Algebraic {\boldmath $K$}-theory and Trace Invariants\vskip 6mm
\vskip 3mm The cyclotomic trace of B\"okstedt-Hsiang-Madsen, the subject of \linebreak B\"okstedt's lecture at the congress in Kyoto, is a map of pro-abelian groups $$K_*(A) \xrightarrow{\operatorname{tr}} \operatorname{TR}_*^{\boldsymbol{\cdot}}(A;p)$$ from Quillen's algebraic $K$-theory to a topological refinement of Connes' cyclic homology. Over the last decade, our understanding of the target and its relation to $K$-theory has been significantly advanced. This and possible future development is the topic of my lecture. The cyclotomic trace takes values in the subset fixed by an operator $F$ called the Frobenius. It is known that the induced map $$K_*(A,\mathbb{Z}/p^v) \xrightarrow{\operatorname{tr}} \operatorname{TR}_*^{\boldsymbol{\cdot}}(A;p,\mathbb{Z}/p^v)^{F=1}$$ is an isomorphism, for instance, if $A$ is a regular local $\mathbb{F}_p$-algebra, or if $A$ is a henselian discrete valuation ring of mixed characteristic $(0,p)$ with a separably closed residue field. It is possible to evaluate $K$-theory by means of the cyclotomic trace for a wider class of rings, but the precise connection becomes slightly more complicated to spell out. The pro-abelian groups $\operatorname{TR}_*^{\boldsymbol{\cdot}}(A;p)$ are typically very large. But they come equipped with a number of operators, and the combined algebraic structure is quite rigid. There is a universal example of this structure --- the de~Rham-Witt complex --- which was first considered by Bloch-Deligne-Illusie in connection with Grothendieck's crystalline cohomology. In general, the canonical map $$W_{\boldsymbol{\cdot}}\,\Omega_A^q \rightarrow \operatorname{TR}_q^{\boldsymbol{\cdot}}(A;p)$$ is an isomorphism, if $q\leq 1$, and the higher groups, too, can often be expressed in terms of the de~Rham-Witt groups. This is true, for example, if $A$ is a regular $\mathbb{F}_p$-algebra, or if $A$ is a smooth algebra over the ring of integers in a local number field. The calculation in the latter case verifies the Lichtenbaum-Quillen conjecture for local number fields, or more generally, for henselian discrete valuation fields of geometric type. \vskip 4.5mm \noindent {2000 Mathematics Subject Classification:} 19D45, 19D50, 19D55, 11S70, 14F30, 55P91. %\noindent {Keywords and Phrases:} Mirror symmetry, D-branes, Floer homology.
0505.tex
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\markboth{Algebraic  $K$-theory and Trace Invariants}{L. Hesselholt}

\title{\bf Algebraic {\boldmath $K$}-theory and Trace Invariants\vskip 6mm}

\author{Lars Hesselholt\vspace*{-0.5cm}\thanks{Massachusetts Institute
of Technology, Cambridge, MA 02139, USA. E-mail:
larsh@math.mit.edu}}

\date{\vspace{-8mm}}

\begin{document}

\maketitle

\begin{center}
\textit{$($Dedicated to Ib Madsen on his sixtieth birthday$)$}
\end{center}

\thispagestyle{first} \setcounter{page}{415}

\begin{abstract}

\vskip 3mm

The cyclotomic trace of B\"okstedt-Hsiang-Madsen, the subject of \linebreak B\"okstedt's lecture at the congress
in Kyoto, is a map of pro-abelian groups
$$K_*(A) \xrightarrow{\operatorname{tr}}
\operatorname{TR}_*^{\boldsymbol{\cdot}}(A;p)$$ from Quillen's algebraic $K$-theory to a topological refinement of
Connes' cyclic homology. Over the last decade, our understanding of the target and its relation to $K$-theory has
been significantly advanced. This and possible future development is the topic of my lecture.

The cyclotomic trace takes values in the subset fixed by an operator $F$ called the Frobenius. It is known that
the induced map
$$K_*(A,\mathbb{Z}/p^v) \xrightarrow{\operatorname{tr}}
\operatorname{TR}_*^{\boldsymbol{\cdot}}(A;p,\mathbb{Z}/p^v)^{F=1}$$ is an isomorphism, for instance, if $A$ is a
regular local $\mathbb{F}_p$-algebra, or if $A$ is a henselian discrete valuation ring of mixed characteristic
$(0,p)$ with a separably closed residue field. It is possible to evaluate $K$-theory by means of the cyclotomic
trace for a wider class of rings, but the precise connection becomes slightly more complicated to spell out.

The pro-abelian groups $\operatorname{TR}_*^{\boldsymbol{\cdot}}(A;p)$ are typically very large. But they come
equipped with a number of operators, and the combined algebraic structure is quite rigid. There is a universal
example of this structure --- the de~Rham-Witt complex
--- which was first considered by Bloch-Deligne-Illusie in
connection with Grothendieck's crystalline cohomology. In general, the canonical map
$$W_{\boldsymbol{\cdot}}\,\Omega_A^q \rightarrow
\operatorname{TR}_q^{\boldsymbol{\cdot}}(A;p)$$ is an isomorphism, if $q\leq 1$, and the higher groups, too, can
often be expressed in terms of the de~Rham-Witt groups. This is true, for example, if $A$ is a regular
$\mathbb{F}_p$-algebra, or if $A$ is a smooth algebra over the ring of integers in a local number field. The
calculation in the latter case verifies the Lichtenbaum-Quillen conjecture for local number fields, or more
generally, for henselian discrete valuation fields of geometric type.

\vskip 4.5mm

\noindent {\bf 2000 Mathematics Subject Classification:} 19D45, 19D50, 19D55, 11S70, 14F30, 55P91.

%\noindent {\bf Keywords and Phrases:} Mirror symmetry, D-branes, Floer homology.
\end{abstract}

\vskip 12mm

\section{Algebraic {\boldmath $K$}-theory}\label{section 1}\setzero
\vskip-5mm \hspace{5mm}

The algebraic $K$-theory of Quillen~\cite{quillen}, inherently, is a
multiplicative theory. Trace invariants allow the study of this theory
by embedding it in an additive theory. It is possible, by this
approach, to evaluate the $K$-theory (with coefficients) of henselian
discrete valuation fields of mixed characteristic. We first recall the
expected value of the $K$-groups of a field $k$.

