives a  map 
$$C^*(EG\times_GX)\tensor_{C^*(BU)}C^*(BS)\lra C^*(EG\times_G (X\times F))$$
natural in $X$. By a lemma of Quillen's this is an equivalence. 
In particular it follows that  $C^*(EG\times_GX)$ is a summand of
$C^*(EG\times_G(X\times F))$ as $C^*(BG)$-modules. 



Taking $X=*$ we conclude that since $C^*(BG)$ 
is a retract of $C^*(EG\times_G F)$ and hence  
$M=M\tensor_{C^*(BG)}C^*(BG)$ is a retract of 
$M\tensor_{C^*(BG)}C^*(EG\times_GF)\simeq 0$.
\end{proof}  

After Blumberg-Cohen-Schlichtkrull

(1)(a)  THH(Z/p; Z/p) = C_*(\Omega S^3) 
and 
(1)(b)  THH(Z; Z/p)= C_*(\Omega S^3<3>)

We know these  are both hGorenstein 
(2) (a): We see THH(Z/p; Z/p) is Gorenstein of shift -3 since C^*(S^3) is (S^3 being a manifold) 
(2) (b): We see THH(Z; Z/p) is Gorenstein of shift -2 since C^*(S^3<3>) is by virtue of the fibration  
K(Z,2)---->S^3<3>--->S^3, and the facts that C^*(S^3) is Gorenstein of shift -3 (as before) and 
C^*(BT) is Gorenstein of shift 1 (eg because C_*(\Omega BT)=C_*(T) is, because T is a Lie group). 

(3) Given a fibre sequence R--->Rbar----->Q  of ring spectra augmented over k=Z/p (ie Q=Rbar\tensor_{R} k)
there is a fibre sequence of ring spectra 
    Q---->T----->Tbar

where T=R\tensor_{R^e} k=THH(R;k) and Tbar=Rbar \tensor_{Rbar^e} k =THH(Rbar; k). [This is just
a standard statement about fibres in maps of squares.] 
Hence [by a fibration lemma] if Tbar and Q are hGorenstein so is T, and the shifts add. 

Hence 
(3) (b) if we take R---->Rbar to be Z---->Z/p, we find Q =C_^*(S^1) and using (2)(a) 
we get a somewhat different proof of (2)(b).  [Again dimension shift 1+(-3)=-2]
(3) (c) if we take R----->Rbar to be lu ---->Z we find Q=C_*(S^{2p-1}), and using (2)(b) or (3)(b)
we deduce THH(lu;Z/p) is Gorenstein of shift (2p-1)+(-2)=2p-3. 

There are details to be filled in, but I think a clear picture has emerged.   
After Blumberg-Cohen-Schlichtkrull

(1)(a)  THH(Z/p; Z/p) = C_*(\Omega S^3) 
and 
(1)(b)  THH(Z; Z/p)= C_*(\Omega S^3<3>)

We know these  are both hGorenstein 
(2) (a): We see THH(Z/p; Z/p) is Gorenstein of shift -3 since C^*(S^3) is (S^3 being a manifold) 
(2) (b): We see THH(Z; Z/p) is Gorenstein of shift -2 since C^*(S^3<3>) is by virtue of the fibration  
K(Z,2)---->S^3<3>--->S^3, and the facts that C^*(S^3) is Gorenstein of shift -3 (as before) and 
C^*(BT) is Gorenstein of shift 1 (eg because C_*(\Omega BT)=C_*(T) is, because T is a Lie group). 

(3) Given a fibre sequence R--->Rbar----->Q  of ring spectra augmented over k=Z/p (ie Q=Rbar\tensor_{R} k)
there is a fibre sequence of ring spectra 
    Q---->T----->Tbar

where T=R\tensor_{R^e} k=THH(R;k) and Tbar=Rbar \tensor_{Rbar^e} k =THH(Rbar; k). [This is just
a standard statement about fibres in maps of squares.] 
Hence [by a fibration lemma] if Tbar and Q are hGorenstein so is T, and the shifts add. 

