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\begin{document}

\title[Cohomological approaches to $SK_1$ and $SK_2$]{Cohomological 
approaches to $SK_1$ and $SK_2$ of central simple algebras}
\author{Bruno Kahn}
\address{Institut de Math\'ematiques de Jussieu\\175--179 rue du
Chevaleret\\75013 Paris\\France}
\email{kahn@math.jussieu.fr}
\date{November 12, 2009}

\begin{abstract} We discuss several constructions of homomorphisms from $SK_1$ and $SK_2$ of
central simple algebras to subquotients of Galois cohomology groups.
\end{abstract}
\maketitle

\hfill To A. Suslin

\tableofcontents

\section*{Introduction}

Given a simple algebra $A$ with centre $F$, the group $SK_i(A)$ is defined
for $i=1,2$ as the kernel of the \emph{reduced norm}
\[\Nrd_i:K_i(A)\to K_i(F).\]

The definition of $\Nrd_1$ is classical, and $\Nrd_2$ was defined by
Suslin in \cite[Cor. 5.7]{suslin}. For further reference, let us recall
these  definitions in a uniform way: let $X$ be the
Severi-Brauer variety of $A$. After Quillen \cite[Th. 8.4]{quillen},
there is an isomorphism
\[\bigoplus_{r=0}^{d-1} K_i(A^{\otimes r})\iso K_i(X)\]
for any $i\ge 0$. The reduced norm is then given by the
composition
\[K_i(A)\to K_i(X)\to H^0(X,\sK_i)\osi K_i(F)\]
where the right isomorphism is obvious for
$i=1$ and is due to Suslin \cite[Cor. 5.6]{suslin} for $i=2$.

Of course, this definition also makes sense for $i=0$: in this case, $\Nrd_0$ is simply
multiplication by the index of $A$:
\[K_0(A)\simeq \Z\>\ind(A)>>\Z\simeq K_0(F)\]
and $SK_0(A)=0$.

[For $i>2$, a reduced norm satisfying reasonable properties cannot exist (Rost, Merkurjev
\cite[p. 81, Prop. 4]{merk}): the right generalisation is in the framework of motivic
cohomology, see \cite{kl}.]

The groups $SK_1(A)$ and $SK_2(A)$ remain mysterious and are known only in
very special cases. Here are a few elementary properties they enjoy:

\begin{enumerate}
\item $SK_i(A)$ is Morita-invariant.
\item $\ind(A) SK_i(A)=0$ (from Morita invariance, reduce to the case
where $A$ is division, and then use a transfer argument thanks to a
maximal commutative subfield of $A$).
\item The cup-product $K_1(F)\otimes K_1(A)\to K_2(A)$ induces a map
\[K_1(F)\otimes SK_1(A)\to SK_2(A).\]
\item Let $v$ be a discrete valuation of rank 1 on $F$, with residue
field $k$, and assume that $A$ spreads as an Azumaya algebra $\sA$ over
the discrete valuation ring $\sO_v$. It can be shown that the map $SK_1(\sA)\to SK_1(A)$
is surjective and that, if $K_2(\sO_v)\to K_2(F)$ is injective, there is a short exact sequence
\[SK_2(\sA)\to SK_2(A)\>\partial>> SK_1(\sA_k)\]
with
\[\partial(\{f\}\cdot x)=v(f)\bar x\]
for $f\in F^*$ and $x\in SK_1(A)$.
\item Let $A(t)=F(t)\otimes_F A$, and similarly $A(x)=F(x)\otimes_F A$
for any closed point $x\in \A^1_F$. Then there is an isomorphism
\[SK_1(A)\iso SK_1(A(t))\]
due to Platonov and an exact sequence
\[0\to SK_2(A)\to SK_2(A(t))\to\bigoplus_{x\in \A^1_F} SK_1(A(x)).\]
\end{enumerate}

From (3) and (4), one deduces that $SK_1(A)$ is a direct summand
of $SK_2(A(t))$ via the map $x\mapsto \{t\}\cdot x$: in particular, the
latter group is nonzero as soon as the former is. More intriguing is the
\emph{Calm\`es symbol}
\begin{gather*}
cal:\Lambda^2\left(\frac{K_1(A)}{\ind(A)K_1(A)}\right)\to
SK_2(A)\\ 
a\wedge b\mapsto \Nrd(a)\cdot b- a\cdot Nrd(b).
\end{gather*}

The image of this symbol is not detected by residues.

Let us now review known results about $SK_1$ and $SK_2$. If $F$ is a
global field, then $SK_i(A)=0$ for
$i=1,2$: this is classical for $i=1$ as a consequence of class field
theory, while for
$i=2$ it is due to Bak and Rehmann using the Merkurjev-Suslin theorem
\cite{br}. In the sequel, I concentrate on more general fields $F$ and always
assume that the index of $A$ is invertible in $F$.

\subsection{$SK_1$} The first one to give an example where $SK_1(A)\ne 0$
was Platonov
\cite{platonov}. In his example, $F$ is provided with a discrete
valuation of rank $2$ and the Brauer group of the second residue field is
nontrivial; in particular, $cd(F)\ge 4$.

Over general fields, a striking and early result for $SK_1$ is Wang's
theorem:

\begin{sloppypar}
\begin{Thm}[Wang \protect{\cite{wang}}]\label{tw} If the index of $A$ is
square-free, then $SK_1(A)=0$.
\end{Thm}
\end{sloppypar}

The most successful approach to
$SK_1(A)$ for other $A$ has been to relate it to Galois cohomology
groups. This approach was initiated by Suslin, who (based on Platonov's
results) conjectured the existence of a canonical homomorphism
\[SK_1(A)\to H^4(F,\mu_n^{\otimes 3})/[A]\cdot H^2(F,\mu_n^{\otimes 2})\]
where $n$ is the index of $A$, supposed to be prime to $\car
F$ \cite[Conj. 1.16]{suslin2}. In
\cite{suslin2}, Suslin was only able to partially carry over this project:
he had to assume that $\mu_{n^3}\subset F$ and then could only construct
twice the expected map, assuming the Bloch-Kato conjecture in degree $3$.

The next result in
this direction is due to Rost in the case of a biquaternion algebra:

\begin{Thm}[Rost \protect{\cite[th. 4]{merk}}]\label{tr} If $A$ is a
biquaternion algebra, there is an exact sequence
\[0\to SK_1(A)\to H^4(F,\Z/2)\to H^4(F(Y),\Z/2)\]
where $Y$ is the quadric defined by an `Albert form' associated to $A$.
\end{Thm}

The surprise here is that Rost gets in particular a finer map than the
one expected by Suslin, as he does not have to mod out by multiples of
$[A]$.

Merkurjev generalised Rost's theorem to the case of a simple algebra of
degree $4$ but not necessarily of exponent $2$:

\begin{Thm}[Merkurjev \protect{\cite[th. 6.6]{merk2}}]\label{tm1} If $A$
has degree
$4$, there is an exact sequence
\[0\to SK_1(A)\to H^4(F,\Z/2)/2[A]\cdot H^2(F,\Z/2)\to H^4(F(Y),\Z/2)\]
where $Y$ is the generalised Severi-Brauer variety $SB(2,A)$, a twisted
form of the Grassmannian $G(2,4)$.
\end{Thm}

Note that the right map makes sense because $A_{F(Y)}$ has exponent $2$. Merkurjev's exact
sequence is obtained from Rost's by descent from $F(Z)$ to $F$, where $Z=SB(A^{\otimes 2})$
(the point is that neither $SK_1(A)$ nor the kernel of the right map in Theorem \ref{tm1}
change when one passes from $F$ to $F(Z)$).

More recently, Suslin revisited his homomorphism of \cite{suslin2} in
\cite{suslin4}, where he constructs an (a priori different) homomorphism
using motivic cohomology rather than Chern classes in $K$-theory. He
compares it with the one of Rost-Merkurjev and proves the following
amazing theorem:

\begin{Thm}[Suslin \protect{\cite[Th. 6]{suslin4}}]\label{tsuslin} For any central simple
algebra
$A$ of degree
$4$, there exists a commutative diagram of isomorphisms
\[\begin{CD}
SK_1(A)\>\phi>\sim> \displaystyle\frac{\Ker(H^4(F,\mu_4^{\otimes 3})\to
H^4(F(X),\mu_4^{\otimes 3}))}{[A]\cdot H^2(F,\mu_4^{\otimes 2})}\\
|| && \V{\tau'}VV\\
SK_1(A)\>\psi>\sim> \displaystyle\frac{\Ker(H^4(F,\mu_2^{\otimes 3})\to
H^4(F(Y),\mu_2^{\otimes 3}))}{2[A]\cdot H^2(F,\mu_2^{\otimes 2})}
\end{CD}\]
where $X=SB(A)$, $Y=SB(2,A)$, $\phi$ is Suslin's homomorphism just mentioned and $\psi$ is Merkurjev's isomorphism from Theorem \ref{tm1}.
\end{Thm}


\subsection{$SK_2$} Concerning $SK_2(A)$, the first result (over an
arbitrary base field) was the following theorem of Rost and Merkurjev:

\begin{Thm}[Rost \protect{\cite{rost3}}, Merkurjev
\protect{\cite{merk5}}]\label{tm2} For any quaternion algebra
$A$, $SK_2(A)=0$.
\end{Thm}

Rost and Merkurjev used this theorem as a step to prove the Milnor
conjecture in degree $3$; conversely, this conjecture and techniques of
motivic cohomology were used in \cite[th. 9.3]{lens} to give a very short
proof of Theorem \ref{tm2}. We revisit this proof in Remark \ref{quick}, in the spirit of the
techniques developed here.

The following theorem is more recent; it was obtained independently by Merkurjev-Suslin
\cite[Th. 2.4]{ms2}. In view of the still fluctuant status of the Bloch-Kato conjecture for odd
primes, we assume its validity in the statement. Of course, it is generally accepted in degree
$\le 2$ (Merkurjev-Suslin \cite{ms}) and for $l=2$ (Voevodsky \cite{voem}).

\begin{Thm}[Kahn-Levine
\protect{\cite[Cor. 2]{kl}}]\label{tmp} Assume the Bloch-Kato conjecture in degree $\le 3$.
For any central simple algebra
$A$ of square-free index, $SK_2(A)=0$.
\end{Thm}


From Theorems \ref{tw} and \ref{tmp}, we get by a well-known d\'evissage
argument a refinement of the elementary property (2) given above: for
any $A$ and $i=1,2$, $\frac{\ind(A)}{\prod l_i} SK_i(A)=0$, where the
$l_i$ are the distinct primes dividing $\ind(A)$.

On the other hand, Baptiste Calm\`es gave a version of
Rost's theorem \ref{tr} for $SK_2$ of biquaternion algebras:

\begin{Thm}[Calm\`es \protect{\cite{calmes2}}] \label{tcalmes} Under the
assumptions of Theorem \ref{tr}, assume further that $F$ contains a separably closed field.
Then there is an exact sequence
\[\Ker(A_0(Z,K_2)\to K_2(F))\to SK_2(A)\to H^5(F,\Z/2)\to H^5(F(Y),\Z/2)\]
where $Z$ is a hyperplane section of $Y$.
\end{Thm}

(Note that in the case of $SK_1$, the corresponding group
$\Ker(A_0(Z,K_1)\allowbreak \to K_1(F))$ is $0$ by a difficult theorem of
Rost.)

Finally, let us mention the construction of homomorphisms \`a la Suslin
\begin{align}
SK_1(A)&\to H^4(F,\Q/\Z(3))/[A]\cdot K_2(F)\label{bl1}\\
SK_2(A)&\to H^5(F,\Q/\Z(4))/[A]\cdot K_3^M(F)\label{bl2}
\end{align}
in \cite[\S 6.9]{kl}, using an \'etale version of the Bloch-Lichtenbaum spectral sequence for
the motive associated to $A$. The second map depends on the Bloch-Kato conjecture in degree
$3$ and assumes, as in Theorem \ref{tcalmes}, that $F$ contains a separably closed field. This
construction goes back to 1999 (correspondence with M. Levine), although the targets of
\eqref{bl1} and \eqref{bl2} were only determined in \cite[Prop. 6.9.1]{kl}.

\subsection{The results} Calm\`es' proof of Theorem \ref{tcalmes} is based in part on the
methods of \cite{motiv}.  In this paper, I propose to generalise his construction to arbitrary
central simple algebras, with the same technique. The methods will also shed some light on 
the difference between Suslin's conjecture and the theorems of Rost and
Merkurjev. The main new results are the following:

\begin{Th}\label{t1} Let $F$ be a field and $A$ a
simple algebra with centre $F$ and index $e$, supposed to be a power of a 
prime $l$ different from $\car F$. Then, for any divisor $r$ of $e$, there
is a complex
\[0\to SK_1(A)\by{\sigma_r^1} H^4(F,\Q/\Z(3))/r[A]\cdot
K_2(F)\to H^0_\Zar(Y^{[r]},\sH^4_\et(\Q/\Z(3)))\] 
where $Y^{[r]}$ is the generalised Severi-Brauer variety
$SB(r,A)$. If the Bloch-Kato conjecture holds in degree $3$ for the prime 
$l$, these complexes refine into complexes
\[0\to SK_1(A)\to H^4(F,\mu_{e/r}^{\otimes 3})/r[A]\cdot
H^2(F,\mu_{e/r}^{\otimes 2})\to H^0_\Zar(Y^{[r]},\sH^4_\et(\mu_{e/r}^{\otimes 3}))).\] 
They are exact for $r=1,e=4$ and for $r=2$ when $A$ is a biquaternion algebra.\\
For $l=2$, the maps starting from $SK_1$ are nontrivial in general,
unless $\ind(A)\le 2$.
\end{Th}

I don't know, and don't 
conjecture, that these complexes are
exact in general.