The groups $K_*(k)$ form a connected anti-commutative graded ring,
there is a canonical isomorphism $\ell\:k^*\xto{\sim}K_1(k)$, and
$\ell(x)\cdot\ell(1-x)=0$. One defines the Milnor $K$-groups
$K_*^M(k)$ to be the universal example of this algebraic
structure~\cite{milnor}. The canonical map
$K_q^M(k) \to K_q(k)$
is an isomorphism, if $q\leq 2$. Let us now fix the attention on the
$K$-groups with finite coefficients. (The rational $K$-groups,
while of great interest, are of a rather different
nature~\cite{dupont1,dupontsah}.) The groups $K_*(k,\Z/m)$ form an
anti-commutative graded $\Z/m$-algebra, at least if $v_2(m)\neq 1,2$
and $v_3(m)\neq 1$. And if $\mu_m\subset k$, there is a canonical
lifting
$$\xymatrix{
{} &
{K_2(k,\Z/m)} \ar[d]^{\beta_m} \cr
{\mu_m} \ar[r]^(.47){\ell} \ar@{-->}[ur]^{b} &
{K_1(k),} \cr
}$$
which to a primitive $m$th root of unity $\zeta$ associates the Bott
element $b_{\zeta}$. Hence, in this case, there is an additional map
of graded rings
$S_{\Z/m}(\mu_m) \to K_*(k,\Z/m).$
The Beilinson-Licthenbaum conjectures predict that the combined map
$$K_*^M(k)\otimes_{\Z}S_{\Z/m}(\mu_m) \to K_*(k,\Z/m)$$
be an isomorphism of graded rings~\cite{beilinson,licthenbaum1}. The
case $m=2^v$ follows from the celebrated proof of the Milnor
conjecture by Voevodsky~\cite{voevodsky}. We here consider the case of
a henselian discrete valuation field of mixed characteristic $(0,p)$
with $p$ odd and $m=p^v$~\cite{hm2,gh2}. The groups $K_*^M(k)/m$
typically are non-zero in only finitely many degrees. Hence, above
this range, the groups $K_*(k,\Z/m)$ are two-periodic. All rings
(resp.~graded rings, resp.~monoids) considered in this paper are
assumed commutative (resp.~anti-commutative, resp.~commutative) and
unital without further notice.

\section{The de~Rham-Witt complex}\label{section 2}\setzero
\vskip-5mm \hspace{5mm}

Let $V$ be a henselian discrete
valuation ring with quotient field $K$ of characteristic zero and
residue field $k$ of odd characteristic $p$. (At this writing, we
further require that $V$ be of geometric type, i.e. that $V$ be
the henselian local ring at the generic point of the special fiber
of a smooth scheme over a henselian discrete valuation ring
$V_0\subset V$ with \emph{perfect} residue field.) A first example of
a trace map is provided by the logarithmic derivative
$$K_*^M(K) \to \Omega_{(V,M)}^*$$
which to the symbol $\{a_1,\dots,a_q\}$ associates the form $d\log
a_1\dots d\log a_q$. The right hand side is the de~Rham complex
with log poles in the sense of Kato~\cite{kato}: A log ring $(A,M)$ is a
ring $A$ and a map of monoids $\alpha\:M\to (A,\,\cdot\,)$; a
log differential graded ring $(E^*,M)$ is a differential graded ring
$E^*$ together with maps of monoids $\alpha\:M\to (E^0,\cdot)$ and
$d\log\:M\to (E^1,+)$ such that $d\circ d\log=0$ and such that
$d\alpha(a)=\alpha(a)d\log a$ for all $a\in M$; the de~Rham complex
$\Omega_{(A,M)}^*$ is  the universal log differential graded ring with
underlying log ring $(A,M)$. We will always consider the ring $V$ with
the \emph{canonical} log structure
$$\alpha\:M=V\cap K^* \hookrightarrow V.$$
(In this case, there are natural short-exact sequences
$$0 \to \Omega_V^q \to \Omega_{(V,M)}^q \to \Omega_k^{q-1} \to 0.)$$
The logarithmic derivative, however, is far from injective. It turns
out that this can be rectified by incorborating the Witt vector
construction which we now recall.

%Every $p$-adic integer $a\in\Zp$ has a unique $p$-adic expansion of
%the form
%$$a=\sum_{i\geq 0}a_ip^i.$$
%The usual choice of digits are the integers $0,1,\dots,p-1$, but any
%choice of coset representatives of $p\Zp\subset\Zp$ will do. It is
%natural to ask for a less arbitrary choice. To give a set of coset
%represenatives is the same as to give a set-theoretic section of the
%canonical projection $\Zp\to\Fp$. It turns out that there is a unique
%multiplicative section --- the Teichm\"uller character --- given by
%$$[\phantom{x}]\:\Fp\to\Zp,\hskip5mm
%[x]=\lim_{s\to\infty}\tilde{x}^{p^s},$$
%where $\tilde{x}\in\Zp$ is any lifting of $x\in\Fp$. One may
%reasonably expect the $p$-adic integers $[x]$ to be a better choice of
%digits. And, indeed, with this choice, the digits of the sum $a+b$
%(resp. the additive inverse $-a$, resp. the product $a\cdot b$) are
%given by polynomials in the digits of $a$ and $b$. Moreover, these
%polynomials have integers coefficients. Hence, they can be used, for
%every ring $A$, to define a (new) ring structure on the set of
%``vectors''
%$$W(A)=\{(a_0,a_1,\dots)\mid a_i\in A\}.$$
%This is the ring of Witt vectors in $A$. The projection $W(A)\to A$,
%which to $(a_0,a_1,\dots)$ associates $a_0$, is a ring homomorphism
%with a multiplicative section
%$$[\phantom{x}]\:A \to W(A),\hskip5mm
%[a]=(a,0,0,\dots).$$
%
%The ring $W(A)$ is equal to the inverse limit of the rings $W_n(A)$ of
%Witt vectors of length $n$. But rather than forming the limit, we
%shall consider the limit system of rings $W\s(A)$ as a pro-ring. There
%is a natural map of pro-rings
%$$F\: W\s(A) \to W_{\boldsymbol{\cdot}-1}(A) \hskip 8 mm
%\text{(Frobenius)}$$
%and a natural map of $W\s(A)$-modules
%$$V\: F_*\,W_{\boldsymbol{\cdot}-1}(A) \to W\s(A) \hskip 6 mm
%\text{(Verschiebung)}.$$
%The former is given, for $\Fp$-algebras, by
%$F(a_0,\dots,a_{n-1})=(a_0^p,\dots,a_{n-2}^p)$; the latter is given,
%in general, by $V(a_0,\dots,a_{n-2})=(0,a_0,\dots,a_{n-2})$, and
%$FV=p$. Finally, we note that if $(A,M)$ is a log ring, then the
%composite
%$$M \xto{\alpha} A \xto{[\phantom{a}]} W\s(A)$$
%makes $(W\s(A),M)$ a pro-log ring.

