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\begin{document}

%%%% title section

\title{Unramified cohomology of quadrics, I}

\author{Bruno Kahn} \address{Institut de Math\'ematiques de Jussieu,
  Equipe Th\'eories G\'eom\'etriques, Universit\'e Paris~7, Case~7012,
  75251~Paris Cedex~05, France} 
\email{kahn@math.jussieu.fr}

\author{Markus Rost} \address{Fakult\"at f\"ur Mathematik,
  Universit\"at Regensburg, Universit\"atsstra\ss e~31, D-93040
  Regensburg, Germany}
\email{markus.rost@mathematik.uni-regensburg.de}

\author{R. Sujatha} \address{School of Mathematics, Tata Institute of
  Fundamental Research, Homi Bhabha Road, Bombay~400005, India}
\email{sujatha@math.tifr.res.in}

\date{November 12, 1997}

\begin{abstract}
  Given a quadric~$X$ over a field~$F$ of characteristic $\neq 2$, we
  compute the kernel and cokernel of the natural map in degree~$4$
  from the mod~$2$ Galois cohomology of~$F$ to the unramified mod~$2$
  cohomology of $F(X)$, when $\dim X>10$ and in several
  smaller-dimensional cases.  Applications of these results to real
  quadrics and to the unramified Witt ring are given.
\end{abstract}

\maketitle

\tableofcontents

%%%% Introduction
\section*{Introduction}

Let $F$ be a field, let $X$ be a smooth, proper, and irreducible
variety over~$F$ and let $n$ be prime to $\car F$.  Consider the
restriction map
\[
H^i(F,\mu_n^{\otimes (i-1)}) \to H^i(F(X),\mu_n^{\otimes (i-1)})
\]
in Galois cohomology.

The kernel of this map has been studied in various cases, for
unirational and notably for homogeneous varieties.  In the case of the
Brauer group ($i=2$) this goes back to Witt~\cite{witt} for conics and
Amitsur~\cite{amitsur} for Severi-Brauer varieties.  For $i=3$ the
kernel has been determined by Arason \cite{arason} for quadrics.
Other computations for $i=3$ and homogeneous varieties
(Suslin~\cite{suslin}, Peyre~\cite{peyre,peyre2},
Merkurjev~\cite{merkurjev}, Esnault et al.~\cite{esnal}) rely on the
Merkurjev-Suslin theorem (see below).  The Merkurjev-Suslin/Rost
theorem makes it possible to study also the case $i=4$ and $n=2$. Here
certain quadrics have been considered by Jacob-Rost \cite{jr},
Szyjewski \cite{sz}, Rost and Merkurjev~\cite{Mecourse}.

\topic{Unramified cohomology} The image of this map sits inside the
\emph{unramified cohomology group}
\[
H^i_{nr}(F(X)/F,\mu_n^{\otimes (i-1)}):
=\Ker\bigl((H^iF(X),\mu_n^{\otimes (i-1)}) \xrightarrow{(\partial_x)}
\bigoplus_{x\in X^{(1)}} H^{i-1}(F_x,\mu_n^{\otimes (i-2)})\bigr).
\]
Here $x$ runs through the codimension~$1$ points of~$X$, and
\[
\partial_x\colon H^i(F(X),\mu_n^{\otimes (i-1)}) \to
H^{i-1}(F_x,\mu_n^{\otimes (i-2)})
\]
is the residue map associated to the discrete valuation ring
${\cal{O}}_{X,x}$.

We study the cokernel of the corresponding map
\[
\eta_n^i\colon H^i(F,\mu_n^{\otimes (i-1)}) \to
H^i_{nr}(F(X)/F,\mu_n^{\otimes (i-1)}).
\]

The group $H^i_{nr}(F(X)/F,\mu_n^{\otimes (i-1)})$ is a birational
invariant and does not change when $X$ is replaced by $X\times\bP^1$.
Moreover for the base field extension from~$F$ to the rational
function field $K=F(t)$ one has an exact sequence
%%\begin{equation}
%%  0\to H^i_{nr}(F(X)/F)\to H^i_{nr}(K(X)/K) \to
%%  \bigoplus_{y\in\af_{(0)}}H^{i-1}_{nr}(F_y(X)/F_y)\to 0.
\begin{multline*}
  \tag{$*$}
  \label{star}
  0\to H^i_{nr}(F(X)/F,\mu_n^{\otimes (i-1)}) \to
  H^i_{nr}(K(X)/K,\mu_n^{\otimes (i-1)}) \to\\\to
  \bigoplus_{y\in\af_{(0)}}H^{i-1}_{nr}(F_y(X)/F_y,\mu_n^{\otimes
    (i-2)})\to 0.
\end{multline*}
%%\end{equation}
(\cf \cite[\S 3.3, th.~4.1.1,~4.1.5]{CTcourse},
\cite[(7.3),(12.10)]{rostch}.)

Saltman~\cite{salt,salt2}, Bogomolov~\cite{bog}, Colliot\shyph
Th\'el\`ene-Ojan\-gu\-ren~\cite{cto}, and Peyre \cite{peyre1}, inter
alia, have studied the unramified cohomology of certain unirational
varieties over an algebraically closed~$F$ (there are many more
references available on this case, for which the reader can consult
for instance~\cite[\S~4]{CTcourse}).  The unramified cohomology of
real varieties has been studied in high cohomological degree by
Colliot\shyph Th\'el\`ene-Parimala~\cite{ctp} and
Scheiderer~\cite{scheiderer}.  For quadrics~$X$ and $i\le 3$, the
cokernel of $\eta_2^i$ was studied by Colliot\shyph
Th\'el\`ene-Sujatha \cite{CTS} ($F=\R$), Peyre \cite{peyre} ($F$
arbitrary, $X$ a conic), Sujatha \cite{suj} ($F$ arbitrary, $X$
defined by a neighbour of a $3$-fold Pfister form) and Kahn
\cite{hcoh} ($F$ arbitrary, $\dim X>2$). It follows from
\cite{witt,arason,hcoh} that $\eta_2^2$ (\resp $\eta_2^3$) is
bijective for $\dim X>2$ (\resp $\dim X>6$) and has kernel and
cokernel of order at most $2$ for all $X$ (\resp for $\dim X> 2$).

In this paper, we show that $\eta_2^4$ is bijective if $X$ is a
quadric of dimension $>14$, and that $\left|\Ker\eta_2^4\right|\le 2$,
$\left|\Coker\eta_2^4\right|\le 4$ for $\dim X>4$.  To our knowledge,
this is the first non-trivial instance where unramified cohomology of
degree $4$ is determined over an arbitrary field.

\topic{The Milnor conjecture} Before a more detailed discussion we
recall the Bloch-Kato conjecture, the Milnor conjecture, and the
related conjecture about existence of cohomological invariants for the
graded Witt ring.

The Bloch-Kato conjecture predicts that if $n$ is invertible in~$F$,
then the norm residue homomorphism $h_i$ of degree~$i$,
\[
h_i\colon K^{\rm M}_i F/n \to H^i(F,\mu_n^{\otimes i})
\]
is an isomorphism for all $i \geq 0$. Here $K_i^{\rm M}F$ denotes the
$i$-th Milnor $K$-group of~$F$ \cite{milnor}. If $n$ is a power
of~$2$, this is called the Milnor conjecture.  The bijectivity
of~$h_i$ holds trivially for $i=0$ and by Kummer theory for $i=1$. The
Merkurjev-Suslin theorem states that $h_2$ is an isomorphism
\cite{ms1} and Merkurjev-Suslin/Rost \cite{ms} proved the Milnor
conjecture for $i =3$.  The second author announced a proof of the
Milnor conjecture for $i=4$. Voevodsky \cite{voem} recently released a
proof of the Milnor conjecture for all~$i$. In the present paper we
use only the Milnor conjecture for $i\leq3$, except in
appendix~\ref{wnr} for an application to the unramified Witt group
which relies on the Milnor conjecture for $i=4$.

Let $W(F)$ denote the Witt ring of~$F$ and let $(I^iF)_{i\geq 0}$ be
the filtration on $W(F)$ given by the higher powers of the fundamental
ideal $I(F)$ (\cite{lam,Scharlau}).  For $a_1$, $a_2$, \dots, $a_i\in
F^*$ let $\Pf{a_1,\dots,a_i}$ denote the \emph{$i$-fold Pfister form}
$\qf{1,-a_1}\otimes\dots\otimes\qf{1,-a_i}$ (\ibid).  A quadratic
form~$q$ is a \emph{neighbour} of an $i$-fold Pfister form~$\varphi$
if $\dim q>2^{i-1}$ and $q$ is similar to a subform of $\varphi$.  For
simplicity, write
\[  
H^iF=H^i(F,\Z/2).
\]
We define a \emph{symbol} to be an element of $H^iF$ of the form
$(a_1,\dots,a_i)$, where $a_j \in F^*$, $(a_j)$ is the class of $a_j$
in $H^1F$ by Kummer theory and
$(a_1,\dots,a_i):=(a_1)\cdot\ldots\cdot(a_i)$ is the cup-product.  Let
$P_i(F)$ denote the set of classes of $i$-fold Pfister forms
in~$W(F)$. There are maps
\[
e^i \colon P_i(F) \to H^iF
\]
which send the class of an $i$-fold Pfister form $\Pf{ a_1,\dots,a_i}$
to the corresponding cohomological symbol $(a_1,\dots,a_i)$. It is
conjectured that these maps extend to well-defined isomorphisms
\[
\overline e^i \colon I^iF/I^{i+1}F \to H^{i}F
\]
for all $i \geq 0$.  For $i \leq 2$, it is easy to see that these maps
are well-defined, being the rank mod~$2$, signed discriminant, and the
Clifford invariant, respectively, for $i=0$, $1$,~$2$.
Arason~\cite{arason} proved that $\overline e^3$ is well-defined and
Jacob-Rost/Szyjewski \cite{jr,sz} proved that $\overline e^4$ is
well-defined.  Granting the existence of~$\overline e^i$, the Milnor
conjecture in degree~$i$ implies the bijectivity of~$\overline e^i$.
Voevodsky has recently announced a proof of the existence and
bijectivity of the homomorphisms~$\overline e^i$ for all~$i$, jointly
with Orlov and Vishik~\cite{ovv}.

\introbreak

We now describe our results. Let $q$ be a (regular) quadratic form
over a field~$F$ with $\car F\neq 2$ and let $X$ be the associated
projective quadric. Unless mentioned otherwise, we assume $\dim q\geq
3$.

Note that if $q$ is isotropic, then the extension $F(X)/F$ is purely
transcendental and therefore $\eta_2^i$ is bijective for all~$i$ (\cf
proposition~\ref{pc5}).

\topic{Kernel of $\eta^i_2$} Here the major result is
\begin{Thm}[\cf theorem~\ref{kersym}]
  \label{kthm}
  Let\/ $i\leq 4$. If\/ $\dim q=2^{i-2}+1$, then every element in\/
  $\Ker\eta^i_2$ is a symbol.
\end{Thm}

For $i=3$ and, in case of a Pfister neighbour, for $i=4$, this theorem
follows easily from the computations of $\Ker\eta^i_2$ by Arason
\cite{arason}, Jacob-Rost \cite{jr}, and Szyjewski \cite[6.1]{sz}.

The proof of theorem~\ref{kthm} for $i=4$ in general is an extension
of the discussions in \cite{jr,Mecourse} to arbitrary $5$-dimensional
quadratic forms. The method used there provides a very general
description of $\Ker\eta^4_2$ for quadratic forms of dimension~$\geq
5$ (see \cite{Mecourse}) as follows:
\begin{equation}
  \label{h4kiso}
  \tag{$**$} \Ker\eta^4_2=\Ker(H^2(X,{\cal K}_3) \to H^2(\overline
  X,{\cal K}_3)).
\end{equation}

For an Albert form with corresponding biquaternion algebra~$A$ (see
below) this yields an isomorphism ${\rm SK_1(A)}=\Ker\eta^4_2$
(\ibid).

For a five dimensional form~$q$, let $\q=q\bot\qf {-d}$, $d=\det(q)$
be the attached Albert form. In this case~\eqref{h4kiso} yields a
surjective homomorphism
\[
\O(\q)/[\O(\q),\O(\q)]{\O(q)}\surj\Ker\eta^4_2.
\]
Theorem~\ref{kthm} follows from a careful computation of this map.

Using this theorem and some quadratic form theory, we show in
section~\ref{generalities}:

\begin{Cor}
  \label{t3} 
  Let\/ $i\leq 4$. Then
  \begin{enumerateT}
  \item A symbol $(a_1,\dots,a_i)\neq 0$ lies in\/ $\Ker\eta^i_2$ if
    and only if\/ $q$ is similar to a subform of\/ $\Pf{
      a_1,\dots,a_i}$.
  \item
    \label{t3.2} 
    $\Ker\eta^i_2$ consists of symbols for\/ $\dim q> 2^{i-2}$.
  \item $\left|\Ker\eta^i_2\right|\le 2$ for\/ $\dim q> 2^{i-1}$.
  \item $\Ker\eta^i_2=0$ for\/ $\dim q> 2^i$.
  \end{enumerateT} 
\end{Cor}

Note that the bound $\dim q> 2^{i-2}$ in~\eqref{t3.2} is optimal: if
$q$ is an $(i-2)$-fold Pfister form $\Pf{ a_1,\dots,a_{i-2}}$, then
$\Ker\eta_2^i$ contains $H^2F\cdot(a_1,\dots,a_{i-2})$, which
obviously contains non-symbolic elements in general.

One may expect that theorem~\ref{kthm} also holds for higher
degree~$i$. Then corollary~\ref{t3} would hold accordingly (using the
existence and bijectivity of the~$\overline e^{i}$).


\topic{Cokernel of $\eta_2^i$} Results on $\Coker\eta_2^i$ are best
understood in terms of $\Coker\eta^i$, where $\eta^i$ is the
restriction map
\[
\eta^i\colon H^i(F,\Q/\Z(i-1)) \xrightarrow{}
H^i_{nr}(K/F,\Q/\Z(i-1)).
\]
Here $\Q/\Z(i-1):=\varinjlim\mu_m^{\oo(i-1)}$, where $m$ runs through
all integers prime to $\car F$. A consequence of Milnor's conjecture
in degree $i-1$ is that $H^iF \to H^i(F,\Q/\Z(i-1))$ is injective;
hence $\Ker\eta_2^i=\Ker\eta^i$ since the latter group is killed by
$2$ (\cf lemma~\ref{lsup}~\Aref{lsup.a}).  To relate $\Coker\eta_2^i$
and $\Coker\eta^i$, define
\[
(\Ker\eta_2^i)_0=\{\,\alpha\in\Ker\eta_2^i\mid (-1)\cdot\alpha=0\in
H^{i+1}F\,\}.
\]
We prove in section~\ref{Coker}:

\begin{Prop}[\cf proposition~\ref{c27}]
  \label{t5} 
  For any quadric~$X$ and\/ $i\le 4$, there is an exact sequence
  \[
  0 \to (\Ker\eta_2^i)_0 \xrightarrow{} \Coker\eta_2^i \to
  \Coker\eta^i.
  \]
\end{Prop}

This way, information on $\Ker\eta_2^i$ and $\Coker\eta^i$ immediately
yields information on $\Coker\eta_2^i$.

%%The general situation about $\Coker\eta^i$ is somewhat
%%mysterious. Our considerations so far suggest the following
%%picture: 
To describe our picture about $\Coker\eta^i$ we formulate:
\begin{Conj}\leavevmode
  \begin{enumerateT}\itshape
  \item
    \label{conj.1}
    $\Coker\eta^i=0$ if\/ $q$ is Pfister neighbour.
  \item
    \label{conj.2}
    $\Coker\eta^i=0$ for\/ $\dim q\leq5$.
  \item
    \label{conj.3}
    $\Coker\eta^i$ is finite for\/ $\dim q>6\cdot2^{i-4}$.
  \item
    \label{conj.4}
    $\Coker\eta^i=0$ for\/ $\dim q>6\cdot2^{i-3}$.
  \end{enumerateT}
\end{Conj}

In appendix~\ref{dim0} we show $\Coker\eta^i=0$ for $\dim q=3$
and---after an appropriate modification of the target group---also for
$\dim q=2$. Here the arguments rely on the Milnor conjecture in
degree~$i-2$ (\resp $i-1$).

For $i\leq 2$ one has (\cf section~\ref{seci2}):
\begin{Thm}
  \label{i2}
  $\Coker\eta^i=0$ for $i\leq 2$.
\end{Thm}

For $i=3$ there appears a peak in the story at $\dim q=6$.  For the
reader's convenience, we recall the various types of $6$-dimensional
quadratic forms.  Let $d_\pm q$ be the signed discriminant and let
$C(q)$ be the Clifford algebra of~$q$~\cite{lam}.  The \emph{pure
  subform} $\Pf{a_1,\dots,a_i}'$ of the Pfister form
$\Pf{a_1,\dots,a_i}$ is defined by
\[
\Pf{a_1,\dots,a_i}=\qf1\bot\Pf{a_1,\dots,a_i}'.
\]
There are four types of anisotropic quadratic forms~$q$ of
dimension~$6$:
\begin{itemize}
\item $d=d_\pm q\neq 1$, $q_{F(\sqrt{d})}$ is anisotropic. For
  convenience, we shall call such forms \emph{virtual Albert forms}.
\item $d_\pm q=1$, \ie $q\in I^2F$. Such forms are called
  (anisotropic) \emph{Albert forms}. An Albert form is similar to a
  form of type
  \[
  \Pf{a,b}'\bot-\Pf{e,f}' \qquad\text{$a$, $b$, $e$, $f\in F^*$.}
  \]
  To an Albert form there is attached the biquaternion algebra
  $A=(a,b)\oo(e,f)$ which is the central simple algebra with
  $C(q)\simeq M_2(A)$. The form~$q$ is isotropic if and only if $A$ is
  not a skewfield.
\item $d=d_\pm q\neq 1$, $q_{F(\sqrt{d})}$ is hyperbolic.  These forms
  are the $6$-dimensional Pfister neighbours (\cf
  \cite[p.~10]{knebusch2}).
\item $d=d_\pm q\neq 1$, $q_{F(\sqrt{d})}$ is isotropic without being
  hyperbolic. Such forms are of type $a\tau\perp b\qf{1,-d}$, where
  $a$, $b\in F^*$ and $\tau$ is a $2$-fold Pfister form.
\end{itemize}

\medbreak

Now for $i=3$ our result is (\cf section~\ref{seci3}):
\begin{Thm}
  \label{pick}
  $\Coker\eta^3=0$, except if\/ $q$ is an anisotropic Albert form, for
  which $\Coker\eta^3\simeq \Z/2$.
\end{Thm}
This result completes those of~\cite{hcoh}.

%%For an Albert form~$q$ the generator~$\alpha$ of $\Coker\eta^3$ can be
%%described as follows (\ibid). Let $K=F(X)$. Then
%%\[
%%q_K=\qf{1,-1}\bot u\Pf{v,w}
%%\]
%%for some $u$, $v$, $w\in K^*$.  A representative of~$\alpha$ is now
%%given by $(u,v,w)\in H^3K$.

%%This example generalizes to higher degree as follows. Let
%%\[
%%q=(\Pf{a,b}'\bot-\Pf{e,f}' )\otimes\Pf{a_1,\dots,a_{i-3}}.
%%\]
%%Similarly as before one has (with $K=F(X)$)
%%\[
%%q_K=(\qf{1,-1}\bot u\Pf{v,w} )\otimes\Pf{a_1,\dots,a_{i-3}}
%%\]
%%for some $u$, $v$, $w\in K^*$. One can show that the element
%%$(u,v,w,a_1,\dots,a_{i-3})\in H^iK$ is unramified.  This way 
%%one may construct examples of quadratic forms~$q$ of
%%dimension~$6\cdot 2^{i-3}$ with nontrivial elements in $\Coker\eta^i$.
%%(For example let $\tilde q$ be an anisotropic Albert form over a field~$k$
%%and consider $q=\tilde q\otimes \Pf{a_1,\dots,a_{i-3}}$ over
%%the iterated power series field $F=k\pseries{a_1}\cdots\pseries{a_{i-3}}$.)

%%In particular the bound on~$\dim q$ in~\eqref{conj.4} above is
%%necessary.

%%One may also use this example to construct $6$-dimensional
%%forms for which $\Coker\eta^4$ is not finite.  Namely if $q$ is an
%%anisotropic Albert form or a virtual Albert form over~$F$, then the
%%exact sequence~\eqref{star} shows that $\Coker\eta^4$ is infinite
%%for~$q_{F(t)}$.

%%\medbreak

%%We now state the main result about $\Coker\eta^4$.
The main results about $\Coker\eta^4$ are:
\begin{Thm}[\cf section~\ref{pth1}]
  \leavevmode
  \label{t1}
  \begin{enumerateT}
  \item
    \label{t1.3}
    For $\dim q>6$ there is an exact sequence
    \[
    0\to \Coker\eta^4\to (\CH^3X)_{\tors}\to \bigl(\CH^1X\oo
    H^3(F,\Q/\Z(2))\bigr)\oplus H^5(F,\Q/\Z(3)).
    \]
    Here $\CH^p X$ denotes the $p$-th Chow group of $X$.
  \item \label{inf} $\left|\Coker\eta^4\right|\le 2$ if\/ $\dim q\neq
    4$, $5$,~$6$.
  \item
    \label{t1.4}
    $\Coker\eta^4=0$ if
    \begin{itemize} 
    \item $q$ is a Pfister neighbour
    \item $\dim q=8$ and $d_\pm q=1$
    \item $\dim q>12$.
    \end{itemize} 
  \end{enumerateT}
\end{Thm}

In the proof of theorem~\ref{t1} we use Karpenko's results on $\CH^3$
of a quadric.  The main computational tool is, as in \cite{hcoh}, to
mix the use of the Hochschild-Serre and Bloch-Ogus \cite{BO} spectral
sequences, here with divisible coefficients. We use in particular the
fact that these two spectral sequences are modules over the Galois
cohomology of~$F$. Divisible coefficients have been used also in
\cite{kato,CT2,peyre,peyre2}.  It is clear that these methods can
also be applied to other homogeneous varieties.

\begin{Thm}
  \label{infinite}
  If $q$ is an anisotropic Albert form or a virtual Albert form
  over~$F$, then $\Coker\eta^4$ is infinite for~$q_{F(t)}$.
\end{Thm}

This theorem is an easy consequence of theorem~\ref{pick} and the
exact sequence~\eqref{star}.