The map of theorem \ref{t1} coincides with those of Rost and
Merkurjev, which is the way we get their nontriviality for $l=2$
\cite{merk3}.

\begin{Th}\label{t2} Let $F$, $A$, $e$ and $Y^{[r]}$ be as in Theorem
\ref{t1}; assume the Bloch-Kato conjecture in degree $\le 3$ at the prime $l$ and that $F$
contains a separably closed  subfield. Then, for any divisor $r$ of $e$, there
is a complex
\[0\to SK_2(A)\by{\sigma_r^2} H^5(F,\Q/\Z(4))/r[A]\cdot
K_3^M(F)\to H^0_\Zar(Y^{[r]},\sH^5_\et(\Q/\Z(4)))\] 
where $Y^{[r]}$ is the generalised Severi-Brauer variety
$SB(r,A)$. If, moreover, the Bloch-Kato conjecture holds in degree $4$ for the prime 
$l$, these complexes refine into complexes
\[0\to SK_2(A)\to H^5(F,\mu_{e/r}^{\otimes 4})/r[A]\cdot
H^3(F,\mu_{e/r}^{\otimes 3})\to H^0_\Zar(Y^{[r]},\sH^5_\et(\mu_{e/r}^{\otimes 4}))).\] 
For $l=2$, the maps starting from $SK_2(A)$ are nontrivial in general for $r=1,2$ (unless
$\ind(A)\allowbreak\le 2$).
\end{Th}

\begin{Th}\label{tSL} For any smooth $F$-variety $X$, define 
\[SK_1(X,A) = \colim \Hom_F(X,\SL_n(A))^\ab\] 
where $\SL_n(A)$ is the reductive group representing the functor $R\mapsto SL_n(A\otimes_F
R)$. Then there exists a natural transformation
\[c_A(X):SK_1(X,A)\to H^5_\et(X,\Z(3)).\]
Restricted to fields, $c_A$ is the universal invariant with values in
$H^5_\et(\Z(3))\allowbreak\simeq H^4_\et(\Q/\Z(3))$ in the sense of Merkurjev \cite{merk2}.
\end{Th}

Loosely speaking, $c_A$ is
defined out of the ``positive" generator of the group
$H^5_\et(\SL_1(A),\Z(3))/H^5_\et(F,\Z(3))$ which turns out to be infinite
cyclic, much like the Rost invariant is defined out of the ``positive"
generator of the infinite cyclic group $H^3_\et(\SL_1(A),\Z(2))\simeq
H^4_\et(B\SL_1(A),\Z(2))$ (see \cite[App. B]{eklv}). This replies  \cite[Rk. 5.8]{merk2} in
the same way as what was done for the Arason invariant in \cite{eklv}.

Theorems \ref{t1}, \ref{t2} and  \ref{tSL} were obtained around 2001/2002, except for the
exactness and nontriviality statements for $r=1$, which follow from the
work of Suslin
\cite{suslin4}. They were presented at the 2002 Talca-Puc\'on conference on quadratic forms
\cite{talca}.

This paper is organised as follows. We set up notation in Section \ref{not}.  In Section
\ref{motup}, we recall the slice spectral sequences in the case of geometrically cellular
varieties.
Sections
\ref{5} to \ref{approx} are technical. In particular, Section \ref{5} recalls the diagrams of
exact sequences from \cite[\S 5]{motiv}, trying to keep track of where the Bloch-Kato
conjecture is used; we deduce a simple proof of Suslin's theorem
\cite[Th. 1]{suslin4}, as indicated by himself in the introduction of \cite{suslin4} (Remark
\ref{r2.1}). In Section
\ref{gencon} we get our first main result, Theorem
\ref{t6}, which constructs functorial injections sending a part of lower $K$-theory of some
projective homogeneous varieties into a certain  subquotient of the Galois cohomology of the base
field. We apply this result in Section
\ref{sflag} to twisted flag varieties, thus getting Theorems \ref{t1} and \ref{t2} (see
Corollary \ref{c8.2}); in Remark
\ref{quick}, we revisit the proof of Theorem
\ref{tm2} given in \cite{lens}.  In Section \ref{klplus1}, we push the main result
of
\cite{kl} one step further. In Section \ref{s:red}, we do some preliminary computations on the
slice spectral sequences associated to a reductive group $G$: the main result is that, if $G$
is simple simply connected of inner type $A_r$ for $r\ge 2$, then the complex $\alpha^*c_3(G)$
of \cite{hk} is isomorphic to $\Z[-1]$ (see Theorem \ref{t5} for a more complete statement).
In section \ref{generic}, we take up the approach of Merkurjev in \cite{merk2} and
prove Theorem \ref{tSL}, see Theorem \ref{t11.2}. We conclude with some incomplete computations
in Section \ref{comp} trying to evaluate the group $SK_1(A_K)/SK_1(A)$, where $K$ is the
function field of $\SL_1(A)$: see Theorem \ref{t11.1} and Corollary \ref{c11.1}.

This paper contains results which are mostly 7 to 8 years old. The main reason why it was
delayed so much was that I tried to compare the 3 ways to construct homomorphisms \`a la Suslin
indicated above: in \eqref{bl1}--\eqref{bl2}, Theorems \ref{t1} and \ref{t2} and  Theorem
\ref{tSL}, and to prove their nontriviality in some new cases. I am sorry to say that I have been mostly unsuccessful. The only easy comparisons
are, for Theorems
\ref{t1} and
\ref{t2}, with the Rost and Calm\`es homomorphisms of Theorems \ref{tr} and \ref{tcalmes}, and
with the new Suslin homomorphism of Theorem
\ref{tsuslin}. All others seem challenging\footnote{Including with the first homomorphism of
Suslin in \protect\cite{suslin2}, a comparison I had initiated in a preliminary version of this
paper.}: I give some comments on these comparison issues in Subsection
\ref{ss:comp} and Remark \ref{r11.1}.

\subsection*{Acknowledgements} This project was started in 2001. I would like to thank Marc Levine,
Baptiste Calm\`es, Annette Huber, Sasha Merkurjev, Evgueny Schinder, Philippe Gille and Tim
Wouters for helpful discussions and  exchanges. Last, I would like to express my admiration for
Suslin's work: the inspiration I have drawn from it will be obvious throughout this paper.

\section{Notation}\label{not}

If $X$ is a projective homogeneous variety, we
denote as in \cite{motiv} by $E_i$ the \'etale $F$-algebra
corresponding to the canonical $\Z$-basis of $CH^i(X_s)$ given by
Schubert cycles, where $X_s=X\otimes_F F_s$ and $F_s$ is a separable closure of $F$.

The motivic cohomology groups used in this paper are (mostly) the Hom groups in Voevodsky's
category $DM_{-,\et}^\eff(F)$ of \cite[\S 3.3]{voetri} (\'etale topology). In particular, the
exponential characteristic $p$ of $F$ is inverted in this category by \cite[Prop.
3.3.3 2)]{voetri}, so that those groups are
$\Z[1/p]$-modules. Very occasionally we shall use Hom groups in the category
$DM_{-}^\eff(F)$ (Nisnevich topology).

Let $(\Q/\Z)'=\bigoplus_{l\ne p} \Q_l/\Z_l$. We abbreviate the \'etale cohomology groups
$H^i_\et(X,(\Q/\Z)'(j))$ with  the notation $H^i(X,j)$, and similarly for $\sH$-cohomology
groups.

Unless otherwise specified, all cohomology groups appearing are \'etale
cohomology groups, with the exception of $\sH$ or $\sK$-cohomology groups which are Nisnevich
(= Zariski) cohomology groups. We shall specify the topology as an index when it seems
necessary for clarity.



\section[Motivic cohomology\dots]{Motivic cohomology of smooth
geometrically cellular varieties updated}\label{motup}

\subsection{The slice spectral sequences} In \cite{motiv}, we constructed spectral sequences for
the \'etale motivic cohomology of smooth geometrically cellular varieties. These results were
limited in two respects:

\begin{enumerate}
\item the ground field $F$ was assumed to be of characteristic $0$;
\item the spectral sequences had a strange abutment, which was
nevertheless sufficient for applications.
\end{enumerate}

The results of \cite{hk} solved both issues. The first one was due to the
fact that \cite{motiv} worked with motives with compact support in
Voevodsky's triangulated category of motives \cite{voetri}, which are
known to be geometric only in characteristic $0$: indeed, it was shown
that the motive with compact supports of a cellular variety $X$ is a pure
Tate motive in the sense of \cite{hk}, from which it was deduced by
duality that the motive of $X$ (without supports) is also pure Tate if
$X$ is smooth. In \cite[Prop. 4.11]{hk}, we prove directly that, over any
field, the motive of $X$ is pure Tate if $X$ is smooth and cellular.

The second issue was more subtle and is discussed in \cite[Remark
6.3]{hk}. The short answer is that by considering a different filtration
than the one used in \cite{motiv}, one gets the ``right" spectral
sequence.

We summarize this discussion by stating the following theorem, which
follows from \cite[(3.2) and Prop. 4.11]{hk} and replaces
\cite[Th. 4.4]{motiv}:

\begin{thm} \label{t4.1} Let $X$ be a smooth,
equidimensional, geometrically cellular
variety over a perfect field $F$. For all $n\ge 0$,
there is a spectral sequence $E(X,n)$:
\begin{equation}\label{e3.1}E_2^{p,q}(X,n)=H^{p-q}_\et(F,CH^q(X_s)\otimes
\Z(n-q))\Rightarrow H^{p+q}_\et(X,\Z(n)).
\end{equation}
Note that, by cellularity, each $CH^q(X_s)$ is a permutation Galois module. These spectral sequences have the following properties:
\begin{thlist}
\item {\em\bf Naturality}. \eqref{e3.1} is covariant in $F$ and
contravariant in $X$ (varying among smooth, equidimensional, geometrically
cellular
varieties) under flat equidimensional maps.
\item {\em\bf Products}. There are pairings of spectral sequences
\[E_r^{p,q}(X,m)\times E_r^{p',q'}(X,n)\to E_r^{p+p',q+q'}(X,m+n)\]
which coincide with the usual cup-product on the $E_2$-terms and the
abutments.
\item {\em\bf Transfer}. For any finite extension $E/F$ and any $n\ge 0$,
there is a morphism of spectral sequences
\[E_r^{p,q}(X_E,n)\to E_r^{p,q}(X,n)\]
which coincides with the usual transfer on the $E_2$-terms and the
abutment.
\item {\em\bf Covariance for closed equidimensional immersions}. For any closed
immersion
$i:Y\inj X$ of pure codimension $c$, where $X$ and $Y$ are smooth, geometrically
cellular, there is a morphism of spectral sequences 
\[E_r^{p-c,q-c}(Y,n-c)\stackrel{i_*}{\longrightarrow} E_r^{p,q}(X,n)\]
``abutting" to the Gysin homomorphisms 
\[H^{p+q-2c}_\et(Y,\Z(n-c))\stackrel{i_*}{\to}
H^{p+q}_\et(X,\Z(n)).\]
\end{thlist}
If $X$ is split, then \eqref{e3.1} degenerates at
$E_2$.
\end{thm}

The only nonobvious point in this theorem is (ii) (products). In
\cite[p. 915]{hk}, it is claimed that there are pairings of slice
spectral sequences for the tensor product of two arbitrary motives $M$
and $N$. This is not true in general: I thank Evgeny Shinder for pointing
out this issue. However, these pairings certainly exist if $M$ or $N$ is a
mixed Tate motive: the argument is essentially the same as the one that
proves that the K\"unneth maps of
\cite[Cor. 1.6]{hk} are isomorphisms in this case \cite[Lemma 4.8]{hk}.
For the reader's convenience, we outline the construction. We take the
notation of \cite{hk}:

Given the way the slice spectral sequence is constructed in \cite[\S
3]{hk} (bottom of p. 914), to get a morphism of filtrations, we need to
get morphisms
\[\nu_{\le q+q'}(M\otimes M')\to \nu_{\le q} M\otimes \nu_{\le q'} M'\]
for two motives $M,M'$ and two integers $q,q'$.

From the canonical maps $M\to \nu_{\le q} M$ and $M'\to \nu_{\le q'} M'$,
we get a morphism
\[M\otimes M'\to \nu_{\le q} M\otimes \nu_{\le q'} M'\]
and we would like to prove that its composition with $\nu^{> q+q'} (M\otimes M')\to M\otimes
M'$ is $0$. This will be true provided 
\begin{multline*}
\nu^{> q+q'} (\nu_{\le q} M\otimes \nu_{\le q'}
M')=\\
\underline{\Hom}(\Z(q+q'+1),\nu_{\le q} M\otimes
\nu_{\le q'} M')(q+q'+1)=0.
\end{multline*}

This is false in general (for example $M=M'=h_1(C)$,
$q=q'=0$, where $C$ is a curve of genus $>0$ over an algebraically closed
field), but it is true if
$M$ or $M'$ is a mixed Tate motive. Indeed, we may reduce to $M=\Z(a)$
for some integer $a$. Then
\[\nu_{\le q} M=\begin{cases}
0&\text{if $q<a$}\\
\Z(a)& \text{if $q\ge a$}
\end{cases} \]
hence
$\underline{\Hom}(\Z(q+q'+1),\nu_{\le q} M\otimes
\nu_{\le q'} M')=0$ if $q<a$, and if $q\ge a$ we get
\begin{multline*}
\underline{\Hom}(\Z(q+q'+1),\nu_{\le q} M\otimes
\nu_{\le q'} M')\\
=\underline{\Hom}(\Z(q+q'+1),\Z(a)\otimes
\nu_{\le q'} M')\\
=\underline{\Hom}(\Z(q+q'+1-a),\nu_{\le q'} M')=0
\end{multline*}
because $q+q'+1-a> q'$.