The ring of Witt vectors associated with a ring $A$ is the set of
``vectors''
$$W(A)=\{(a_0,a_1,\dots)\mid a_i\in A\}$$
with a new ring structure, see~\cite{hm3}. The ring operations are
polynomial in the coordinates. The projection $W(A)\to A$, which to
$(a_0,a_1,\dots)$ associates $a_0$, is a natural ring homomorphism
with the unique natural multiplicative section
$$[\phantom{x}]\:A \to W(A), \hskip5mm [a]=(a,0,0,\dots).$$
If $F$ is a perfect field of characteristic $p>0$, then $W(F)$ is the
unique (up to unique isomorphism) complete discrete valuation ring
of mixed characteristic $(0,p)$ such that $W(F)/p\xto{\sim}F$. In
general, the ring $W(A)$ is equal to the inverse limit of the rings
$W_n(A)$ of Witt vectors of length $n$. But rather than forming the
limit, we shall consider the limit system of rings $W\s(A)$ as a
pro-ring. There is a natural map of pro-rings $F\: W\s(A) \to
W_{\boldsymbol{\cdot}-1}(A)$, called the Frobenius, and a natural map
of $W\s(A)$-modules $V\: F_*\,W_{\boldsymbol{\cdot}-1}(A) \to W\s(A)$,
called the Verschiebung. The former is given, as the ring structure,
by certain polynomials in the coordinates; the latter is given by
$V(a_0,\dots,a_{n-2})=(0,a_0,\dots,a_{n-2})$, and $FV=p$. Finally, we
note that if $(A,M)$ is a log ring, then the composite
$$M \xto{\alpha} A \xto{[\phantom{a}]} W\s(A)$$
makes $(W\s(A),M)$ a pro-log ring.

There is a natural way to combine differential forms and Witt
vectors; the result is called the de~Rham-Witt complex. It was
considered first for $\Fp$-algebras by
Bloch-Deligne-Illusie~\cite{bloch,illusie} in  connection with the
crystalline cohomology of Berthelot-Grothendieck~\cite{berthelot}. A
generalization to log-$\Fp$-algebras was constructed by
Hyodo-Kato~\cite{hyodokato}. The following extension to
log-$\Z_{(p)}$-algebras was obtained in collaboration with Ib
Madsen~\cite{hm3,hm2}: Let $(A,M)$ be a log ring such that $A$ is a
$\Z_{(p)}$-algebra with $p$ odd. A \emph{Witt complex} over $(A,M)$
is:
\vskip1mm

(i) a pro-log differential graded ring $(E\s^*,M_E)$ and a map of
pro-log rings
$$\lambda\:(W\s(A),M)\to (E\s^0,M_E);$$

(ii) a map of pro-log graded rings
$$F\:E\s^*\to E_{\boldsymbol{\cdot}-1}^*,$$
such that $\lambda F=F\lambda$ and such that
$$\begin{aligned}
Fd\log_n a & = d\log_{n-1}a, \hskip15.2mm
\text{for all $a\in M$,} \cr
Fd\,\lambda[a]_n & =\lambda[a]_{n-1}^{p-1}d\,\lambda[a]_{n-1},
\hskip5mm \text{for all $a\in A$;}\cr
\end{aligned}$$

(iii) a map of pro-graded modules over the pro-graded ring $E^*\s$,
$$V\:F_*\,E_{\boldsymbol{\cdot}-1}^*\to E\s^*,$$
such that $\lambda V=V\lambda$, $FV=p$ and $FdV=d$.

A map of Witt complexes over $(A,M)$ is a map of pro-log differential
graded rings which commutes with the maps $\lambda$, $F$ and $V$.
Standard category theory shows that there exists a universal Witt
complex over $(A,M)$. This, by defintion, is the de~Rham-Witt complex
$W\s\,\Omega_{(A,M)}^*$. (The canonical maps $W\s(A) \to
W\s\,\Omega_{(A,M)}^0$ and $\Omega_{(A,M)}^* \to W_1\,\Omega_{(A,M)}^*$
are isomorphisms, so the construction really does combine differential
forms and Witt vectors.) We  lift the logarithmic derivative to
a map
$$K_q^M(K) \to W_n\,\Omega_{(V,M)}^q$$
which to the symbol $\{a_1,\dots,a_q\}$ associates $d\log_na_1\dots
d\log_na_q$. This trace map better captures the Milnor $K$-groups.
Indeed, the following result was obtained in collaboration with
Thomas Geisser~\cite{gh2}:

\begin{theorem}\label{kdrw} Suppose that $\mu_{p^v}\subset K$ and that
$k$ is separably closed. Then the trace map induces an isomorphism of
pro-abelian groups
$$K_q^M(K)/p^v \xto{\sim}
\big(W\s\,\Omega_{(V,M)}^q/p^v\big)^{F=1}.$$
\end{theorem}

To prove this, we first show that $W_n\,\Omega_{(V,M)}^q/p$ has a
(non-canonical) $k$-vector space structure and find an explicit
basis. The dimension is
$$\operatorname{dim}_k\big(W_n\,\Omega_{(V,M)}^q/p\big) =
\binom{r+1}{q}\,e\,\sum_{s=0}^{n-1}p^{rs},$$
where $|k:k^p|=p^r$ and $e$ the ramification index of $K$. It is not
difficult to see that this is an upper bound for the dimension. The
proof that it is also a lower bound is more involved and uses a
formula for the de~Rham-Witt complex of a polynomial extension by
Madsen and the author~\cite{hm3}. We then evaluate the kernel of $1-F$
and compare with the calculation of $K_q^M(K)/p$ by
Kato~\cite{kato1,blochkato}. The assumption that the residue field $k$
be separably closed is not essential. In the general case, one instead
has a short-exact sequence
$$0 \to \big(W\s\,\Omega_{(V,M)}^{q-1}\otimes\mu_{p^v}\big)_{F=1}
\to K_q^M(K)/p^v \to \big(W\s\,\Omega_{(V,M)}^q/p^v\big)^{F=1} \to
0,$$
where the superscript (resp.~subscript) ``$F=1$'' indicates Frobenius
invariants (resp.~coinvariants).