Hence 
(3) (b) if we take R---->Rbar to be Z---->Z/p, we find Q =C_^*(S^1) and using (2)(a) 
we get a somewhat different proof of (2)(b).  [Again dimension shift 1+(-3)=-2]
(3) (c) if we take R----->Rbar to be lu ---->Z we find Q=C_*(S^{2p-1}), and using (2)(b) or (3)(b)
we deduce THH(lu;Z/p) is Gorenstein of shift (2p-1)+(-2)=2p-3. 

There are details to be filled in, but I think a clear picture has emerged.   

-------------------------------
As context, we 
suppose that $S\lra R \lra Q$ is a cofibre sequence of commutative
algebras augmented over $k$. We consider the Gorenstein ascent
question: when does the fact that $S$ is Gorenstein imply that $R$ is
Gorenstein. It is natural to assume that $Q$ is Gorenstein, and we
show that very often this is sufficient. 

\begin{lemma}
\label{lem:proxyascent}
Suppose that $S$ is proxy-regular with Koszul complex $K_S$ and that $Q$ has a Koszul complex of
the special form $K_Q=K(Q; q_1, \ldots , q_n)$ where $q_i\in \pi_*(Q)$ lifts to $r_i
\in \pi_*(R)$ for $i=1, \ldots , n$. Then $R$ is proxy-regular with Koszul complex
$K_R:=K_S\tensor_S K(R;r_1, \ldots, r_n)$. 
\end{lemma}
\begin{proof} There are three things to prove.

 Since $S\finbuilds K_S$ and $R\finbuilds K(R;r_1, \ldots, r_n)$, it
follows that 
$$R=S\tensor_SR\finbuilds K_S\tensor_S K(R;r_1, \ldots, r_n)=K_R.$$ 

 Since $Q\simeq k\tensor_SR$, we find firstly 
$$k\finbuilds K_Q= k\tensor_S K(R;r_1, \ldots , r_n)\finbuilds
K_S\tensor_S K(R;r_1, \ldots , r_n)=K_R$$
and secondly 
$$K_R=K_S\tensor_SK(r_1, \ldots , r_n) \builds k\tensor_SK(r_1, \ldots
, r_n) =K_Q\builds k .$$
This completes the proof. 
\end{proof}


The proof in \cite{DGI1} gives a sufficient condition for Gorenstein
ascent.  

\begin{lemma}
\label{lem:Gorascent}
If the natural map $\nu: \Hom_S(k,S)\tensor_S R\lra  \Hom_S(k,R)$ is an equivalence
then 
$$\Hom_R(k,R)\simeq \Hom_{Q}(k, Hom_S(k,S)\tensor_k Q). $$
In particular, if all three rings are proxy regular and $S$ and $Q$
are Gorenstein, so is $R$, and the shifts add up: $\shift (R)=\shift
(S)+\shift (Q)$. 
\end{lemma}
\begin{proof}
The proof in \cite{DGI1} gives this conclusion. 
\end{proof}

We now want to show that $\nu$ is an equivalence in cases of relevance
to us. The most familiar case is when $R$ is small over $S$ (or
equivalently, when $Q$ is finitely built from $k$). 
We emphasize that the hypothesis on $\nu$ in Lemma \ref{lem:Gorascent}
only depends on $R$ as a {\em module} over $S$, and we will obtain a
useful generalization by approximating $R$ by $S$-modules for which 
$\nu$ is an equivalence. 


\begin{lemma}
\label{lem:Gorascentlim}
Suppose that $\pi_*S$ is connective and Noetherian and that 
 $\pi_*(\Hom_S(k,S))$ is a finitely generated module over $\pi_*(S)$.
If $R\simeq \ilim_n R_n$ for uniformly bounded below small $S$-modules $R_n$
then the hypothesis  of Lemma \ref{lem:Gorascent} applies, so 
$$\Hom_R(k,R)\simeq \Hom_{Q}(k, Hom_S(k,S)\tensor_k Q) $$
and Gorenstein ascent holds for the cofibre sequence $S\lra R\lra Q$.    
\end{lemma} 

\begin{proof}
First, 
$$\Hom_S(k, R) \simeq \Hom_S(k, \holim_n R_n) \simeq \holim_n
\Hom_S(k,  R_n).  $$ 
Now, since $R_n$ is a small $S$-module,  $\Hom_S(k,R_n)\simeq \Hom_S(k,S)\tensor_S R_n$.