From corollary \ref{t3}, proposition \ref{t5}, and theorem \ref{t1}
one easily deduces:

\begin{Cor}\leavevmode
  \label{C1}
  \begin{enumerateT}
  \item $\Coker\eta_2^2=0$, unless $q$ is a neighbour of a $2$-fold
    Pfister form $\Pf{ a,b}$. In this case, it is $0$ or\/~$\Z/2$
    according as $(-1,a,b)\in H^3F$ is nonzero or not.
  \item If\/ $q$ is an anisotropic Albert form, $(\Ker\eta_2^3)_0=0$
    and\/ $\Coker\eta_2^3\simeq \Z/2$. In all other cases,
    $\Coker\eta_2^3\simeq (\Ker \eta_2^3)_0$. In particular:
    \begin{enumerate}
    \item If\/ $q$ is a neighbour of a $3$-fold Pfister form $\Pf{
        a,b,c}$, then $\Coker\eta_2^3=0$ or\/~$\Z/2$ according as
      $(-1,a,b,c)\in H^4F$ is nonzero or not.
    \item If\/ $q$ is not an Albert form and does not embed into a
      $3$-fold Pfister form \upp(up to similarity\upp), then
      $\Coker\eta_2^3=0$.
    \end{enumerate}
  \item \begin{enumerate}
    \item If\/ $\dim q>12$ and $q$ is not a neighbour of a $4$-fold
      Pfister form, then $\Coker\eta_2^4=0$.
    \item If\/ $q$ is a neighbour of a $4$-fold Pfister form $\Pf{
        a,b,c,d}$, then $\Coker\eta_2^4=0$ or\/~$\Z/2$ according as
      $(-1,a,b,c,d)\in H^5F$ is nonzero or not.
    \item For\/ $\dim q>8$, $\left|\Coker\eta_2^4\right|\le 4$.
    \end{enumerate}
  \end{enumerateT}
\end{Cor}

In the case $F=\R$, these results immediately imply that
$\Coker\eta_2^i=0$ for all quadrics when $i\leq4$. In
appendix~\ref{app.1}, we also get partial results on the $\cal
H$-cohomology of such quadrics.

\topic{The unramified Witt ring} As in \cite[corollary]{hcoh}, we wish
to apply the above results to the \emph{unramified Witt ring}.  Denote
by $W_{nr}(K/F)$ the unramified Witt ring of a finitely generated
extension $K$ of $F$ (\cf \cite{cto}) and, for $i\ge 0$, by
$I^i_{nr}(K/F)$ the intersection $I^iK\cap W_{nr}(K/F)$.  Let
\[
\chi^i\colon I^iF/I^{i+1}F \to I^i_{nr}(K/F)/I^{i+1}_{nr}(K/F)
\]
be the natural map analogous to $\eta_2^i$ and $\eta^i$.  The
homomorphism $\bar e^i$ induces a homomorphism
\[
I^i_{nr}K/I^{i+1}_{nr}K \to H^i_{nr}K
\]
and hence a homomorphism
\[
\Coker\chi^i \to \Coker\eta_2^i.
\]

Using the Milnor conjecture in degree~$i=4$, we find:

\begin{Thm}
  \label{Witt} 
  Let\/ $i\le 4$. For any quadric $X$, the composition
  \[
  \Coker\chi^i \to \Coker\eta_2^i \to\Coker\eta^i
  \]
  is injective.
\end{Thm}

From corollary \ref{t3}, theorem \ref{t1}, and theorem \ref{Witt} we
deduce:

\begin{Cor}\leavevmode
  \label{C2}
  \begin{enumerateT}
  \item The map $W(F)/I^3F \to W_{nr}(F(X)/F)/I_{nr}^3(F(X)/F)$ is
    surjective. It is bijective for\/ $\dim q>4$.
  \item The map $W(F)/I^4F \to W_{nr}(F(X)/F)/I_{nr}^4(F(X)/F)$ is
    surjective, except if\/ $q$ is an anisotropic Albert form. In this
    case, its cokernel is $\simeq \Z/2$ \upp(by \cite[remark~3,
    p.~249]{hcoh}\upp).  It is bijective for\/ $\dim q>8$.
  \item \label{I5} The map $W(F)/I^5F \to
    W_{nr}(F(X)/F)/I_{nr}^5(F(X)/F)$ is surjective in the following
    cases
    \begin{itemize}
    \item $q$ is a Pfister neighbour.
    \item $\dim q=8$ and $d_\pm q=1$.
    \item $\dim q>12$.
    \end{itemize}
    It is bijective for\/ $\dim q>16$. \qed
  \end{enumerateT} 
\end{Cor}

\begin{rk} 
  Voevodsky's work yields also the computation of $\Ker\eta_2^i$ for
  $i$-fold Pfister forms.  A consequence of \cite{voem} and~\cite{ovv}
  is the validity of proposition~\ref{t5} and of theorem~\ref{Witt}
  for all~$i$.
  
  However these announcements do not imply a generalization of
  theorem~\ref{kthm} to higher degree or a proof of the conjecture in
  an obvious way.
\end{rk}

In a sequel to this paper, we plan to complete the present results by
determining $\Ker\eta^4$ and $\Coker\eta^4$ in the cases pending here,
and by refining results on real quadrics. In particular, we shall show
that $\Ker\eta^4_2$ is always generated by its symbols.  For this, we
shall use a method different from that of this paper, based on the
following result:

\begin{thmA}[\cite{ks}]
  Let\/ $X$ be a quadric and let\/ $C$ be a conic such that there
  exist a morphism $C\to X$.  Let\/ $D$ be the quaternion algebra
  associated to $C$. Then there is a canonical isomorphism
  \[
  \Ker\eta^4_2\iso \frac{H^0(X,K_2^D)}{K_2D}.
  \]
\end{thmA}
Here $H^0(X,K_2^D)$ is the intersection of the kernels of the residue
maps
\[
K_2(D\oo_F F(X)) \to K_1(D\oo_F F(x)),
\]
where $x$ runs through the points of codimension~$1$ in~$X$.

\introbreak

This paper is organized as follows. In part~\ref{part.1} we recall
some known results on Chow groups, in particular of Karpenko, and of
$\cal{K}$- and $\cal H$-cohomology of quadrics, supplying proofs when
they don't appear in the literature.  Part~\ref{part.2} is devoted to
the proof of theorem~\ref{kthm} and corollary~\ref{t3}, and
part~\ref{part.3} to the proof of proposition~\ref{t5}, theorem
\ref{pick}, and theorem~\ref{t1}.  In appendix~\ref{dim0} we consider
$\Coker\eta^i$ for quadrics of dimension $0$ and~$1$. In
appendix~\ref{app.1} we consider the case $F=\R$.  In
appendix~\ref{wnr} we prove theorem~\ref{Witt}.


\begin{notconv}
  For any scheme $X$, $H^iX$ denotes the $i$-th \'etale cohomology
  group of~$X$ with coefficients $\Z/2$. We use the abbreviation
  $H^i(X,j)$ for $H^i_{\et}(X,\Q/\Z(j))$ and similarly $H^p(X,{\cal
    H}^q(j))$ for the $E_2$-term of the coniveau spectral sequence
  with divisible coefficients. If $X$ is smooth over a field $F$, we
  denote by $\CH^iX$ its $i$-th Chow group, the group of cycles of
  codimension~$i$ over~$X$ modulo rational equivalence. We denote by
  $cl^i_X$ (\resp $cl'{}^i_X$) the cycle map with $\Z/2$ coefficients
  $\CH^iX/2 \to H^{2i}X$ (\resp with divisible coefficients $\CH^iX\oo
  \Q/\Z \to H^{2i}(X,\Q/\Z(i))$). We set $\overline X=X\times_F F_s$,
  where~$F_s$ is a separable closure of~$F$. The word ``quadric''
  means smooth, projective and absolutely irreducible quadric.  If $X$
  is an even-dimensional quadric over~$F$ (assumed of characteristic
  $\neq 2$), we denote by $d(X)\in F^*/F^{*2}$ the (signed)
  discriminant $d_\pm q$ of any quadratic form $q$ defining $X$.  If
  $\varphi\colon
  A \to B$ is a homomorphism between two abelian groups, we sometimes
  allow ourselves to write $B/A$ or $\displaystyle \frac BA$ for
  $\Coker \varphi$, even if $\varphi$ is not injective.
\end{notconv}

\begin{ack}
  The first author wishes to thank Andrei Suslin for a conversation
  which catalyzed the use of divisible coefficients in this paper, and
  gratefully acknowledges the hospitality of the Tata Institute of
  Fundamental Research, where this work was completed.  The third
  author acknowledges the partial support of a NATO grant.
\end{ack}

%%%% part 1
\part{Collected results on quadrics}
\label{part.1}

In this part, we state general results on Chow groups, ${\cal H}$- and
$\cal K$-cohomology of quadrics, most of which are ``classical''. The
reader is advised to skip it and to use it only for reference
purposes. We merely give references when published proofs are
available, and provide proofs otherwise.

We say that a quadric $X$ is \emph{anisotropic} (\resp
\emph{isotropic}, \emph{split}) if a quadratic form defining $X$ is
anisotropic (\resp is isotropic, has maximal index).

\section{Chow groups}

The following proposition is mainly a reformulation of results of N.
Karpenko~\cite{K}.

\begin{prop}
  \label{pc3}
  Let $X$ be a quadric over a field $F$ of characteristic $\neq 2$.
  Let $h$ denote the class of a hyperplane section in $\CH^1X$.
  \begin{enumerateA}
  \item
    \label{pc3.a}
    For any $i\ge 0$, the natural map $\CH^iX \to \CH^i\overline X$
    has kernel the torsion subgroup of $\CH^iX$ \upp(which is
    $2$-primary\upp).
  \item
    \label{pc3.b}
    For $i<\dim X/2$, $\CH^iX/\tors$ is generated by $h^i$; $\CH^iX
    \to \CH^i\overline X$ is surjective.
  \item
    \label{pc3.c}
    For $i>\dim X/2$, $\CH^iX/\tors$ is generated by $h^i$ if $i\le
    \dim X-n$ and by the class of a rational linear subspace of
    codimension $i$ if $i> \dim X-n$. Here $n$ is the Witt index of a
    quadratic form defining $X$.  $\Coker(\CH^iX \to \CH^i\overline
    X)\simeq\Z/2$ if $i\le \dim X-n$ and\/ $\CH^iX \to \CH^i\overline
    X$ is surjective if $i> \dim X-n$.
  \item
    \label{pc3.d}
    For $i=\dim X/2$ and $d(X)=1$ one has
    \[
    \CH^iX/\tors\simeq \Z\oplus \Z,
    \] 
    with basis $(h^i,2^tl)$, where $t\ge 0$ and\/~$l$ is the class of
    some maximal linear subspace of $\overline X$.  Moreover
    $\Coker(\CH^iX \to \CH^i\overline X) \simeq\Z/2^t$.  We have $t=
    1$ if $i=1$ and $X$ is anisotropic, and $t= 2$ if $i=2$ and $X$ is
    anisotropic.
  \item
    \label{pc3.e}
    For $i=\dim X/2$ and $d(X)\neq 1$ one has
    \[
    \CH^iX/\tors\simeq \Z
    \]
    generated by $h^i$.  The map $\CH^iX \to (\CH^i\overline X)^{G_F}$
    is surjective, where $G_F$ is the absolute Galois group of~$F$.
  \end{enumerateA}
\end{prop}

\begin{proof}
  \Aref{pc3.a}, \Aref{pc3.b}, and \Aref{pc3.c} follow easily from
  \cite[(2.1), (2.2), and (2.7)]{K}; \Aref{pc3.d} and~\Aref{pc3.e}
  follow from the same references plus \cite[(2.4) and (5.5)]{K}.
\end{proof}

\begin{cor}
  \label{cc3}
  Let $X$ be a quadric over $F$, $D$ a divisible group and $i\ge 0$.
  If $i\neq \dim X/2$ or $i=\dim X/2$ and $d(X)=1$, there is an exact
  sequence
  \[
  0 \to \Tor(D,\Coker(\CH^iX \to \CH^i\overline X)) \to D\oo \CH^iX
  \to D\oo \CH^i\overline X \to 0.
  \]
\end{cor}

This is clear, since $\CH^i\overline X$ is torsion-free and the map
$\CH^iX \to \CH^i\overline X$ has torsion kernel and cokernel.

\begin{cor}
  \label{cc2}
  Let $X$ be a quadric of dimension\/ $2i$ over $F$ such that
  $d=d(X)\neq 1$, $E=F(\sqrt d)$ and $\Gamma=\Gal(E/F)$. Then the
  $\Gamma$-module $\CH^iX_E/\tors$ is isomorphic to\/
  $\Ind_1^\Gamma\Z$ if $X_E$ is split, and to\/ $\Z\oplus
  \Z(\varepsilon)$ otherwise, where $\Z(\varepsilon)$ is the
  $\Gamma$-module of support\/ $\Z$ with the nontrivial action of\/
  $\Gamma$. On the other hand, the $G_F$-module $\CH^i\overline X$ is
  always isomorphic to $\Ind_{G_E}^{G_F}\Z$. The maps
  \[
  \CH^iX/\tors \to (\CH^i\overline X)^{G_F}
  \]
  and
  \[
  \CH^iX\oo\Q/\Z \to H^0(F,\CH^i\overline X\oo\Q/\Z)
  \]
  are isomorphisms.
\end{cor}

\begin{proof}
  By prop.~\ref{pc3}~\Aref{pc3.d}, $\CH^iX_E/\tors$ is a free
  $\Z$-module with basis $(h^i,2^tl)$ for some $t\ge 0$. It is clear
  that $t=0$ if and only if $l$ is rational over $E$, \ie iff $X_E$ is
  split.  In this case, another basis of $\CH^iX_E/\tors$ is $(l,l')$
  where $l'=h^i-l$ is the class of another linear section, and this
  basis is permuted by $\Gamma$, hence the first claim. If $t>0$,
  another basis of $\CH^iX_E/\tors$ is given by
  $(h^i,2^{t-1}h^i-2^tl)$, and $2^{t-1}h^i-2^tl$ is a $-1$-eigenvector
  for $\Gamma$. This proves the second claim. The isomorphism
  $\CH^iX/\tors \iso (\CH^i\overline X)^{G_F}$ has been seen in
  prop.~\ref{pc3}~\Aref{pc3.a} and~\Aref{pc3.e}.  To get the other, we
  consider the short exact sequence of $G_F$-modules
  \[
  0 \to \CH^i\overline X \to \CH^i\overline X\oo\Q \to \CH^i\overline
  X\oo\Q/\Z \to 0
  \]
  obtained by tensoring the exact sequence $0 \to \Z \to \Q \to \Q/\Z
  \to 0$ by the flat $\Z$-module $\CH^i\overline X$. Since
  $\CH^i\overline X\simeq \Ind_{G_E}^{G_F}\Z$, we have
  $H^1(F,\CH^i\overline X)=0$ by Shapiro lemma, hence we get an
  isomorphism
  \[
  \CH^iX\oo\Q/\Z\iso (\CH^i\overline X)^{G_F}\oo\Q/\Z\iso
  H^0(F,\CH^i\overline X\oo\Q/\Z).  \qed
  \]
  \noqed\end{proof}

\begin{cor}
  \label{cc1}
  Let $X$ be an anisotropic quadric over $F$, $K=F(X)$ and $i_1(X)$
  the Witt index of $q_K$, where $q$ is a quadratic form defining $X$.
  \begin{enumerateA}
  \item
    \label{cc1.a}
    For $i<\dim X/2$, $\CH^iX/\tors \to \CH^i X_K/\tors$ is bijective.
  \item
    \label{cc1.b}
    For $i>\dim X/2$, $\CH^iX/\tors \to \CH^i X_K/\tors$ is bijective
    if $i\le \dim X-i_1(X)$ and\/ $\Coker(\CH^iX/\tors \to \CH^i
    X_K/\tors)\simeq\Z/2$ if $i> \dim X-i_1(X)$.
  \item
    \label{cc1.c}
    For $i=\dim X/2$, $\CH^iX/\tors \to \CH^i X_K/\tors$ is bijective
    if $d(X)\neq 1$. If $d(X)=1$, $\Coker(\CH^iX/\tors \to \CH^i
    X_K/\tors)$ is cyclic of order\/ $2^s$ for some $s\le t$, where
    $t$ is as in prop.\ \ref{pc3}~\Aref{pc3.d}.  For $i=1$ or\/~$2$ we
    have $s=1$.
  \item
    \label{cc1.d}
    For $i=\dim X/2$ and $d=d(X)\neq 1$, let $E=F(\sqrt{d})$. Then
    \[
    \CH^iX_E/\tors\to \CH^iX_{KE}/\tors
    \]
    is bijective if $X_E$ is isotropic and has cokernel as
    in~\Aref{cc1.c} otherwise.
  \end{enumerateA}
\end{cor}

\begin{proof}
  Everything directly follows from proposition~\ref{pc3}, except for
  the last sentence of \Aref{cc1.c} and for~\Aref{cc1.d}.  Assume
  $\dim X=2$. Then $X$ is defined by an anisotropic quaternion form,
  (= a $2$-fold Pfister form).  hence $i_1(X)=2$, which implies $s=1$
  in this case by proposition~\ref{pc3}~\Aref{pc3.d}. Assume $\dim
  X=4$. Then $X$ is defined by an anisotropic Albert form~$\varphi$,
  hence $i_1(X)=1$.  Let $Y$ be the quadric over~$K$ given by the
  anisotropic part of~$\varphi_K$ ($\dim Y=2$). By \cite[(2.2)]{K}
  (\cf lemma~\ref{lpc2} below), there is an isomorphism $\CH^1Y\iso
  \CH^2X_K$. This isomorphism is natural under change of base field,
  as can be seen from the proof of lemma~\ref{lpc2}.  In particular,
  the diagram
  \[
  \begin{CD}
    \CH^1 Y@>\sim>>\CH^2X_K \\ @VVV @VVV\\ \CH^1\overline Y@>\sim>>
    \CH^2\overline{X_K}
  \end{CD}
  \]
  commutes, which shows that $\Coker(\CH^2X_K \to \CH^2\overline{
    X_K})\simeq \Z/2$. Moreover, since $\Coker(\CH^2X \xrightarrow{}
  \CH^2\overline X)\simeq \Z/4$ and $\CH^2\overline X\iso
  \CH^2\overline{ X_K}$, this indeed shows that $s=1$.  Finally,
  \Aref{cc1.d} is obvious since, if $X_E$ is isotropic, the extension
  $KE/E$ is purely transcendental (compare \cite[proof of (3.12)]{K}).
\end{proof}

\begin{prop}
  \label{pc1}
  Let $X$ be a quadric over $F$, where $F$ is separably closed \upp(of
  characteristic $\neq 2$\upp). Then $H^jX=0$ for $j$ odd and the
  cycle maps
  \[
  cl_X^i\colon \CH^iX/n \to H^{2i}(X,\mu_n^{\oo i})
  \]
  are isomorphisms for any\/ $n$ prime to\/ $\car F$ and\/ $i\ge 0$.
\end{prop}

\begin{proof}
  This follows from proposition \ref{pc2}~\Aref{pc2.a} and the
  computation of $H^*_{\et}(X,\Z_l)$ \cite[th.~3.3]{deligne}.
\end{proof}

\begin{prop}
  \label{pc4}
  Let $X$ be a quadric over $F$.
  \begin{enumerateA}
  \item \iname{classical}$\CH^1X$ is torsion-free.
  \item \iname{\cite[(6.1)]{K}} $(\CH^2X)_{\tors}\simeq\Z/2$ if $X$ is
    defined by a neighbour of an an\-iso\-tro\-pic $3$-fold Pfister
    form, and is $0$ otherwise.
  \item \iname{\cite[(2.6)]{K},\cite{K2},\cite{K3}}
    $(\CH^3X)_{\tors}\simeq 0$ or\/~$\Z/2$. If\/ $\dim X=3$ or\/ $\dim
    X>10$, $(\CH^3X)_{\tors}=0$.
  \end{enumerateA}
\end{prop}

\section{$\cal{K}$- and $\cal{H}$-cohomology}

We denote by $H^i(X,{\cal K}_j)$ and $H^i(X,{\cal H}^j(\mu_m^{\oo
  k}))$ the cohomology groups of the localization complexes with terms
\[
\bigoplus_{x\in X^{(i)}} K_{j-i}F(x) \qquad\resp\qquad \bigoplus_{x\in
  X^{(i)}} H^{j-i}(F(x),\mu_m^{\oo(k-i)}).
\]
These cohomology groups are defined for any equidimensional, 
not necessarily smooth variety. 
For the long exact excision sequences and for homotopy
invariance we refer to~\cite{rostch}. The interpretation for
smooth~$X$ as Zariski cohomology groups is not needed here.