We discuss a global version of theorem~\ref{kdrw}. Let $V_0$ be a
henselian discrete valuation ring with quotient field $K_0$ of
characteristic zero and \emph{perfect} residue field $k_0$ of odd
characteristic $p$. Let $\mathfrak{X}$ be a smooth $V_0$-scheme, and
let $i$ (resp. $j$) denote the inclusion of the special
(resp. generic) fiber as in the cartesian diagram
$$\xymatrix{
{X\;} \ar@{^{(}->}[r]^{j} \ar[d] &
{\mathfrak{X}} \ar[d]^{f} &
{\;Y} \ar@{_{(}->}[l]_{i} \ar[d] \cr
{\Spec K_0\;} \ar@{^{(}->}[r] &
{\Spec V_0} &
{\;\Spec k_0.} \ar@{_{(}->}[l] \cr
}$$
Suppose that $\mu_{p^v}\subset K_0$. Then the proof of
theorem~\ref{kdrw} shows that there is a short-exact sequence of
sheaves of pro-abelian groups on $Y$ for the \'etale topology
$$0 \to i^*R^qj_*\Z/p^v(q) \to
i^*(W\s\,\Omega_{(\mathfrak{X},M)}^q/p^v) \xto{1-F}
i^*(W\s\,\Omega_{(\mathfrak{X},M)}^q/p^v) \to 0.$$
The left hand term is the sheaf of $p$-adic vanishing cycles.

\section{The cyclotomic trace}\label{section 3}\setzero
\vskip-5mm \hspace{5mm}

We now turn to Quillen $K$-theory. The
analog of the logarithmic derivative is the topological Dennis trace
with values in topological Hochschild homology,
$$K_*(\mathcal{C}) \to \THH_*(\mathcal{C}),$$
defined by B\"okstedt~\cite{bokstedt}. It is a refinement of earlier
trace maps by Dennis~\cite{dennis} and Waldhausen~\cite{waldhausen1}.
% see also~\cite{kassel,dundasmccarthy1}.
We will use a variant of the construction due to
Dundas-McCarthy~\cite{dundasmccarthy,mccarthy} that can be
applied to a category with cofibrations and weak equivalences in the
sense of Waldhausen~\cite{waldhausen}. The
category $\mathcal{C}$ we consider is the category of bounded chain
complexes of finitely generated projective $V$-modules. The
cofibrations are the degree-wise monomorphisms, and the weak
equivalences are the chain maps $C\to C'$ such that
$K\otimes_VC\to K\otimes_VC'$ is a quasi-isomorphism. The
$K$-theory of this category is canonically isomorphic to Quillen's
$K$-theory of the field $K$. We showed in~\cite{hm2} that the groups
$$\THH_*(V|K)=\THH_*(\mathcal{C})$$
form a log differential graded ring with underlying log ring
$(V,M)$, where the structure map $d\log$ is given by the composite
$$M=V\cap K^* \xto{\ell} K_1(K) \to \THH_1(V|K).$$
The canonical map from the de~Rham complex
$$\Omega_{(V,M)}^q \to \THH_q(V|K)$$
is compatible with the trace maps and is an isomorphism, if $q\leq 2$.
The topological Dennis trace, again, is far from injective. This can be
rectified by a construction which, in retrospect, can be seen as
incorporating Witt vectors. The result is the cyclotomic trace of
B\"okstedt-Hsiang-Madsen~\cite{bhm} which we now recall. The reader is
referred to~\cite{hm2,gh,dundasmccarthy} for details.

The topological Dennis trace, we recall, is defined as the map of
homotopy groups induced from a continuous map of spaces
$$K(\mathcal{C}) \to \THH(\mathcal{C}).$$
As a consequence of Connes' theory of cyclic sets, the right hand
space is equipped with a continuous action by the circle group
$\mathbb{T}$. Moreover, the image of the trace map is point-wise fixed
by the $\mathbb{T}$-action. Let
$$\TR^n(\mathcal{C};p)=\THH(\mathcal{C})^{C_{p^{n-1}}}$$
be the subset fixed by the subgroup $C_{p^{n-1}}\subset\mathbb{T}$ of
the indicated order. It turns out that, as $n$ and $q$ varies, the
homotopy groups
$$\TR_q^n(V|K;p)=\pi_q(\TR^n(\mathcal{C};p))$$
form a Witt complex over $(V,M)$; see~\cite{hm,h,hm2}. The map $F$ is
induced from the obvious inclusion map, and the map $V$ is the
accompanying transfer map. The structure maps in the limit system and
the map $\lambda$, however, are more difficult to define. The former
was defined in~\cite{bhm} and the latter in~\cite{hm}. The topological
Dennis trace induces a map of pro-abelian groups
$$K_q(K) \to \TR_q\cs(V|K;p).$$
This is the cyclotomic trace. It takes values in the subset fixed
by the Frobenius operator. The canonical map
$$W_n\Omega_{(V,M)}^q \to \TR_q^n(V|K;p)$$
is compatible with the trace maps from Milnor $K$-theory and Quillen
$K$-theory, respectively, and is an isomorphism, if $q\leq 2$. The
following is a combination of results obtained in collaboration with
Thomas Geisser~\cite{gh} and Ib Madsen~\cite{hm,hm2}.

\begin{theorem}\label{trace}Suppose that $k$ is separably closed. Then
the cyclotomic trace induces an isomorphism of pro-abelian groups
$$K_q(K,\Z/p^v) \xto{\sim} \TR_q\cs(V|K;p,\Z/p^v)^{F=1}.$$
\end{theorem}

We briefly outline the steps in the proof: We proved in~\cite{gh} that
the sequence
$$0 \to K_q(k,\Z/p^v) \to \TR_q\cs(k;p,\Z/p^v) \xto{1-F}
\TR_q\cs(k;p,\Z/p^v) \to 0$$
is exact. This uses~\cite{blochkato, geisserlevine,h}. Given this, the
theorem by McCarthy~\cite{mccarthy1} that for nilpotent extensions,
relative $K$-theory and relative topological cyclic homology agree,
and the continuity results of Suslin~\cite{suslin1} for $K$-theory and
Madsen and the author~\cite{hm} for $\TR$ show that also the sequence
$$0 \to K_q(V,\Z/p^v) \to \TR_q\cs(V;p,\Z/p^v) \xto{1-F}
\TR_q\cs(V;p,\Z/p^v) \to 0$$
is exact. Theorem~\ref{trace} follows by comparing the localization
sequence of Quillen~\cite{quillen}
$$\cdots\to K_q(k,\Z/p^v) \xto{i^!} K_q(V,\Z/p^v) \xto{j_*}
K_q(K,\Z/p^v) \to \cdots$$
to the corresponding sequence by Madsen and the author~\cite{hm2}
$$\cdots\to \TR_q^n(k;p,\Z/p^v) \xto{i^!} \TR_q^n(V;p,\Z/p^v)
\xto{j_*} \TR_q(V|K;p,\Z/p^v) \to \cdots.$$
Again, the assumption in the statement of theorem~\ref{trace} that the
residue field $k$ be separably closed is not essential. The general
statement will be given below. It is also not necessary for
theorem~\ref{trace} to assume that $V$ be of geometric type.