 Dealing with
the spectral sequences for
\'etale motivic cohomology, it will suffice that
$M$ or $N$ is geometrically mixed Tate in the sense of \cite[\S 5]{hk} to
have these products.

\begin{rk}\label{r2.2} As stressed in \S \ref{not}, the spectral sequences of Theorem \ref{t4.1}
are spectral sequences of $\Z[1/p]$-modules, where $p$ is the exponential characteristic of $F$.
Thus all results of this paper are ``away from $p$". It is nevertheless possible to extend the
methods to $p$-algebras in characteristic $p$, at some cost: this is briefly discussed in
Appendix \ref{cancel}. I am grateful to Tim Wouters for a discussion leading to this
observation.
\end{rk}

\subsection{Vanishing of $E_2$-terms} Since this issue may be confusing, we include here an estimate
in the case of the spectral sequences \eqref{e3.1} and of the coniveau spectral sequences,
which will be used in the next section (compare \cite[p. 161]{motiv}). It shows that these two
spectral sequences live in somewhat complementary regions of the $E_2$-plane.

\begin{prop}\label{p2.1} a) In the spectral sequence \eqref{e3.1}, we have
$E_2^{a,bq}(X,n)\allowbreak=0$ in the following cases:
\item[(ai)] $a\le b$, $b\ge n-1$, except $a=b=n$.
\item[(aii)] $a=n+1$ under the Bloch-Kato conjecture in degree $n-b$.\\ 
Moreover, $E_2^{a,b}(X,n)$ is uniquely divisible for $a\le b$ and $b<n-1$,
under the Bloch-Kato conjecture in degree $n-b$.\\
b) Let $X$ be a smooth variety. In the coniveau spectral sequence for \'etale motivic cohomology
\[E_1^{a,b} = \bigoplus_{x\in X^{(a)}} H^{b-a}(k(x),\Z(n-a))\Rightarrow H^{a+b}(X,\Z(n))\]
we have $E_1^{a,b}=0$ in the following cases:
\item[(bi)] $a\ge b$, $a\ge n-1$, except $a=b=n$.
\item[(bii)] $b=n+1$ under the Bloch-Kato conjecture in degree $n-a$.\\ 
Moreover, $E_2^{a,b}(X,n)$ is uniquely divisible for $a\ge b$ and $a<n-1$,
under the Bloch-Kato conjecture in degree $n-a$.\\
Finally, for $b=n$, the natural map
\[H^a(X,\sK_n^M)[1/p]\to E_2^{a,n}\]
is surjective under the Bloch-Kato conjecture in degrees $\le n-a$, and bijective under the
Bloch-Kato conjecture in degrees $\le n-a+1$.
\end{prop}

\begin{proof} For (ai), we use that $E_2^{a,b}(X,n)=H^{a-b}_\et(F,CH^b(X_s)\otimes
\Z(n-b))\simeq H^{a-b-1}_\et(F,CH^b(X_s)\otimes
\Q/\Z(n-b))$ for $n-b<0$ (by definition of $\Z_\et(n-b)$ for $n-b<0$, see \cite[Def. 3.1]{hk}),
and also that $\Z(0)=\Z$ and $\Z(1)=\bG_m[-1]$. (aii) follows from Hilbert 90 in degree
$n-b$. The proofs of (bi) and (bii) are similar. The divisibility claims are also
known consequences of the Bloch-Kato conjecture. Finally, the last claim follows from a diagram
chase in the comparison map between the Gersten complexes for Nisnevich and \'etale cohomology
with $\Z(n)$ coefficients.
\end{proof}

%\enlargethispage*{20pt}

\section{Weight $3$ and weight $4$ \'etale motivic cohomology}\label{5}

\subsection{Weight $3$} Let $X$ be a projective homogeneous $F$-variety.
In
\cite[\S 5.4]{motiv}, we drew a commutative diagram with some exactness
properties, by mixing the coniveau spectral sequence and the spectral
sequence of \cite[Th. 4.4]{motiv} for \'etale motivic cohomology in weight
$3$. We can now use the
spectral sequence \eqref{e3.1} to get the same diagram over any perfect
field. To get the diagram of \cite[\S 5.4]{motiv}, we made the blanket assumption in
\cite{motiv} that all groups were localised at  $2$, because calculations relied on the
Bloch-Kato conjecture in degree $3$, which was only proven for $l=2$.

In this paper, we are also interested in getting results independent from
this conjecture, at least for $SK_1$. How much exactness remains in this
diagram if we don't wish to use it? Using Proposition \ref{p2.1}, we see that at least the
following part of the diagram of
\cite[\S 5.4]{motiv} remains exact by only using the Bloch-Kato conjecture in degree $\le 2$ (=
the Merkurjev-Suslin theorem): the exponential characteristic $p$ is implicitly inverted in
this diagram as well as in the next one, \eqref{eq13}.
\begin{equation}\begin{CD}\label{eq14}
&&H^4(F,\Z(3))\\
&&\V{}VV\\
0\to H^1(X,\sK_3^M)\>>> H^4(X,\Z(3))\>>> H^0(X,\sH^4(\Z(3)))\\
&&\V{}VV\\
&&K_2(E_1)\\
&&\V{}V{d_2^{3,1}(3)}V\\
H^0(X,\sH^4(\Z(3))) &&H^4(F,3)\\
\V{}VV\Big\downarrow &\searrow\scriptstyle\eta^4\\
H^2(X,\sK_3^M)\>>> H^5(X,\Z(3))&\to&
H^0(X,\sH^4(3))\\ 
&\searrow\scriptstyle \xi^4&\V{}VV\V{}VV\\
&&E_2^*&&CH^3(X)\\
&&&&\V{}VV\\ 
&&&&H^6(X,\Z(3)).
\end{CD}
\end{equation}

The group
$H^0(X,\sH^4(\Z(3)))$, which appears twice in this diagram, is of course torsion, as well as
$H^4(F,\Z(3))$, and their
$l$-primary components are $0$ under the Bloch-Kato conjecture in degree $3$ for the
prime $l$.


\subsection{Weight 4} In weight $4$, we cannot avoid using the
Bloch-Kato conjecture in degree $3$. There is a commutative diagram, which
was only written down in a special case in \cite{motiv}:
\begin{equation}
\begin{CDS}\label{eq13}
H^5(X,\Z(4))\to K_3(E_2)_{\ind}&&K_3^M(E_1)\\
&\SE{}E{d_3^{3,2}(4)}E \V{}V{d_2^{4,1}(4)}V\\
H^0(X,\sH^5(\Z(4)))&& H^5(F,4)\\
\V{}VV&\V{}VV \SE{}E{\eta^5}E\\
H^2(X,\sK_4^M)\>>> H^6(X,\Z(4))&\to& H^0(X,\sH^5(4))\\
&\SE{}E{\xi^5}E\V{}VV &\V{}VV\\
&&K_2(E_2)&&H^3(X,K_4^M)\\
&\SW{}W{d_3^{4,3}(4)}W\V{}V{d_2^{4,3}(4)}V&\V{}VV\\
H^6(F,4) &&H^4(E_1,3)&&H^7(X,\Z(4)).
\end{CDS}
\end{equation}

In this diagram, the differentials appearing correspond to the spectral
sequence \eqref{e3.1} in weight $4$. The path snaking from $H^0(X,\sH^5(\Z(4)))$ to
$H^7(X,\Z(4))$ is exact (it comes from the coniveau spectral sequence for weight $4$ \'etale
motivic cohomology: see Proposition \ref{p2.1}). The differential
$d_3^{4,3}(4)$ is only defined on the kernel of
$d_2^{4,3}(4)$ and the differential $d_3^{3,2}(4)$ takes values
in the cokernel of $d_2^{3,2}(4)$. The column is a complex, exact at
$H^6(X,\Z(4))$; its exactness properties at $H^5(F,4)$ and
$K_2(E_2)$ involve the differentials $d_3$ in an obvious sense. 

All these
exactness properties depend on the fact that the natural map $K_i^M(E)\to
H^i_\et(E,\Z(i))$ is an isomorphism for any field $E$ and any $i\le 3$,
and also on the vanishing of $H^{i+1}_\et(E,\Z(i))$ under the same
conditions. Both statements (for all fields $E$) are equivalent to the
Bloch-Kato conjecture in degree $i$.

The map $\eta^5$ is the natural map from the Galois cohomology of the
ground field to the unramified cohomology of $X$. 

\begin{defn}\label{d1}
For $i=1,2$, we denote
by $\overline{\Ker}\,\ \eta^{i+3}$ the homology of the complex
\[K_{i+1}^M(E_1)\>d_2^{i+2,1}(X,i+2)>>
H^{i+3}(F,i+2)\>\eta^{i+3}>>H^0(X,\sH^{i+3}(i+2)).\]
\end{defn}

Diagram \eqref{eq14} yields an exact sequence
\[H^0(X,\sH^4(\Z(3)))\to \Ker \xi^4\to \overline{\Ker}\,\
\eta^4\to 0\]
hence an isomorphism 
\begin{equation}\label{eq15.3}
\Ker\xi^4\iso \overline{\Ker}\,\
\eta^4
\end{equation}
under the Bloch-Kato conjecture in degree $\le 3$.

If $F$ contains an algebraically closed subfield, then $K_3(E_2)_{\ind}$
is divisible and the differential $d_3^{3,2}(4)$ is $0$ since it is a
priori torsion \cite[Prop. 4.6]{motiv}. Then diagram
\eqref{eq13} yields an exact sequence
\[H^0(X,\sH^5(\Z(4)))\to \Ker \xi^4\to \overline{\Ker}\,\
\eta^4\to 0\]
under the Bloch-Kato conjecture in degree $\le 3$ and an isomorphism
\begin{equation}\label{eq15.4}
\Ker \xi^5\iso \overline{\Ker}\,\ \eta^5
\end{equation}
under the Bloch-Kato conjecture in degree $\le 4$.

\begin{rk}\label{r2.1} Let us recover  Suslin's theorem \cite[Th. 1]{suslin4} from 
\eqref{eq15.3}. The point is simply that the coniveau spectral sequence for Nisnevich motivic
cohomology yields an isomorphism
\[H^2(X,\sK_3^M)\iso H^5_\Nis(X,\Z(3))\]
(\cf \cite[Lemma 9]{suslin4}). The differential $d_2^{3,1}(3)$ was computed in \cite[Th.
7.1]{motiv} for Severi-Brauer varieties.
\end{rk}


\section{$H^1(X,\sK_3)$ and $H^0(X,\sK_3)$}\label{3}



\subsection{$H^1(X,\sK_3)$} Recall from \cite[Prop. 4.5]{motiv} that 
\begin{equation}\label{eq4.1}
H^i(X,\sK_3^M)\iso H^i(X,\sK_3) \text{ for } i>0.
\end{equation}

We have:

\begin{prop}\label{p1} Let $X$ be a projective homogeneous variety over
$F$, and $K/F$ a regular extension. Under the Bloch-Kato conjecture in degree $3$,  the map
\[H^1(X,\sK_3)\to H^1(X_K,\sK_3)\]
has $p$-primary torsion kernel, where $p$ is the exponential
characteristic of $F$. More precisely, the kernel of this map is torsion
and its
$l$-primary part vanishes for $l\ne p$ if the Bloch-Kato conjecture
holds at the prime $l$ in degree $3$. 
\end{prop}

\begin{proof} Up to passing to its perfect closure, we may assume $F$ 
perfect. By Diagram \eqref{eq14} and \eqref{eq4.1}, there is a canonical map
\[H^1(X,\sK_3)\to K_2(E_1)\]
where $E_1$ is a certain \'etale $F$-algebra associated to $X$, whose
kernel is contained in $H^4_\et(F,\Z(3))$: hence the $l$-primary part of
this kernel vanishes under the condition in Proposition \ref{p1}. The
result now follows from \cite[th. 3.6]{suslin}.\end{proof}

\subsection{$H^0(X,\sK_3)$}


Let $X$ be a projective homogeneous $F$-variety. As in \cite[\S
5.1]{motiv}, for all $i\ge 0$ we write $E_i$ for the \'etale $F$-algebra
determined by the Galois-permutation basis of $CH^i(X_s)$ given by
Schubert cycles (see \S \ref{not}). 

\begin{thm}\label{t3} a) For $i\le 2$, the map $K_i(F)\to H^0(X,\sK_i)$ is bijective.\\
b) Under the Bloch-Kato conjecture in degree $3$, the
cokernel of the homomorphism
\[K_3(F)\to H^0(X,\sK_3)\]
is torsion, and its prime-to-the-characteristic part is
\begin{enumerate}
\item finite if $F$ is finitely generated over its prime subfield;
\item $0$  in the following
cases:
\begin{thlist}
\item $F$ contains a separably closed subfield;
\item the map $CH^1(X_{E_1})\to CH^1(X_s)$ is surjective.
\end{thlist}
\end{enumerate}
More precisely, under the Bloch-Kato conjecture in degree $3$ for the
prime $l$, the above is true after localisation at $l$.
\end{thm}

\begin{proof} a) is well-known and is quoted for reference purposes: it is obvious for $i=0,1$
(since $X$ is proper), and for $i=2$ it is a theorem of Suslin \cite[Cor. 5.6]{suslin}.

 b) After \cite[Th. 3
a)]{QL2} (see also
\cite[Th. 16.4]{levine}), the homomorphism $K_3^M(K)\to K_3(K)$ is injective for any field $K$. 
Consider the commutative  diagram with exact rows
\[\begin{CD}
0\>>>K_3^M(F)\>>> K_3(F)\>>> K_3(F)_{\ind}&\to& 0\\
&&\V{}VV \V{}VV \V{}VV\\
0\>>> H^0(X,\sK_3^M)\>>> H^0(X,\sK_3)\>>> H^0(X,\sK_3^{\ind}).
\end{CD}\]

As $X$ is a rational variety, the right vertical map is bijective
\cite[lemma 6.2]{eklv}. It therefore suffices to prove the claims of
theorem \ref{t3} for the left vertical map. 