It therefore remains to show that the natural map 
$$ M\tensor_S [\holim_n R_n] \lra \holim_n[M\tensor_S R_n] $$
is an equivalence when $M=\Hom_S(k,S)$. The hypotheses ensure that
$M$ is of upward finite type,  and by looking at
homotopy in each degree we see that the map is an equivalence. 
\end{proof}

To show when $R$ may be approximated in this way it is convenient to
introduce some terminology.

\begin{defn}
We say that a map $R \lra Q$ is {\em semi-split} if $\pi_*(Q)$ is
finitely generated as a module over $\pi_*(R)$. A cofibration $S\lra R
\lra Q$ is {\em semi-split} if the map $R\lra Q$ is semi-split. 
\end{defn}


\begin{cor}
\label{cor:GorascentNoeth}
Suppose that $\pi_*S$ is connective and Noetherian and that
$\pi_*(\Hom_S(k,S))$ is a finitely generated $S$-module. If in addition 
$\pi_*Q$ is connective and Noetherian and the cofibration is semi-split
 then  
$$\Hom_R(k,R)\simeq \Hom_{Q}(k, Hom_S(k,S)\tensor_k Q) $$
and Gorenstein ascent holds for the cofibre sequence $S\lra R\lra Q$.    
\end{cor}

--------------------------


\section{Poincar\'e and Gorenstein duality algebras.}
There are two basic sources of Gorenstein algebras in topology:
cochains $C^*(M)$ on an orientable manifold $M$ (giving Poincar\'e duality algebras) and chains
$C_*(G)$ on a topological group $G$ (giving Frobenius algebras). These generate
many others by use of Morita theory and cofibre lemmas. 

Experience shows it is valuable to be clear about where the various
structure maps come from and how they are related, particularly when
we consider examples arising from   a compact Lie group $G$, where we have
the Poincar\'e duality algebra structure on $C^*(G)$ and the Frobenius 
algebra structure on $C_*(G)$, and where these are dual vector
spaces. 




\subsection{Hopf algebras}
The notion of Hopf algebra axiomatises the properties the space of 
functions on a group. More precisely, a group has maps
$$G\stackrel{\Delta}\lra G\times G \stackrel{\mu}\lra G, 
 G\stackrel{i}\lra G
\mbox{ and } 1 \stackrel{e}\lra G \stackrel{\pi}\lra 1, $$
and the group axioms can be stated as commutative diagrams
in terms of these. Since the multiplication $\mu$ and inversion $i$ are maps
of sets, they are compatible with the diagonal and projection.
The axioms of a Hopf algebra are obtained by replacing the 
categorical product $\times$ by a symmetric monoidal product
$\tensor$, and evidently applying any monoidal functor to a 
group will give a Hopf algebra. 

In short, a Hopf algebra $H$ is a sextuple $(H, m, \eta, \delta, \eps, \chi)$
specified by 
(i) a multiplication $m : H\tensor H \lra H$ and unit   $\eta: k\lra
H$ making it an algebra (ii) a comultiplication $\delta : H\lra
H\tensor H$ and counit   $\eta: H\lra k$ making it a coalgebra 
(iii) an antipode $\chi : H \lra H$. It is required that $m$ is a 
coalgebra map (or equivalently that $\delta$ is an algebra map), 
and that $\chi$ behaves like an inverse. 

\begin{example}
If $G$ is a topological group then the suspension spectrum of $G_+$
is a Hopf algebra in spectra, and $H_*(G)$ is a Hopf algebra in
$k$-modules.  
\end{example}

\subsection{Poincar\'e duality algebras.}
The notion of Poincar\'e duality algebra $A$ of formal  cohomological
dimension $n$ (or formal dimension $d=-n$) is an axiomatization of 
the cohomology ring of a compact  connected orientable $n$-manifold $M$. This time the relevant maps
of spaces are the diagonal map $\Delta : M \lra M\times M$ and 
$\pi : M \lra 1$ giving the algebra structure,  together with a map 
arising as cap product with the fundamental class.  