\section{The Tate spectral sequence}\label{section 4}\setzero
\vskip-5mm \hspace{5mm}

If $G$ is a finite group and $X$ a
$G$-space, it is usually not possible to evaluate the groups
$\pi_*(X^G)$ from knowledge of the $G$-modules $\pi_*(X)$. At first
glance, this is the problem that one faces in evaluating the groups
$$\TR_q^n(\mathcal{C};p) =
\pi_q\big(\THH(\mathcal{C})^{C_{p^{n-1}}}\big).$$
However, the mapping fiber of the structure map
$\TR^n(\mathcal{C};p) \to \TR^{n-1}(\mathcal{C};p)$, it turns out,
is given by the Borel construction
$\borel(C_{p^{n-1}},\THH(\mathcal{C}))$ whose homotopy groups are the
abutment of a (first quadrant) spectral sequence
$$E_{s,t}^2=H_s(C_{p^{n-1}},\THH_t(\mathcal{C}))
\Rightarrow \pi_{s+t}\borel(C_{p^{n-1}},\THH(\mathcal{C})).$$
This suggests that the groups $\TR_q^n(\mathcal{C};p)$ can be
evaluated inductively starting from the case $n=1$. However, it is
generally difficult to carry out the induction step. In addition, the
absence of a multiplicative structure makes the spectral sequence
above difficult to solve. The main vehicle to overcome these
problems, first employed by B\"okstedt-Madsen
in~\cite{bokstedtmadsen}, is the following diagram of fiber sequences
$$\xymatrix{
{\borel(C_{p^{n-1}},\THH(\mathcal{C}))} \ar[r] \ar@{=}[d] &
{\TR^n(\mathcal{C};p)} \ar[r] \ar[d]^{\Gamma} &
{\TR^{n-1}(\mathcal{C};p)} \ar[d]^{\hat{\Gamma}} \cr
{\borel(C_{p^{n-1}},\THH(\mathcal{C}))} \ar[r] &
{\coborel(C_{p^{n-1}},\THH(\mathcal{C}))} \ar[r] &
{\tate(C_{p^{n-1}},\THH(\mathcal{C}))} \cr
}$$
together with a multiplicative (upper half-plane) spectral sequence
$$E_{s,t}^2 = \hat{H}^{-s}(C_{p^{n-1}},\THH_t(\mathcal{C}))
\Rightarrow \pi_{s+t}\tate(C_{p^{n-1}},\THH(\mathcal{C}))$$
starting from the Tate cohomology of the (trivial)
$C_{p^{n-1}}$-module $\THH_t(\mathcal{C})$. The lower fiber sequence
is the Tate sequence; see Greenlees and May~\cite{greenleesmay}
or~\cite{hm2}. In favorable cases, the maps $\Gamma$ and
$\hat{\Gamma}$ induce isomorphisms of homotopy groups in non-negative
degrees. Indeed, this is true in the case at hand (if $k$ is
perfect). The differential structure of the spectral sequence
$$E_{s,t}^2 = \hat{H}^{-s}(C_{p^{n-1}},\THH_t(V|K,\Z/p))
\Rightarrow \pi_{s+t}(\tate(C_{p^{n-1}},\THH(V|K)),\Z/p)$$
was determined in collaboration with Ib Madsen~\cite{hm2} in the
case where the residue field $k$ is perfect. This is the main
calculational result of the work reported here. The following
result, for perfect $k$, is a rather immediate consequence. The
extension to non-perfect $k$ is given in~\cite{hm3}.

\begin{theorem}\label{main}Suppose that $\mu_{p^v}\subset K$. Then the
canonical map is an isomorphism of pro-abelian groups
$$W\s\,\Omega_{(V,M)}^*\otimes_{\Z}S_{\Z/p^v}(\mu_{p^v}) \xto{\sim}
\TR_*\cs(V|K;p,\Z/p^v).$$
\end{theorem}

We can now state the general version of theorem~\ref{trace} which does
not require that the residue field $k$ be separably closed. The second
tensor factor on the left hand side in the statement of
theorem~\ref{main} is the symmetric algebra on the $\Z/p^v$-module
$\mu_{p^v}$, which is free of rank one. Spelling out the statement for
the group in degree $q$, we get an isomorphism of pro-abelian groups
$$\bigoplus_{s\geq
0}W\s\,\Omega_{(V,M)}^{q-2s}\otimes\mu_{p^v}^{\otimes s}
\xto{\sim}\TR_q\cs(V|K;p,\Z/p^v).$$
In the case of a separably closed residue field, theorem~\ref{trace}
idenfies the Frobenius fixed set of the common pro-abelian group
with $K_q(K,\Z/p^v)$. In the general case, one has instead a
short-exact sequence
$$0 \to \bigoplus_{s\geq 1}
\big(W\s\,\Omega_{(V,M)}^{q+1-2s} \otimes
\mu_{p^v}^{\otimes s}\big)_{F=1}\!\! \to
K_q(K,\Z/p^v) \to
\bigoplus_{s\geq 0}
\big(W\s\,\Omega_{(V,M)}^{q-2s} \otimes
\mu_{p^v}^{\otimes s}\big)^{F=1}\!\! \to 0,$$
valid for all integers $q$. (There is a similar sequence for the
topological cyclic homology group
$\operatorname{TC}_q\cs(V|K;p,\Z/p^v)$~\cite{hm2} which includes the
summand ``$s=0$'' on the left.) Comparing with the general version of
theorem~\ref{kdrw}, we obtain the following result promised
earlier~\cite{hm2,gh2}.

\begin{theorem}\label{bl}Suppose that $\mu_{p^v}\subset K$. Then the
canonical map
$$K_*^M(K)\otimes_{\Z}S_{\Z/p^v}(\mu_{p^v}) \xto{\sim}
K_*(K,\Z/p^v)$$
is an isomorphism.
\end{theorem}