\begin{lemma}
  \label{lpc2}
  Let $X$ be an isotropic quadric of dimension $n\ge 2$ over a field
  $F$ of characteristic $\neq 2$; write $q\simeq q'\perp x_nx_{n+1}$
  for a quadratic form defining $X$ and $Z\subseteq X$ for the
  codimension $2$ subquadric with equation $q'=0$. Then, for\/
  $0<i<n$, $j\ge i$ and $k\in\Z$, there are isomorphisms:
  \begin{align*}
    K_jF&\iso H^0(X,{\cal K}_j),\\ H^{i-1}(Z,{\cal K}_{j-1})&\iso
    H^i(X,{\cal K}_j),\\ K_{j-n}F&\iso H^n(X,{\cal K}_j),
  \end{align*}
  \begin{align*}
    H^{j}(F,\mu_m^{\oo k})&\iso H^0(X,{\cal H}^j(\mu_m^{\oo k})),\\ 
    H^{i-1}(Z,{\cal H}^{j-1}(\mu_m^{\oo (k-1)}))&\iso H^i(X,{\cal
      H}^j(\mu_m^{\oo k})),\\ H^{j-n}(F,\mu_m^{\oo (k-n)})&\iso
    H^n(X,{\cal H}^j(\mu_m^{\oo k})).
  \end{align*}
  These isomorphisms commute with change of base field, transfer and
  cup-product by $K_*F$ \upp(\resp $H^*(F,\mu_m^{\oo l})$\upp).
\end{lemma}

\begin{rk} If $n=2$, the ``subquadric'' $Z$ is $\Spec E$, with 
  $E=F[t]/t^2-d$, where $d=d(X)$ is the (signed) discriminant of the
  $2$-dimensional form $q'$. $E$ is a quadratic extension of~$F$ if
  $d\neq 1$ and $F\times F$ otherwise.\end{rk}

\begin{proof} 
  (compare \cite[lemma~3.3]{jr}, \cite[prop.~3.2.1]{sz} for $\dim
  X=3$, and \cite[(2.2)]{K} for $i=j$). We do it only for $\cal
  K$-cohomology, the case of $\cal H$-cohomology being identical. Let
  $Y$ be the hyperplane section $x_{n+1}=0$ and $P=(0:\dots:0:1:0)$.
  We have excision exact sequences:
  \begin{multline*}
    \dots \to H^{i-1}(Y,{\cal K}_{j-1}) \to H^i(X,{\cal K}_j) \to
    H^i(X\setminus Y,{\cal K}_j) \to H^i(Y,{\cal K}_{j-1}) \to \cdots
  \end{multline*}
  \begin{multline*}
    \dots \to H^{i-n}(P,{\cal K}_{j-n}) \to H^{i-1}(Y,{\cal K}_{j-1})
    \to H^{i-1}(Y\setminus P,{\cal K}_{j-1})\to \\ \to
    H^{i-n+1}(P,{\cal K}_{j-n}) \to \cdots
  \end{multline*}
  
  On the other hand, $X\setminus Y$ is isomorphic to the affine space
  and $P$ makes the projection $Y\setminus P \to Z$ a vector bundle
  with affine line fibres. Homotopy invariance of ${\cal
    K}$-cohomology yields new exact sequences and isomorphisms:
  \begin{gather}
    H^{i-1}(Y,{\cal K}_{j-1})\iso H^i(X,{\cal K}_j)\quad \text{if
      $i>1$,}\\ 
    \label{*}0 \to H^0(X,{\cal K}_j) \to K_jF \to H^0(Y,K_{j-1}) 
    \to H^1(X,{\cal K}_j) \to 0
  \end{gather}
  and
  \begin{gather}
    H^{i-1}(Z,{\cal K}_{j-1})\iso H^{i-1}(Y\setminus P,{\cal K}_{j-1})
    \quad \text{if $i<n-2$,}\\ 
    \label{**} 
    0 \to H^{n-2}(Y,{\cal K}_{j-1}) \to H^{n-2}(Z,{\cal K}_{j-1}) \to
    K_{j-n}F \to H^{n-1}(Y,{\cal K}_{j-1}) \to 0.
  \end{gather}
  
  The composition $K_jF \to H^0(X,{\cal K}_j) \to K_jF$ is the
  identity, so that the exact sequence \eqref{*} splits as two
  isomorphisms. Similarly, for $n>0$, $K_{j-n}F \to H^{n-1}(Y,{\cal
    K}_{j-1})\iso H^n(X,{\cal K}_j)$ is induced by the inclusion of
  the closed point $P$ in $X$ and composition with the norm
  $H^n(X,{\cal K}_j) \xrightarrow{{N}} K_{j-n}F$ is the identity, so
  the exact sequence \eqref{**} also splits as two isomorphisms.
  Putting all this together, we get the isomorphisms of
  lemma~\ref{lpc2}. The naturality statements are clear from the
  construction of these isomorphisms.
\end{proof}

\begin{prop}
  \label{pc2}
  Let $X$ be a split quadric over a field $F$ of characteristic $\neq
  2$, and $h\in \CH^1X$ the class of a hyperplane section.
  \begin{enumerateA}
  \item
    \label{pc2.a}
    $\CH^iX\simeq \Z$ for $i\neq \dim X/2$, and $\CH^iX\simeq
    \Z\oplus\Z$ if $i=\dim X/2$; $h^i$ is a generator of $\CH^iX$ for
    $i<\dim X/2$ and twice a generator for $i>\dim X/2$. For $i=\dim
    X/2$, a basis of $\CH^iX$ is given by the classes $l$, $l'$ of two
    maximal linear subspaces of $X$ such that $h^i=l+l'$.
  \item
    \label{pc2.b}
    The natural maps
    \[
    K_{j-i}F\oo \CH^iX \to H^i(X,{\cal K}_j)
    \]
    given by the Brown-Gersten-Quillen spectral sequence and
    cup-product are isomorphisms.
  \item
    \label{pc2.c}
    The natural maps
    \[
    H^{j-i}(F,\mu_m^{\oo k})\oo \CH^iX \to H^i(X,{\cal
      H}^j(\mu_m^{\oo(i+k)}))
    \]
    given by the coniveau spectral sequence and cup-product are
    isomorphisms for all $m\ge 1$.
  \end{enumerateA}
\end{prop}

\begin{proof} 
  \Aref{pc2.a} follows from proposition~\ref{pc3}. Let us prove
  \Aref{pc2.b} (as above, the proof of \Aref{pc2.c} is identical and
  omitted). (See \cite[prop.~1 and prop.~3]{merkurjev} for another
  approach.) The quadric $X$ is isomorphic to
  \[
  \begin{array}{lc}
    X_n:&\begin{cases}x_0x_1+\dots+x_nx_{n+1}=0&\text{if $n$ is
        even}\\ x_0^2+x_1x_2+\dots+x_nx_{n+1}=0&\text{if $n$ is odd}
    \end{cases}
  \end{array}
  \]
  where $n=\dim X$. As a consequence of lemma~\ref{lpc2}, the diagram
  \[
  \begin{CD}
    H^{i-1}(X_{n-2},{\cal K}_{j-1})@>\sim>> H^i(X_n,{\cal K}_j)\\ 
    @A{\cup}AA @A{\cup}AA\\ K_{j-i}F\oo
    \CH^{i-1}X_{n-2}@>\sim>>K_{j-i}F\oo \CH^iX_n
  \end{CD}\quad\text{for $0<i<n$}
  \]
  commutes for $i\le j$. Inductively we get the commutative diagrams
  \[
  \begin{CD}
    H^{0}(X_{n-2i},{\cal K}_{j-i})@>\sim>> H^i(X_n,{\cal K}_j)\\ 
    @A{\cup}AA @A{\cup}AA\quad\text{for $2i\le n$,}\\ K_{j-i}F\oo
    \CH^{0}X_{n-2i}@>\sim>>K_{j-i}F\oo \CH^iX_n \vadjust{\vskip1ex}\\ 
    H^{2i-n}(X_{2i-n},{\cal K}_{j+i-n})@>\sim>> H^i(X_n,{\cal K}_j)\\ 
    @A{\cup}AA @A{\cup}AA\quad\text{for $2i\ge n$.}\\ K_{j-i}F\oo
    \CH^{2i-n}X_{2i-n}@>\sim>>K_{j-i}F\oo \CH^iX_n
  \end{CD}
  \]
  Still by lemma~\ref{lpc2}, $K_{j-i}F\oo \CH^{0}X_{n-2i}
  \xrightarrow{\cup} H^{0}(X_{n-2i},{\cal K}_{j-i})$ and
  \[
  K_{j-i}F\oo \CH^{2i-n}X_{2i-n} \xrightarrow{{\cup}}
  H^{2i-n}(X_{2i-n},{\cal K}_{j+i-n})
  \]
  are isomorphisms (note that for $2i=n$, $X_0$ is a disjoint union of
  two points so $\CH^0X_0\simeq \Z\oplus\Z$). The claim follows.
\end{proof}

\begin{lemma}
  \label{lc1}
  Let $i>0$ and $X$ be a quadric over $F$, with\/ $\dim X\neq 2i$ or\/
  $\dim X= 2i$ and\/ $d(X)=1$. Then, for all\/ $j\ge i$, the map
  $K_{j-i}F\oo \CH^iX \to H^i(X,{\cal K}_j)$ has $2$-primary torsion
  kernel and cokernel. If\/ $\dim X=2i$ and\/ $d(X)=d\neq 1$, the
  composition
  \[
  (K_{j-i}E\oo \CH^iX_E)_\Gamma \to H^i(X_E,{\cal K}_j)_\Gamma
  \xrightarrow{N} H^i(X,{\cal K}_j)
  \]
  has $2$-primary torsion kernel and cokernel, where $E=F(\sqrt d)$,
  $\Gamma=\Gal(E/F)$ and~$N$ is the transfer in $\cal K$-cohomology.
\end{lemma}

\begin{proof} 
  Assume first that $\dim X\neq 2i$ or $\dim X=2i$ and $d(X)=1$.  Let
  $K$ be a splitting field for $X$ that is finite and Galois over $F$,
  with $[K\mathop:F]$ a power of $2$ (some multiquadratic extension
  will do) and $G=\Gal(K/F)$.  In the commutative diagram
  \[
  \begin{CD}
    K_{j-i}F\oo \CH^iX@>>> H^i(X,{\cal K}_j)\\ @V{\Res}VV @V{\Res}VV\\ 
    (K_{j-i}K\oo \CH^iX_K)^G@>>> H^i(X_K,{\cal K}_j)^G
  \end{CD}
  \]
  the bottom horizontal map is an isomorphism by proposition~\ref{pc2}
  and the right vertical map has $2$-primary torsion kernel by the
  usual transfer argument. Therefore, to prove the claim, it is enough
  to show that the left vertical map has $2$-primary torsion kernel
  and cokernel. But this map decomposes as
  \begin{multline*}
    K_{j-i}F\oo \CH^iX \to K_{j-i}F\oo \CH^iX_K\to \\ \to
    (K_{j-i}K)^G\oo \CH^iX_K= (K_{j-i}K\oo \CH^iX_K)^G
  \end{multline*}
  since $\CH^iX_K\simeq \Z$ or $\Z\oplus \Z$ with trivial Galois
  action by propositions~\ref{pc3} and~\ref{pc2}.  The same
  proposition shows that the left map has $2$-primary torsion kernel
  and cokernel, while the transfer argument shows the same for the
  right map. This implies the same for their composition.
  
  Assume now that $\dim X=2i$ and $d(X)=d\neq 1$. By the above, the
  left map in the composition of lemma~\ref{lc1} has $2$-primary
  torsion kernel and cokernel, and the same holds for the right one
  still by a transfer argument. This concludes the proof of
  lemma~\ref{lc1}.
\end{proof}

\begin{prop}
  \label{l2}
  Let $i>0$ and $X$ be a quadric over $F$, with $\dim X\neq 2i-2$ or
  $\dim X= 2i-2$ and $d(X)=1$. Then there is an exact sequence:
  \[
  H^1(F,1)\oo \CH^{i-1}X \to H^{i-1}(X,{\cal H}^i(i)) \to
  (\CH^iX)_{\tors} \to 0.
  \]
  If\/ $\dim X=2i-2$ and $d(X)=d\neq 1$, there is an exact sequence:
  \[
  (H^1(E,1)\oo \CH^{i-1}X_E)_\Gamma \to H^{i-1}(X,{\cal H}^i(i)) \to
  (\CH^iX)_{\tors} \to 0,
  \]
  where $E=F(\sqrt d)$, $\Gamma=\Gal(E/F)$ and the left map is
  cup-product followed by transfer.
\end{prop}

\begin{proof}
  Tensoring the maps of lemma~\ref{lc1} with $\Q/\Z$ we get
  surjections:
  \[
  K_1F\oo \CH^{i-1}X\oo \Q/\Z\Surj H^{i-1}(X,{\cal K}_i)\oo \Q/\Z
  \]
  if $\dim X\neq 2i-2$ or $\dim X= 2i-2$ and $d(X)=1$,
  \begin{multline*}(K_1E\oo \CH^{i-1}X_E\oo \Q/\Z)_\Gamma=(K_1E\oo
    \CH^{i-1}X_E)_\Gamma\oo \Q/\Z\\ \Surj H^{i-1}(X,{\cal K}_i)\oo
    \Q/\Z\end{multline*} if $\dim X=2i-2$ and $d(X)=d\neq 1$.
  
  By Kummer theory, $K_1F\oo\Q/\Z\simeq H^1(F,1)$. The claim now
  follows from the short exact sequence \cite[3.6]{CT}
  $$
  0 \to H^{i-1}(X,{\cal K}_i)\oo\Q/\Z \to H^{i-1}(X,{\cal H}^i(i)) \to
  (\CH^iX)_{\tors} \to 0.\eqno{\qed}
  $$\noqed
\end{proof}

\begin{prop}\leavevmode
  \label{pc5}
  \begin{enumerateA}
  \item
    \label{pc5.a}
    \iname{\cite[prop.~1.2]{cto}} If the quadric $X$ is isotropic over
    $F$, then $\eta_2^i$ and $\eta^i$ are isomorphisms for all\/ $i\ge
    0$.
  \item
    \label{pc5.b}
    In general, $\Ker \eta^i$ and\/ $\Coker\eta^i$ have exponent $\le
    2$.
  \item
    \label{pc5.c}
    \iname{\cf \cite[lemma~1.3]{CTS}} Suppose $X$ and $X'$ are two
    quadrics such that $X'$ becomes isotropic over $F(X)$. Then there
    is a natural commutative diagram for all\/~$i$,~$k$:
    \[
    \begin{CD}
      &&H^i_{nr}(F(X)/F,k)\\ &\scriptstyle
      \eta^i_X\displaystyle\nearrow\\ H^i(F,k) &&@AAA\\ &\scriptstyle
      \eta^i_{X'}\displaystyle\searrow\\ &&H^i_{nr}(F(X')/F,k)
    \end{CD}
    \]
    so that\/ $\Ker\eta^i_{X'}\subseteq\Ker\eta^i_X$ and\/
    $\Coker\eta^i_{X'}$ maps to\/ $\Coker\eta^i_X$. If $X$ also
    becomes isotropic over $F(X')$, then\/
    $\Ker\eta^i_X=\Ker\eta^i_{X'}$ and\/
    $\Coker\eta^i_X\simeq\Coker\eta^i_{X'}$.
  \end{enumerateA}
\end{prop}

\begin{proof}
  \Aref{pc5.a} comes from the fact that the extension $F(X)/F$ is
  purely transcendental if $X$ is isotropic (compare
  \cite[th.~4.1.5]{CTcourse}).  \Aref{pc5.b} follows from \Aref{pc5.a}
  and a transfer argument, using a quadratic extension of $F$ which
  makes $X$ isotropic. We give a proof of \Aref{pc5.c} which is more
  elementary than \cite[lemma~1.3]{CTS}.  Consider the diagram of
  fields
  \[
  \begin{CD}
    &&F(X)\\ &\nearrow&&\searrow\\ F\hskip.8em &&&&F(X\times_F X')\\ 
    &\searrow&&\nearrow\\ &&F(X')
  \end{CD}
  \]
  There are natural maps
  \[
  H^i_{nr}(F(X)/F,k) \to H^i_{nr}(F(X\times_F X')/F,k)
  \]
  and
  \[
  H^i_{nr}(F(X')/F,k)\to H^i_{nr}(F(X\times_F X')/F,k).
  \]
  If $X'$ becomes isotropic over $F(X)$, then $F(X\times_F X')/F(X)$
  is a purely transcendental extension. Therefore,
  by~\cite[prop.~1.2]{cto} or~\cite[th.~4.1.5]{CTcourse}, the first
  quoted map is an isomorphism, which proves the first claim. If $X$
  is also purely transcendental over $F(X')$, then
  $H^i_{nr}(F(X')/F,k) \to H^i_{nr}(F(X)/F,k)$ is clearly the inverse
  of $H^i_{nr}(F(X)/F,k) \to H^i_{nr}(F(X')/F,k)$, hence the second
  claim.
\end{proof}

%%%% part 2
\part{$\Ker\eta_2^4$ for 
  higher-dimensional quadrics}\label{kernel}
\label{part.2}

\section{Generalities on $\Ker\eta_2^i$}
\label{generalities}

We fix a number~$i$ and assume that the invariant $\overline e^r_F$ is
is well-defined for $r\leq i-1$ and an isomorphism for $r\leq i-2$
(for all fields $F$).

\begin{lemma}
  \label{la2-x} 
  A symbol $(a_1,\dots,a_i)\neq 0$ lies in $\Ker\eta^i_2$ if and only
  if\/ $q$ is similar to a subform of\/ $\Pf{ a_1,\dots,a_i}$.
\end{lemma}
\begin{proof}
  Let $\varphi=\Pf{ a_1,\dots,a_i}$. Since $(a_1,\dots,a_i)_{F(X)}=0$,
  \cite[remark p.~555]{jr} implies $\varphi_{F(X)}\sim 0$, and the
  claim follows by the Cassels-Pfister theorem~\cite[th.~IX.1.3]{lam}.
\end{proof}

As observed in the introduction, for $\dim q\leq 2^{i-2}$ the group
$\Ker\eta^i_2$ does not consist of symbols in general.  The following
proposition and corollary indicate that this bound on $\dim q$ might
be sharp. Recall that two $n$-fold Pfister forms $\varphi$, $\psi$ are
\emph{linked} if there exist $a_1$, \dots, $a_{n-1}$, $a_n$, $b_n\in
F^*$ such that $\varphi\simeq \Pf{ a_1,\dots,a_{n-1},a_n}$ and
$\psi\simeq \Pf{ a_1,\dots,a_{n-1},b_n}$. We define the notion of
linked symbols (in Galois cohomology) similarly.

\begin{prop}\label{pa4} 
  Let\/ $\dim q\ge 2^{i-2}+1$, let $X$ be the projective quadric with
  equation $q=0$ and let $\varphi$, $\psi\in \Ker\bigl(W(F) \to
  W(F(X))\bigr)$ be two $i$-fold Pfister forms. Then $\varphi$ and
  $\psi$ are linked, hence $\varphi\perp -\psi$ is Witt-equivalent to
  a third\/ $i$-fold Pfister form \upp(up to similarity\upp).
\end{prop}

\begin{proof}
  \iname{\cf Fitzgerald \cite[proof of th.~2.1]{fitz}} The forms
  $\varphi$ and $\psi$ are multiplicative and split over~$F(X)$.
  Therefore $q$ is isomorphic to subforms of~$\varphi$ and of~$\psi$
  (assuming $q$ represents~$1$).  Hence the split form $q\perp-q$ is a
  subform of $\varphi\perp -\psi$ and the Witt index
  $n=i(\varphi\perp-\psi)$ is $\ge 2^{i-2}+1$. By \cite[th.~4.5]{el},
  $n$ is a power of $2$, hence $n\ge 2^{i-1}$.  Therefore $\varphi$
  and~$\psi$ are linked, again by \loccit.
\end{proof}

\begin{lemma}
  \label{ca2}
  For\/ $\dim q\ge 2^{i-2}+1$ the following are equivalent:
  \begin{enumerateA}
  \item $\Ker\eta_2^i$ consists of symbols;
  \item
    \label{ca2.b}
    $\Ker\eta_2^i$ is additively generated by symbols.
  \end{enumerateA}
\end{lemma}

\begin{proof}
  Let $\alpha$, $\beta\in \Ker\eta_2^i$ be two symbols and
  $\tilde\alpha$, $\tilde\beta$ Pfister forms such that
  $e^i(\tilde\alpha)=\alpha$, $e^i(\tilde\beta)=\beta$. By
  \cite[remark p.~555]{jr}, $\tilde\alpha$, $\tilde\beta\in \Ker(W(F)
  \to W(F(X)))$. By proposition~\ref{pa4}, $\tilde\alpha$ and
  $\tilde\beta$ are linked, so $\alpha$ and $\beta$ are linked and
  $\alpha+\beta$ is a symbol.
\end{proof}


\begin{prop}
  \label{t3.X}
  Assume that for\/ $\dim q =2^{i-2}+1$ the group\/ $\Ker\eta_2^i$
  consists of symbols. Then
  \begin{enumerateT}
  \item
    \label{t3.X.2}
    $\Ker\eta^i_2$ consists of symbols for\/ $\dim q> 2^{i-2}$.
  \item
    \label{t3.X.3}
    $\left|\Ker\eta^i_2\right|\le 2$ for\/ $\dim q> 2^{i-1}$.
  \item
    \label{t3.X.4}
    $\Ker\eta^i_2=0$ for\/ $\dim q> 2^i$.
  \end{enumerateT} 
\end{prop}

\begin{rk}
  The hypothesis of proposition~\ref{t3.X} is satisfied for $i\leq3$
  by \cite{arason}. For $i=4$ it will be proved in the next three
  sections (theorem~\ref{kersym}).
\end{rk}

\begin{proof}
  \iname{\ref{t3.X.2}} Let $\dim q' \ge 2^{i-2}+1$ and let $q$ be a
  subform of~$q'$ of dimension $2^{i-2}+1$.  The claim follows now
  from proposition \ref{pc5}~\Aref{pc5.c} applied to the associated
  quadrics $X'$ \resp~$X$.
  
  \eqref{t3.X.3} $+$ \iname{\ref{t3.X.4}} If $\dim q> 2^{i-1}$, for
  any nonzero symbol $(a_1,\dots,a_i)\in \Ker\eta_2^i$, $q$ is a
  neighbour of the Pfister form $\Pf{ a_1,\dots,a_i}$ by
  lemma~\ref{la2-x}.  But $q$ can be a neighbour of at most one
  Pfister form, up to isomorphism, \cf \cite[7.4]{knebusch2}. Moreover
  if $\Pf{ a_1,\dots,a_i}\simeq \Pf{b_1,\dots,b_i}$, then
  $(a_1,\dots,a_i)=(b_1,\dots,b_i)$ \cite[Satz~1.6]{arason}.  Hence
  $\left|\Ker\eta^i_2\right|\le 2$, and if it is nontrivial then $\dim
  q\le 2^i$.
\end{proof}

\section{$6$-dimensional spinors}
\label{spinors}

For generalities on Clifford algebras and the special Clifford group
see~\cite{Scharlau}.