A Poincar\'e duality algebra $A$ of formal dimension $d$ is a quadruple 
$(A, m, \eta,  \theta)$ specified by 
(i) a multiplication $m : A\tensor A \lra A$ and unit   $\eta: k\lra
A$ making it an algebra (ii) a  trace   $\theta: A\lra \Sigma^n k$. 
It is required that the adjunct 
$A\lra \Sigma^{n} A^{\vee$ of the pairing 
$A\tensor A \stackrel{\mu}\lra A \stackrel{\theta}\lra \Sigma^nk$
gives an isomorphism $A^{\vee}\cong \Sigma^{-n} A$ of $A$-modules. 

\begin{remark} A {\em Frobenius algebra} is an algebra with a non-degenerate bilinear form 
so that $(a,bc)=(ab,c)$. In particular a Poincar\'e duality algebra as
defined here is an example. A {\em quasi-Frobenius algebra} doesn't have a
given  bilinear form but does have $A$ injective (on one side) as a module over
itself. In particular, a commutative quasi-Frobenius algebra is a 0-dimensional
Gorenstein ring. 
\end{remark}

\begin{example}
If $M$ is a compact $n$-manifold then applying $H^*(\cdot ;
k)$
to $1\lla M \lra M \times M$ gives the cohomology ring of $M$, and
choosing
a fundamental class $[M]\in H_n(M;k)$ gives a trace
$\theta : H^*(M)\lra \Sigma^{-n}k$ making $H^*(M)$ into a Poincar\'e
duality algebra. The only point requiring comment is that to avoid
confusion we use homological grading, so that $H^*(M) $ (which is 
concentrated in cohomological degrees from $0$ to $n$) is concentrated
in homological degrees from $0$ to $-n$. 
\end{example}

\begin{example}
Adding disjoint basepoints above, we obtain the maps
$$S^0=1_+ \lla M_+ \lra (M\times M)_+=M_+\sm M_+. $$ 
Applying functional duals, we see that $DM_+$ is commutative ring in
spectra. By Atiyah duality $DM_+\simeq M^{\nu}$ where $\nu$ is the 
stable normal bundle of $M$. If $M$ has trivial spherical normal 
bundle then $M^{\nu}\simeq \Sigma^{-n}M_+$, and in that case we can 
take $\theta : DM_+\simeq \Sigma^{-n}M_+\lra \Sigma^{-n}1_+=S^{-n}$ to
be a degree 1 map. 
The statement of Poincar\'e duality  is that this
map $\theta$ does define an equivalence $DM_+ \simeq \Sigma^{-n}M_+$
of $DM_+$-modules. 
\end{example}


\begin{example}
Another basic example of a Poincar\'e duality algebra of formal
dimension is the group ring $A=kG$ of a finite
group. Here the trace is obtained by evaluation at the identity
$e$. This example is best understood in the context of the next
subsection. 
\end{example}

\subsection{The chains on a group. }
Now if we view a finite group $G$ as a manifold there are two distinct
Poincar\'e duality algebras that we may form. 

We may view it as a 0-dimensional manifold and  apply cochains, so that
 the product comes from the diagonal map of spaces. This is the
 ring $k^G$ (product of copies of $k$), and (for example since the
 fundamental class is the sum of all the points) the trace is given by the
 map adding up the values of functions. 

Alternatively, we may take chains and use the group multiplication to
give the group ring (much more interesting). The trace in this case is
evaluation at the identity. 