\section{Galois descent}\label{section 5}\setzero
\vskip-5mm \hspace{5mm}

We now assume that the residue field $k$ be perfect. In homotopy
theoretic terms, theorem~\ref{main} states that the pro-spectrum
$\TR\cs(V|K;p)$ is equivalent to the $(-1)$-connected cover of its
localization with respect to complex periodic $K$-theory,
see~\cite{bousfield}. This suggests the possibility
of completely understanding the homotopy type of this pro-spectrum. We
expect that this, in turn, is closely related to the following question.
Let $\bar{K}$ be an algebraic closure of $K$ with Galois group $G_K$,
and let $\bar{V}$ be the integral closure of $V$ in $\bar{K}$. (The
ring $\bar{V}$ is a valuation ring with value group the additive group
of rational numbers.)

\begin{conjecture}If $k$ is perfect then for all $q>0$, the canonical map
$$\TR_q\cs(V|K;p,\Qp/\Zp) \to
\TR_q\cs(\bar{V}|\bar{K};p,\Qp/\Zp)^{G_K}$$
be an isomorphism of pro-abelian groups and that the higher continuous
cohomology groups
$H_{\operatorname{cont}}^i(G_K,\TR_q^n(\bar{V}|\bar{K};p,\Qp/\Zp))$
vanish.
\end{conjecture}

It follows from Tate~\cite{tate} that the groups
$H_{\operatorname{cont}}^i(G_K,\TR_q^n(\bar{V}|\bar{K};p,\Qp))$
vanish for $i\geq 0$ and $q>0$. One may hope that these methods will
help shed some light on the structure of the groups
$H_{\operatorname{cont}}^i(G_K,\TR_q^n(\bar{V}|\bar{K};p,\Qp/\Zp))$.
We now describe the structure of these $G_K$-modules; proofs will
appear elsewhere.

The group $\TR_q^n(\bar{V}|\bar{K};p,\Qp/\Zp)$ is divisible, if $q>0$,
and uniquely divisible, if $q>0$ and even. The Tate module
$T_p\TR_1^n(\bar{V}|\bar{K};p)$ is a free module of rank one over
$\TR_0^n(\bar{V}|\bar{K};p,\Zp)$, and the canonical map an
isomorphism:
$$S_{\TR_0^n(\bar{V}|\bar{K};p,\Zp)}(T_p\TR_1^n(\bar{V}|\bar{K};p))
\xto{\sim}\TR_*^n(\bar{V}|\bar{K};p,\Zp)$$
(note that  $\TR_q^n(\bar{V}|\bar{K};p,\Qp/\Zp) \xto{\sim}
\TR_q^n(\bar{V}|\bar{K};p,\Zp)\otimes\Qp/\Zp$). We note the formal
analogy with the results on $K_*(\bar{K})$ by
Suslin~\cite{suslin,suslin1}.

The structure of the ring
$\TR_0^n(\bar{V}|\bar{K};p,\Zp)=W_n(\bar{V})\f$ is well-understood
(unlike that of $W_n(V)$): Following Fontaine~\cite{fontaine}, we
let $R_{\bar{V}}$ be the inverse limit of the diagram $\bar{V}/p
\leftarrow \bar{V}/p \leftarrow \cdots$ with the Frobenius as
structure map. This is a perfect $\Fp$-algebra and an integrally
closed domain whose quotient field is algebraically closed. There is a
surjective ring homomorphism
$\theta_n\: W(R_{\bar{V}}) \twoheadrightarrow W_n(\bar{V})\f$
whose kernel is a principal ideal. If $\e=\{\e^{(v)}\}_{v\geq 1}$ is a
compatible sequence of primitive $p^{v-1}$st roots of unity considered
as an element of $R_{\bar{V}}$, and if $\e_n$ is the unique $p^n$th
root of $\e$, then $([\e]-1)/([\e_n]-1)$ is a generator. Moreover, as
$n$ varies, the maps $\theta_n$ constitute a map of pro-rings
compatible with the Frobenius maps.

The Bott element $b_{\e,n}\in T_p\TR_1^n(\bar{V}|\bar{K};p)$
determined by the sequence $\e$ is not a generator (so the statement
of theorem~\ref{main} is not valid for $\bar{K}$). Instead there is a
generator $\alpha_{\e,n}$ such that
$b_{\e,n}=([\e_n]-1)\cdot\alpha_{\e,n}$. The structure map of the
pro-abelian group $T_p\TR_1\cs(\bar{V}|\bar{K};p)$ (resp. the
Frobenius) takes $\alpha_{\e,n}$ to
$([\e_{n-1}]-1)/([\e_n]-1)\cdot\alpha_{\e,n-1}$ (resp. to
$\alpha_{\e,n-1}$), and the action of the Galois group is given by
$$\alpha_{\e,n}^{\sigma} =
\chi(\sigma)\frac{[\e_n]-1}{[\e_n^{\sigma}]-1}\cdot\alpha_{\e,n},$$
where $\chi\:G_K \to \operatorname{Aut}(\mu_{p^{\infty}})=\Zp^*$ is
the cyclotomic character.

\bigskip

\noindent{\bf Acknowledgments} The research reported here was
supported in part by grants from the National Science Foundation
and by an Alfred P.~Sloan Fellowship.

\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\begin{thebibliography}{10}

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\bibitem{berthelot}
P.~Berthelot, \emph{Cohomologie cristalline des schemas de caracteristique
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\bibitem{bloch}
S.~Bloch, \emph{Algebraic {$K$}-theory and crystalline cohomology}, Publ. Math.
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\bibitem{blochkato}
S.~Bloch and K.~Kato, \emph{$p$-adic etale cohomology}, Publ. Math. IHES
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\bibitem{bokstedt}
M.~B\"okstedt, \emph{Topological {H}ochschild homology}, Preprint 1985,
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\bibitem{bhm}
M.~B\"okstedt, W.-C. Hsiang, and I.~Madsen, \emph{The cyclotomic trace and
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\bibitem{bokstedtmadsen}
M.~B\"okstedt and I.~Madsen, \emph{Topological cyclic homology of the
  integers}, {$K$}-theory (Strasbourg, 1992), Ast\'erisque, vol. 226, 1994,
  57--143.

\bibitem{bousfield}
A.~K. Bousfield, \emph{The localization of spectra with respect to homology},
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\bibitem{dennis}
K.~Dennis, \emph{Algebraic {$K$}-theory and {H}ochschild homology}, Algebraic
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\bibitem{dundasmccarthy}
B.~I. Dundas and R.~McCarthy, \emph{Topological {H}ochschild homology of ring
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\bibitem{dupont1}
J.~L. Dupont, \emph{Algebra of polytopes and homology of flag complexes}, Osaka
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\bibitem{dupontsah}
J.~L. Dupont and C.-H. Sah, \emph{Scissors congruences, {II}}, J. Pure Appl.
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\bibitem{fontaine}
J.-M. Fontaine, \emph{Le corps des p\'eriodes {$p$}-adiques}, P\'eriodes
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  59--111.