Let us first assume $F$ perfect: then we can use Theorem
\ref{t4.1}. Mixing the weight $3$ coniveau
spectral sequence for
\'etale motivic cohomology with the spectral sequence \eqref{e3.1} in
weight $3$, we get modulo the Bloch-Kato conjecture in degree $3$ the
following commutative diagram with exact rows:
\[\begin{CDS}
&&&&0\\
&&&&\V{}VV\\
&&&&K_3^M(F)\\
&&&&\V{}VV\SE{}E{\alpha}E\\
0&\to& H^1(X,\sH^2(\Z(3)))\>>> H^3(X,\Z(3))&\to& H^0(X,\sK_3^M)&\to& 0\\
&&& \SE{}E{\beta}E \V{}VV\\
&&&& K_3(E_1)_{\ind}\\
&&&&\V{}VV\\
&&&&0.
\end{CDS}\]

For the reader's convenience, let us explain where the Bloch-Kato conjecture in degree $3$ is
necessary. The weight
$3$ spectral sequence \eqref{e3.1} gives a priori an exact
sequence
\begin{multline*}
H^0(E_1,\Z(2))\>d_2^{1,1}(X,3)>> H^3(F,\Z(3))\to H^3(X,\Z(3))\\
\to H^1(E_1,\Z(2))\to H^4(F,\Z(3)).
\end{multline*}

The group $H^0(E_1,\Z(2))$ is conjecturally $0$; the Merkurjev-Suslin
theorem and \cite{sv} or \cite{gl} (see also \cite{glrev}) imply that it
is uniquely divisible in any case. Since the differential $d_2^{1,1}(X,3)$
is torsion (proof as in \cite[Prop. 4.6]{motiv}), it must be $0$.
Similarly, the identification of
$H^1(E_1,\Z(2))$ with
$K_3(E_1)_\ind$ only depends on the Merkurjev-Suslin theorem. On the
other hand, the bijectivity of $K_3^M(F)\to H^3(F,\Z(3))$ and the
vanishing of $H^4(F,\Z(3))$ depend on the Bloch-Kato conjecture in degree
$3$ (recall that all groups are \'etale cohomology groups here). This
takes care of the vertical exact sequence. Similarly, the Bloch-Kato conjecture in degree $3$ is
necessary to identify the last term of the horizontal exact sequence (stemming from the coniveau
spectral sequence) with $H^0(X,\sK_3^M)$.

The diagram above gives an isomorphism
\[\Coker\alpha\simeq\Coker\beta.\]

Let us show that $\Coker\beta$ is $m$-torsion for some $m>0$. 
The group
$K_3(E_1)_{\ind}$ appearing in the diagram is really
\[H^0(F,CH^1(X_s)\otimes H^1(F_s,\Z(2)))\]
via Shapiro's lemma, the isomorphism $H^1(K,\Z(2))\simeq K_3(K)_{\ind}$
for any field and Galois descent for $K_3(K)_{\ind}$ \cite{MS,levine2}. A
standard computation shows that the corestriction map
\[H^0(E_1,CH^1(X_s)\otimes H^1(F_s,\Z(2)))\>\Cores>>
H^0(F,CH^1(X_s)\otimes H^1(F_s,\Z(2)))\]
is split surjective. On the other hand, since $CH^1(X_s)$ is finitely
generated, there exists a finite extension $E/F$ such that $CH^1(X_E)\to
CH^1(X_s)$ is surjective. Without loss of generality, we may assume
that $E$ contains all the residue fields of the \'etale algebra $E_1$. A
transfer argument then shows that the map
$CH^1(X_{E_1})\to CH^1(X_s)$ has cokernel killed by some integer
$m>0$. Hence the composition
\begin{multline*}
CH^1(X_{E_1})\otimes H^1(E_1,\Z(2))\to CH^1(X_s)\otimes
H^1(E_1,\Z(2))\\
\iso CH^1(X_s)\otimes H^0(E_1,H^1(F_s,\Z(2)))\\
\simeq
H^0(E_1,CH^1(X_s)\otimes H^1(F_s,\Z(2)))
\end{multline*}
has cokernel killed by $m$, and the same holds for the
composition
\begin{multline*}
CH^1(X_{E_1})\otimes H^1(E_1,\Z(2))\to
H^0(E_1,CH^1(X_s)\otimes H^1(F_s,\Z(2)))\\
\>\Cores>>H^0(F,CH^1(X_s)\otimes H^1(F_s,\Z(2))).
\end{multline*}

But this composition factors via cup-product as
\begin{multline*}
CH^1(X_{E_1})\otimes H^1(E_1,\Z(2))=H^1(X_{E_1},\sH^2(\Z(1)))\otimes
H^1(E_1,\Z(2))\\\to H^1(X_{E_1},\sH^2(\Z(3)))
\>\Cores>>H^1(X,\sH^2(\Z(3)))\\\>\beta>> H^0(F,CH^1(X_s)\otimes
H^1(F_s,\Z(2)))
\end{multline*}
which proves the claim. 

Coming back the the case where $F$ is not necessarily perfect, let $F'$
be its perfect (radicial?) closure and $\alpha'$ the map $\alpha$ ``viewed
over
$F'$". Then a transfer argument shows that the natural map
$\Coker\alpha\to \Coker \alpha'$ has $p$-primary torsion kernel and
cokernel, where $p$ is the exponential characteristic of $F$. In
particular, $\Coker\alpha$ is torsion, and its prime-to-$p$ part is
killed by some $m$.

The integer $m$ equals $1$ provided
$CH^1(X_{E_1})\to CH^1(X_s)$ is surjective, which proves 2) (ii)
in Theorem \ref{t3}. In general, the map
\[K_3(F_0)_{\ind}/m\to K_3(F)_{\ind}/m\]
is bijective, where $F_0$ is the field of constants of $F$
\cite{MS,levine2}. If
$F_0$ is separably closed, then $K_3(F_0)_{\ind}/m=0$ (ibid.), which
proves 2) (i); if
$F$ is finitely generated, then $F_0$ is a finite field or a number field
with ring of integers $A$ and $K_3(F_0)_{\ind}$ is a quotient of
$K_3(A)$; in both cases it is finitely generated, which proves 1).
\end{proof}

\begin{ex} $X$ is a conic curve. Then $\Coker\beta$ is isomorphic to the
cokernel of the map
\[\bigoplus_{x\in X^{(1)}} K_3(F(x))_{\ind}\>(N_{F(x)/F})>>
K_3(F)_{\ind}.\]

Even in the case $F=\Q$, $K_3(\Q)_{\ind}\simeq\Z/24$, I am not able
either to produce an example where this map is not onto, or to prove that
it is always onto. As a first try, one might restrict to points of degree
$2$ on $X$. To have an idea of how complex the situation is, the reader may refer to \cite[\S
8]{descente}. In particular, Theorem 8.1 (iv) of {\it loc. cit.} shows that the map is onto
provided
$X$ has a quadratic splitting field of the form $\Q(\sqrt{-p})$, where
$p$ is prime and $\equiv -1\pmod{8}$. If $X$ corresponds to the Hilbert symbol $(a,b)$, with
$a,b$ two coprime integers, the theorem of the arithmetic progression shows that there are
infinitely many $p\equiv -1\pmod{8}$ such that $p\nmid ab$ and
$\displaystyle\left(\frac{-p}{l}\right)=-1$ for all primes $l\mid ab$. Since $-p$ is a square in
$\Q_2$, this implies that
$(a,b)_{\Q(\sqrt{-p})}=0$ if and only if $(a,b)_{\Q_2}=0$. Thus the above map is surjective if
$X(\Q_2)\neq \emptyset$, but I don't know the answer in the other case.
\end{ex}

\section{$H^i(X,\sK_4^M)$ and $H^i(X,\sK_4)$}\label{4}

\begin{thm} \label{t4} a) For any smooth variety $X$, the natural map 
\[\phi_i:H^i(X,\sK_4^M)\to H^i(X,\sK_4)\]
is bijective for $i\ge 3$ and surjective for $i=2$ with kernel killed by
$2$.\\
b) Suppose that $F$ contains a separably closed
subfield. Then $\phi_2$ is bijective. 
\end{thm}

\begin{proof} a) By Gersten's conjecture, we may compute both groups as cohomology
groups of the respective Gersten complexes
\[\dots\to\bigoplus_{x\in X^{(i)}} K_{4-i}^M(F(x))\to\dots\]
\[\dots\to\bigoplus_{x\in X^{(i)}} K_{4-i}(F(x))\to\dots\]

Therefore, Theorem \ref{t4} is obvious for $i\ge 3$, and
$\phi_2$ is surjective. Using the Adams operations
on algebraic $K$-theory, we see that, for any field $K$, the exact
sequence
\[0\to K_3^M(K)\to K_3(K)\to K_3(K)_{\ind}\to 0\]
is split up to $2$-torsion. It follows that $2\Ker\phi_2=0$.

b) We have an exact sequence
\[\bigoplus_{x\in X^{(1)}} K_{3}(F(x))_\ind\>\psi>>
H^2(X,\sK_4^M)\>\phi_2>> H^2(X,\sK_4)\to 0.\] 

By assumption, each group $K_{3}(F(x))_\ind$ is divisible (compare the
proof of Theorem \ref{t3}). Since their images in $H^2(X,\sK_4^M)$ are
killed by $2$, they are $0$.\end{proof}

\begin{rk} I don't know if the condition on $F$ is necessary for the
bijectivity of $\phi_2$. Note that $\psi$ factors through the group
$H^1(X,\sH^2(\Z(3)))$ appearing in the proof of Theorem \ref{t3}.
\end{rk}

\section{An approximation of  cycle cohomology}\label{approx}

Let $M_*$ be a cycle module in the sense of Rost \cite{rost} and let $X$
be projective homogeneous. There are cup-products
\begin{equation}\label{eq1}
CH^p(X)\otimes M_{q-p}(F)\to A^{p}(X,M_q).
\end{equation}
which are isomorphisms when $X$ is split, by \cite[Prop. 3.7]{eklv}.
 
Assume now that $X$ is not necessarily split. Let $Y$
be a splitting variety for
$X$: if
$X_s=G_s/P$ where $G$ is a semi-simple $F$-algebraic group and $P$ is a
parabolic subgroup of $G_s$, we may take $Y$ such that $Y_s=G_s/B$ for
$B$ a Borel subgroup contained in $P$. Then $X_{F(y)}$ is cellular for any
point $y\in Y$. It is possible to define a map
\begin{equation}\label{eq6.1}
A^p(X,M_q)\>\tilde\xi^{p,q}>> A^0(Y_{E_p},M_{q-p})
\end{equation}
which is an isomorphism after tensoring with $\Q$ and corresponds to the
inverse of \eqref{eq1} when $X$ is split. When
$q-p\le 2$ and $M_*=K_*^M$, this map refines into a map
\begin{equation}\label{eq7.1}
A^p(X,K^M_q)\>\xi^{p,q}>> K^M_{q-p}(E_p)
\end{equation}
thanks to Suslin's theorem \cite[Cor. 5.6]{suslin} for $q-p=2$ and trivially
for $q-p=0,1$. In this paper, we shall only construct such a map in the
substantially simpler inner case where all algebras $E_p$ are split, which 
is sufficient for our needs.

We note that, if $X$ is split, the functor $K\mapsto CH^p(X_K)$ from
field extensions of $F$ to abelian groups is constant, with finitely
generated free value. When $X$ is arbitrary, we shall authorise ourselves
of this to denote by $CH^p(X_s)$ the common value of $CH^p(X_K)$ for all
splitting fields $K$ of $X$.

For $Y$ a splitting variety of $X$ as above, consider the Rost spectral
sequence \cite[\S 8]{rost}
\[E_2^{p,q}=A^p(Y,R^q p_* M_*)\Rightarrow A^{p+q}(X\times Y,M_*)\]
where $p$ is the projection $X\times Y\to Y$ and the $R^q p_*M_*$ are the
higher direct images of $M_*$ in the sense of Rost \cite[\S 7]{rost}.  Using the fact that
\eqref{eq1} is an isomorphism in the split case, we get canonical
isomorphisms
\[R^q p_*M_*=CH^q(X_s)\otimes M_{*-q}\]
hence an edge homomorphism
\[A^p(X\times Y,M_q)\to E_2^{0,p}=CH^p(X_s)\otimes A^0(Y,M_{q-p}).\]

In the inner case, the composition of this map with the obvious map
$A^p(X,M_q)\to A^p(X\times Y,M_q)$ is the desired map $\tilde\xi^{p,q}$ of \eqref{eq6.1}.