Let $q\colon V\to F$ be $6$-dimensional, let $C_0(q)$ be the even
Clifford algebra of~$q$, and let $Z(q)$ be the center of~$C_0(q)$.
Then $Z(q)$ is a quadratic extension of~$F$ and $C_0(q)$ is an Azumaya
algebra of degree~$4$ over~$Z(q)$.  The standard involution~$x\mapsto
x^t$ is an involution of second kind.  Denote by
\[
\Nrd\colon C_0(q)\to Z(q)
\]
the reduced norm.

Let
\[
\SG(q)=\{\,x\in C_0(q)^*\mid xVx^{-1}=V\,\}
\]
be the special Clifford group and let
\[
\GG(q)=\{\,x\in C_0(q)^*\mid xx^t\in F^*\,\}.
\]
Then $\SG(q)\subset\GG(q)$ and there is an exact sequence
\[
1\to\Spin(q)\to \SG(q)\xrightarrow{\sn} F^*
\]
where $\sn(x)= xx^t$ is the spinor norm.

Suppose $d_\pm(q)=1$. Then $Z(q)=F\times F$ and $C_0(q)=A\times A'$
where $A$ and~$A'$ are algebras of degree~$4$ over~$F$. The standard
involution identifies $A'$ with $A^\op$, so that there is an
isomorphism
\begin{equation}
  \label{CAAop}
  C_0(q)=A\times A^\op
\end{equation}
under which the standard involution reads as $(a,b)^t=(b,a)$.  The
reduced norm $C_0(q)\to Z(q)$ is then componentwise the reduced norm
on~$A$. The group~$\GG(q)$ becomes the group of pairs
\[
\{\,(a,a^{-1}f)\in A\times A^\op \mid a\in A^*,\,f\in F^*\,\}.
\]
Let $\pi\colon C_0(q)\to A$ be the projection onto the first factor.
Then
\[
\GG(q)=A^*\times F^*
\]
via $x\mapsto (\pi(x),xx^t)$.

We call an element~$x\in \SG(q)$ \emph{plane} if $x\in \SG(\tilde q)$
where $\tilde q$ is a regular $2$-dimensional subform of~$q$. If $q$
is anisotropic, then $x$ is plane if and only if $x$ is the
product~$vw$ of two vectors $v$, $w\in V\setminus 0$.


\begin{lemma} 
  \label{Nrdsn}
  $\Nrd(x)=(xx^t)^2$ for\/ $x\in\SG(q)$.
\end{lemma}
\begin{proof}
  The action of~$\SG(q)$ on~$V$ induces an exact sequence
  \begin{equation}
    \label{clifort}
    1 \to F^* \to \SG(q) \to \SO(q) \to 1.
  \end{equation}
  Since $\O(q)$ is generated by reflections, it follows that $\SG(q)$
  is generated by products~$x=vw$ where $v$, $w\in V$ are anisotropic
  vectors. For generic $v$,~$w$, the subspace of~$V$ generated by
  $v$,~$w$ is regular of dimension~$2$.  Therefore it suffices to
  check the claim for plane elements.
  
  So suppose $x\in C_0(\tilde q)$ for a regular $2$-dimensional
  subform~$\tilde q$.  Write $L=C_0(\tilde q)$.  The standard
  involution leaves~$L$ invariant and restricts to the canonical
  involution of~$L/F$.  Hence $x\mapsto xx^t$ restricts on~$L$ to the
  norm of~$L/F$. The claim follows now from the fact that $C_0(q)$ is
  of degree~$4$ and $L\otimes Z(q)\subset C_0(q)$ is of degree~$2$
  over~$Z(q)$.
\end{proof}

\begin{prop}
  \label{A3D3}
  \leavevmode
  \begin{enumerateA}
  \item
    \label{A3D3.1}
    $\qquad\phantom{\Spin(q)}\llap{$\SG(q)$}=\{\,x\in C_0(q)^*\mid
    xx^t\in F^*,\,\Nrd(x)=(xx^t)^2\,\}$, \vspace{1ex}
  \item
    \label{A3D3.2}
    $\qquad\Spin(q)=\{\,x\in C_0(q)^*\mid xx^t=1,\, \Nrd(x)=1 \,\}$.
  \end{enumerateA}
\end{prop}

%%\begin{prop}
%%%%  \leavevmode
%%  \label{A3D3}
%%  \begin{align}\tag{a}
%%    \label{A3D3.1}
%%    \SG(q)&=\{\,x\in C_0(q)^*\mid xx^t\in
%%    F^*,\,\Nrd(x)=(xx^t)^2\,\},\\ \tag{b}
%%    \label{A3D3.2}
%%    \Spin(q)&=\{\,x\in C_0(q)^*\mid xx^t=1,\, \Nrd(x)=1 \,\}.
%%  \end{align}
%%\end{prop}

\begin{proof}
  The inclusions ``$\subset$'' are clear from lemma~\ref{Nrdsn}.
  
  We may assume that $F$ is algebraically closed. In this case it easy
  to see that $\SG(q)=\Spin(q)\cdot F^*$ and similarly for the
  righthand sides.  Hence \Aref{A3D3.2}~$\Rightarrow$~\Aref{A3D3.1}.
  
  The righthand side of~\Aref{A3D3.2} becomes under~\eqref{CAAop} the
  group of pairs
  \[
  \{\,(a,a^{-1})\in A\times A^\op \mid a\in
  A^*,\,\Nrd(a)=1\,\}=\SL(1,A).
  \]
  This way~\Aref{A3D3.2} reads as the classical isomorphism
  $\Spin(6)=\SL(4)$ \cite{VdW,DiEQ,involutions}.
\end{proof}

\begin{cor}
  \label{SGNf}
  If\/ $d_\pm(q)=1$ and\/ $A$ is as above, there is an exact sequence
  \[
  1\to \SG(q)\to A^*\times F^*\xrightarrow{\omega} F^*
  \]  
  where\/ $\omega(a,f)=\Nrd(a)/f^2$.
\end{cor}

\begin{cor}
  \label{bi3}
  Let $A$ be a biquaternion skewfield and let\/ $a\in A^*$ and\/ $f\in
  F^*$ with
  \[
  \Nrd(a)=f^2.
  \]
  Then there exist\/ $a_1$, $a_2$, $a_3\in A^*$ with the following
  properties:
  \begin{itemize}
  \item $a=a_1a_2a_3$,
  \item Every\/ $a_i$ lies in a quadratic subfield\/ $L_i\subset A$,
  \item $N_{L_1/F}(a_1)N_{L_2/F}(a_2)N_{L_3/F}(a_3)=f$.
  \end{itemize}
\end{cor}

\begin{proof}
  If $A=(a,b)\otimes(c,d)$, let $q=\qf{-a,-b,ab,c,d,-cd}$.  We are
  then in the situation of corollary~\ref{SGNf}.  The form~$q$ is
  anisotropic. Since every element in~$\SO(q)$ is a product of~$6$
  reflections, it follows that every element in~$\SG(q)$ is a product
  of~$3$ plane elements~$x_i$.  Their images $a_i=\pi(x_i)$ in~$A^*$
  do the job.
\end{proof}

We will have to make use of the following variant of the
Cartan-Dieudonn\'e theorem for anisotropic forms.

\begin{lemma}
  \label{CartDieu}
  Let\/ $q\colon V\to F$ be anisotropic of dimension~$n$ and let\/
  $V_1\subset V_2\subset \dots \subset V_n=V$ be a sequence of
  subspaces with\/ $\dim V_i=i$. Then every $\alpha\in\O(q)$ is a
  product
  \[
  \alpha=\tau_{i_k}\cdots\,\tau_{i_1}
  \]
  where $1\leq i_1<\cdots<i_k\leq n$ and\/ $\tau_j$ is a reflection
  in\/ $\O(q{|V_j})$.
\end{lemma}
\begin{proof}
  By induction on~$n$. Let $v\neq0$ be orthogonal to $V_{n-1}$.  If
  $\alpha\notin\O(q{|V_{n-1}})$, let $\tau$ be the reflection at the
  hyperplane orthogonal to $\alpha(v)-v$. Then
  $\tau\alpha\in\O(q{|V_{n-1}})$.
\end{proof}

\section{$\Ker \eta^4_2$ and $H^2(X,\cal K_3)$}
\label{H2K4}

The following theorem reduces the computation of~$\Ker \eta^4_2$
for~$\dim q\geq5$ to a $\cal K$-cohomology computation.  The statement
is essentially a reformulation of arguments in \cite{Mecourse}, where
the result has been applied to Albert forms.  The principal idea has
already been used in \cite{jr} for $5$-dimensional Pfister neighbours.

\begin{thm}
  \label{H2K4.thm}
  Let\/ $X$ be a smooth quadric of dimension\/~$\geq3$.  Then there
  exist isomorphisms
  \begin{align*}
    \rho_3&\colon \Ker(H^2(X,{\cal K}_2) \to H^2(\overline X,{\cal
      K}_2))\,\iso\, \Ker(H^3F\to H^3F(X)),\\ \rho_4&\colon
    \Ker(H^2(X,{\cal K}_3) \to H^2(\overline X,{\cal K}_3))\,\iso\,
    \Ker(H^4F\to H^4F(X)).
  \end{align*}
  These are functorial with respect to base change $F\to F'$ and with
  respect to transfer for finite extensions~$F'/F$.
  
  Moreover they commute with multiplication with~$K_1$ in the
  following sense:
  \[
  \rho_4(\{a\}\cdot x)=(a)\cdot\rho_3(x)
  \]
  for\/ $a\in F^*$ and\/ $x\in \Ker(H^2(X,{\cal K}_2) \to
  H^2(\overline X,{\cal K}_2))$.
\end{thm}
\begin{proof}
  The (inverse of the) map~$\rho_4$ has been described in
  \cite[p.~71]{Mecourse} in terms of a diagram chase. It is also
  proven that the map is an isomorphism (it should be noted that
  condition~(7) in \cite[p.~71]{Mecourse} is superfluous).  From this
  definition the compatibility with respect to base change and
  transfer is easy to see.
  
  One proceeds in a similar way for~$\rho_3$ (see also \cite[diagrams
  (3.5), (4.2)]{jr}), just by repeating the diagram chase in one
  degree less. The product formula is then obvious.
\end{proof}

\begin{rk}
  It can be shown that the maps~$\rho_i$ are functorial with respect
  to linear morphisms $j\colon X\subset X'$ of quadrics.  Namely then
  $\Ker\eta^i_{X'}\subset\Ker\eta^i_X$
  (proposition~\ref{pc5}~\Aref{pc5.c}) and the maps~$\rho_i$ commute
  with these inclusions and with the pull back maps~$j^*$
  on~$H^2(\emptyvar,{\cal K}_{i-1})$.
\end{rk}

Our aim is to compute the map~$\rho_4$ for $\dim X=3$.

Let $q$ be a form defining~$X$ and let $A=C_0(q)$ be its even Clifford
algebra (a biquaternion algebra).  We denote by $h\in\CH^1(X)$ the
class of a hyperplane section.  If $A$ is split, then $X$ is split. In
this case $\CH^2(X)=\Z$, generated by the class~$l$ of a line and one
has $h^2=2l$.

\begin{thm}
  \label{sur}
  For\/ $\dim X=3$ there exist surjections
  \begin{align*}
    \theta_3&\colon K_0A\oplus K_0F \surj H^2(X,{\cal K}_2),\\ 
    \theta_4&\colon K_1A\oplus K_1F \surj H^2(X,{\cal K}_3),
  \end{align*}
  functorial with respect to base change and transfer, compatible with
  multiplication with\/~$K_1$, and such that\/ $\theta_3$ has the
  following properties:
  \begin{enumerateA}
  \item
    \label{sur.1}
    $\theta_3(0,1)=-h^2$.
  \item
    \label{sur.2}
    If\/ $A$ is split, let $y\in K_0A$ be the element with\/
    $\Nrd(y)=1$. Then\/ $\theta_3(y,0)=l$.
  \end{enumerateA}
\end{thm}
\begin{proof}
  All the essential arguments are in \cite[p.~74--76]{Mecourse}.  In
  fact, in the $5$-dimensional case the situation is somewhat simpler.
  By \cite[prop.~2]{Mecourse},
  \[
  H^2(X,{\cal K}_3)= K_1(X)^{(2)}/ K_1(X)^{(3)}
  \]
  where the superscripts refer to the filtration coming from the
  Brown-Gersten-Quillen (BGQ) spectral sequence~\cite{quillen}.  In
  our case one finds $K_1(X)^{(2)}=K_1A\oplus K_1F$.  (We don't have
  to care about a computation of~$K_1(X)^{(3)}$).
\end{proof}

Let
\[
\Res_{\overline F/F}\colon H^2(X,{\cal K}_{i})\to H^2(\overline
X,{\cal K}_{i})
\]
be the restriction to a separable closure.  Theorem \ref{sur}
\Aref{sur.1},~\Aref{sur.2} shows
\begin{equation*}
  \Res_{\overline F/F}\circ \,\theta_3(y,m)=(\Nrd(y)-2m)\cdot l \in
  \CH^2(\overline X)=\Z\cdot l.
\end{equation*}
The compatibility of $\theta_3$, $\theta_4$ with multiplication
with~$K_1$ yields
\begin{equation*}
  \Res_{\overline F/F}\circ \,\theta_4(z,\{f\})=(\Nrd(z)-2\{f\})\cdot
  l \in H^2(\overline X,\cal K_3)=K_1\overline F\cdot l.
\end{equation*}
This together with theorem~\ref{H2K4.thm} and theorem~\ref{sur} yields
\begin{cor}\label{c5.1} There are surjections
  \begin{align*}
    \omega_3&\colon \Ker(K_0A\oplus K_0F\xrightarrow{(\Nrd,-2)}K_0F)
    \,\surj\, \Ker\eta_2^3,\\ \omega_4&\colon \Ker(K_1A\oplus
    K_1F\xrightarrow{(\Nrd,-2)}K_1F) \,\surj\, \Ker\eta_2^4,
  \end{align*}
  compatible with base change, transfer and multiplication
  with~$K_1$.\qed
\end{cor}

\section{Computations for $\dim X=3$}

Let $q\colon W\to F$ be $5$-dimensional and $A=C_0(q)$.  We assume
that the determinant of~$q$ is $-1$.  Put
\begin{gather*}
  \q\colon F\oplus W\to F\\ \q(t,w)=t^2+q(w).
\end{gather*}
Then $C_0(\q)=A\times A^\op$ as in~\eqref{CAAop}.

We first compute $\omega_3$, $\omega_4$ in the case when $\q$ is
isotropic.  In this case $q$ represents~$-1$ and therefore
\[
q\simeq \qf{-1}\bot c\Pf{a,b}
\]
for some $a$, $b$, $c\in F^*$.  Moreover $A=M_2(F)\otimes D$ where $D$
is the quaternion algebra associated to $a$,~ $b$.

\begin{prop}
  \label{p6.1}
  Let $y\in K_0A=K_0D$ be the element represented by the
  $D$-module~$D$ \upp(\ie $\Nrd(y)=2$\upp) and let\/
  $\varepsilon=(y,1)$.
  \begin{enumerateT}
  \item
    \label{p6.1.4}
    The sequences
    \begin{gather*}
      K_0F\xrightarrow{\cdot\, \varepsilon}K_0A\oplus
      K_0F\xrightarrow{(\Nrd,-2)}K_0F,\\ K_1F\xrightarrow{\cdot\,
        \varepsilon}K_1A\oplus K_1F\xrightarrow{(\Nrd,-2)}K_1F
    \end{gather*}
    are exact.
  \item
    \label{p6.1.2} 
    One has
    \begin{align*}
      \omega_3(\varepsilon)&=(a,b,c),\\ 
      \omega_4(\{e\}\cdot\varepsilon)&=(e,a,b,c),\qquad e\in F^*.
    \end{align*}
  \end{enumerateT}
\end{prop} 
\begin{proof} \eqref{p6.1.4}  follows
  from the injectivity of the reduced norm $K_iD\to K_iF$ for $i=0$, $1$.
  
  Corollary \ref{c5.1} shows that $\omega_3(\varepsilon)$
  generates~$\Ker\eta_2^3$. Since $(a,b,c)\in \Ker\eta_2^3$, we must
  have the first formula in~\eqref{p6.1.2}.  (If $(a,b,c)=0$, then $q$
  is isotropic and $\Ker\eta_2^3=0$.)  The second formula follows from
  the compatibility with multiplication.
\end{proof}

We now assume that $\q$ is anisotropic.  Via corollary~\ref{SGNf}, the
map~$\omega_4$ of corollary~\ref{c5.1} reads as a map
\[
\tilde\omega_4\colon \SG(\q)\surj \Ker\eta_2^4.
\]
We compute this map for plane elements $x\in\SG(\q)$.

Suppose $x\in L=C_0(\tilde q)$ for a $2$-dimensional subform~$\tilde
q$ of~$\q$.  The map
\[
L^*=C_0(\tilde q)^*=\SG(\tilde q)\subset\SG(\q)\to A^*\times F^*
\]
coincides with the map
\[
K_1L\to K_1A\oplus K_1F
\]
given by the inclusion $L\subset A$ and the norm for $L/F$.

The form $\q_L$ is isotropic. Denote by
\[
\varepsilon_L\in \Ker(K_0(A\otimes L)\oplus
K_0L\xrightarrow{(\Nrd,-2)}K_0L)
\]
the generator considered above.

By compatibility with transfer for $L/F$ one finds
\[
\tilde\omega_4(x)=\Cores_{L/F}((x)\cdot\omega_3(\varepsilon_L))
\]
Note that if $\tilde q$ is a subform of~$q$, then $q_L$ is isotropic
and therefore $\omega_3(\varepsilon_L)=0$.

\medbreak We compute $\omega_4(u)$ for arbitrary $u\in\SG(\q)$.  If
$u\in\SG(q)$, then $u$ is a product of plane elements in $\SG(q)$ and
$\tilde\omega_4(u)=0$ by the last remark. In general we have

\begin{lemma} 
  \label{l6.1} 
  For any\/ $u\in S\Gamma(q)$, there exist a one-dimensional
  subspace~$H$ of\/~$W$ and a plane element\/ $x\in S\Gamma(\hat q|
  (F\oplus H))$ such that $u\in x\cdot S\Gamma(q)$.
\end{lemma}

\begin{proof} 
  There exist by lemma~\ref{CartDieu} and the exact sequence
  \eqref{clifort} a vector $v_1\in F\oplus W$ and vectors $v_2$,
  \dots, $v_6\in W$ such that $u=v_1\cdots v_6$.
  
  Write $v_1=s(1,0)+th$ with $s$,~$t\in F^*$, $0\neq h\in W$, and let
  $H$ be the subspace generated by~$h$.  Then $x=v_1h$ has the desired
  properties.
\end{proof}

Since $\omega_4|\SG(q)=0$, we are reduced to
computing~$\tilde\omega_4$ on plane elements $x\in S\Gamma(\hat q|
F\oplus H)$ as in lemma \ref{l6.1}. Let $h$ be a basis vector of~$H$
and let $a=-q(h)$.  Then $L=C_0(\tilde q)=F(\sqrt a)$ and $q$ is of
the form
\[
q\simeq\qf{-a,a_1,a_2,a_3,aa_1a_2a_3}.
\]
Over~$L$ this gives
\[
q_L\simeq\qf{-1,a_1,a_2,a_3,a_1a_2a_3} \simeq \qf{-1}\bot
a_1\Pf{-a_1a_2,-a_1a_3}.
\]
Hence
\[
\omega_3(\varepsilon_L)=\Res_{L/F}((a_1,-a_1a_2,-a_1a_3))
\]
comes from the ground field (although $\varepsilon_L$ does not).  So
\[
\tilde\omega_4(x)=\Cores_{L/F}((x)\cdot\omega_3(\varepsilon_L))
=(N_{L/F}(x),a_1,-a_1a_2,-a_1a_3)
\]
is a symbol.

We have proved:
\begin{thm}
  \label{kersym}
  If\/ $\dim X=3$, then\/ $\Ker \eta^4_2$ consists of symbols.\qed
\end{thm}

\begin{rk}
  The homomorphism~$\tilde\omega_4$ is obviously trivial on the
  commutator subgroup~$[\SG(\q),\SG(\q)]$. Moreover it is easy to see
  that the map
  \[
  \SG(\q)/\SG(q) \to \O(\q)/\O(q)
  \]
  is bijective. Hence the map $\tilde\omega_4$ actually induces a
  surjective homomorphism
  \[
  \Omega\colon \O(\q)/[\O(\q),\O(\q)]{\O(q)}\surj \ker(H^4F\to
  H^4F(X))
  \]
  Merkurjev and Tignol have described this map in terms of symplectic
  involutions~\cite{merk3,tignol}.
  
  The kernel of the spinor norm
  \[
  \O(\q)/[\O(\q),\O(\q)]\to F^*/(F^*)^2,
  \]
  can be identified with ${\it SK}_1A$. The restriction of~$\Omega$ to
  ${\it SK}_1A$ depends only on $\q$ (and not on the choice of
  $5$-dimensional subform $q$). It fits into an exact
  sequence~\cite{Mecourse}
  \[
  0\to {\it SK}_1A\xrightarrow{\Omega} H^4F\to H^4F(\widehat X)
  \]
  where $\widehat X$ is the quadric for~$\q$.
  