There is a natural generalization of this  group ring example. 

\begin{example}
If $G$ is a compact Lie group of dimension $d$ we may apply 
$H_*(G)$ to get an algebra. We give this a trace by capping with 
the fundamental class of the identity component. 
This gives a Poincar\'e duality algebra of formal dimension $d$ {\bf [[Proof?]]}

Similarly, the suspension spectrum $G_+$ is a ring spectrum, and 
we may give it a trace $\theta: G_+ \lra S^d$ by collapsing all 
non-identity components to the basepoint and using a degree 1
map on the identity component. {\bf [[When does this give a duality?]]}
\end{example}

\subsection{The suspension spectrum of a compact Lie group.}
It rather clarifies things  to consider all available structure at once, working
in the category of spectra. Although a compact
Lie group is parallelizable, we
need to take account of the action of the group on it. In these terms
the tangent bundle is the adjoint representation, and the essential 
question is how the component group $\pi_0(G)$ acts on the fundamental
class of the identity, which is to say the orientation character
$\pi_0(G) \lra \Z^{\times}$. We then have $DG_+\simeq
S^{-ad}\sm G_+$, and indeed, this is true $G$-equivariantly, where the 
structure  of $ad$ as a representation (and not just its determinant)
is important. 

For $G_+$ we have 

$\pi :  G_+\lra S^0$ and its dual $D\pi : S^0\lra DG_+\simeq
S^{-ad}\sm G_+$

$e : S^0 \lra G_+$ and its dual $De : S^{-ad} \sm G_+\simeq DG_+ \lra S^0$

$\Delta : G_+\lra G_+ \sm G_+$ and its dual $D\Delta :  S^{-2ad} \sm
G_+\sm G_+ \simeq DG_+ \sm
DG_+\lra DG_+ \simeq S^{-ad}\sm G_+$ 

$\mu  : G_+ \sm G_+\lra G_+$ and its dual $D\mu : S^{-2ad} \sm G_+ \sm
G_+\simeq DG_+\lra DG_+ \sm
DG_+\simeq S^{-ad} \sm G_+. $

To make sense of this let us make some observations.

The composite
$$S^0 \stackrel{D\pi }\lra S^{-ad} \sm G_+ \stackrel{\pi} \lra S^0$$
is multiplication by $\chi (G)$. 

The composite
$$S^0 \stackrel{e }\lra G_+ \stackrel{De} \lra S^{ad}$$
is inclusion of $0$ in $ad$. It is an isomorphism if $G$ is
finite and 0 otherwise. 

\begin{example}
To obtain classical Poincar\'e duality algebra of a manifold we apply cochains to 
$\Delta, \pi$ to get the algebra, and we apply cochains to $D\pi$ to 
get the trace. Equivalently, we apply chains to 
$D\Delta, D\pi$ to get the algebra, and we apply chains to $\pi$ to 
get the trace. As is expected, none of these use the group structure. 
\end{example}
\begin{example}
To obtain the Poincar\'e duality algebra of the group
we apply chains to $\mu, e$ to get the algebra, and we apply chains 
to $De$ to get the trace. Equivalently, we apply cochains to 
$D\mu, De$ to get the algebra, and we apply cochains to $e$ to 
get the trace. 
\end{example}

\subsection{Gorenstein duality algebras}

We may want to axiomatize the setup of an $r$-dimensional 
graded commutative local Noetherian ring $R$ which is Gorenstein in a
similar style to the 0-dimensional case. 

In other words, we begin with a multiplication $m:R\tensor R \lra R$
and a unit $\eta : k\lra R$. We also need an augmentation $\eps :
R\lra l$, with kernel $\fm$, and a trace map 
$$\theta : R\tensor_k H^r_{\fm}(R)\lra \Sigma^{a+r}k, $$
for which the adjunct
$$H^r_{\fm}(R)\stackrel{\cong}\lra \Sigma^{a+r}R^{\vee} $$
is an isomorphism. In other words, if we view $\fm$ as given, a 
Gorenstein duality algebra of Krull dimension $r$ and shift $a$
is a quadruple $(R, m, \eta, \theta)$. Evidently, with this definition a
Poincar\'e duality algebra is a 0-dimensional Gorenstein duality 
algebra. 