In the special case $M_* = K_*^M$, a functoriality argument
shows that the map $\xi^{2,3}$ (\resp $\xi^{2,4}$) of \eqref{eq7.1} coincides with the map
$\xi^4$ of Diagram
\eqref{eq14} (\resp with the map
$\xi^5$ of Diagram
\eqref{eq13}).

\begin{comment}
We shall need the following strengthening of proposition \ref{p1} below:

\begin{prop}\label{p1bis} Let $X$ be a projective homogeneous variety over
$F$, and
$Y$ a unirational variety over $F$. Denote by $\sH^i(X\times?,\sK_j)$ the
Zariski sheaf over $Y$ associated to the presheaf $U\mapsto
\sH^i(X\times U,\sK_j)$ Then, for $i\le 3$, the map
\[H^1(X,\sK_i)\to H^0(Y,\sH^1(X\times ?,\sK_i))\]
has odd torsion kernel and cokernel; it is bijective for $i\le 2$.
\end{prop}

\begin{proof} If $X$ is split, then $CH^1(X)$ is free abelian and the map
$CH^1(X)\otimes K_{i-1}(F)\to H^1(X,\sK_i)$ is bijective.
\end{comment}

\section{A general $K$-theoretic construction}\label{gencon}

Let $X$ be projective homogeneous, and let $K$ be a splitting field for $X$ such that $K/F$ is
geometrically rational (for example, take for $K$ the function field of the corresponding full
flag variety, see beginning of \S \ref{approx}). We assume as in the previous section that the
associated algebras $E_p$ are split: this is probably not essential. We write $K_*(X)^{(i)}$ for
the coniveau filtration on
$K_*(X)$, and
$K_*(X)^{(i/i+1)}$ for its successive quotients.

\subsection{The first steps of the coniveau filtration}

\begin{thm}\label{t6} For $i\le 2$,\\
a) The map 
\[K_i(F)\oplus K_i(X)^{(1)}\to K_i(X)\]
is an isomorphism.\\
b)  The maps 
\begin{multline*}
\Ker(K_i(X)^{(2)}\to K_i(X_K)^{(2)})\to \Ker(K_i(X)^{(1)}\to K_i(X_K)^{(1)})\\
\to\Ker(K_i(X)\to K_i(X_K))
\end{multline*}
are isomorphisms. (For $i=2$, we assume the Bloch-Kato conjecture in degree $3$ for the torsion
primes of $X$.)\\ 
c) There are canonical monomorphisms
\[\Ker(K_i(X)^{(2/3)}\to K_i(X_K)^{(2/3)})\Inj  \overline{\Ker}\,\eta^{i+3}\]
where $\overline{\Ker}\,\eta^{i+3}$ was introduced in Definition \ref{d1}. (If $i=2$, we assume the
Bloch-Kato conjecture in degree
$3$ for the torsion primes of $X$, and also that
$F$ contains a separably closed field.) These homomorphisms are contravariant in $X$.
\end{thm}

\begin{proof} a) By Theorem \ref{t3} a), the composition
\[K_i(F)\to K_i(X)\to H^0(X,\sK_i)\]
is bijective; hence this composition yields a splitting to the
exact sequence
\[0\to K_i(X)^{(1)}\to K_i(X)\to H^0(X,\sK_i).\]

b) It suffices to show that the maps
$K_i(X)^{(j/j+1)}\to K_i(X_K)^{(j/j+1)}$ are injective for $j=0,1$. For $j=0$, this is clear
from a) (reapplying Theorem \ref{t3} a)). 

For $j=1$, by the BGQ spectral sequence it suffices to show that
the map
\[H^1(X,\sK_{i+1})\to H^1(X_K,\sK_{i+1})\]
is injective. For $i=0$, the statement (concerning $\Pic$) is classical; for $i=1$, it follows
from
\cite[Theorem]{me} and for  $i=2$ it follows from Proposition
\ref{p1}.

c) The BGQ spectral sequence gives a map
\[
K_i(X)^{(2/3)}\iso
E_\infty^{2,-i-2}
\inj \Coker(H^0(X,\sK_{i+1})\>d_2^{0,-i-1}>>
H^2(X,\sK_{i+2})).
\] 

The differential $d_2^{0,-i-1}$ is $0$ by Theorem \ref{t3}. Therefore, we get an injection
\[\Ker(K_i(X)^{(2/3)}\to K_i(X_K)^{(2/3)})\inj \Ker(H^2(X,\sK_{i+2})\to H^2(X_K,\sK_{i+2})).\]

Clearly, the right-hand-side kernel is equal to $\Ker\xi^{2,i+2}$, where $\xi^{2,i+2}$ is the map
defined in the previous section. As observed at the end of this section, this map coincides with
the map $\xi^{i+3}$ of diagrams \eqref{eq14} and \eqref{eq13} (for $i=1,2$; similarly for $i=0$).
The result then follows from \eqref{eq15.3} and \eqref{eq15.4} (and their analogue for $i=0$).
\end{proof}

\subsection{The reduced norm and projective homogeneous varieties}

\begin{prop}\label{p7.1} Let $B$ be a central simple $F$-algebra, and let $\sF$ be a locally free
sheaf on
$X$, provided with an action of $B$. For $i\le 2$, consider the map
\[u_\sF:K_i(B)\to K_i(X)\]
induced by the exact functor
\begin{align}\label{eq7.2}
P(B)&\to P(X)\\
M&\mapsto \sF\otimes_B M\notag
\end{align}
where $P(B)$ (\resp $P(X)$) denotes the category of finitely generated [projective] $B$-modules
(\resp of locally free $\sO_X$-sheaves of finite rank).\\
a) The composition
\[K_i(B)\by{u_\sF} K_i(X)\to H^0(X,\sK_i)\osi K_i(F)\]
equals $\rank_B(\sF)\Nrd_B$, where $\rank_B(\sF):= \displaystyle\frac{\rank(\sF)}{\deg(B)}$.\\
b) The map
\[\tilde u_\sF:K_i(B)\to K_i(X)\]
defined by $x\mapsto u_\sF(X) - \rank_B(\sF)\Nrd_B(x)$ has image contained in $K_i(X)^{(1)}$. The
composition
\[K_i(B)\by{\tilde u_\sF} K_i(X)^{(1)}\to H^1(X,\sK_{i+1})\by{\xi^{1,i+1}}
K_i(E_1)=CH^1(X)\otimes K_i(F)\] 
where $\xi^{1,i+1}$ is as in Section \ref{approx}, equals $c_1(\sF)\otimes \Nrd_B$.
\end{prop}

\begin{proof} Observe that $\Nrd_B$ is characterised
by the commutation of the diagram
\[\begin{CD}
K_i(B_L)\>\sim >> K_i(L)\\
\A{}AA \A{}AA\\
K_i(B)\>\Nrd_B>> K_i(F)
\end{CD}\]
for any extension $L/F$ that splits $B$ and such that $L=F(Y)$, where $Y$ is a smooth
projective  geometrically rational $F$-variety and the upper isomorphism is given by Morita
theory. Indeed, this diagram then refines to a diagram of the form
\[\begin{CD}
H^0(Y,K_i(B\otimes_F\sO_Y))\>\sim >> H^0(Y,\sK_i)\\
\A{}AA \A{\wr}AA\\
K_i(B)\>\Nrd_B>> K_i(F)
\end{CD}\]
see \cite[Cor. 5.6]{suslin} for the right vertical isomorphism.

It is therefore sufficient to check Proposition \ref{p7.1} after extending
scalars to $L=K(Y)$, where $Y$ is the Severi-Brauer variety of $B$. Thus, we may assume $X$ and
$B$ split. 

By Morita, $u_\sF$ then corresponds to the map $K_i(F)\to K_i(X)$ given by cup-product with
$[\sF\otimes_B S]\in K_0(X)$, where $S$ is a simple $B$-module. a) is now obvious, the first
statement of b) follows, and the second is also obvious since $\xi^{i,1}$ commutes with products
in the split case.
\end{proof}

From Proposition \ref{p7.1} and Theorem \ref{t6} a), it follows that the restriction of $u_\sF$ and
$\tilde u_\sF$ to $SK_i(B)$ induce the same map: $SK_i(B)\to K_i(X)^{(2)}$, that we shall still
denote by $u_\sF$. If $L/F$ is chosen as in the proof of Proposition \ref{p7.1}, then clearly the
composition $SK_i(B)\to K_i(X)^{(2)}\to K_i(X_L)^{(2)}$ is $0$. This yields:

\begin{defn}\label{d7.1} Let $L/F$ be a geometrically rational extension splitting both $X$ and
$B$. We denote by
$\sigma_\sF^i:SK_i(B)\to
\overline{\Ker}\eta^{i+3}$ the composition
\[SK_i(B)\by{u_\sF} \Ker(K_i(X)^{(2/3)}\to K_i(X_L)^{(2/3)})\Inj
\overline{\Ker}\,\eta^{i+3}\]
where the second map is that of Theorem \ref{t6} c).
\end{defn}


\section{Twisted flag varieties}\label{sflag}

In this section, we define maps from $SK_i(A)$ to Galois cohomology as promised in the
introduction. We use the results of the previous section. In order to get these maps, it is
enough to deal with generalised Severi-Brauer varieties (twisted Grassmannians); however, we
start with the apparently greater generality of twisted flag varieties. The reason for doing
this is the hope to be able to compare the various maps with each other in the future, see
Subsection \ref{ss:comp}.

\subsection{$K$-theory of twisted flag varieties}\label{ss:twisted} Let $A$ be a simple
algebra of degree $d$, with centre
$F$. For
$\underline{r}=(r_1,\dots,r_k)$ with $d\ge r_1>\dots > r_k\ge 0$, let
$Y^{[\underline{r}]}=SB(\underline{r};A)$ be the twist of the flag variety
$G(r_1,\dots,r_k;d)$ by a $1$-cocycle defining $A$: its function field is generic
among extensions $K/F$ such that $A_K$ acquires a chain $I_1\supset\dots \supset I_k$ of
left ideals of respective $K$-dimensions $dr_1,\dots,dr_k$. If $\underline{s}$ is a subset of
$\underline{r}$, there is an obvious projection
\[Y^{[\underline{r}]}\to Y^{[\underline{s}]}.
\]

The variety $Y^{[\underline{r}]}$ carries a chain of locally free sheaves
\begin{equation}\label{eq8.1}
A_{Y^{[\underline{r}]}}\Surj\sJ_{r_1}\Surj\dots
\Surj\sJ_{r_k}
\end{equation}
where $A_{Y^{[\underline{r}]}}$ is the constant sheaf with value $A$: if $A$ is split,
\eqref{eq8.1}  corresponds by Morita to the tautological flag $\A^d_{Y^{[\underline{r}]}}\surj
V_{r_1}\dots\surj V_{r_k}$ on
$G(r_1,\dots,r_k;d)$ ($\sJ_{r_j}$ is the quotient of $\End(\A^d)_{Y^{[\underline{r}]}}$ by the
sheaf of ideals consisting of endomorphisms vanishing on $\Ker(\A^d_{Y^{[\underline{r}]}}\to
V_{r_j})$).

There is an action of $A$ on this chain. More generally, for any partition $\alpha=(\alpha_1,
\dots, \alpha_m)$ of $|\alpha|= \sum\alpha_i$ with $\alpha_1\ge \dots\ge \alpha_m \ge 0$, with
associated Schur functor
$S^\alpha$, the sheaf $S^\alpha(V_{r_j})$ on $G(r_1,\dots,r_k;d)$ defines by faithfully flat
descent a sheaf $S^\alpha(\sJ_{r_j})$ of $A^{\otimes |\alpha|}$-algebras on
$Y^{[\underline{r}]}$ \cite[\S 4]{lsw}.

By Levine-Srinivas-Weyman \cite[Th. 4.6]{lsw}, we have an isomorphism
\begin{equation}\label{eq9}
\bigoplus_\alpha K_*(A^{\otimes |\alpha|})\by{(u_\alpha)}
K_*(Y^{[\underline{r}]})
\end{equation} 
where $\alpha = (\alpha^1,\dots,\alpha^k)$ is a family of partitions,
with $0\le \alpha_i^j\le r_i-r_{i+1}$, $|\alpha|=\sum|\alpha^j|$ and $u_\alpha$ is induced by
the exact functor
\begin{align*}
P(A^{|\alpha]})&\to P(Y^{[\underline{r}]})\\
M&\mapsto S^\alpha(\sJ)\otimes_{A^{|\alpha|}} M
\end{align*}
with $S^\alpha(\sJ)=S^{\alpha^1}(\sJ_1)\otimes\dots \otimes S^{\alpha_k}(\sJ_k)$. Actually our
choice of generators is not the one of \cite{lsw}, but rather the same as in Panin \cite[Th.
7.1]{panin}, who proves the same results by a different method.