  Further details will appear in~\cite{involutions}.
\end{rk}

%%%% part 3
\part{$\Coker\eta_2^i$ for higher-dimensional quadrics}
\label{part.3}

\section{A relationship between  $\Ker \eta_2^i$ and
  $\Coker \eta_2^i$}
\label{Coker}

In this section, we prove a general result which we then apply to
quadrics. To start with, let $F$ have arbitrary characteristic
exponent $p$ and $X$ be a smooth variety over~$F$. We denote by
$\delta=\delta(X/F)$ the prime-to-$p$ part of the $gcd$ of degrees of
closed points of~$X$. Let $n$ be some integer divisible by~$\delta$
and prime to $p$, $i\ge 1$ and let
\begin{gather*}
  \eta_n^{i}\colon H^{i}(F,\mu_n^{\oo (i-1)}) \to H^0(X,{\cal H}^i
  (\mu_n^{\oo (i-1)}))\\ \eta^{i}\colon H^{i}(F,i-1) \to H^0(X,{\cal
    H}^i(i-1))
\end{gather*} 
be the natural maps.

\begin{prop}
  \label{p1}
  Let $m$ be another integer divisible by\/ $\delta$ and prime to~$p$.
  For $i\le 3$, there is an exact sequence
  \[
  0 \to \Ker\eta_m^{i}\cap\IM\left(H^i(F,\mu_{mn}^{\oo (i-1)})
    \xrightarrow{n} H^i(F,\mu_m^{\oo (i-1)})\right) \to \Coker
  \eta_n^{i} \to \Coker\eta^{i}
  \]
  If\/ $m$, $n$ are powers of\/~$2$, this result also holds for\/
  $i=4$.
\end{prop}

\begin{proof} 
  We shall freely use the following fact, which is a consequence of
  Kummer theory for $i=2$ (\resp the Merkurjev-Suslin theorem for
  $i=3$, the Merkurjev-Suslin/Rost theorem for $i=4$): the map
  $H^i(F,\mu_n^{\oo (i-1)}) \xrightarrow{{\iota_*}} H^i(F,i-1)$ is
  injective, where $\iota'\colon \mu_n^{\oo (i-1)} \to \Q/\Z(i-1)$ is
  the natural inclusion.  We need:
  
  \begin{lemma}
    \label{lsup} 
    Under the assumptions of proposition~\ref{p1},
    \begin{enumerateA}
    \item
      \label{lsup.a} 
      $\Ker \eta_n^{i}$ and $\Ker \eta_{mn}^{i}$ are killed by\/
      $\delta$.
    \item
      \label{lsup.b} 
      $\Ker \eta_n^{i} \to \Ker\eta_{mn}^{i}$ is an isomorphism.
    \item
      \label{lsup.c} 
      $\Ker(\Coker\eta_n^{i} \to \Coker\eta_{mn}^{i})= \Ker(\Coker
      \eta_n^{i} \to \Coker\eta^i)$.
    \end{enumerateA}
  \end{lemma}
  
  \begin{proof}
    \Aref{lsup.a} If $X(F)\neq \emptyset$, this follows from a
    classical specialization argument; in general, it follows from
    this case and a transfer argument.
    
    \Aref{lsup.b} follows from \Aref{lsup.a}.
    
    \Aref{lsup.c} We have a commutative diagram with exact rows:
    \[
    \begin{CD}
      0&\to& H^i(F,\mu_n^{\oo (i-1)})&\to& H^i(F,i-1)
      @>{n}>>H^i(F,i-1)\\ &&@V{\eta_n^{i}}VV @V{\eta^i}VV
      @V{\eta^i}VV\\ 0&\to& H^0(X,{\cal H}^i(\mu_n^{\oo (i-1)}))&\to&
      H^0(X,{\cal H}^i(i-1)) @>{n}>>H^0(X,{\cal H}^i(i-1))
    \end{CD}
    \]
    where the exactness of the bottom row follows from a diagram chase
    and the injectivity of $H^{i-1}(F(x),\mu_n^{\oo (i-2)}) \to
    H^{i-1}(F(x),i-2)$ for all codimension $1$ points of~$X$. Let
    $x\in \Ker(\Coker \eta_n^{i} \to \Coker\eta^i)$, $\tilde x$ a lift
    of $x$ in $H^0(X,{\cal H}^i(\mu_n^{\oo (i-1)}))$ and pick
    $\alpha\in H^i(F,i-1)$ such that $\eta^i\alpha = $ image of
    $\tilde x$. Since $n\alpha\in \Ker \eta^i$, we have $\delta
    n\alpha=0$ by \Aref{lsup.a}, hence $mn\alpha=0$. If $\beta$ is the
    corresponding element of $H^i(F,\mu_{mn}^{\oo (i-1)})$, it is
    clear that $\eta_{mn}^{i}\beta$ is the image of $\tilde x$ in
    $H^0(X,{\cal H}^i(\mu_{mn}^{\oo (i-1)}))$.
  \end{proof}
  
  Let $\iota\colon \mu_n^{\oo (i-1)} \to \mu_{mn}^{\oo (i-1)}$ (\resp
  $\pi\colon \mu_{mn}^{\oo (i-1)} \to \mu_m^{\oo (i-1)}$) denote the
  natural inclusion (\resp projection). Consider the commutative
  diagram with exact rows:
  \[
  \begin{cd}
    0@>>> H^i(F,\mu_n^{\oo (i-1)})@>{\iota_*}>> H^i(F,\mu_{mn}^{\oo
      (i-1)})@>{\pi_*}>>H^i(F,\mu_m^{\oo (i-1)})_0 \ \longrightarrow
    0\\ &&@V{\eta_n^{i}}VV@V{\eta_{mn}^{i}}VV@V{\eta_m^{i}}VV\\ 0@>>>
    H^0(X,{\cal H}^i(\mu_n^{\oo (i-1)}))@>{\iota_*}>> H^0(X,{\cal
      H}^i(\mu_{mn}^{\oo (i-1)}))@>{\pi_*}>>H^0(X,{\cal
      H}^i(\mu_m^{\oo (i-1)}))\phantom{\longrightarrow}
  \end{cd}
  \]
  where $H^i(F,\mu_m^{\oo (i-1)})_0= \IM\pi_*$. From
  lemma~\ref{lsup}~\Aref{lsup.b} and the snake lemma applied to this
  diagram, we derive an exact sequence
  \[
  0 \to \Ker\eta_m^{i}\cap\IM\left(H^i(F,\mu_{mn}^{\oo (i-1)})
    \xrightarrow{n} H^i(F,\mu_m^{\oo (i-1)})\right) \to \Coker
  \eta_n^{i} \to \Coker\eta_{mn}^{i}.
  \]
  
  Proposition~\ref{p1} now follows from
  lemma~\ref{lsup}~\Aref{lsup.c}.
\end{proof}

\begin{prop}\label{p1.2}
  Let\/ $\car F\neq 2$, $i\ge 0$ and $\alpha\in H^iF$ be a
  decomposable element \upp(\ie a sum of symbols\upp). Then $\alpha\in
  \IM(H^i(F,\mu_4^{\oo(i-1)}) \xrightarrow{p} H^iF)$ if and only if
  $(-1)\cdot\alpha=0\in H^{i+1}F$. Here $p$ is the map induced by the
  surjective homomorphism of Galois modules $\mu_4^{\oo(i-1)}\to
  \mu_2^{\oo(i-1)}\simeq\Z/2$.
\end{prop}

\begin{proof} 
  Let $\partial$ (\resp $\partial'$) be the boundary map $H^iF \to
  H^{i+1}F$ corresponding to the short exact sequence $0 \to \Z/2 \to
  \mu_4^{\oo i} \to \Z/2 \to 0$ (\resp $0 \to \Z/2 \to \mu_4^{\oo
    (i-1)} \to \Z/2 \to 0$). It follows from \cite[lemma 1]{scharlau}
  that $\partial\alpha -\partial'\alpha= (-1)\cdot\alpha$.  If
  $\alpha$ is decomposable, then it obviously comes from
  $H^i(F,\mu_4^{\oo i})$ (reduce to a symbol
  $(a_1)\cdot\ldots\cdot(a_i)$, where $a_k\in F^*$, and observe that
  each $(a_k)$ comes from the Kummer class of $a_k$ in
  $H^1(F,\mu_4)$), hence $\partial\alpha=0$ and the lemma follows.
\end{proof}

\begin{prop}
  \label{c27}
  Suppose $\car F\neq 2$ and $\delta=2$ in proposition~\ref{p1}
  \upp(\eg $X$ is a quadric\upp). For $i\le 4$, the group
  \[
  \Ker\eta_2^{i}\cap\IM\left(H^i(F,\mu_{4}^{\oo (i-1)})
    \xrightarrow{{2}} H^iF\right)
  \]
  coincides with the group $(\Ker\eta_2^i)_0$ of the introduction. We
  have an exact sequence
  \[
  0 \to (\Ker\eta_2^i)_0\to \Coker\eta_2^i \to \Coker\eta^i.
  \]
\end{prop}
\begin{proof}
  Let $q$ be a quadratic form defining $X$ and $q'=\qf{a,b}$ be any
  binary subform of $q$.  Then $\Ker\eta_2^i\subseteq \Ker(H^iF \to
  H^iF(\sqrt{-ab}))=(-ab)\cdot H^{i-1}F$. Since $H^{i-1}F$ is
  generated by symbols \cite{ms1,ms}, we see that every element of
  $\Ker\eta_2^i$ is decomposable.  Finally, the last claim follows
  from this and proposition \ref{p1}.
\end{proof}

\begin{rk} 
  The arguments in this section show in fact that
  proposition~\ref{c27} follows for a given~$i$ from the Milnor
  conjecture in degree~$i$ (see the remark at the end of
  introduction).
\end{rk}

\section{Constructing some maps}\label{pair}

In this section, we show how the Chow ring acts on \'etale and
$\cal{H}$-cohomology with divisible coefficients. This action will be
repeatedly used in the sequel.

Let $X$ be a smooth variety over $F$, $r\ge 1$, $i$, $j\ge 0$ and $k$,
$k'\in\Z$. We first recall that Galois cohomology of~$F$ acts on the
Hochschild-Serre and coniveau spectral sequences
\begin{gather*}
  \HS_2^{p,q}(\Z/2^r(k)):=H^p(F,H^q(\overline X,\Z/2^r(k)))\Rightarrow
  H^{p+q}(X_{\et},\Z/2^r(k))\\ \BO_1^{p,q}(\Z/2^r(k)):=\coprod_{x\in
    X^{(p)}}H^{q-p}(F(x),\Z/2^r(k-p))\Rightarrow
  H^{p+q}(X_{\et},\Z/2^r(k))
\end{gather*}
in the following sense. There are pairings
\begin{gather*}
  H^n(F,\Z/2^r(j))\oo \HS_2^{p,q}(\Z/2^r(k)) \to
  \HS_2^{p+n,q}(\Z/2^r(j+k))\\ H^n(F,\Z/2^r(j))\oo
  \BO_1^{p,q}(\Z/2^r(k)) \to \BO_1^{p,q+n}(\Z/2^r(j+k))
\end{gather*}
coming from the cup-product in Galois cohomology (and the restriction
maps $H^*(F) \to H^*(F(x))$ for the coniveau spectral sequence). These
pairings commute with differentials and induce compatible pairings
\[
\begin{gathered}
  H^n(F,\Z/2^r(j))\oo \HS_s^{p,q}(\Z/2^r(k)) \to
  \HS_s^{p+n,q}(\Z/2^r(j+k))\\ H^n(F,\Z/2^r(j))\oo
  \BO_s^{p,q}(\Z/2^r(k)) \to \BO_s^{p,q+n}(\Z/2^r(j+k))
\end{gathered}
\quad\text{for $s\le \infty$}
\]
which are also compatible at the $E_\infty$-level with the pairings on
the abutments
\[
H^n(F,\Z/2^r(j))\oo H^{p+q}(X_{\et},\Z/2^r(k)) \to
H^{p+q+n}(X_{\et},\Z/2^r(j+k))
\]
(obtained from the maps $H^i(F,\Z/2^r(k)) \to H^i(X,\Z/2^r(k))$ and
cup-product in the \'etale cohomology of~$X$). This is classical for
the Hochschild-Serre spectral sequence; for the coniveau spectral
sequence, it is immediate from its exact couple construction out of
\'etale cohomology with supports (this supersedes the sophisticated
argument of \cite[end of introduction]{hcoh}).

If $E/F$ is a finite extension, there are also compatible transfer
maps.

We now play with these pairings a little bit. Using the cycle map
\[
\CH^jX/2^r \to H^{2j}_{\et}(X,\Z/2^r(j))
\]
we get composite pairings
\begin{multline*}
  H^i(F,\Z/2^r(k))\oo \CH^{j}X \to H^i(F,\Z/2^r(k))\oo
  H^{2j}_{\et}(X,\Z/2^r(j))\\ \xrightarrow{\cup}
  H^{i+2j}_{\et}(X,\Z/2^r(k+j))\end{multline*} hence, at the inductive
limit:
\begin{equation}\label{pair4}
  H^i(F,k)\oo \CH^{j}X \to H^{i+2j}(X,k+j).
\end{equation}

Recall now that $\BO_2^{j,j}(\Z/2^r(j))\simeq \CH^jX/2^r$. We get
therefore a pairing (similar to those in proposition~\ref{pc2}):
\[
H^i(F,\Z/2^r(k))\oo \CH^{j}X \to
H^{j}\bigl(X,{\cal{H}}^{i+j}(\Z/2^r(k+j))\bigr).
\]

Passing to the inductive limit, we get pairings:
\begin{equation}\label{pair3}
  H^i(F,k)\oo \CH^{j}X \to H^{j}(X,{\cal{H}}^{i+j}(k+j)).
\end{equation}

It is not difficult to check that the maps of proposition~\ref{l2} are
special cases of~\eqref{pair3}.

We have:

\begin{lemma}
  \label{lpair3}
  Let\/ $(H^*(X,k+j)^{(m)})_{m\ge 0}$ be the decreasing filtration
  on\/ $H^*(X,k+j)$ defined by the coniveau spectral sequence. Then
  \begin{enumerateA}
  \item
    \label{lpair3.a}
    The image of the map \eqref{pair4} is contained in
    $H^{i+2j}(X,k+j)^{(j)}$.
  \item
    \label{lpair3.b}
    The image of the map~\eqref{pair3} consists of universal cycles.
  \item
    \label{lpair3.c}
    The diagram
    \[
    \begin{CD}
      H^i(F,k)\oo \CH^{j}X@>>> H^{j}(X,{\cal{H}}^{i+j}(k+j))_{d=0}\\ 
      @VVV @VVV\\ H^{i+2j}(X,k+j)^{(j)}@>>> \BO_\infty^{j,i+j}
    \end{CD}
    \]
    commutes, where the subscript $d=0$ denotes universal cycles and
    $\BO_\infty^{j,i+j}$ denotes the $E_\infty$-term of the coniveau
    spectral sequence.
  \item
    \label{lpair3.d}
    For $i=0$, $1$, the triangle
    \[
    \begin{CD}
      H^i(F,k)\oo \CH^{j}X@>>> H^{j}(X,{\cal{H}}^{i+j}(k+j))\\ @VVV
      \llap{$\swarrow$\hskip5.5em}\\ H^{i+2j}(X,k+j)
    \end{CD}
    \]
    commutes, where the vertical map is~\eqref{pair4}, the horizontal
    map is~\eqref{pair3} and the diagonal map is the edge homomorphism
    from the coniveau spectral sequence.
  \end{enumerateA}
\end{lemma}

Recall that, in a spectral sequence $(E_r)$, an element $x\in
E_r^{p,q}$ is called a \emph{universal cycle} if $d_s^{p,q} x=0$ for
all $s\ge r$ (this definition makes sense inductively).

\begin{proof}
  It is enough to prove this for the maps defined at the finite
  coefficients level. By the multiplicativity of the coniveau spectral
  sequence, we are reduced to the case $i=k=0$. Then \Aref{lpair3.a},
  \Aref{lpair3.b}, and~\Aref{lpair3.c} follow from the fact that
  $\BO_1^{p,q}=0$ for $p>q$. Finally, \Aref{lpair3.d} follows
  from~\Aref{lpair3.c} and the definition of the edge homomorphism.
\end{proof}

Let $A$ be a discrete topological $G_F$-module and $M$ an abelian
group.  There is a pairing
\[
H^i(F,A)\oo M \to H^i(F,A\oo M)
\]
for all $i\ge 0$, coming from the natural map $M \to \Hom_{G_F}(A,A\oo
M)$.

\begin{lemma}\leavevmode
  \label{cheveux}
  \begin{enumerateA}
  \item
    \label{cheveux.a}
    If $M$ is torsion-free, the above pairing is an isomorphism.
  \item
    \label{cheveux.b}
    If $A$ is divisible, it is surjective.
  \item
    \label{cheveux.c}
    If $A$ and $H^i(F,A)$ are divisible, it is bijective.
  \end{enumerateA}
\end{lemma}

\begin{proof} 
  \Aref{cheveux.a} Reduce to the case where $M$ is finitely generated,
  hence free.
  
  \Aref{cheveux.b} Reduce to the case where $M$ is torsion-free by
  using the commutative diagram:
  \[
  \begin{CD}
    H^i(F,A)\oo M&\Surj& H^i(F,A)\oo (M/\tors)\\ @VVV @V{\wr}VV\\ 
    H^i(F,A\oo M) @>\sim>> H^i(F,A\oo (M/\tors)).
  \end{CD}
  \]
  
  \Aref{cheveux.c} Use the same diagram, noting that now the top
  horizontal map is bijective as well.
\end{proof}

Define a map
\begin{equation}\label{pair1}
  \alpha=\alpha^{i,j,k}\colon H^i(F,k)\oo \CH^{j}X \xrightarrow{}
  H^i(F,H^{2j}(\overline X,j+k))
\end{equation}
as the composite
\begin{multline*}
  H^i(F,k)\oo \CH^{j}X \to H^i(F,\CH^{j} X\oo\Q/\Z(k))\\ \to
  H^i(F,\CH^{j}\overline X\oo\Q/\Z(k))
  \xrightarrow{H^i(F,cl'{}^{j}_{\overline X}(k))}
  H^i(F,H^{2j}_{\et}(\overline X,j+k)).
\end{multline*}
We have:

\begin{lemma}
  \label{lpair2}
  Let\/ $(H^*(X,k+j)^{[m]})_{m\ge 0}$ be the decreasing filtration
  on\/ $H^*(X,k+j)$ defined by the Hochschild-Serre spectral sequence.
  Then
  \begin{enumerateA}
  \item
    \label{lpair2.a}
    The image of the map \eqref{pair4} is contained in
    $H^{i+2j}(X,k+j)^{[i]}$.
  \item
    \label{lpair2.b}
    The image of\/ $\alpha^{i,j,k}$ consists of universal cycles.
  \item
    \label{lpair2.c}
    The diagram
    \[
    \begin{CD}
      H^i(F,k)\oo \CH^{j}X@>\alpha^{i,j,k}>> H^i(F,H^{2j}(\overline
      X,k+j))_{d=0}\\ @VVV @VVV\\ H^{i+2j}(X,k+j)^{[i]}@>>>
      \HS_\infty^{i,2j}
    \end{CD}
    \]
    commutes, where the subscript $d=0$ denotes universal cycles and
    $\HS_\infty^{i,2j}$ denotes the $E_\infty$-term of the
    Hoch\-schild-Serre spectral sequence.
  \item
    \label{lpair2.d}
    Suppose that\/ $H^{2j+1}(\overline X,k+j)=0$. Then, for $i=1$, the
    triangle
    \[
    \begin{CD}
      H^1(F,k)\oo \CH^{j}X@>\alpha^{1,j,k}>> H^1(F,H^{2j}(\overline
      X,k+j))\\ @VVV \llap{$\nearrow$\hskip5.5em}\\ H^{1+2j}(X,k+j)
    \end{CD}
    \]
    commutes, where the vertical map is \eqref{pair4} and the diagonal
    map is the edge homomorphism from the Hochschild-Serre spectral
    sequence.
  \end{enumerateA}
\end{lemma}

\begin{proof} 
  We argue as above: it is enough to prove this for the maps defined
  at the finite coefficients level (we could have defined $\alpha$ as
  a limit, as before). By the multiplicativity of the
  Hoch\-schild-Serre spectral sequence, we are reduced to the case
  $i=k=0$. Then \Aref{lpair2.a} is trivial, and \Aref{lpair2.b} and
  \Aref{lpair2.c} hold because $\alpha^{0,j,0}$ factors through the
  ($F$-rational) cycle map.  Finally, \Aref{lpair2.d} follows as in
  the proof of lemma~\ref{lpair3} from~\Aref{lpair2.c} and the
  definition of the edge homomorphism.
\end{proof}

The map $\alpha$ is the most important of the three maps introduced
above in \eqref{pair4}, \eqref{pair3}, and~\eqref{pair1}. We are
interested in its kernel and cokernel.

\begin{lemma}\label{lpair5}
  If the geometric divisible cycle map is bijective, then
  \[
  \Coker\alpha^{i,j,k}\simeq \Coker\bigl(H^i(F,\CH^{j} X\oo\Q/\Z(k))
  \to H^i(F,\CH^{j}\overline X\oo\Q/\Z(k))\bigr).
  \]
  If moreover $G_F$ acts trivially on $\CH^j\overline X$, this
  cokernel is a quotient of
  \[
  \Coker(H^i(F,k)\oo \CH^{j} X \to H^i(F,k)\oo \CH^{j} \overline X).
  \]
\end{lemma}

\begin{proof}
  The first claim follows from the definition of $\alpha^{i,j,k}$ and
  lemma~\ref{cheveux}. The second one follows from the commutative
  diagram
  \[
  \begin{CD}
    H^i(F,k)\oo \CH^{j} X&\Surj& H^i(F,\CH^{j} X\oo\Q/\Z(k))\\ @VVV
    @VVV\\ H^i(F,k)\oo \CH^{j} \overline X&\Surj&
    H^i(F,\CH^{j}\overline X\oo\Q/\Z(k)).\qed
  \end{CD}
  \]
  \noqed
\end{proof}

\begin{lemma}
  \label{lpair1}
  Let $X$ be a quadric and\/ $k=i\le 3$. Then
  \begin{enumerateA}
  \item
    \label{lpair1.a}
    If\/ $\dim X\neq 2j$ or\/ $\dim X=2j$ and\/ $d(X)=1$,
    $\alpha^{i,j,i}$ is surjective.
  \item
    \label{lpair1.b}
    Suppose $\dim X=2j$ and $d(X)=d\neq 1$. Let $E=F(\sqrt{d})$.  Then
    the composition
    \[
    H^i(E,i)\oo \CH^{j}X_E \xrightarrow{\alpha^{i,j,i}_E}
    H^i(E,H^{2j}(\overline X,i+j)) \xrightarrow{\Cores_{E/F}}
    H^i(F,H^{2j}(\overline X,i+j))
    \]
    is surjective. For $i=0$, $\alpha^{0,j,0}$ is itself surjective.
  \item
    \label{lpair1.c}
    If\/ $\dim X>2j$, $\alpha^{i,j,i}$ is bijective.
  \end{enumerateA}
\end{lemma}

\begin{proof} 
  In case \Aref{lpair1.a}, the geometric cycle map is bijective by
  proposition \ref{pc1} and $G_F$ acts trivially on $\CH^{j}\overline
  X$ by proposition~\ref{pc3}, hence, by lemma~\ref{lpair5}, it is
  enough to see that the map
  \[
  H^i(F,i)\oo \CH^{j}X \to H^i(F,i)\oo \CH^{j}\overline X
  \]
  is surjective. But, since $i\le 3$, the group $H^i(F,i)$ is
  divisible by bijectivity of the $2$-primary Galois symbol in this
  case.  Surjectivity now follows from corollary~\ref{cc3}, applied
  with $D=H^i(F,i)$.
  