\bibitem{gh2}
T.~Geisser and L.~Hesselholt, \emph{On the {$K$}-theory of a henselian discrete
  valuation field with non-perfect residue field}, Preprint 2002.

\bibitem{gh}
\bysame, \emph{Topological cyclic homology of schemes}, {$K$}-theory (Seattle,
  1997), Proc. Symp. Pure Math., vol.~67, 1999, 41--87.

\bibitem{geisserlevine}
T.~Geisser and M.~Levine, \emph{The {$K$}-theory of fields in characteristic
  {$p$}}, Invent. Math. \textbf{139} (2000), 459--493.

\bibitem{greenleesmay}
J.~P.~C. Greenlees and J.~P. May, \emph{Generalized {T}ate cohomology}, vol.
  113, Mem. Amer. Math. Soc., no. 543, 1995.

\bibitem{h}
L.~Hesselholt, \emph{On the $p$-typical curves in {Q}uillen's {$K$}-theory},
  Acta Math. \textbf{177} (1997), 1--53.

\bibitem{hm3}
L.~Hesselholt and I.~Madsen, \emph{On the de {R}ham-{W}itt complex in mixed
  characteristic}, Preprint 2002.

\bibitem{hm2}
\bysame, \emph{On the {$K$}-theory of local fields}, Ann. Math. (to appear).

\bibitem{hm}
\bysame, \emph{On the {$K$}-theory of finite algebras over {W}itt vectors of
  perfect fields}, Topology \textbf{36} (1997), 29--102.

\bibitem{hyodokato}
O.~Hyodo and K.~Kato, \emph{Semi-stable reduction and crystalline cohomology
  with logarithmic poles}, P\'eriodes $p$-adiques, Asterisque, vol. 223, 1994,
  221--268.

\bibitem{illusie}
L.~Illusie, \emph{Complexe de de {R}ham-{W}itt et cohomologie cristalline},
  Ann. Scient. \'Ec. Norm. Sup. (4) \textbf{12} (1979), 501--661.

\bibitem{kato1}
K.~Kato, \emph{Galois cohomology of complete discrete valuation fields},
  Algebraic {$K$}-theory, {P}art {II} ({O}berwolfach, 1980), Lecture Notes in
  Math., vol. 967, Springer, {B}erlin-{N}ew {Y}ork, 1982, 215--238.

\bibitem{kato}
\bysame, \emph{Logarithmic structures of {F}ontaine-{I}llusie}, Algebraic
  analysis, geometry, and number theory, Proceedings of the JAMI Inaugural
  Conference (Baltimore, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989,
  191--224.

\bibitem{licthenbaum1}
S.~Licthenbaum, \emph{Values of zeta-functions at non-negative integers},
  Number theory, Lecture Notes in Math., vol. 1068, Springer-Verlag, 1983,
  127--138.

\bibitem{mccarthy}
R.~McCarthy, \emph{The cyclic homology of an exact category}, J. Pure Appl.
  Alg. \textbf{93} (1994), 251--296.

\bibitem{mccarthy1}
\bysame, \emph{Relative algebraic {$K$}-theory and topological cyclic
  homology}, Acta Math. \textbf{179} (1997), 197--222.

\bibitem{milnor}
J.~Milnor, \emph{Algebraic {$K$}-theory and quadratic forms}, Invent. Math.
  \textbf{9} (1970), 318--344.

\bibitem{quillen}
D.~Quillen, \emph{Higher algebraic {$K$}-theory {I}}, Algebraic {$K$}-theory I:
  Higher {$K$}-theories (Battelle Memorial Inst., Seattle, Washington, 1972),
  Lecture Notes in Math., vol. 341, Springer-Verlag, 1973.

\bibitem{suslin}
A.~A. Suslin, \emph{On the {$K$}-theory of algebraically closed fields},
  Invent. Math. \textbf{73} (1983), 241--245.

\bibitem{suslin1}
\bysame, \emph{On the {$K$}-theory of local fields}, J. Pure Appl. Alg.
  \textbf{34} (1984), 304--318.

\bibitem{tate}
J.~Tate, \emph{{$p$}-divisible groups}, Proc. conf. local fields (Driebergen,
  1966), Springer-Verlag, 1967, 158--183.

\bibitem{voevodsky}
V.~Voevodsky, \emph{The {M}ilnor conjecture}, Preprint 1996, Max Planck
  Institut, Bonn.

\bibitem{waldhausen1}
F.~Waldhausen, \emph{Algebraic {$K$}-theory of topological spaces. {II}},
  Algebraic topology (Aarhus, 1978), Lecture Notes in Math., vol. 763,
  Springer-Verlag, 1979, 356--394.

\bibitem{waldhausen}
\bysame, \emph{Algebraic {$K$}-theory of spaces}, Algebraic and geometric
  topology (New Brunswick, N.J., 1983), Lecture Notes in Math., vol. 1126,
  Springer-Verlag, 1985, 318--419.

\end{thebibliography}

\label{lastpage}

\end{document}

math0304297.extracted_bib
\begin{thebibliography}{10}

\bibitem{beilinson}
A.~A. Beilinson, \emph{Height pairing between algebraic cycles}, {$K$}-theory,
  arithmetic and geometry (Moscow, 1984--1986), Lecture Notes in Math., vol.
  1289, Springer-Verlag, 1987, 1--25.

\bibitem{berthelot}
P.~Berthelot, \emph{Cohomologie cristalline des schemas de caracteristique
  {$p>0$}}, Lecture Notes in Math., vol. 407, Springer-Verlag, 1974.

\bibitem{bloch}
S.~Bloch, \emph{Algebraic {$K$}-theory and crystalline cohomology}, Publ. Math.
  I.H.E.S. \textbf{47} (1977), 187--268.

\bibitem{blochkato}
S.~Bloch and K.~Kato, \emph{$p$-adic etale cohomology}, Publ. Math. IHES
  \textbf{63} (1986), 107--152.

\bibitem{bokstedt}
M.~B\"okstedt, \emph{Topological {H}ochschild homology}, Preprint 1985,
  Universit\"at Bielefeld.

\bibitem{bhm}
M.~B\"okstedt, W.-C. Hsiang, and I.~Madsen, \emph{The cyclotomic trace and
  algebraic {$K$}-theory of spaces}, Invent. Math. \textbf{111} (1993),
  465--540.