\subsection{Maps from $SK_i$ to Galois cohomology} We now apply Definition \ref{d7.1} with
$\sF=\sJ_{r_j}$ for each $j$: in the above notation, this corresponds to the case 
$\alpha^{j'}=0$ for $j'\ne j$ and
$\alpha^j=(1,0,\dots)$. We find maps
\begin{equation}\label{eq8.2}
\sigma_{r_j}^i:SK_i(A)\to \overline{\Ker}\eta^{i+1}_{Y^{[\underline{r}]}}.
\end{equation}

We now proceed to compute the differential $d_2^{i+2,1}(Y^{[\underline{r}]},i+2)$ involved in
Definition \ref{d1}. Using the multiplicativity of \eqref{e3.1} (Th. \ref{t4.1}
(ii)), we reduce to computing the differential
$d_2^{1,1}(Y^{[\underline{r}]},1)$ (\cf
\cite[lemma 6.1]{motiv}). We have an exact sequence \cite[5.2]{motiv}
\[CH^1(Y^{[\underline{r}]}_s)^{G_F}\>d_2^{1,1}(Y^{[\underline{r}]},1)>> Br(F)\to
Br(Y^{[\underline{r}]}).\]

The group $CH^1(Y^{[\underline{r}]}_s)$ has a basis consisting of the first Chern classes of the
bundles $V_{r_j}$: in particular, $G_F$ acts trivially on it. For
$j\in [1,k]$, write $Y^{[r_j]}$ for the twisted Grassmannian (generalised Severi-Brauer variety)
corresponding to $r_j$. Then we have a commutative diagram
\begin{equation}\label{eq8.3}
\begin{CD}
CH^1(Y^{[\underline{r}]}_s)\>d_2^{1,1}(Y^{[\underline{r}]},1)>> Br(F)\>>>
Br(Y^{[\underline{r}]})\\
\A{}AA ||&& \A{}AA \\
\Z=CH^1(Y^{[r_j]}_s)\>d_2^{1,1}(Y^{[r_j]},1)>> Br(F)\>>>
Br(Y^{[r_j]}).
\end{CD}
\end{equation}

This shows that $CH^1(Y^{[\underline{r}]}_s)$ is generated by the images of the maps
$CH^1(Y^{[r_j]}_s)\to CH^1(Y^{[\underline{r}]}_s)$ for $j=1,\dots,k$, and thus there is no loss of
generality in assuming $k=1$ for the computation of the differential, which we do now. Let us
simplify the notation by writing $r$ for $r_j$.

The following lemma is well-known:

\begin{lemma}\label{l8.3} $\Ker(Br(F)\to Br(Y^{[r]}))=\langle
r[A]\rangle$.
\end{lemma}

\begin{proof} By a Morita argument, we may assume that $r\mid d$. Since $K=F(X)$ splits
$Y^{[r]}$, where $X$ is the Severi-Brauer variety of
$A$, we have $\Ker(Br(F)\to Br(Y^{[r]}))\subseteq \Ker(Br(F)\to Br(X))$.
The latter group is generated by $[A]$ (Amitsur \cite{amitsur}). The
conclusion now follows from the fact that $\ind(A_{F(Y^{[r]})})$ divides
$r$, plus a transfer argument using a $0$-cycle of degree $d/r$ on
$Y^{[r]}$.\end{proof}

Hence we get $d_2^{1,1}(Y^{[r]},1)(1)=r[A]$ (up to a unit), and therefore from Diagram
\eqref{eq8.3}:
\[d_2^{1,1}(Y^{[\underline{r}]},1)(V_{r_j})=r_j[A] \text{ (up to a unit).}\]

We conclude:

\begin{cor}\label{c8.1} a) The maps \eqref{eq8.2} give rise to commutative diagrams of complexes
($i=1,2$):
\[\begin{CD}
0&\to& SK_i(A)&\by{\sigma_{r_j}^i}&\displaystyle \frac{H^{i+4}(F,\Z(i+2))}{gcd(r_j) [A]\cdot
H^{i+1}(F,\Z(i+1))}&\to& H^0(Y^{[\underline{r}]}),\sH^{i+4}(\Z(i+2)))\\
&&||&& \A{}AA \A{p^*}AA \\
0&\to& SK_i(A)&\by{\sigma_{r_j}^i}&\displaystyle \frac{H^{i+4}(F,\Z(i+2))}{r_j [A]\cdot
H^{i+1}(F,\Z(i+1))}&\to& H^0(Y^{[r_j]}),\sH^{i+4}(\Z(i+2)))
\end{CD}\]
where $Y^{[r_j]}=SB(r_j,A)$ is the generalised Severi-Brauer variety of ideals of rank
$r_j$, and the middle vertical map is the natural surjection.\\ 
b) If $j=k$ and $r_k$ divides the other $r_j$, then both vertical maps are isomorphisms.
\end{cor}


\begin{proof} The only thing to remain proven is b). The generic fibre of $p:Y^{[\underline{r}]}\to Y^{[r_k]}$ is then easily seen to be the split flag variety $G(r_1-r_k,\dots,r_{k-1}-r_k;d)$; in
particular it is rational and the claim follows.
\end{proof}


\begin{rk} \label{quick} By construction, this homomorphism for $i=2$ factors through an injection
\[SK_2(A)\Inj K_2(Y^{[\underline{r}]})^{(2)}.\]

If $A$ is a quaternion algebra, the only choice for $Y^{[\underline{r}]}$ is the conic
corresponding to $A$ and $K_2(Y^{[\underline{r}]})^{(2)}=0$. This is a variant of the proof of
Theorem \ref{tm2} given in \cite{lens}.
\end{rk}

As seen above, for
$i=1$, the definition of $\sigma_{r_j}^i$ only involves the Merkurjev-Suslin theorem, while for
$i=2$ it involves the Bloch-Kato conjecture in degree $3$ (for the primes dividing $d$). If we
are ready to grant the Bloch-Kato conjecture one degree further, we get a refinement of these
maps:

\begin{cor} \label{c8.2} Assume the Bloch-Kato conjecture in degree $i+2$ ($i=1,2$).
Assume also for simplicity that $r_j$ divides $d$. The the complexes on the bottom row of
Corollary
\ref{c8.1} refine into complexes
\begin{equation}\label{eq11}SK_1(A)\to H^4(F,\mu_{d/r_j}^{\otimes
3})/r_j[A]\cdot H^2(F,\mu_{d/r_j}^{\otimes 2})\to
H^4(F(Y^{[r_j]}),\mu_{d/r_j}^{\otimes 3})
\end{equation}
\begin{equation}\label{eq11bis}SK_2(A)\to H^5(F,\mu_{d/r_j}^{\otimes 4})/r_j[A]\cdot
H^3(F,\mu_{d/r_j}^{\otimes 3})\to H^5(F(Y^{[r_j]}),\mu_{d/r_j}^{\otimes 4}).
\end{equation}
\end{cor}

\begin{proof} Use the fact that $d/r \Ker \eta^i=0$ (transfer argument), and that the map 
$H^4(F,\mu_{d/r_j}^{\otimes 3})\to H^4(F,\Q/\Z(3))=H^5(F,\Z(3))$ (\resp the map
$H^5(F,\mu_{d/r_j}^{\otimes 4})\to H^5(F,\Q/\Z(4))=H^6(F,\Z(4))$) is injective under the
Bloch-Kato conjecture in degree $3$ (\resp $4$).
\end{proof}



\subsection{Examples: maps \`a la Suslin and \`a la Rost-Merkurjev}\label{suslin} The case of
Suslin corresponds to $r_j=1$ for any $A$. More precisely, the way Suslin constructs his map in
\cite[\S 3]{suslin4} shows that it coincides with the one here for $r_j=1$, compare Remark
\ref{r2.1}.  Similarly, the cases of Rost-Merkurjev correspond to $d=4$, $r_j=2$. Using the
work of Calm\`es \cite[\S 2.5]{calmes2}, one can check that in the case of a biquaternion
algebra we get back Rost's map for $SK_1$ (\resp Calm\`es' map for $SK_2$). This implies:

\begin{cor}\label{c8.3} a) For $i=1$, the bottom sequence in Corollary \ref{c8.1} is exact for
$r_j=1$ and $\deg(A) = 4$, or for $r_j=2$ if $A$ is a biquaternion algebra. \\ b) The maps
$\sigma^1_1$ and $\sigma_2^1$ are nonzero in general if $4\mid \ind(A)$.
\end{cor}

\begin{proof} a) Let us first assume $r_j=1$. Then, as explained above, the map $\sigma^1_1$
coincides with Suslin's map in \cite[\S 3]{suslin4}, and the exactness is loc. cit., Th. 3. If
now $A$ is a biquaternion algebra, the exactness is Rost's theorem \cite[Th. 4]{merk}.

b) This follows from a) by a standard argument, \cf \cite{merk3}.
\end{proof}

\subsection{Some properties of the maps $\sigma_r^i$} For simplicity, we replace $r_j$ by $r$;
we still assume that $r$ divides $d$.

\begin{lemma} If $r= d$, the maps \eqref{eq11} and \eqref{eq11bis} are
$0$.
\end{lemma}

\begin{proof} In this case the variety $Y^{[r]}$ has a rational point,
hence the two kernels are $0$. (Alternately, the coefficients of the cohomology groups
involved in Corollary \ref{c8.2} are $0$!)\end{proof}

\begin{prop}\label{p5} Let $a\in F^*$. Then, for all $r\mid d$, the diagram
\[\begin{CD}
SK_1(A)\>\sigma_r^1>> H^4(F,\mu_{d/r}^{\otimes 3})/r[A]\cdot
H^2(F,\mu_{d/r}^{\otimes 2})\\
\V{\cdot \{a\}}VV \V{\cdot \{a\}}VV\\
SK_2(A)\>\sigma_r^2>> H^5(F,\mu_{d/r}^{\otimes 4})/r[A]\cdot
H^3(F,\mu_{d/r}^{\otimes 3})
\end{CD}\]
commutes, where the vertical maps are cup-product by $\{a\}$ and the
horizontal maps are those of \eqref{eq11} and \eqref{eq11bis}.
\end{prop}

\begin{proof} Since the spectral sequences of \cite[Th. 4.4]{motiv} are
multiplicative, it suffices to chek that the diagram
\[\begin{CD}
SK_1(A)\>\sigma_r^1>> \Ker\xi^{4}_{Y^{[r]}}\\
\V{\cdot \{a\}}VV \V{\cdot \{a\}}VV\\
SK_2(A)\>\sigma_r^2>> \Ker\xi^{5}_{Y^{[r]}}
\end{CD}\]
commutes. This in turn reduces to the compatibility of the BGQ spectral
sequence and the isomorphisms \eqref{eq9} with products.\end{proof}

Similarly:

\begin{prop}\label{p8.1} Let $A$ be a discrete valuation $F$-algebra, with quotient
field $K$ and residue field $E$. Then the diagrams
\[\begin{CD}
SK_2(A_K)\>\sigma_r^2>> H^5(K,\mu_{d/r}^{\otimes 4})/r[A]\cdot
H^3(K,\mu_{d/r}^{\otimes 3})\\
\V{\partial}VV \V{\partial}VV\\
SK_1(A_E)\>\sigma_r^1>> H^4(E,\mu_{d/r}^{\otimes 3})/r[A]\cdot
H^2(E,\mu_{d/r}^{\otimes 2})
\end{CD}\]
commutes, where the homomorphisms $\partial$ are induced by the residue
maps in $K$-theory and Galois cohomology respectively.
\end{prop}

\begin{proof} Similar.\end{proof}

Using Corollary \ref{c8.3} b), Proposition \ref{p5} and Proposition \ref{p8.1}, we find that $\sigma_1^2$ and $\sigma_2^2$ are nontrivial when $4\mid \ind(A)$.

\subsection{A refinement}\label{ref}

In this subsection, where we keep the previous notation, we assume that $A$ is a division algebra,
$d$ is a power of a prime $l$ and
$r[A]=0$: for $r$ strictly dividing $d$, this is possible if and only if
the exponent $\epsilon$ of $A$ is smaller than $d$ (and then we may
choose for $r$ any $l$-power between $\epsilon$ and $d/l$). Then we can
compute $K_1(X)^{(1/2)}$ and extend the map 
\[SK_i(A)\to K_i(X)^{(2)}\]
of the previous section to a map
\[K_i(A)\to K_i(X)^{(2)}.\]

This approach corresponds to that of Rost in the case where $A$ is a
biquaternion algebra \cite{merk}.

Let $H$ be the class of a hyperplane section in
$K_0(Y^{[r]})$.

\begin{prop} For $i\le 2$,\\
a) The composition
\[K_i(F)\>\cdot H>> K_i(Y^{[r]})^{(1)}\to
H^1(Y^{[r]},\sK_{i+1})\>\xi^{1,i+1}>> K_i(F)\]
is the identity.\\
b) The induced map
\[K_i(F)\to K_i(Y^{[r]})^{(1/2)}\]
is an isomorphism.\\
c) Let $\sJ$ be the tautological bundle on $Y^{[r]}$. Then the image of the map
\begin{align*}
\Phi^{[r]}:K_i(A)&\to K_i(Y^{[r]})^{(1)}\\
x&\mapsto \tilde u_\sJ(x)-\Nrd(x)\cdot H
\end{align*}
(see Proposition \ref{p7.1} b)) sits into $K_i(X)^{(2)}$.
\end{prop} 

\begin{proof} By Lemma \ref{l8.3}, the map
\[CH^1(Y^{[r]})\to CH^1(Y^{[r]}_s)\]
is bijective. In particular, $c_1(H)=h$ in $CH^1(Y^{[r]}_s)$. We then get
a) by multiplicativity. b) follows
from a) and the fact that the maps
\[K_i(Y^{[r]})^{(1/2)}\to H^1(Y^{[r]},\sK_{i+1})\>\xi^{1,i+1}>> K_i(F)\]
are injective. c) follows immediately from a).\end{proof}

\subsection{The comparison issue}\label{ss:comp} For $s\mid r\mid d$,
let $Y^{[r,s]}=SB(r,s,A)$ be as in \ref{ss:twisted} with the two projections
\[\begin{CDS}
&&Y^{[r,s]}\\
&\SW{}W{p_r}W& \SE{}E{p_s}E\\
Y^{[r]}&&&& Y^{[s]}.
\end{CDS}\]

We have corresponding diagrams ($i=1,2$)
\[\begin{CDS}
&&&&\Ker\eta^{i+3}_{Y^{[r]}}\\
&&\NE{\sigma^i_r}EE&&\SE{p_r^*}EE\\ 
SK_i(A)&&&&&&\Ker\eta^{i+3}_{Y^{[r,s]}}\\
&&\SE{\sigma^i_s}EE&&\NE{p_s^*}EE\\
&&&& \Ker\eta^{i+3}_{Y^{[s]}}
\end{CDS}
\]

The comparison issue is to know whether this diagram commutes: if this is the case, then the
maps $\sigma^i_r$ and $\sigma^i_s$ are compatible in an obvious sense thanks to Corollary
\ref{c8.1} b). In view of Theorem \ref{t6} c), this commutation is equivalent to the
commutation of the diagram
 \[\begin{CDS}
&&&&K_i(Y^{[r]})^{(2)}\\
&&\NE{}E{u_{\sJ_r}}E&&\SE{p_r^*}EE\\ 
SK_i(A)&&&&&&K_i(Y^{[r,s]})^{(2)}\\
&&\SE{}E{u_{\sJ_s}}E&&\NE{p_s^*}EE\\
&&&& K_i(Y^{[s]})^{(2)}
\end{CDS}\]
or to the vanishing of the map
\[u_{\sJ_r}-u_{\sJ_s}:SK_i(A)\to K_i(Y^{[r,s]})^{(2)}.\]

We may also consider the sheaf $\sI_{r,s} = \Ker(\sI_r\to \sI_s)$; then the above amounts to
the vanishing of the map
\[u_{\sI_{r,s}}:K_i(A)\to K_i(Y^{[r,s]})\]
on the subgroup $SK_i(A)$. In \cite[Th. 4]{suslin4}, Suslin obtains this commutation (or
vanishing) for $(s,r,d)=(1,2,4)$ in a very sophisticated and roundabout way. I have no idea how
to prove it in general.