This is supposed to correspond to a Gorenstein ring spectrum $R$ of shift $a$, so that 
$$\Hom_R(k, R)\simeq \Sigma^a k$$
and if it is orientable 
$$\Gamma_{\fm}R\simeq \Sigma^a R^{\vee}. $$

-----------------
\section{Gorenstein duality}

We have shown that $T=THH(R;k)$ is proxy-regular and Gorenstein in
many cases. The examples we have been explicit about are elementary in 
the sense that the coefficients $\pi_*(T)$ themselves are
Gorenstein. Indeed, if $\pi_*(T)$ is known to be 
Gorenstein, the spectral sequence for calculating the homotopy of 
$\Hom_T(k,T)$  immediately shows that $T$ is Gorenstein. However this
situation is unusual and even when $T$ is Gorenstein as a ring
spectrum, its homotopy $\pi_*T$ will not usually be Gorenstein as a
ring. On the other hand $\pi_*(T)$ will have some
useful duality properties. 


To describe the duality we need standard notation for 
the stable Koszul complex.  For elements $x_1, \ldots, x_n\in R_*$,
we may  form the stable Koszul complex 
$$K_{\infty}(x_1, \ldots, x_n)=\fibre(R\lra R[1/x_1])\tensor_R \cdots \tensor_R \fibre (R\lra R[1/x_n]). $$
Up to equivalence this only depends on the radical $I$ of the ideal $(x_1,
\ldots , x_n)$. Now define
$$\Gamma_{I}(M):=K_{\infty}(x_1, \ldots , x_n)\tensor_R M$$
as usual and note that there is a natural map 
$$\Gamma_{I}M\lra M. $$
By definition of local cohomology, there is a spectral sequence
$$H^*_{I}(R_*; M_*)\Rightarrow \pi_*\Gamma_I(M).$$

If  the maximal ideal $\fm$ of $R_*$ is radically
generated by finitely many elements, we may take $I=\fm$, and we note 
$$\Gamma_{\fm}M\lra M $$
is $k$-cellularization. Indeed, $\Gamma_{\fm}(R)$ is a homotopy direct limit of finite Koszul
complexes and is therefore $k$-cellular. The map is a $k$-equivalence since $\Hom_R(k,
R[1/x])\simeq 0$ for any $x\in \fm$. It follows that
$\Gamma_{\fm}(R)\simeq \cell_kR$, so the spectral
sequence reads
$$H^*_{\fm}(R_*; M_*)\Rightarrow \pi_*(\cell_kM).$$


\begin{prop}
Suppose $T$ is a proxy-regular connected $k$-algebra
and $\pi_*(T)$ is Noetherian with $\pi_0(T)=k$ and 
maximal ideal $\fm$ of positive degree elements.  If $T$ is 
Gorenstein of shift $a$, then $T$ it is automatically orientable and so 
has Gorenstein duality in the sense that 
$$\Gamma_{\fm} T \simeq \cell_k(T)\simeq \Sigma^a T^{\vee}. $$
Accordingly there is a local cohomology spectral sequence
$$H^*_{\fm}(THH_*(R;k))\Rightarrow \Sigma^a THH_*(R;k)^{\vee}. $$
\end{prop}




\begin{proof} 
First we argue that if $T$ is Gorenstein, it is automatically
orientable. Indeed, we show that $\cE=\Hom_T(k,k)$ has a
unique action on $k$. Since $T$ is a $k$-algebra, the
action of $\cE$ on $k$ factors through 
$$\cE=\Hom_T(k,k) \lra \Hom_k(k,k)=k,$$
so since $k$ is in degree 0,  the action is through $\pi_0(\cE)$. Now we observe that since $T$ is connected, 
$\Ext_{T_*}^s(k,k)$ is in degrees $\leq -s$, so that the spectral
sequence for calculating $\pi_*(\Hom_T(k,k))$ shows $\cE$ is
coconnective with $\pi_0(\cE)=k$ which must act trivially on $k$. 