\bibitem{bokstedtmadsen}
M.~B\"okstedt and I.~Madsen, \emph{Topological cyclic homology of the
  integers}, {$K$}-theory (Strasbourg, 1992), Ast\'erisque, vol. 226, 1994,
  57--143.

\bibitem{bousfield}
A.~K. Bousfield, \emph{The localization of spectra with respect to homology},
  Topology \textbf{18} (1979), 257--281.

\bibitem{dennis}
K.~Dennis, \emph{Algebraic {$K$}-theory and {H}ochschild homology}, Algebraic
  {$K$}-theory, Evanston, IL., 1976 (unpublished lecture).

\bibitem{dundasmccarthy}
B.~I. Dundas and R.~McCarthy, \emph{Topological {H}ochschild homology of ring
  functors and exact categories}, J. Pure Appl. Alg. \textbf{109} (1996),
  231--294.

\bibitem{dupont1}
J.~L. Dupont, \emph{Algebra of polytopes and homology of flag complexes}, Osaka
  J. Math. \textbf{19} (1982), 599--641.

\bibitem{dupontsah}
J.~L. Dupont and C.-H. Sah, \emph{Scissors congruences, {II}}, J. Pure Appl.
  Alg. \textbf{25} (1982), 159--195.

\bibitem{fontaine}
J.-M. Fontaine, \emph{Le corps des p\'eriodes {$p$}-adiques}, P\'eriodes
  {$p$}-adiques (S\'eminaire de {B}ures, 1988), Ast\'erisque, vol. 223, 1994,
  59--111.

\bibitem{gh2}
T.~Geisser and L.~Hesselholt, \emph{On the {$K$}-theory of a henselian discrete
  valuation field with non-perfect residue field}, Preprint 2002.

\bibitem{gh}
\bysame, \emph{Topological cyclic homology of schemes}, {$K$}-theory (Seattle,
  1997), Proc. Symp. Pure Math., vol.~67, 1999, 41--87.

\bibitem{geisserlevine}
T.~Geisser and M.~Levine, \emph{The {$K$}-theory of fields in characteristic
  {$p$}}, Invent. Math. \textbf{139} (2000), 459--493.

\bibitem{greenleesmay}
J.~P.~C. Greenlees and J.~P. May, \emph{Generalized {T}ate cohomology}, vol.
  113, Mem. Amer. Math. Soc., no. 543, 1995.

\bibitem{h}
L.~Hesselholt, \emph{On the $p$-typical curves in {Q}uillen's {$K$}-theory},
  Acta Math. \textbf{177} (1997), 1--53.

\bibitem{hm3}
L.~Hesselholt and I.~Madsen, \emph{On the de {R}ham-{W}itt complex in mixed
  characteristic}, Preprint 2002.

\bibitem{hm2}
\bysame, \emph{On the {$K$}-theory of local fields}, Ann. Math. (to appear).

\bibitem{hm}
\bysame, \emph{On the {$K$}-theory of finite algebras over {W}itt vectors of
  perfect fields}, Topology \textbf{36} (1997), 29--102.

\bibitem{hyodokato}
O.~Hyodo and K.~Kato, \emph{Semi-stable reduction and crystalline cohomology
  with logarithmic poles}, P\'eriodes $p$-adiques, Asterisque, vol. 223, 1994,
  221--268.

\bibitem{illusie}
L.~Illusie, \emph{Complexe de de {R}ham-{W}itt et cohomologie cristalline},
  Ann. Scient. \'Ec. Norm. Sup. (4) \textbf{12} (1979), 501--661.

\bibitem{kato1}
K.~Kato, \emph{Galois cohomology of complete discrete valuation fields},
  Algebraic {$K$}-theory, {P}art {II} ({O}berwolfach, 1980), Lecture Notes in
  Math., vol. 967, Springer, {B}erlin-{N}ew {Y}ork, 1982, 215--238.

\bibitem{kato}
\bysame, \emph{Logarithmic structures of {F}ontaine-{I}llusie}, Algebraic
  analysis, geometry, and number theory, Proceedings of the JAMI Inaugural
  Conference (Baltimore, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989,
  191--224.

\bibitem{licthenbaum1}
S.~Licthenbaum, \emph{Values of zeta-functions at non-negative integers},
  Number theory, Lecture Notes in Math., vol. 1068, Springer-Verlag, 1983,
  127--138.

\bibitem{mccarthy}
R.~McCarthy, \emph{The cyclic homology of an exact category}, J. Pure Appl.
  Alg. \textbf{93} (1994), 251--296.

\bibitem{mccarthy1}
\bysame, \emph{Relative algebraic {$K$}-theory and topological cyclic
  homology}, Acta Math. \textbf{179} (1997), 197--222.

\bibitem{milnor}
J.~Milnor, \emph{Algebraic {$K$}-theory and quadratic forms}, Invent. Math.
  \textbf{9} (1970), 318--344.

\bibitem{quillen}
D.~Quillen, \emph{Higher algebraic {$K$}-theory {I}}, Algebraic {$K$}-theory I:
  Higher {$K$}-theories (Battelle Memorial Inst., Seattle, Washington, 1972),
  Lecture Notes in Math., vol. 341, Springer-Verlag, 1973.

\bibitem{suslin}
A.~A. Suslin, \emph{On the {$K$}-theory of algebraically closed fields},
  Invent. Math. \textbf{73} (1983), 241--245.

\bibitem{suslin1}
\bysame, \emph{On the {$K$}-theory of local fields}, J. Pure Appl. Alg.
  \textbf{34} (1984), 304--318.

\bibitem{tate}
J.~Tate, \emph{{$p$}-divisible groups}, Proc. conf. local fields (Driebergen,
  1966), Springer-Verlag, 1967, 158--183.

\bibitem{voevodsky}
V.~Voevodsky, \emph{The {M}ilnor conjecture}, Preprint 1996, Max Planck
  Institut, Bonn.

\bibitem{waldhausen1}
F.~Waldhausen, \emph{Algebraic {$K$}-theory of topological spaces. {II}},
  Algebraic topology (Aarhus, 1978), Lecture Notes in Math., vol. 763,
  Springer-Verlag, 1979, 356--394.

\bibitem{waldhausen}
\bysame, \emph{Algebraic {$K$}-theory of spaces}, Algebraic and geometric
  topology (New Brunswick, N.J., 1983), Lecture Notes in Math., vol. 1126,
  Springer-Verlag, 1985, 318--419.

\end{thebibliography}