\section{Motivic cohomology of some Severi-Brauer varieties}\label{klplus1}

In this section, unlike in the rest of the paper, we write $H^*(X,\Z(n))$ (\resp
$H^*_\et(X,\Z(n))$ for motivic cohomology of some smooth variety $X$ computed in the Nisnevich
(\resp \'etale) topology.

\begin{thm} Let $A$ have prime index $l$, and let $X$ be its Severi-Brauer variety. Let $\Z_A$ be
the Nisnevich sheaf with transfers defined in \cite[5.3]{kl}. Let $n\ge 0$, and assume the
Bloch-Kato conjecture in degrees $\le n+1$.  Then:\\  
a) There is an exact sequence
\begin{multline*}
0\to H^n(F,\Z_A(n))\by{\Nrd} H^n(F,\Z(n))\by{\cdot [A]} H^{n+3}_\et(F,\Z(n+1))\\
\to H^0(X,\sH^{n+3}_\et(\Z(n+1)))\to 0.
\end{multline*}
b) There is a cross of exact sequences
\[\begin{CD}
&&&& 0\\
&&&&\V{}VV\\
&&&& H^1(X,\sH^{n+3}_\et(\Z(n+1)))\\
&&&&\V{}VV\\
0&\to& H^{n+4}_\et(F,\Z(n+1))&\to& H^{n+4}(X,\bar \Z(n+1))&\to& H^0(X,\sH^{n+2}_\et(\Z(n))\\
&&&&\V{}VV \V{\cdot [A]}VV\\
&&&& H^0(X,\sH^{n+4}_\et(\Z(n+1)))&& H^{n+5}_\et(F,\Z(n+1))
\end{CD}\]
where $\bar \Z(n)$ is the cone of the morphism $\Z(n)\to R\alpha_*\alpha^*\Z(n)$, with $\alpha$
the projection of the big \'etale site onto the big Nisnevich site.
\end{thm}

\begin{proof} This is an extension of \cite[Th. 8.1.4 and 8.2.2]{kl}, and it is proven by the
same method. The exact sequence of a) is part 2 of Theorem 8.1.4 of loc. cit. (where the
differential is identified with the cup-product with $[A]$ in 8.2), except that in \cite[Th.
8.1.4 (2)]{kl}, the last term is $H^{n+3}_\et(F(X),\Z(n+3))$ and there is no surjectivity
claimed.

To prove a) and b) we look at the spectral sequence (8.1.4) of \cite{kl}. Let $d=\dim X (=l-1)$. In
the proof of Theorem 8.1.4 and in 8.2, the following was established:
\begin{itemize}
\item $E_2^{p,q}=0$ for $-q\notin [0,d]$, $p<d-1$, $p=d$ or $(p,q)=(d-1,-d)$.
\item The differential 
\begin{multline*}
d_2:\Coker(H^n(F,\Z_A(n))\to H^n_\et(F,\Z(n)))\simeq E_2^{d-1,1-d}\\
\to
E_2^{d+1,-d}\simeq H^{n+3}_\et(F,\Z(n+1))
\end{multline*} 
is injective, and induced by the cup-product $H^n_\et(F,\Z(n))\by{\cdot
[A]}H^{n+3}_\et(F,\Z(n+1))$.
\end{itemize}

The abutment of this spectral sequence on the diagonal $p+q=N$ is
\[\Hom(\Z(d)[2d],\bar M(X)(n+1)[n+2+N])\]
computed in $DM^\eff(F)$, where
\[\bar M(X) = cone(M(X)\to R\alpha_*\alpha^* M(X)).\]

Note that $\bar M(X)(n+1)\simeq M(X)\otimes \bar \Z(n+1)$ (by a projection formula). Hence the
abutment may be rewritten (by Poincar\'e duality)
\[H^{n+2+N}(X,\bar \Z(n+1)).\]

The Bloch-Kato conjecture in degree $n+1$ identifies $\bar \Z(n+1)$ with
$\tau_{>n+2}(R\alpha_*\alpha^*\Z(n))$. The hypercohomology spectral sequence then gives
\begin{align*}
H^{n+2+N}(X,\bar \Z(n+1))&=0 \text{ for } N\le 0\\
H^{n+3}(X,\bar \Z(n+1))&\simeq H^0(X,\sH_\et^{n+3}(\Z(n+1)))\\
\end{align*}
and for $N=2$ an exact sequence
\begin{multline*}
0\to H^1(X,\sH^{n+3}_\et(\Z(n+1)))\to H^{n+4}(X,\bar \Z(n+1))\\
\to H^0(X,\sH^{n+4}_\et(\Z(n+1))).
\end{multline*}

Consider the differentials $d_2^{d-1,q}:E_2^{d-1,q}\to E_2^{d+1,q-1}$ for $-q\le d-1$. We have
\[E_2^{p,q} = \Hom(\Z,\bar\Z_{A^{\otimes(-q+1)}}(n+1-d-q)[n+2-2d+p-q])\]
where $\bar\Z_{A^{\otimes(-q+1)}}=cone(\Z_{A^{\otimes(-q+1)}}\to
R\alpha_*\alpha^*\Z_{A^{\otimes(-q+1)}})$. Therefore
\begin{multline*}
E_2^{d-1,q}=\Hom(\Z,\bar\Z_{A^{\otimes(-q+1)}}(n+1-d-q)[n+1-d-q])\\=
\Coker(H^{n+1-d-q}(F,\Z_A(n+1-d-q))\to H^{n+1-d-q}(F,\Z(n+1-d-q)))
\end{multline*}
and
\begin{multline*}
E_2^{d+1,q-1}=\Hom(\Z,\bar\Z_{A^{\otimes(-q+2)}}(n+2-d-q)[n+4-d-q])\\=
H^{n+4-d-q}_\et(F,\Z(n+2-d-q))).
\end{multline*}

The computation of \cite[8.2]{kl} identifies $d_2^{d-1,q}$ with the map induced by cup-product by
$[A]$. By the above, we get that $d_2^{d-1,q}$ is \emph{injective}. The computation of
\cite[8.2]{kl} also identifies $d_2^{d+1,q-1}$ with the cup-product by
$[A]$. This gives both a) and b).
\end{proof}

\section{\'Etale motivic cohomology of reductive groups}\label{s:red}


\subsection{The slice spectral sequence for a reductive group} Let $X$ be a smooth
$F$-variety. There are spectral sequences
\cite[(3.1), (3.2)]{hk}, similar to those of Theorem \ref{t4.1}:
\begin{equation}\label{eq5Nis}
E_2^{p,q}(X,n)_\Nis=\Hom_{DM_{-}^\eff(F)}(c_q(X),
\Z(n-q)[p-q])\Rightarrow H^{p+q}_\Nis(X,\Z(n))
\end{equation}  
\begin{equation}\label{eq5}
E_2^{p,q}(X,n)_\et=\Hom_{DM_{-,\et}^\eff(F)}(\alpha^*c_q(X),
\Z(n-q)[p-q])\Rightarrow H^{p+q}_\et(X,\Z(n))
\end{equation}  
where  $c_q(X)$ are complexes of
Nisnevich sheaves with transfers associated to $X$ (canonically in the
derived category) and $\alpha$ is the projection from the \'etale site of smooth
$F$-varieties to the Nisnevich site. These spectral sequences have the same formal
properties as \eqref{e3.1}: transfers, and  products if the motive of $X$ is mixed Tate (\resp
geometrically mixed Tate), \cf discussion in the proof of Th. \ref{t4.1} (ii).

Let $X=G$ be a connected
reductive group over $F$, with maximal torus $T$ defined over $F$. Set $Y=G/T$. Assume first
$G$ and $T$ split.  In \cite[Prop. 9.3]{hk}, it was shown that $c_q(G)$ is dual, in the derived
category, to the complex of constant Nisnevich sheaves 
$c^q(G)$ (denoted by $K(G,q)$ in \emph{loc. cit.}) given by
\begin{multline}\label{eq2}
0\to \Lambda^q(T^*)\to \Lambda^{q-1}(T^*)\otimes CH^1(Y)\to\dots\\
\dots\to T^*\otimes CH^{q-1}(Y)\to CH^q(Y)\to 0
\end{multline}
in which $T^*$ is the group of characters of $T$, $CH^q(Y)$ is in
degree $0$ and the maps are induced by intersection products and the
characteristic map $\gamma:T^*\to CH^1(X)$ (compare
\cite[3.14]{eklv}). Thus \eqref{eq5Nis} may be rewritten in this case as
\[E_2^{p,q}(G,n)_\Nis=H^{p-q}_\Nis(F,c^q(G)\otimes
\Z(n-q))\Rightarrow H^{p+q}_\Nis(G,\Z(n)).\]

Since $c^q(G)$ is concentrated in degrees $\le 0$, $c^q(G)\otimes
\Z(n-q)$ is concentrated in degrees $\le n-q$ and $E_2^{p,q}(G,n)_\Nis=0$ for $p>n$. We also
have $E_2^{p,q}(G,n)_\Nis=0$ for $q>n$, since $\Z(n-q)=0$ in this case. For $(p,q)=(n,n)$ this
yields

\begin{lemma}[\cf Grothendieck \protect{\cite[p. 21, Rem. 2]{grothendieck}}] \label{lgroth} If
$G$ is split, we have isomorphisms $E_2^{n,n}(G,n)_\Nis \simeq E_\infty^{p,q}(G,n)_\Nis \simeq
H^{2n}(G,\Z(n))$, hence an exact sequence
\[T^*\otimes CH^{n-1}(Y)\to CH^n(Y)\to CH^n(G)\to 0.\]
\end{lemma}

We shall also use:

\begin{lemma} \label{lgroth2} Suppose $G$ split, simply connected and absolutely simple. Then,
for all $n> 0$, $CH^n(G)$ is killed by $(n-1)!$ and by the torsion index $t_G$ of $G$
\cite[\S 5]{demazure}. In particular, $CH^i(G)=0$ for $i=1, 2$. If $G$ is of type $A_r$ or
$C_r$, $CH^i(G)=0$ for all $i>0$.
\end{lemma}

\begin{proof} The first fact follows from $K_0(G)=\Z$, \cf \cite[Proof of Prop. 3.20
(iii)]{eklv}. For the second one, Demazure proves in \cite[Prop. 5]{demazure} that the
cokernels of the characteristic maps $\gamma^n:\bfS^n(T^*)\to CH^n(Y)$ are killed by $t_G$: the
claim then follows from Lemma \ref{lgroth} and a small diagram chase. The last fact follows
from \cite[Lemme 5]{demazure}, which says that $t_G=1$ for $G$ of type $A_r$ or $C_r$. (This
also follows from Suslin \cite[Th. 2.7 and 2.12]{suslin3}.)
\end{proof}

We now relax the assumption that $G$ is split, and would like to study the spectral sequences
\eqref{eq5}. If we knew that 
\begin{equation}\label{dualet}
\alpha^*c_q(G)\simeq c_q(G_s)
\end{equation} 
in the derived category of  complexes of \'etale sheaves (or $G_F$-modules),
this would allow us to rewrite \eqref{eq5} in the form
\[E_2^{p,q}(G,n)_\et=H^{p-q}_\et(F,c^q(G_s)\otimes
\Z(n-q))\Rightarrow H^{p+q}_\et(G,\Z(n))\]
as for the split case, in the Nisnevich topology.

I don't know how to prove \eqref{dualet}, except in the trivial case where the cohomology of
$c^p(G_s)$ is concentrated in at most one degree. We shall therefore make-do with \eqref{eq5}
and be saved by the fact that, for low values of $q$ and for the groups $G$ we are interested
in, the latter fact is true. For simplicity, we shall write
\[\Hom_{DM_{-,\et}^\eff(F)}(\alpha^*c_q(G),
\Z(n-q)[p-q])=\Ext^{p-q}_\et(\alpha^*c_q(G),
\Z(n-q)).\]

We always have $c^0(G_s)=CH^0(Y_s)=\Z^{\pi_0(G_s)}$. Suppose that $G$ is
semi-simple, simply connected. Then
$c$ is bijective and one finds \cite{eklv}
\begin{align}
\label{eq3}c^1(G_s)&=0\\
\label{eq4}c^2(G_s)&=\bfS^2(T_s^*)^W[1]
\end{align}
where $W$ is the Weyl group of $G_s$. If $G$ is absolutely simple, then
$\rank \bfS^2(T_s^*)^W=1$ (with trivial Galois action).