  In case \Aref{lpair1.b}, $\alpha_E$ is surjective
  by~\Aref{lpair1.a}.  It remains to prove that
  \[
  H^i(E,H^{2j}(\overline X,i+j)) \xrightarrow{\Cores_{E/F}}
  H^i(F,H^{2j}(\overline X,i+j))
  \]
  is surjective. By the geometric cycle map, the coefficients can be
  identified with $\CH^{j}\overline X\oo\Q/\Z(i)\simeq
  \Ind_{G_E}^{G_F}\Q/\Z(i)$ by corollary~\ref{cc2}. Using Shapiro's
  lemma, the corestriction map can therefore be identified with
  \[
  H^i(E,i)\oplus H^i(E,i) \xrightarrow{\Sigma} H^i(E,i)
  \]
  which is obviously surjective. Finally, the last claim is
  corollary~\ref{cc2}.
  
  In case \Aref{lpair1.c}, the three maps out of which $\alpha$ is
  defined are bijective: the first one is by lemma
  \ref{cheveux}~\Aref{cheveux.c}, the second one by proposition
  \ref{pc3}~\Aref{pc3.b} and the last one by proposition~\ref{pc1}.
\end{proof}

In section~\ref{i4} we shall need the following lemma and its
corollary for $(j,k)=(2,1)$.

\begin{lemma}\label{lpair6}
  Let $X$ be a quadric over $F$. Then:
  \begin{enumerateA}
  \item If $F$ contains all\/ $2$-primary roots of unity,
    $\alpha^{0,j,k}$ is surjective for any $j$,~$k$. The same holds
    if\/ $k$ is even and $F(\sqrt{-1})$ contains all\/ $2$-primary
    roots of unity.
  \item In general, $\Coker\alpha^{0,j,k}$ is
    \begin{enumerate}
    \item $0$ if\/ $\dim X> 2j$;
    \item cyclic of order $\le 2$ if\/ $\dim X<2j$;
    \item cyclic of order $\le 2^t$ if\/ $\dim X=2j$ and $d(X)=1$,
      where $t$ is the integer of proposition \ref{pc3}~\Aref{pc3.d}.
    \end{enumerate}
  \end{enumerateA}
\end{lemma}

\begin{proof}
  By lemma~\ref{lpair5},
  \[
  \Coker\alpha^{0,j,k}\simeq\Coker\bigl(H^0(F,\CH^{j} X\oo\Q/\Z(k))
  \to H^0(F,\CH^{j}\overline X\oo\Q/\Z(k))\bigr).
  \]
  Assume first that we are not in the case $\dim X=2j$ and $d=d(X)\neq
  1$. Applying proposition~\ref{pc3} and corollary~\ref{cc3} with
  $D=\Q/\Z(k)$, we see that the map on coefficients
  \[
  \CH^jX\oo\Q/\Z(k) \to \CH^j\overline X\oo\Q/\Z(k)
  \]
  can be identified with multiplication by $\left|\Coker(\CH^jX \to
    \CH^j\overline X)\right|$ on $\Q/\Z(k)$, except if $\dim X=2j$ and
  $d(X)=1$, where it is $\diag(1,2^t)$ on $\Q/\Z(k)\oplus\Q/\Z(k)$
  with~$t$ as in prop.~\ref{pc3}~\Aref{pc3.d}.  If $F$ contains all
  $2$-primary roots of unity, this shows that $\alpha^{0,j,k}$ is
  surjective, and similarly if $k$ is even and $F(\sqrt{-1})$ contains
  all $2$-primary roots of unity. For arbitrary $F$, if $\dim X>2j$
  then $\CH^jX \to \CH^j\overline X$ is surjective (prop.~\ref{pc3})
  hence $\alpha^{0,j,k}$ is an isomorphism. If $\dim X<2j$ (\resp
  $\dim X=2j$ and $d(X)=1$), the map $\CH^jX \to \CH^j\overline X$ has
  cokernel cyclic of order $\le 2$ (\resp $\le 2^t$), so that
  $\Coker\alpha^{0,j,k}$ is cyclic of order $\le 2$ (\resp $\le 2^t$).
  
  Assume now that $\dim X=2j$ and $d=d(X)\neq 1$; let $E=F(\sqrt d)$.
  By corollary~\ref{cc2}, $\alpha^{0,j,k}$ can be identified with the
  map $H^0(F,k) \to H^0(E,k)$, so is surjective when $F$ and $E$ have
  the same number of $2$-primary roots of unity, in particular if $F$
  contains all $2$-primary roots of unity; similarly if $k$ is even
  and $F(\sqrt{-1})$ contains all $2$-primary roots of unity.
\end{proof}

\begin{cor}\label{cpair1}
  Let $X$ be a quadric over $F$. Consider the composition
  \[
  \gamma\colon H^j(X,{\cal{H}}^j(j+k)) \to H^{2j}(X,j+k) \to
  H^0(F,H^{2j}(\overline{X},j+k))
  \]
  where the first \upp(\resp second\upp) map is an edge homomorphism
  of the coniveau \upp(\resp Hoch\-schild-Serre\upp) spectral
  sequence.  Then:
  \begin{enumerateA}
  \item If $F$ contains all\/ $2$-primary roots of unity or if\/ $k$
    is even and $F(\sqrt{-1})$ contains all\/ $2$-primary roots of
    unity, $\gamma$ is surjective.
  \item In general, $\Coker\gamma$ is:
    \begin{enumerate}
    \item $0$ if\/ $\dim X> 2j$;
    \item cyclic of order $\le 2$ if\/ $\dim X<2j$;
    \item cyclic of order $\le 2^t$ if\/ $\dim X=2j$ and $d(X)=1$,
      where $t$ is the integer of proposition \ref{pc3}~\Aref{pc3.d}.
    \item cyclic of order $\le 2^{t_E}$ if\/ $\dim X=2j$ and
      $d(X)=d\neq 1$, where $t_E$ is the integer of proposition
      \ref{pc3}~\Aref{pc3.d} for $X_E$, with $E=F(\sqrt{d})$.
    \end{enumerate}
  \end{enumerateA}
\end{cor}

\begin{proof} 
  By lemmas \ref{lpair3} and \ref{lpair2}, the composition
  \[
  H^0(F,k)\oo \CH^jX \to H^j(X,{\cal{H}}^j(j+k)) \xrightarrow{\gamma}
  H^0(F,H^{2j}(\overline{X},j+k))
  \]
  coincides with $\alpha^{0,j,k}$. Therefore corollary~\ref{cpair1}
  follows directly from lemma~\ref{lpair6}, except when $\dim X=2j$,
  $d(X)=d\neq 1$ and $F$ does not contain enough roots of unity. But
  in this case, we have a commutative diagram:
  \[
  \begin{CD}
    H^0(E,k)\oo \CH^jX_E@>>> H^j(X_E,{\cal{H}}^j(j+k))@>\gamma_E>>
    H^0(E,H^{2j}(\overline{X},j+k))\\ &&@V{\Cores_{E/F}}VV
    @V{\Cores_{E/F}}VV\\ H^0(F,k)\oo \CH^jX@>>>
    H^j(X,{\cal{H}}^j(j+k))@>\gamma>> H^0(F,H^{2j}(\overline{X},j+k))
  \end{CD}
  \]
  where the left $\Cores$ is best described from the $E_1$-term of the
  coniveau spectral sequence. By lemma~\ref{lpair6}, $\Coker\gamma_E$
  is cyclic of order $\le 2^{t_E}$. But
  \[
  \Cores_{E/F}\colon H^0(E,H^{2j}(\overline{X},j+k)) \to
  H^0(F,H^{2j}(\overline{X},j+k))
  \]
  is surjective: this follows from the bijectivity of the geometric
  cycle map and the fact that the Galois module $\CH^j\overline X$ is
  isomorphic to $\Ind_{G_E}^{G_F}\Z$. This concludes the proof of
  corollary~\ref{cpair1}.
\end{proof}

\section{The case $i=2$: Proof of theorem~\ref{i2}}
\label{seci2}

Let $X$ be a smooth variety over $F$ such that $H^1(\overline
X,\Q/\Z)=0$. We have a cross of exact sequences
\[
\let\s\scriptstyle
\begin{cd}[1.5]
  @.@. 0\\ @.@.@VVV\\ @.@.H^2(F,1)\\ @.@.\Big\downarrow&
  \searrow\raise3pt\rlap{$\s\eta^2$}\\ 0@>>>H^1(X,{\cal H}^1(1))@>>>
  H^2(X,1)@>>> H^0(X,{\cal H}^2 (1))@>>> 0 \\ &&&
  \llap{$\s\alpha^{0,1,0}$}\searrow&@VVV\\ &&&&H^0(F,H^2(\overline
  X,1))\\ &&&&@VVV\\ &&&&H^3(F,1)\\ &&&&@VVV\\ &&&&H^3(X,1)
\end{cd}
\]
where the horizontal (\resp vertical) line is given by the coniveau
(\resp Hoch\-schild-Serre) spectral sequence with coefficients
$\Q/\Z(1)$. The group $H^1(X,{\cal H}^1(1))$ can be identified with
$\CH^1X\oo\Q/\Z$ and $H^1(X,{\cal H}^1(1)) \to H^2(X,1)$ with
$cl'{}^1_X$.  The composite
\[
H^1(X,{\cal H}^1(1)) \xrightarrow{} H^2(X,1) \to H^0(F,H^2({\overline
  X},1))
\]
defined by the above cross can therefore be identified with the map
$\alpha^{0,1,0}$ \eqref{pair1} of section~\ref{pair}.

\begin{prop}
  For $X$ as above, there is an exact sequence
  \[
  0 \to \Coker \eta^{2} \to\Coker\alpha^{0,1,0} \to H^3(F,1) \to
  H^3(X,1).
  \]
\end{prop}

\begin{proof}
  This follows from the ``lemma of the 700th''~\cite{MT}.
\end{proof}

\begin{cor}
  \label{c1}
  For any \upp(projective\upp) quadric $X$ over $F$, $\eta^{2}$ is
  surjective.
\end{cor}

\begin{proof}
  $\alpha^{0,1,0}$ is surjective for all quadrics by lemma
  \ref{lpair1}.
\end{proof}

\section{The case $i=3$: Proof of theorem~\ref{pick}}
\label{seci3}

Let $X$ be a smooth variety over $F$ such that $H^1(\overline
X,\Q/\Z)=H^3(\overline X,\Q/\Z) =0$.  In the following cross of exact
sequences the (broken) horizontal sequence comes from the coniveau
spectral sequence and the vertical sequence from the
Hoch\-schild-Serre spectral sequence with coefficients~$\Q/\Z(2)$.
\[
\let\s\scriptstyle
\begin{CD}
  &&&&H^3(F,2)\\ &&&&\Big\downarrow&
  \searrow\raise3pt\rlap{$\s\eta^3$}\\ 0@>>> H^1(X,{\cal H}^2(2))@>>>
  H^3(X,2)@>>> H^0(X,{\cal H}^3 (2)) \\ 
  &&&\llap{$\s\beta$}\searrow&@VVV@VVV\\ &&&&H^1(F,H^2(\overline
  X,2))&&H^2(X,{\cal H}^2(2))\\ &&&&@VVV@VVV\\ 
  &&&&H^4(F,2)&&H^4(X,2)\\ &&&&@VVV\\ &&&&H^4(X,2).
\end{CD}
\]
In this diagram, one can identify the group $H^2(X,{\cal H}^2(2))$
with $\CH^2X\oo \Q/\Z$ and accordingly the map $H^2(X,{\cal H}^2(2))
\to H^4(X,2)$ with $cl'{}^2_X$. The above cross defines a composite
homomorphism:
\[
\beta\colon H^1(X,{\cal H}^2(2)) \xrightarrow{} H^3(X,2) \to
H^1(F,H^2({\overline X},2)).
\]

\begin{prop}
  \label{p20}
  For $X$ as above, there is a natural surjection
  \[
  \Coker \eta^{3} \to \Ker cl'{}^2_X \to 0.
  \]
  Denote by\/ $(\Coker\eta^{3})^1$ the kernel of this surjection. Then
  there is a natural exact sequence
  \[
  0 \to (\Coker\eta^{3})^1 \to \Coker\beta \to H^4(F,2) \to H^4(X,2).
  \]
\end{prop}

\begin{proof}
  The first claim is obvious and the second again follows from the
  lemma of the 700th~\cite{MT}.
\end{proof}

\begin{cor}
  \label{c2}
  For any \upp(projective\upp) quadric $X$ over $F$, there is a
  natural isomorphism\/ $\Coker \eta^{3}\iso \Ker cl'{}^2_X$.
\end{cor}

\begin{proof} 
  We prove that $\beta$ is surjective for all quadrics. For this, we
  remark that the composition
  \[
  H^1(F,1)\oo \CH^1X \to H^1(X,{\cal H}^2(2)) \xrightarrow{\beta}
  H^1(F,H^2({\overline X},2)),
  \]
  where the first map is \eqref{pair3}, coincides with
  $\alpha^{1,1,1}$ by lemmas \ref{lpair3} and \ref{lpair2}. If $\dim
  X\neq 2$ or $\dim X=2$ and $d(X)=1$, lemma~\ref{lpair1} says that
  $\alpha^{1,1,1}$ is surjective, hence $\beta$ is surjective. If
  $\dim X=2$ and $d=d(X)\neq 1$, we use the same trick as in the proof
  of corollary~\ref{cpair1}: the composite
  $\Cores_{E/F}\circ\alpha^{1,1,1}_E$ is surjective, still by
  lemma~\ref{lpair1}, where $E=F(\sqrt d)$. But there is a transfer
  map $H^1(X_E,{\cal H}^2(2)) \xrightarrow{\Cores_{E/F}} H^1(X,{\cal
    H}^2(2))$ making the following diagram commute:
  \[
  \begin{CD}
    H^1(E,1)\oo \CH^1X_E @>>> H^1(X_E,{\cal H}^2(2)) @>\beta_E>>
    H^1(E,H^2({\overline X},2))\\ &&@V{\Cores_{E/F}}VV
    @V{\Cores_{E/F}}VV\\ H^1(F,1)\oo \CH^1X @>>> H^1(X,{\cal H}^2(2))
    @>\beta>> H^1(F,H^2({\overline X},2)),
  \end{CD}
  \]
  which concludes the proof in this case.
\end{proof}

\begin{prop}[\cf \Cite{th.~3}{hcoh}]
  \label{p27} For a quadric $X$ over a field of
  characteristic $\neq 2$, the cycle map
  \[
  cl'{}^2_X\colon \CH^2X\oo \Q/\Z \to H^4(X,2)
  \]
  is injective, except if $X$ is defined by an anisotropic Albert
  form.  In this case, $\Ker cl'{}^2_X\simeq\Z/2$.
\end{prop}

\begin{proof} 
  In view of corollary~\ref{c2}, we may assume $X$ anisotropic
  (prop.~\ref{pc5}). If $\dim X>4$ or $\dim X=4$ and $d(X)\neq 1$, the
  map
  \[
  \CH^2X\oo \Q/\Z \to H^0(F,\CH^2\overline X\oo \Q/\Z)
  \]
  is an isomorphism by proposition~\ref{pc3} and the claim follows
  from the injectivity of the geometric cycle map
  (proposition~\ref{pc1}). If $\dim X=1$, $\CH^2X=0$ and the claim is
  trivial. If $\dim X=3$ and $Y$ is a hyperplane section of $X$, then
  the restriction map $\CH^2X\oo \Q/\Z \to \CH^2Y\oo \Q/\Z$ is an
  isomorphism by proposition~\ref{pc3} again, so that injectivity for
  $\dim X=2$ implies injectivity for $\dim X=3$. Let us assume $\dim
  X=2$. As in \cite {hcoh}, we may assume $d(X)=1$ since, if
  $d=d(X)\neq 1$, the map $\CH^2X \to \CH^2X_{F(\sqrt d)}$ is
  bijective (prop.~\ref{pc3} and \ref{pc4}). In this case, $\CH^2X$ is
  generated by $h^2$, where $h\in \CH^1X$ is the class of a hyperplane
  section (\ibid).  Since $\Ker cl'{}^2_X$ is killed by $2$ (cor.
  \ref{c2} and prop.~\ref{pc5}), it is enough to prove:
  
  \begin{lemma}
    If $X$ is an anisotropic quaternion quadric \upp{(=} defined by a
    $2$-fold Pfister form\upp), the image of\/ $cl^2_X(h^2)\in H^4X$
    in $H^4(X,2)$ is\/ $\neq 0$.
  \end{lemma}
  
  \begin{proof}
    Indeed, $cl^2_X(h^2)=c\cdot cl^1_X(h)$ by \cite[lemma~2]{hcoh},
    where $c$ is the Brauer class of the quadric, and $cl^1_X(h)\neq
    0$ by \cite[lemma 5.3.2~c)]{sz}. Therefore $cl^2_X(h^2)$ maps to a
    nonzero element of $H^2(F,H^2\overline X)\simeq H^2F$. Its image
    in $H^2(F,H^2(\overline X,2))\simeq H^2(F,1)$ is not $0$ since the
    map $H^2F \to H^2(F,1)$ is injective (compare beginning of proof
    of prop.~\ref{p1}). Therefore the image of $cl^2_X(h^2)$ in
    $H^4(X,2)$ cannot be $0$.
  \end{proof}
  
  It remains to deal with the case of an anisotropic Albert quadric.
  In this case we have $\Ker\eta_2^{3}=0$ \cite[Satz 5.6]{arason} and
  $\Coker\eta_2^{3}\simeq\Z/2$ \cite[theorem 2~d)]{hcoh}, hence from
  proposition~\ref{c27} and corollary~\ref{c2} we get an injection
  \begin{equation}\label{nou}\Z/2\simeq\Coker\eta_2^{3}\hookrightarrow \Ker
    cl'{}^2_X.\end{equation}
  
  But propositions \ref{pc1} and \ref{pc3}~\Aref{pc3.d} yield a
  commutative diagram with exact row
  \[
  \begin{CD}
    0@>>>\Z/4@>>>\CH^2X\oo \Q/\Z@>>>\CH^2\overline X\oo \Q/\Z @>>>0\\ 
    &&&&@V{cl'{}^2_X}VV @V{cl'{}^2_{\overline X}}V{\wr}V\\ 
    &&&&H^4(X,2)@>>>H^4(\overline X,2)\\ 
  \end{CD}
  \]
  
  It follows that $\Ker cl'{}^2_X$ is cyclic; as it is of exponent $2$
  by corollary~\ref{c2} and proposition \ref{pc5}~\Aref{pc5.b}, the
  map \eqref{nou} is an isomorphism.
\end{proof}

Theorem~\ref{pick} follows from corollary \ref{c2} and proposition
\ref{p27}.

\goodbreak

\section{The case $i=4$}\label{i4}

This case is similar to the former two, but more complicated.

For any smooth variety $X$ over $F$ define
\[
\overline H^4(X,3)=\frac{H^4(X,3)}{ H^4(F,3)+H^2(X,{\cal
    H}^2(3))+\displaystyle\bigoplus_{[E{\mathop:}F]<+\infty}H^2(E,2)\oo
  \CH^1X_E},
\]
where $H^2(X,{\cal H}^2(3)) \to H^4(X,3)$ is the edge homomorphism of
the coniveau spectral sequence and the map
$\bigoplus_{[E{\mathop:}F]<+\infty}H^2(E,2)\oo \CH^1X_E \to H^4(X,3)$
comes from~\eqref{pair4} via transfer.

Suppose that $H^1(\overline X,\Q/\Z)=H^3(\overline X,\Q/\Z) =0$.  We
then have two exact sequences
\begin{multline}
  \label{bo4}
  0 \to H^1(X,{\cal H}^3(3)) \to {H^4(X,3)\over H^2(X,{\cal H}^2(3))}
  \to H^0(X,{\cal H}^4(3))\longrightarrow \\ \xrightarrow{d_2^{0,4}}
  H^2(X,{\cal H}^3(3)) \to H^5(X,3)
\end{multline} 
(coniveau)
\begin{multline}
  \label{hs4}
  0 \to\Ker\Bigl(H^2(F,H^2(\overline X,3)) \xrightarrow{d_3^{2,2}}
  H^5(F,3)\Bigr)\to\\ \to{H^4(X,3)\over{H^4(F,3)}} \to
  H^0(F,H^4(\overline X,3))
\end{multline}
(Hochschild-Serre).