We now apply Morita theory to  deduce 
$$\Gamma_{\fm}T \simeq \cell_k(T)\simeq \Sigma^a T^{\vee} $$
as required. 
\end{proof}

\begin{cor} \cite{ringlct}
Suppose $T$ has Gorenstein duality of shift $a$, that $\pi_*T$ is
Noetherian of Krull dimension $r$ and Hilbert series $p(t)=\sum_s\dim_k(T_s)t^s$.

If $\pi_*(T) $ is Cohen-Macaulay it is also Gorenstein, and the
Hilbert series satisfies 
$$p(1/t)=(-1)^rt^{r-a}p(t).$$

If $\pi_*(T) $ is almost Cohen-Macaulay it is also almost Gorenstein, and the
Hilbert series satisfies 
$$p(1/t)-(-1)^rt^{r-a}p(t)=(-1)^{r-1}(1+t)q(t) \mbox{ and } q(1/t)=(-1)^{r-1}t^{a-r+1}q(t).$$ 

In any case $\pi_*(T)$ is Gorenstein in codimension 0 and almost
Gorenstein in codimension 1. 
\end{cor}


1406.2162.extracted_bib
{AHL V.Angeltveit, M.Hill and T.Lawson ``Topological Hochschild homology of $\ell$ and $ko$.'' American J Math {132 (2010) 297-330
1406.2162.extracted_bib
{AR V.Angeltveit and J. Rognes ``Hopf algebra structure on topological Hochschild homology.'' Algebraic and Geometric Topology {5 (2005) 1223-1290
1406.2162.extracted_bib
{Ausoni C.Ausoni ``Topological Hochschild homology of connective complex $K$-theory.'' Amer. J. Math. 127 (2005), no. 6, 1261–1313.
1406.2162.extracted_bib
{AR C. Ausoni and J. Rognes, ``Algebraic K-theory of topological K-theory'', Acta Mathematica {188 (2002), 1-39.
1406.2162.extracted_bib
{Bass H. Bass ``On the ubiquity of Gorenstein rings.'' Math. Z. {82 (1963) 8–28.
1406.2162.extracted_bib
{BCS A. Blumberg, R.Cohen and C.Schlichtkrull ``The topological Hochschild homology of Thom spectra and the free loop space.'' arXiv 2008
1406.2162.extracted_bib
{Bok M. B\"okstedt, ``Topological Hochschild homology of $F_p$, and $\Z$'', Preprint, Bielefeld University.
1406.2162.extracted_bib
{DGI1 W.G.Dwyer, J.P.C.Greenlees and S.B.Iyengar ``Duality in algebra and topology'' Advances in Maths. {200 (2006) 357-402
1406.2162.extracted_bib
{ringlct J.P.C.Greenlees and G.Lyubeznik ``Rings with a local cohomology theorem and applications to cohomology rings of groups.'' J. Pure and Applied Algebra {149 (2000) 267-285.
1406.2162.extracted_bib
{HM M.J.Hopkins and M.E.Mahowald ``From elliptic curves to homotopy theory'' Preprint (1998)
1406.2162.extracted_bib
{LM A.Lindenstrauss and I. Madsen ``Topological Hochschild homology of number rings.'' Trans AMS {352 (2000) 2179-2204
1406.2162.extracted_bib
{Mahowald M.E.Mahowald ``Ring spectra which are Thom complexes.'' Duke Math. J., {46 (1979) 549–559
1406.2162.extracted_bib
{MS J.E.McClure and R.E.Staffeldt ``On the topological Hochschild homology of $bu$ I'' AJM {118 (1993) 1005-1012
1406.2162.extracted_bib
{Malm E.J.Malm `` String topology and the based loop space'' Preprint (2011) arXiv:1103.6198
1406.2162.extracted_bib
{AM Akhil Matthew ``The homology of tmf'' Preprint (2013) arXiv:1305.6100
1406.2162.extracted_bib
{Veen T.Veen ``Detecting Periodic Elements in Higher Topological Hochschild Homology Preprint (2014) arXiv:1312.5699