The exact sequence \eqref{hs4} identifies $\Ker d_3^{2,2}$ to a
subgroup of $\displaystyle{H^4(X,3)\over{H^4(F,3)}}$. For the purpose
of stating proposition~\ref{p2} below, let $H^2(X,{\cal H}^2(3))_0$
denote its intersection with $\IM\left(H^2(X,{\cal H}^2(3)) \to
  H^4(X,3)\to \displaystyle{H^4(X,3)\over{H^4(F,3)}}\right)$ (the
first map in this composition being an edge homomorphism of the
coniveau spectral sequence).

\begin{prop}
  \label{p2}
  For $X$ as above, there are exact sequences
  \[
  \overline H^4(X,3) \to \Coker \eta^{4} \to H^2(X,{\cal H}^3(3)) \to
  H^5(X,3)
  \]
  and
  \[
  {\Ker\Bigl(H^2(F,H^2(\overline X,3)) \xrightarrow{d_3^{2,2}}
    H^5(F,3)\Bigr)\over{H^2(X,{\cal H}^2(3))_0+
      \displaystyle\bigoplus_{[E{\mathop:}F]<+\infty}H^2(E,2)\oo
      \CH^1X_E}} \to \overline H^4(X,3)\to {H^0(F,H^4(\overline
    X,3))\over{ H^2(X,{\cal H}^2(3))}},
  \]
  %%  \begin{multline*}
  %%    0 \to {\Ker\Bigl(H^2(F,H^2(\overline X,3)) \xrightarrow{d_3^{2,2}}
  %%      H^5(F,3)\Bigr)\over{H^2(X,{\cal H}^2(3))_0+
  %%        \displaystyle\bigoplus_{[E{\mathop:}F]<+\infty}H^2(E,2)\oo
  %%        \CH^1X_E}} \to \overline H^4(X,3)\to \\ \to
  %%    {H^0(F,H^4(\overline X,3))\over{ H^2(X,{\cal H}^2(3))}},
  %%  \end{multline*} 
  where the factor
  \[
  \bigoplus_{[E{\mathop:}F]<+\infty}H^2(E,2)\oo \CH^1X_E
  \]
  comes from
  \[
  \displaystyle\sum_{[E{\mathop:}F]<+\infty}\Cores_{E/F}{}\circ
  \alpha_E^{2,1,2}.
  \]
\end{prop}

\begin{proof}
  This follows from the exact sequences \eqref{bo4} and \eqref{hs4},
  using lemma~\ref{lpair2}.
\end{proof}

Consider the cross of exact sequences
\begin{equation}\label{di3}
  \let\s\scriptstyle
  \begin{CD}
    &&&&H^1(F,1)\otimes \CH^2X\\ &&&&\Big\downarrow&
    \llap{$\s\nu$}\displaystyle\searrow\\ \overline H^4(X,3) @>>>
    \Coker \eta^{4} @>>> H^2(X,{\cal H}^3(3)) @>>> H^5(X,3)\\ 
    &&&&@VVV\\ &&&&(\CH^3X)_{\tors}\\ &&&&@VVV\\ &&&&0
  \end{CD}\end{equation}
coming from propositions \ref{l2} (for $i=3$) and \ref{p2}. From
diagram \eqref{di3} and the lemma of the 700th, we deduce a complex
\begin{equation}\label{ch3}
  \overline H^4(X,3)\to\Coker \eta^{4} \to (\CH^3X)_{\tors} \to
  \Coker\nu
\end{equation}
which is exact at $(\CH^3X)_{\tors}$ and whose homology at $\Coker
\eta^{4}$ is a quotient of $\Ker\nu$.

\begin{lemma}
  \label{l1}
  Let $X$ be a quadric. Then\/ $\overline H^4(X,3)$ is cyclic of
  order\/ $\le 4$; it is\/ $0$ if $\dim X=1$, $\dim X>4$ or if $F$
  contains all\/ $2$-primary roots of unity.
\end{lemma}

\begin{proof} 
  By lemma~\ref{lpair1}, $\alpha^{2,1,2}$ is surjective, except if
  $\dim X=2$ and $d=d(X)\neq 1$; in this case,
  $\Cores_{E/F}\circ\alpha_E^{2,1,2}$ is surjective. It follows that
  for any quadric, the group on the left of $\overline H^4(X,3)$ in
  the second exact sequence of proposition~\ref{p2} is~$0$.  Therefore
  $\overline H^4(X,3)$ injects into $\dfrac{H^0(F,H^4(\overline
    X,3))}{H^2(X,{\cal H}^2(3))}$. But the latter group is cyclic of
  order as in the statement of lemma~\ref{l1}, by
  corollary~\ref{cpair1} and proposition~\ref{pc3}.
\end{proof}

\begin{lemma}
  \label{l3}
  If\/ $\dim X=1$ or\/ $\dim X>4$, the map\/ $\nu$ of diagram
  \eqref{di3} is injective. If\/ $\dim X>4$, there is a canonical
  isomorphism
  \[
  \Coker\nu\simeq H^5(F,3)\oplus \bigl(H^3(F,2)\otimes \CH^1X\bigr).
  \]
\end{lemma}

\begin{proof} 
  The case $\dim X=1$ is obvious, since then $\CH^2X=0$.  Suppose
  $\dim X>4$. By lemma \ref{lpair3}~\Aref{lpair3.d}, the map $\nu$
  coincides with~\eqref{pair4}. By lemma~\ref{lpair1}~\Aref{lpair1.c},
  $\alpha^{1,2,1}$ is bijective. Therefore, by lemma
  \ref{lpair2}~\Aref{lpair2.d}, $\nu$ is (split-) injective. This
  argument shows that, in the Hochschild-Serre spectral sequence for
  coefficients $\Q/\Z(3)$, the term~$E_2^{1,4}$ is isomorphic to
  $H^1(F,1)\oo \CH^2X$ and consists of universal cycles. The same
  holds for all the terms $E_2^{p,2}\simeq H^p(F,2)\oo \CH^1X$ by the
  same argument, since $\CH^1X\iso \CH^1\overline X$ for $\dim X\ge
  3$.  Finally the term~$E_2^{0,4}$ also consists of universal cycles
  by lemma~\ref{l1}. Hence we get a canonical isomorphism:
  \[
  H^5(X,3)\simeq H^5(F,3)\oplus \bigl(H^3(F,2)\oo \CH^1X\bigr)\oplus
  \bigl(H^1(F,1)\oo \CH^2X\bigr)
  \]
  and the second part of the lemma follows.
\end{proof}

\goodbreak

\section{Proof of theorems \ref{t1} and~\ref{infinite}}
\label{pth1}

\begin{proof}[Proof of theorem~\ref{t1}]
  Claim~\eqref{t1.3} follows from~\eqref{ch3}, lemma~\ref{l1} and
  lemma~\ref{l3}.  We now check \eqref{inf} and~\eqref{t1.4}.  Let $q$
  be a quadratic form defining the quadric $X$.
  \begin{enumerateA}
  \item
    \label{a}
    If $\dim q=3$, $\Coker \eta^4=0$ by
    \eqref{ch3}, lemma~\ref{l1} and lemma~\ref{l3}.
  \item
    \label{b}
    If $\dim q>6$, part~\eqref{t1.3} of theorem~\ref{t1} implies that
    $\Coker\eta^4$ injects into the group $(\CH^3X)_{\tors}$, hence
    has order $\le 2$ by Karpenko's results (proposition~\ref{pc4}).
    If $\dim q>12$, $(\CH^3X)_{\tors}=0$ by the same results, and so
    $\Coker\eta^4=0$.
  \item
    \label{c}
    If $\dim q=8$, $d_\pm q=1$, and $q$ is not similar to a Pfister
    form, then $\CH^3X$ is torsion-free by \cite[th.~8.3]{K} and
    $\Coker\eta^4=0$ by~\eqref{t1.3}.
  \end{enumerateA}
  
  The case of a $2$-fold Pfister neighbour and that of an $i$-fold
  Pfister neighbour for $i\geq 4$ is covered by \Aref{a} and \Aref{b}
  respectively.
  
  It remains to prove claim~\eqref{t1.4} in the case of a $3$-fold
  neighbour. We may assume the quadric to be of dimension at least~$5$
  by proposition \ref{pc5}~\Aref{pc5.c}. Then the claim follows from:
  \noqed
\end{proof}

\begin{lemma}
  \label{voisine} 
  Suppose\/ $\dim X>4$ and $X$ is defined by a neighbour of a $3$-fold
  Pfister form. Then the map of theorem~\ref{t1}~\eqref{t1.3}
  \[
  (\CH^3X)_{\tors}\to H^5(F,3)\oplus \bigl(H^3(F,2)\oo \CH^1X\bigr)
  \]
  is injective.
\end{lemma}

\begin{proof} 
  By the same argument as in the proof of lemma~\ref{l3}, we get a
  canonical isomorphism
  \[
  H^3(X,2)\simeq H^3(F,2)\oplus \bigl(H^1(F,1)\otimes \CH^1X\bigr)
  \]
  compatible with the maps of section~\ref{pair}.  The diagram
  analogous to \eqref{di3}
  \[
  \begin{CD}
    &&H^1(F,1)\oo \CH^1X\\ &&@VVV\\ 0@>>> H^1(X,{\cal H}^2(2)) @>>>
    H^3(X,2)\\ &&@VVV\\ &&(\CH^2X)_{\tors}\\ &&@VVV\\ &&0
  \end{CD}
  \]
  yields an injection
  \[
  (\CH^2X)_{\tors}\Inj H^3(F,2).
  \]
  
  Tensoring this injection by $\CH^1X$ yields another injection (since
  $\CH^1X$ is torsion-free)
  \[
  (\CH^2X)_{\tors}\oo \CH^1X\Inj H^3(F,2)\oo \CH^1X.
  \]
  The way these maps are defined shows that the diagram
  \[
  \begin{CD}
    (\CH^2X)_{\tors}\oo \CH^1X&\Inj& H^3(F,2)\oo \CH^1X\\ @VVV @VVV\\ 
    (\CH^3X)_{\tors}@>>>H^5(F,3)\oplus (H^3(F,2)\oo \CH^1X)
  \end{CD}
  \]
  commutes, where the left vertical map is multiplication, the right
  vertical map is inclusion, the top horizontal map has just been
  defined and the bottom horizontal map comes from theorem
  \ref{t1}~\eqref{t1.3}.  Since $\CH^1X\simeq \Z$ and
  $(\CH^2X)_{\tors}\simeq \Z/2$ by propositions \ref{pc3} and
  \ref{pc4}, this shows that the latter map is nonzero. Since, by
  proposition~\ref{pc4}, $(\CH^3X)_{\tors}$ has order at most~$2$,
  this map is injective, which is what we wanted (moreover, we get
  that $\CH^1X\oo (\CH^2X)_{\tors}\to (\CH^3X)_{\tors}$ is bijective,
  which fits with what Karpenko's method gives~\cite{K2}).
\end{proof}

\begin{proof}[Proof of theorem~\ref{infinite}]
  In the case where $X$ is defined by an anisotropic Albert form~$q$,
  we define a homomorphism
  \[
  \rho\colon H^1F \xrightarrow{} \Coker\eta_2^4
  \]
  by cup-product with a representative of $\Coker\eta_2^3\simeq\Z/2$.
  
  Recall \cite{hcoh} that such a representative $e$ is constructed as
  follows: let $K=F(X)$ and $q_K\sim a\tau$, where $a\in K^*$ and
  $\tau$ is a $2$-fold Pfister form. Then $e=(a)\cdot e^2(\tau)$.
  
  The composition $H^1F \to \Coker\eta_2^4 \to \Coker\eta^{4}$ has
  infinite image if we pass from~$F$ to the rational function
  field~$F(t)$. Indeed, $F$ is infinite since $q$ is
  an\-iso\-trop\-ic. Then, for any $\alpha\in F$, the element
  \[
  (t-\alpha)\cdot e\in H^4F(X)(t)\inj H^4(F(X)(t),3)
  \]
  is unramified over $F(t)$, and its residue at $\alpha$ is $e\in
  H^3(F(X),2)$. This shows that the classes of the $(t-\alpha)\cdot e$
  in $\Coker \eta^4_{F(t)}$ are linearly independent.
  
  In the case of a virtual Albert form $q$ of discriminant $d$, we
  define a homomorphism
  \[
  \rho'\colon E^* \xrightarrow{} \Coker\eta_2^4
  \]
  simply as the composite
  \[
  E^* \xrightarrow{\rho} \Coker\eta_{2,q_E}^4
  \xrightarrow{\Cores_{E/F}} \Coker\eta_2^4
  \]
  where $\Coker\eta_{2,q_E}^4$ is the cokernel attached to the form
  $q_E$ ($E=F(\sqrt{d})$).  We show just as above that
  $\Coker\eta_{F(t)}^4$ is infinite, now using rational points
  of~$\af_E$ not defined over~$F$.
  
  These arguments can be summarized by noting that the groups
  $\Coker\eta^i$ are cycle modules in the sense of~\cite{rostch} and
  using the exact sequence
  \[
  0\to \Coker\eta^4_F \to \Coker\eta^4_{F(t)}\to
  \bigoplus_{y\in\af_{(0)}} \Coker\eta^3_{F_y} \to 0
  \]
  \cf the exact sequence~\eqref{star} in the introduction.
\end{proof}

\begin{exo} 
  Let $X$ be an anisotropic quadric, $a\in F^*$ and $E=F(\sqrt a)$ a
  quadratic extension such that $X_E$ is isotropic.
  \begin{enumerateA}
  \item Show that extension of scalars defines a homomorphism
    \[
    \theta^n_E\colon H^0(X,{\cal{H}}^n) \to H^nE
    \]
    for all $n\ge 0$.
  \item
    \label{exo.b}
    Define a $0$-sequence
    \[
    H^3F \xrightarrow{{\cdot(a)}} \Ker\theta_E^4 \xrightarrow{}
    \Ker(\CH^3X \to \CH^3X_E)
    \]
    (proceed by a diagram chase, as in \cite{Mecourse} for the
    definition of the map $\Ker \eta_2^4 \to \Ker(H^2(X,{\cal{K}}_3)
    \to H^2(X_E,{\cal{K}}_3))$).
  \item Compare the map $\Ker\theta_E^4 \to \Ker(\CH^3X \to \CH^3X_E)$
    of \Aref{exo.b} with the map $\Coker\eta_2^4 \to (\CH^3X)_{\tors}$
    deduced from theorem~\ref{t1}.
  \end{enumerateA}
\end{exo}

\appendix


\section{Quadrics of dimension $0$ and $1$}
\label{dim0}

\begin{prop}
  \label{conics}
  Let\/ $\dim X=1$ and suppose the Milnor conjecture holds in
  degree~$i-2$. Then\/ $\Coker\eta^i=0$.
\end{prop}

\begin{proof}
  It follows from the cross of exact sequences
  \[
  \begin{array}{ccc}
    &H^i(F,i-1)&\\ &\downarrow&\\ 0 \to H^1(X,{\cal H}^{i-1}(i-1))
    \to& H^i(X,i-1)& \to H^0(X,{\cal H}^i(i-1)) \to 0\\ &\downarrow&\\ 
    &H^{i-2}(F,H^2(\overline X,i-2))&
  \end{array}
  \]
  (coniveau and Hochschild-Serre), by using the fact that the
  composition
  
  \[
  H^{i-2}(F,i-2)\oo \CH^1X \to H^1(X,{\cal H}^{i-1}(i-1)) \to
  H^{i-2}(F,H^2(\overline X,i-2))
  \]
  is surjective. As above, this is deduced from the divisibility of
  $H^{i-2}(F,i-2)$ and corollary~\ref{cc3}.
\end{proof}

In the following we present analogues of proposition~\ref{conics} and
proposition~\ref{t5} for $0$-dimensional quadrics.  Let
$L=F(\sqrt{d})$ be a quadratic extension with Galois group
$G=\Gal(L/F)$.

\begin{prop}
  Suppose the Milnor conjecture holds in degree~$i-1$. Then
  \begin{enumerateA}
  \item
    \label{d=0.a}
    The restriction map
    \[
    H^i(F,i-1)\to (H^i(L,i-1))^G
    \]
    is surjective.
  \item
    \label{d=0.b}
    The restriction map
    \[
    H^iF \to (H^i L)^G
    \]
    has kernel\/ $(d)\cdot H^{i-1}F$ and cokernel isomorphic to
    \[
    \{\,x\in (d)\cdot H^{i-1}F\mid (-1)\cdot x=0\,\}.
    \]
  \end{enumerateA}
\end{prop}


\begin{proof} 
  \Aref{d=0.a} Let $\sigma$ be the generator of $G$ and consider the
  exact sequence of Galois-modules
  \[
  0 \to \Z \to \Z[G] \xrightarrow{{1-\sigma}} \Z[G] \to \Z \to 0.
  \]
  
  Tensoring with $\Q/\Z(i-1)$ gives a spectral sequence converging to
  $0$, with $E_1$-term:
  \[
  \let\s\scriptstyle \newcommand{\TO}{@>>>\s}
  \newcommand{\DOTS}{\raise2pt\hbox{$\cdots$}} \scriptstyle
  \begin{cd} 
    \s 0 \TO H^{0}(F,i-1) @>\Res>> \s H^{0}(L,i-1)@>{1-\sigma}>>\s
    H^{0}(L,i-1)@>\Cores>>\s H^{0}(F,i-1) \TO 0\\ &&&\DOTS \\\s 0 \TO
    H^{i-1}(F,i-1)@>\Res>> \s H^{i-1}(L,i-1)@>{1-\sigma}>>\s
    H^{i-1}(L,i-1)@>\Cores>>\s H^{i-1}(F,i-1) \TO 0\\ \s 0 \TO
    H^{i}(F,i-1) @>\Res>> \s H^{i}(L,i-1)@>{1-\sigma}>> \s
    H^{i}(L,i-1)@>\Cores>>\s H^{i}(F,i-1) \TO 0\\ &&&\DOTS
  \end{cd}
  \]
  Here the $d^2$-differentials go $2$ steps right and $1$ step up.
  The norm map
  \[
  H^{i-1}(L,i-1) \xrightarrow{\Cores} H^{i-1}(F,i-1)
  \]
  is computed as the norm map
  \[
  K_{i-1}^{\rm M} L\otimes\Q/\Z \xrightarrow{N} K_{i-1}^{\rm M}
  F\otimes\Q/\Z,
  \]
  which is obviously surjective. Hence
  \[
  H^{i}(F,i-1) \xrightarrow{\Res} H^{i}(L,i-1)
  \xrightarrow{{1-\sigma}} H^{i}(L,i-1)
  \]
  is exact, \ie $\eta^i$ is surjective.
  
  This discussion holds for any cyclic extension $L/F$ with the degree
  $[L\mathop:F]$ prime to $\car F$.
  
  \Aref{d=0.b} The statement on the kernel is classical \cite{arason};
  the statement on the cokernel follows from this one and
  \Aref{d=0.a}, by using proposition~\ref{c27}, which is shown to
  apply in this case just as in section~\ref{Coker}.
\end{proof}


\section{Application to real quadrics}\label{app.1}

In this section, we examine the case when the ground field is the
field~$\R$ of real numbers. The real anisotropic quadric of dimension
$d$, denoted~$Q_d$, is defined by the $(d+2)$-dimensional form
$(d+2)\qf{1}$. As all anisotropic quadratic forms over~$\R$ are
Pfister neighbours, the anisotropic quadrics can be grouped according
to the level of their function fields (\cf \cite[\S 3]{CTS}). As
observed in~\cite[\S 3]{CTS}, the unramified cohomology groups are
invariant for the set of quadrics grouped in this manner.
In~\cite{CTS}, it was proved that the unramified cohomology group
$H^0(Q_d,{\cal H}^3)$ is $0$ for $d \leq 6$ and that the map $ \Z/2
\simeq H^3(\R) \to H^0(Q_d,{\cal H}^3)$ is bijective for $d >7$. This
result was used to show that the cycle map in codimension two
\[
\CH^2Q_d/2 \to H^4Q_d
\]
is injective for all quadrics~$Q_d$ (compare also \cite[th.~3]{hcoh}).

We extend these results in this section, and use them to get extra
information on the cohomology of real quadrics. Concerning unramified
cohomology, results can be expressed concisely in a striking way:

\begin{thm}
  \label{t2} Let $F=\R$ and $i\le 4$. Then, for any $d>0$, the map 
  $H^i(\R) \to H^0(Q_d,{\cal H}^i)$ is surjective. If\/ $d> 2^i-2$ it
  is bijective; if\/ $d\le 2^i-2$, $H^0(Q_d,{\cal H}^i)=0$.
\end{thm}

\begin{proof} 
  For $i=2$ it is classical and for $i=3$ this is the main result of
  \cite{CTS}. Note however that these cases also follow directly from
  proposition~\ref{c27}, theorem~\ref{t1} and the fact that there are
  no anisotropic Albert quadrics over $\R$ in view of
  
  \begin{lemma}
    \label{l4}
    With notation as above, if $F=\R$ then\/ $(\Ker\eta_2^i)_0=0$ for
    all quadrics $X$ and all\/ $i\ge 1$.
  \end{lemma}
  
  \begin{proof}
    Cup-product by $(-1)$ is injective on $H^*\R$.
  \end{proof}
  
  Let us now prove theorem~\ref{t2} in the case $i=4$. For $d>14$ and
  for $6<d\le 14$, this follows from theorem~\ref{t3.X} and
  corollary~\ref{C1}.  For $3< d\le 6$, we may assume $d=3$; then
  $cd(\R(Q_3))=3$ \cite{CG}, hence $H^4(\R(Q_3))=0$ and \emph{a
    fortiori} $H^0(Q_3,{\cal H}^4)=0$. The argument is the same for
  $d\le 2$.
\end{proof}

Theorem \ref{t2} implies in particular that, in the coniveau spectral
sequence
\[
E_2^{p,q}=H^p(Q_d,{\cal H}^q)\Rightarrow H^{p+q}Q_d,
\]
$E_2^{0,q}$ consists of universal cycles for $q\le 4$. This in turn
implies:

\begin{thm}\leavevmode
  \label{t4}
  \begin{enumerateA}
  \item For any\/ $d\ge 1$, we have:
    \begin{gather*}
      \dim H^1(Q_d,{\cal H}^2)+\dim H^0(Q_d,{\cal H}^3)=\dim H^3Q_d;\\ 
      \dim \CH^2Q_d\otimes\Z/2+\dim H^1(Q_d,{\cal H}^3)+\dim
      H^0(Q_d,{\cal H}^4)=\dim H^4Q_d.
    \end{gather*}
  \item The ``edge homomorphism'' $H^2(Q_d,{\cal H}^3) \to H^5Q_d$ is
    injective.
  \end{enumerateA}\qed
\end{thm}

\begin{cor}
  We have:
  \[
  \dim H^1(Q_d,{\cal H}^2)=\begin{cases} 0& \text{for $d=1$}\\ 1&
    \text{for $d=2$}\\ 2& \text{for $3\le d\le 6$}\\ 1& \text{for
      $d>6$}
  \end{cases}
  \]
  and
  \[
  \dim H^1(Q_d,{\cal H}^3)=\begin{cases} 0& \text{for $d\le 3$}\\ 1&
    \text{for $4\le d\le 6$}\\ 2& \text{for $7\le d\le 14$}\\ 1&
    \text{for $d>14$.}
  \end{cases}
  \]
\end{cor}
\begin{proof}[Sketch of proof]
  An inductive argument allows one to compute $\dim H^*Q_d$. The
  method is to first compute the cohomology of the corresponding
  affine quadric by reduction to the complex case. The answer is
  \cite{ks}:
  \[
  \dim H^nQ_d=\begin{cases} \left[\displaystyle {n\over {2}}\right]+1&
    \text{if $d\ge n$}\\ &\\ \left[\displaystyle {2d-n\over
        {2}}\right]+1& \text{if $d\le n$.}
  \end{cases}
  \]
  The corollary follows from this fact, from theorem~\ref{t2}, theorem
  \ref{t4}, and from (for the value of $\dim \CH^2Q_d\otimes\Z/2$)
  propositions \ref{pc3} and \ref{pc4}.
\end{proof}

Further results on $H^2(Q_d,{\cal H}^3)$, $H^1(Q_d,{\cal H}^4)$ and
the cycle map $cl^3_{Q_d}$ can be obtained at a higher cost \cite{ks}.

\section{The unramified Witt ring}
\label{wnr}

In this appendix, we prove theorem~\ref{Witt}, which we now recall:
%%
\begin{thm}
  \label{Witt.X}
  Let $i\le 4$. For any quadric $X$, the composition
  \[
  \Coker\chi^i \to \Coker\eta_2^i \to\Coker\eta^i
  \]
  is injective, where
  \[
  \chi^i\colon I^iF/I^{i+1}F \xrightarrow{}
  I^i_{nr}(F(X)/F)/I^{i+1}_{nr}(F(X)/F)
  \]
  is the map induced by extension of scalars and the first map in the
  composition is induced by the cohomological invariant\/ $\bar e^i$.
\end{thm}

\begin{lemma} \label{wl1} 
  Let\/ $C$ be a conic over $F$ and $K=F(C)$. Let $c\in H^2F$ be the
  class of the quaternion algebra corresponding to\/ $C$. Then, for\/
  $i\le 4$,
  \begin{enumerateA} 
  \item
    \label {wl11} $\Ker \eta_2^i= c\cdot H^{i-2}F$.
  \item
    \label{wl12} 
    The natural map
    \[
    I^{i+1}F \to I^{i+1}_{nr}(K/F)
    \]
    is surjective.
  \end{enumerateA}
\end{lemma}

\begin{proof} 
  \Aref{wl11} has been given earlier in the paper for $i\le 3$ and is
  proven in \cite[prop.~3.15]{Mebi} for $i=4$. Let us prove
  \Aref{wl12}. By a result of Parimala \cite{P1,vangeel,CTS}, the map
  $W(F) \to W_{nr}(K/F)$ is surjective.  Let $\varphi\in
  I^{i+1}_{nr}(K/F)$ and, by induction on~$i$, let $\psi\in I^iF$ be
  such that $\psi_K=\varphi$. We have $e^i(\psi)\in \Ker\eta_2^i$,
  hence $e^i(\psi)=c\cdot x$ for some $x\in H^{i-2}F$ by \Aref{wl11}.
  By the surjectivity of $e^{i-2}$ and the injectivity of $\bar e^i$,
  we can write $\psi=\tau\oo\theta\perp \psi'$, where $\tau$ is the
  quaternion form corresponding to $c$, $e^{i-2}(\theta)=x$ and
  $\psi'\in I^{i+1}F$. Then $\psi'_K=\varphi$, as desired.
\end{proof}

\begin{notat}
  We write $\bar I^iF=I^iF/I^{i+1}F$ and, for a finitely generated
  extension $L/F$, $\bar
  I^i_{nr}(L/F)=I^i_{nr}(L/F)/I^{i+1}_{nr}(L/F)$.
\end{notat}

\begin{lemma}\label{wl2} 
  Let $X$ be a quadric \upp(of dimension $> 0$\upp), and let $C$ be a
  conic lying on $X$. Let $\tau$ be the quaternion form associated
  to\/ $C$. Then, for\/ $i\le 4$, there is a surjection
  \[
  \tau\bar I^{i-2}F(X)\cap \bar I^i_{nr}(F(X)/F)\Surj \Coker \chi^i.
  \]
\end{lemma}

\begin{proof} 
  Let $K=F(C)$. Consider the commutative diagram
  \[
  \begin{CD}
    \bar I^i_{nr}(F(X)/F) @>>>\bar I^i_{nr}(K(X)/F)\\ @A{\chi^i}AA
    @A{\wr}AA\\ \bar I^iF &\Surj& \bar I^i_{nr}(K/F)
  \end{CD}
  \]
  
  In this diagram, the lower horizontal map is surjective by lemma
  \ref{wl1}~\Aref{wl12} and the right vertical one is an isomorphism
  because $K(X)/K$ is purely transcendental. A diagram chase then
  yields a surjection
  \[
  \Ker(\bar I^i_{nr}(F(X)/F) \xrightarrow{} \bar
  I^i_{nr}(K(X)/F))\surj \Coker \chi^i.
  \]
  
  To conclude the proof, it suffices to observe that $\Ker(\bar
  I^iF(X) \xrightarrow{} \bar I^iK(X))=\tau \bar I^{i-2}F(X)$ by lemma
  \ref{wl1}~\Aref{wl11} and the bijectivity of $\bar e^{i-2}$ and
  $\bar e^i$.
\end{proof}

\begin{proof}[Proof of theorem~\ref{Witt.X}] 
  Consider the commutative diagram
  \[
  \begin{CD}
    \Coker\chi^i@>\bar e^i>>\Coker\eta^i_2@>\lambda>>\Coker\eta^i\\ 
    @AAA @AAA @AAA\\ \bar I^i_{nr}(F(X)/F)@>\bar
    e^i>>H^i_{nr}(F(X)/F)@>\lambda>>H^i_{nr}(F(X)/F,2)\\ @A{\chi^i}AA
    @A{\eta^i_2}AA @A{\eta^i}AA\\ \bar I^iF@>\bar
    e^i>>H^iF@>\lambda>>H^i(F,2)
  \end{CD}
  \]
  where $\lambda$ is the natural map. Let $x\in\bar I^i_{nr}(F(X)/F)$.
  Choose $C$ as in lemma~\ref{wl2}.  By this lemma, we can modify $x$
  by an element from $I^iF$ so that $x=\tau\oo\theta$. Let $y\in
  H^i(F,2)$ be such that $\lambda\bar e^i(x)=\eta^i(y)$. Let $E/F$ be
  a quadratic extension such that $C_E$, hence $X_E$, is isotropic.
  Then
  \[
  \tau_E\sim 0
  \]
  hence
  \[
  x_{E(X)}=0
  \]
  and
  \[
  \eta^i(y)_{E(X)}=\eta^i(y_E)=0.
  \]
  
  Since $X_E$ is isotropic, $E(X)/E$ is purely transcendental and
  $y_E=0$ (prop.\ \ref{pc5}~\Aref{pc5.a}).  Hence $2y=0\in H^i(F,2)$
  by the usual transfer argument, and $y=\lambda(z)$ for some $z\in
  H^iF$. By the surjectivity of $\bar e^i$, write $z=\bar e^i(t)$ for
  $t\in \bar I^iF$. Then
  \[
  \lambda \bar e^i(x)=\eta^i\lambda \bar e^i(t)=\lambda \bar
  e^i(\chi^i(t))
  \]
  hence $x=\chi^i(t)$ by the injectivity of $\bar e^i$ and $\lambda$.
\end{proof}

%%%% bibliography
\begin{thebibliography}{10}
  \raggedbottom
  
\bibitem{arason} J.Kr.~Arason, \newblock {\it Cohomologische
    Invarianten quadratischer Formen}, \newblock J. Alg. {\bf 36}
  (1975), 448--491.
  
\bibitem{amitsur} S.A.~Amitsur, \newblock {\it Generic splitting
    fields and central simple algebras}, \newblock Ann. of Math. {\bf
    62} (1955), 8--43.
  
\bibitem{bog} F.A. Bogomolov, {\it The Brauer group of quotient spaces
    by linear group actions}, Izv. Akad. Nauk. SSSR {\bf 51} (1987),
  485--516. English translation: Math. USSR Izv. {\bf 30} (1988),
  455--485.
  
\bibitem{BO} S.~Bloch and A.~Ogus, \newblock {\it Gersten's conjecture
    and the homology of schemes}, \newblock Ann. Sci. Ec. Norm. {\bf
    7} (1974), 181--202.
  
\bibitem{CT} J.-L.~Colliot-Th\'el\`ene, {\it Cycles alg\'ebriques de
    torsion et $K$-th\'eorie alg\'ebrique}, Arithmetical algebraic
  geometry (Trento, 1991), E. Ballico, ed., Lect. Notes in Math. {\bf
    1553}, Springer, Berlin-Heidelberg-New York, 1993, 1--49.
  
\bibitem{CT2} J.-L.~Colliot-Th\'el\`ene, \newblock {\it On the
    reciprocity sequence in higher class field theory of function
    fields}, \newblock {\it in} Algebraic $K$-theory and algebraic
  topology (P.G. Goerss and J.F. Jardine, eds), NATO ASI Series, Ser.
  C {\bf 407} (1993), 35--55.
  
\bibitem{CTcourse} J.-L.~Colliot-Th\'el\`ene, \newblock {\it
    Birational invariants, purity and the Gersten conjecture},
  \newblock in: W.  Jacob and A. Rosenberg (ed.), $K$-theory and
  algebraic geometry: connections with quadratic forms and division
  algebras, Proceedings of Symposia in Pure Mathematics {\bf 58} (I)
  (1995), 1--64.
  
\bibitem{cto} J.-L.~Colliot-Th\'el\`ene and M.~Ojan\-gu\-ren,
  \newblock {\it Vari\'et\'es unirationnelles non rationnelles:
    au-del\`a de l'exemple d'Artin et Mumford}, \newblock Invent.
  Math. {\bf 97} (1989), 141--158.
  
\bibitem{ctp} J.-L.~Colliot-Th\'el\`ene and R.~Parimala, {\it Real
    components of algebraic varieties and \'etale cohomology}, Invent.
  Math.  {\bf 105} (1991), 221--245.
  
\bibitem{CTS} J.-L.~Colliot-Th\'el\`ene and R.~Sujatha, \newblock {\it
    Unramified Witt groups of real anisotropic quadrics}, \newblock
  in: W. Jacob and A. Rosenberg (ed.), $K$-theory and algebraic
  geometry: connections with quadratic forms and division algebras,
  Proceedings of Symposia in Pure Mathematics {\bf 58} (II) (1995),
  127--147.
  
\bibitem{deligne} P.~Deligne, \newblock {\it Quadriques}, \newblock
  Groupes de Monodromie en g\'eom\'etrie alg\'ebrique (SGA 7),
  expos\'e XII, \newblock Lect. Notes in Math. {\bf 340}, Springer,
  Berlin-Heidelberg-New York (1973), 62--81.
  
\bibitem{DiEQ} J.~Dieudonn{\'e}, \emph{Les extensions quadratiques des
    corps non commutatifs et leurs applications}, Acta Math.
  \textbf{87} (1952), 175--242.
  
\bibitem{el} R.~Elman and T.Y.~Lam, \newblock {\it Pfister forms and
    $K$-theory of fields}, \newblock J. Alg. {\bf 23} (1972),
  181--213.
  
\bibitem{esnal} H. Esnault, B. Kahn, M. Levine, E. Viehweg,
  \newblock {\it The Arason invariant and\/ $\rm mod\,2$ algebraic cycles},
  to appear in J. Amer. Math. Soc.

\bibitem{fitz} R.W. Fitzgerald, {\it Witt kernels of function field
    extensions}, Pacific J. Math. {\bf 109} (1983), 89--106.
  
\bibitem{jr} W.~Jacob and M.~Rost, \newblock {\it Degree four
    cohomological invariants for quadratic forms}, \newblock Invent.
  Math. {\bf 96} (1989), 551--570.
  
\bibitem{hcoh} B.~Kahn, \newblock {\it Lower $\cal H$-cohomology of
    higher-dimensional quadrics}, \newblock Archiv Math. (Basel) {\bf
    65} (1995), 244--250.
  
\bibitem{scharlau} B.~Kahn, \newblock {\it On the Scharlau transfer},
  \newblock Rocky Mountain J. Math. {\bf 19} (1989), 741--747.
  
\bibitem{ks} B.~Kahn, R.~Sujatha, {\it Unramified cohomology of
    quadrics, II}, in preparation.
  
\bibitem{K} N.~Karpenko, \newblock {\it Algebro-geometric invariants
    of quadratic forms} (in Russian), \newblock Algebra-i-Analiz {\bf
    2} (1990), 141--162, 262.  \newblock English translation:
  Leningrad Math. J. {\bf 2} (1991), 119--138.
  
\bibitem{K2} N.~Karpenko, \newblock {\it Cycles of codimension $3$ on
    a projective quadric} (in Russian), \newblock Zapiski Nauch. Sem.
  {\bf 191} (1991), 114--123. English translation: Journal of Soviet
  Mathematics {\bf 63} (1993), 678--682.
  
\bibitem{K3} N.~Karpenko, {\it Chow groups of quadrics and index
    reduction formula}, Nova J. of Algebra and Geometry 3 (1995),
  357--379.
  
\bibitem{kato} K. Kato, {\it A Hasse principle for two-dimensional
    global fields}, J. reine angew. Math. {\bf 366} (1986), 142--183.
  
\bibitem{knebusch2} M.~Knebusch \newblock {\it Generic splitting of
    quadratic forms, II}, \newblock Proc. London Math. Soc. {\bf 34}
  (1977), 1--31.
  
\bibitem{involutions} M. Knus, A.S. Merkurjev, M. Rost, J.-P. Tignol,
  {\it The book of involutions}, in preparation.
  
\bibitem{lam} T.Y. Lam, The Algebraic theory of quadratic forms (2nd
  edition), Benjamin, New York, 1980.
  
%%\bibitem{MeSK1} A.S.~Merkurjev, \newblock {\it Generic element in $SK\ 
%%    1$ for simple algebras} $K$-Theory {\bf 7} (1993), no. 1, 1--3.

\bibitem{Mecourse} A.S.~Merkurjev, \newblock {\it $K$-theory of simple
    algebras}, \newblock in: W. Jacob and A. Rosenberg (ed.),
  $K$-theory and algebraic geometry: connections with quadratic forms
  and division algebras, Proceedings of Symposia in Pure Mathematics
  {\bf 58} (I) (1995), 65--83.
  
%%\bibitem{merkurjev2} A.S.~Merkurjev, {\it On the norm residue symbol
%%    of degree $2$}, Dokl. Akad. Nauk SSSR {\bf 261} (1981), 542--547.
%%  English translation: Soviet Math. Dokl. {\bf 24} (1981), 546--551.
  
\bibitem{merk3} A.S.~Merkurjev, {\it $SK_1$ for biquaternion
    algebras}, preprint, 1994.
  
\bibitem{merkurjev} A.S.~Merkurjev, {\it The group
    $H^1(X,{\cal{K}}_2)$ for projective homogeneous varieties},
  Algebra i Analiz. English translation: Leningrad (Saint-Petersburg)
  Math. J. {\bf 7} (1995), 136--164.
  
\bibitem{Mebi} A.S. Merkurjev, {\it On the norm residue homomorphism
    for fields}, Mathematics in St. Petersburg, 49--71, Amer. Math.
  Soc. Transl. Ser. 2, {\bf174}, Amer. Math. Soc., Providence, RI,
  1996.
  
\bibitem{ms1} A.S.~Merkurjev and A.A.~Suslin, \newblock {\it $\cal
    K$-cohomology of Severi-Brauer varieties and norm residue
    homomorphism} (in Russian), \newblock Izv. Akad. Nauk SSSR {\bf
    46} (1982), 1011--1046.  \newblock English translation: Math USSR
  Izv. {\bf 21} (1983), 307--340.
  
\bibitem{ms} A.S.~Merkurjev and A.A.~Suslin, \newblock {\it The norm
    residue homomorphism of degree $3$} (in Russian), \newblock Izv.
  Akad. Nauk SSSR {\bf 54} (1990), 339--356.  \newblock English
  translation: Math. USSR Izv. {\bf 36} (1991), 349--368.
  
\bibitem{MT} A.S.~Merkurjev and J.-P. Tignol, \newblock {\it Galois
    cohomology of biquadratic extensions}, \newblock Comment. Math.
  Helv. {\bf 68} (1993), 138--169.
  
\bibitem{milnor} J.W.~Milnor, {\it Algebraic $K$-theory and quadratic
    forms}, Invent. Math.  {\bf 9} (1969/70), 318--344.
  
\bibitem{ovv} D. Orlov, A. Vishik, V. Voevodsky, {\it Motivic
    cohomology of Pfister quadrics}, in preparation.
  
\bibitem{Panin:84}
  I.~Panin, \emph{Applications of {$K$-theory} in algebraic geometry}, 1984,
  Thesis, LOMI, Leningrad.

\bibitem{P1} R.~Parimala, \newblock {\it Witt groups of anisotropic
    conics, elliptic and hyperelliptic curves}, \newblock J. Number
  Theory {\bf 28} (1988), 69--93.
  
\bibitem{peyre1} E. Peyre, {\it Unramified cohomology and rationality
    problems}, Math. Ann. {\bf 296} (1993), 247--268.
  
\bibitem{peyre} E.~Peyre, \newblock {\it Products of Severi-Brauer
    varieties and Galois cohomology}, \newblock {\it in}: W. Jacob and
  A. Rosenberg (ed.), $K$-theory and algebraic geometry: connections
  with quadratic forms and division algebras, Proceedings of Symposia
  in Pure Mathematics {\bf 58} (2) (1995), 369--401.
  
\bibitem{peyre2} E. Peyre, {\it Corps de fonctions de vari\'et\'es
    homog\`enes et cohomologie galoisienne}, C. R. Acad. Sci. Paris
  {\bf 321} (1995), 891--896.
  
\bibitem{quillen} D.~Quillen, {\it Higher algebraic $K$-theory, I},
  Lect. Notes in Math. {\bf 341} (1973), Springer, Berlin, 83--147.
  
%%\bibitem{rost} M.~Rost, \newblock {\it Hilbert's theorem 90 for
%%    $K_3^M$ for degree-two extensions}, \newblock preprint,
%%  Regensburg, 1986.
  
\bibitem{rostch} M.~Rost, {\it Chow groups with coefficients},
  Documenta Math. {\bf 1} (1996), 319--393.
  
%%\bibitem{rost2} M.~Rost, {\it Cohomological invariants}, in
%%  preparation.
  
\bibitem{salt} D.J. Saltman, {\it Noether's problem over an
    algebraically closed field}, Invent. Math. {\bf 77} (1984),
  71--84.
  
\bibitem{salt2} D.J. Saltman, {\it Brauer groups and the center of
    generic matrices}, J. of Alg. {\bf 97} (1985), 53--67.
  
\bibitem{Scharlau} W. Scharlau Quadratic and hermitian forms, Grundl.
  Math. Wiss. {\bf 294}, Springer, Berlin, 1985.
  
\bibitem{scheiderer} C. Scheiderer, Real and \'etale cohomology, Lect.
  Notes in Math. {\bf 1588}, Springer, Berlin, 1994.
  
\bibitem{CG} J-P.~Serre, \newblock {\it Sur la dimension cohomologique
    des groupes profinis}, \newblock Topology {\bf 3} (1965),
  413--420.
  
\bibitem{suj} R. Sujatha, {\it Unramified cohomology and Witt groups
    of anisotropic Pfister quadrics}, Trans. Amer. Math. Soc. {\bf
    349} (1997), no. 6, 2341--2358.
  
%%\bibitem{su} A. Suslin, {\it The quaternion homomorphism for the
%%    function field of a conic} (en russe), Dokl. Akad. Nauk SSSR {\bf
%%    265} (1982), 292--296. English translation: Soviet Math. Dokl.
%%  {\bf 26} (1982), 72--77.
  
\bibitem{suslin} A. Suslin, {\it $K$-theory and $\cal{K}$-cohomology
    of certain group varieties}, Algebraic $K$-theory (A. Suslin,
  ed.), Adv. Soviet Math. {\bf 4}, AMS, Providence, 1991, 53--74.
  
\bibitem{Swan:89} R.G. Swan, \emph{Zero cycles on quadric
    hypersurfaces}, Proc. Amer. Math. Soc.  \textbf{107} (1989),
  43--46.

%%\bibitem{swan} R.G.~Swan, {\it $K$-theory of quadric hypersurfaces},
%%  Ann. of Math. {\bf 121} (1985), 113--153.
  
\bibitem{sz} M.~Szyjewski, \newblock {\it The fifth invariant of
    quadratic forms} (in Russian), \newblock Algebra Anal. {\bf 2}
  (1990), 213--234.  \newblock English translation: Leningrad Math. J.
  {\bf 2} (1991), 179--198.
  
\bibitem{tignol} J.-P.~Tignol, {\it $SK_1$ for biquaternion algebras,
    after M. Rost and A.  Merkurjev}, secret paper, 1994.
  
\bibitem{vangeel} J.~van~Geel, {\it Applications of the Riemann-Roch
    theorem for curves to quadratic forms and division algebras},
  preprint, Universit\'e catholique de Louvain, 1991.
  
\bibitem{voem} V. Voevodsky, {\it The Milnor conjecture}, preprint,
  Max-Planck-Institute for Mathematics, Bonn, 1996, {\tt
    http://www.math.uiuc.edu/K-theory/0170},
  
\bibitem{VdW} B.L. van der Waerden, \emph{Gruppen von linearen
    {Transformationen}}, Ergebnisse Math.  Wiss., {\bf 4},
  Springer-Verlag, Berlin, 1935.

\bibitem{witt} E. Witt, {\it \"Uber ein Gegenbeispiel zum Normensatz},
  Math. Zeit. {\bf 39} (1935), 462--467.

\end{thebibliography}

\end{document}