We
note that the unit section of $G$ splits off from
\eqref{eq5} spectral sequences
\[\tilde E_2^{p,q}(G,n)\Rightarrow \tilde H^{p+q}_\et(G,\Z(n))\]
with 
\[\tilde E_2^{p,q}(G,n)=
\begin{cases}
\Ext^{p-q}_\et(\alpha^*c_q(G),
\Z(n-q))&\text{for $q>0$}\\
0&\text{for $q=0$}
\end{cases}\]
and $H^{p+q}_\et(G,\Z(n))=H^{p+q}_\et(F,\Z(n))\oplus \tilde
H^{p+q}_\et(G,\Z(n))$ via the unit section. These spectral sequences are
modules over \eqref{eq5}.

From the above spectral sequence in weight $3$, the corresponding
coniveau spectral sequence, \eqref{eq3} and \eqref{eq4}, we get a
commutative diagram analogous to \eqref{eq14}:
\begin{equation}\label{eq6}
\begin{CD}
&&&&0\\
&&&&\V{}VV\\
0&\to& H^2(G,\sK_3^M)\>>> \tilde H^5(G,\Z(3))&\to&
\tilde H^0(G,\sH^4(3))\\
&&&&\V{}VV\V{\alpha}VV\\ 
&&&&\Ext^{-1}_\et(\alpha^*c_3(G),\Z)&&CH^3(G)\\
&&&&\V{\tilde d_2^{2,3}(G,3)}VV\V{}VV\\
&&H^2(F,\bG_m\otimes
\bfS^2(T_s^*)^W) \>\sim>> \Ext^{2}_\et(\alpha^*c_2(G),\Z(1))&& \tilde H^6(G,\Z(3)) \\
&&&&\V{}VV\\
&&&&\tilde H^6(G,\Z(3))\\
&&&&\V{}VV\\
&&&&\Ext^{0}_\et(\alpha^*c_3(G),\Z)
\end{CD}
\end{equation}

In this diagram, the column and the row forking downwards are both exact.
The groups marked with a $\tilde{}$\ are, as above, the direct summands of
the corresponding groups without a $\tilde{}$\ defined by the unit section
of
$G$.

\subsection{An invariant computation}
In this subsection, we want to compute $\bfS^3(T_s^*)^W$ when $G$ is 
absolutely simple simply connected. We start with the case of type $A_r$. It is
then convenient to think of $G_s$ as $\SL_{r+1}$ embedded into $\GL_{r+1}$.
The maximal torus $T_s$ of $G_s$ is then a subtorus of a maximal
torus $S$ of $\GL_{r+1}$, conjugate to its canonical maximal subtorus. The
character group $S^*$ is free of rank
$l+1$, with basis $(e_1,\dots,e_{r+1})$, and $T_s^*$ is the
quotient of
$S^*$ by $\Z\sigma_1$, with $\sigma_1=\sum e_i$. 

The Weyl group $W$ of $G_s$ coincides with that of $\GL_{r+1}$; it is
isomorphic to $\fS_{r+1}$ and permutes the $e_i$. Let $\sigma_i$
be the $i$-th symmetric function in the $e_i$: by the symmetric
functions theorem, we have
\[\bfS(S^*)^W=\Z[\sigma_1,\dots,\sigma_{r+1}].\]

It is clear that the sequence
\begin{equation}\label{eq15}
0\to \sigma_1 \bfS(S^*)\to \bfS(S^*)\to \bfS(T_s^*)\to 0
\end{equation}
is exact.

\begin{lemma}\label{l10} If $r\ge 2$, the map $\bfS^3(S^*)^W\to
\bfS^3(T_s^*)^W$ is surjective; $\bfS^3(T_s^*)^W$ is free of rank $1$,
with basis the image $\bar\sigma_3$ of $\sigma_3$. If $r=1$,
$\bfS^3(T_s^*)^W=0$.
\end{lemma}

\begin{proof} Suppose first $r\ge 2$. In view of \eqref{eq15}, for the first
assertion it suffices to check that $H^1(W,\bfS^2(S^*))=0$. A basis of
$\bfS^2(S^*)$ is given by
$(e_1^2,\dots,e_{r+1}^2,e_1e_2,\dots)$. The
group $W$ permutes the squares and the rectangular products transitively;
the isotropy group of $e_1^2$ is $\fS_r$ while the isotropy group
of $e_1e_2$ is $\fS_{r-1}$. By Shapiro lemma, we get
\[H^1(W,\bfS^2(S^*))\simeq H^1(\fS_r,\Z)\oplus H^1(\fS_{r-1},\Z)=0.\]

For the second assertion, we use \eqref{eq15} again and get an exact
sequence (thanks to the symmetric functions theorem)
\[0\to \sigma_1 \langle \sigma_1^2,\sigma_2\rangle\to \langle
\sigma_1^3,\sigma_1\sigma_2,\sigma_3\rangle\to \bfS(T_s^*)^W\to 0.\]

If $r=1$, the same calculation gives the result.\end{proof}

In the other cases, an application of the theory of exponents \cite[V.6.2, Prop. 3 and tables of Ch. VI]{bbkilie} gives

\begin{lemma}\label{l10bis} If $G$ is not of type $A_r$, 
$\bfS^3(T_s^*)^W=0$.\qed
\end{lemma}


\subsection{Some facts about the $c^q(G_s)$} Part a) of the following theorem is a version of S.
Gille's theorem \cite[th. 1.5]{gille}\footnote{For $q=3$ and $G$ of type $A_r$, it was obtained
in 2001/2002. The general case was inspired by Gille's work.}:

\begin{thm}\label{t5} Let $G$ be semi-simple and simply connected. Then:\\
a) For $q\ge 3$, $H^r(c^q(G_s))=0$ for $r=-q,-q+1$, and $H^{-q+2}(c^q(G_s))$ is torsion-free.\\
b) Suppose $G$ simple. For $q=3$, $H^{-1}(c^3(G_s))\simeq \bfS^3(T_s^*)^W$ and
$H^0(c^3(G_s))\allowbreak \simeq CH^3(G_s)$.\\ 
c) If $G$ is simple of type  $A_r$, with $r\ge 2$, then $c^3(G_s)\simeq \Z(\chi)[1]$,
generated by
$\bar\sigma_3$ (see Lemma \ref{l10}) where $\chi$ is the (quadratic) character of
$G_F$ corresponding to its (possibly trivial) outer action on the Dynkin
diagram of $G$. If $G$ is of type $A_1$, $c^3(G_s)=0$. If $G$ is not of type $A_r$, $c^3(G_s) =
CH^3(G_s)[0]$.
\end{thm}

\begin{proof} a) For two split reductive groups $G,H$ and $n\ge 0$, we have the K\"unneth formula
\begin{equation}\label{eq13.1}
c^n(G\times H)\simeq \bigoplus_{p+q=n} c^p(G)\oo^L c^q(H)
\end{equation}
in the derived category \cite[Lemma 4.8]{hk}, since $M(G)$ and $M(H)$ are mixed Tate motives.
Thus we may assume $G$ to be simple. Consider now the commutative diagram
\[
\begin{CD}
&&&& \s \Lambda^{q-2}(T_s^*)\otimes \bfS^2(T_s^*)^W&\overset{e}{\to}& \s 
\Lambda^{q-3}(T_s^*)\otimes \bfS^2(T_s^*)^W\otimes T_s^*\\ 
&&&&\V{}VV \V{f}VV\\
\s\Lambda^q(T_s^*) &\to& \s \Lambda^{q-1}(T_s^*)\otimes T_s^* &\to& \s
\Lambda^{q-2}(T_s^*)\otimes \bfS^2(T_s^*) &\to&
\s \Lambda^{q-3}(T_s^*)\otimes \bfS^3(T_s^*)\\
\V{}V{||}V \V{1\otimes \gamma}V{\wr}V \V{1\otimes \gamma^2}VV \V{1\otimes \gamma^3}VV\\
\s\Lambda^q(T_s^*) &\to& \s \Lambda^{q-1}(T_s^*)\otimes CH^1(Y_s) &\to& \s 
\Lambda^{q-2}(T_s^*)\otimes CH^2(Y_s) &\to&\s \Lambda^{q-3}(T_s^*)\otimes CH^3(Y_s)
\end{CD}
\]
where the bottom row is the beginning of $c^q(G_s)$, the middle row is the $q$-th Koszul
complex for $T_s^*$, $\gamma^i$ are induced by the characteristic map, the top row is
$\bfS^2(T_s^*)^W$ tensored with the beginning of the $(q-2)$-nd Koszul complex for $T_s^*$, the
left column is obtained by tensoring the exact sequence of free abelian groups
\[0\to \bfS^2(T_s^*)^W\to \bfS^2(T_s^*)\to CH^2(Y_s)\to 0\]
with $\Lambda^{q-3}(T_s^*)$ and, finally, $f$ is induced by the product
$\bfS^2(T_s^*)^W\otimes T_s^*\to
\bfS^3(T_s^*)$. The middle row is universally exact as the Koszul complex of a free module,
and the middle column is (split) short exact. 

Since $G$ is simple, $\bfS^2(T_s^*)^W$ is a rank $1$ direct summand of $\bfS^2(T_s^*)$, which
implies that
$f$ is injective and remains so after tensoring with $\Z/m$ for any $m$. The same is true for
$e$ by the acyclicity of Koszul complexes. A diagram chase then gives the result.

b) For $q=3$, let us rewrite part of the above diagram, for clarity:
\[\begin{CD}
0&\to& \Lambda^3(T_s^*)&\to& \Lambda^{2}(T_s^*)\otimes T_s^*&\to&
 T_s^*\otimes \bfS^2(T_s^*)&\to& \bfS^3(T_s^*)&\to& 0\\
&&\V{||}VV\V{1\otimes \gamma}VV\V{1\otimes \gamma^2}VV\V{\gamma^3}VV\\
0&\to& \Lambda^3(T_s^*)&\to& \Lambda^{2}(T_s^*)\otimes CH^1(Y_s)&\to&
 T_s^*\otimes CH^{2}(Y_s)&\to& CH^3(Y_s)&\to& 0.
\end{CD}\]

The two left vertical maps are isomorphisms; by \eqref{eq4}, $1\otimes
\gamma^2$ is surjective, with kernel $T_s^*\otimes \bfS^2(T_s^*)^W$; also, by  \cite[p. 292,
Cor. 2]{demazure} $\Ker \gamma^3$ is the $\Q$-span of $T_s^*
\bfS^2(T_s^*)^W+\bfS^3(T_s^*)^W$ in
$\bfS^3(T_s)^*$. Using Lemma \ref{lgroth}, tt follows that
\[H^i(c^3(G_s))=
\begin{cases}
0&\text{for $i=-3$}\\
\Ker\phi&
\text{for $i=-2$}\\
\Coker\phi&
\text{for $i=-1$}\\
CH^3(G_s)&\text{for $i=0$}
\end{cases}\]
where $\phi$ is the map
\[T_s^*\otimes \bfS^2(T_s^*)^W\to \langle T_s^* \bfS^2(T_s^*)^W+\bfS^3(T_s^*)^W\rangle_\Q,\]
$\langle - \rangle_\Q$ denoting the $\Q$-span. We have seen in a) that $\Ker\phi=0$ and
$\Coker\phi$ is torsion-free.  We may factor $\phi$ as a composition
\[T_s^*\otimes \bfS^2(T_s^*)^W\by{\tilde \phi}T_s^*
\bfS^2(T_s^*)^W+\bfS^3(T_s^*)^W\Inj \langle T_s^*
\bfS^2(T_s^*)^W+\bfS^3(T_s^*)^W\rangle_\Q.\]

Thus $\Coker\phi$ is an
extension of the finite group 
\[\frac{\langle T_s^*
\bfS^2(T_s^*)^W+\bfS^3(T_s^*)^W\rangle_\Q}{T_s^*
\bfS^2(T_s^*)^W+\bfS^3(T_s^*)^W}\] 
by a group isomorphic to $\bfS^3(T_s^*)^W/\bfS^3(T_s^*)^W\cap T_s^*
\bfS^2(T_s^*)^W$; but
\[\bfS^3(T_s^*)^W\cap T_s^* \bfS^2(T_s^*)^W\subseteq (T_s^*
\bfS^2(T_s^*)^W)^W= T_s^{*W}
\bfS^2(T_s^*)^W=0.\]

Thus, the map $\bfS^3(T_s^*)^W\to \Coker \tilde\phi$ is bijective. To conclude, we use the fact
that
$\bfS^3(T_s^*)^W$ is pure in $\bfS^3(T_s^*)$ (the quotient is torsion-free), which follows
from Lemmas \ref{l10} and \ref{l10bis}: since $\Coker
\phi$ is torsion-free, this implies that it is isomorphic to $\bfS^3(T_s^*)^W$.

c) now follows from b), Lemmas \ref{l10}, \ref{l10bis} and \ref{lgroth2}. For $G$ of type
$A_r$ with $r\ge 2$, the claim on the Galois action follows from the well-known fact that the
nontrivial outer automorphism of the Dynkin diagram of $G_s$ maps $\bar e_i$ to $-\bar
e_{r+1-i}$, where $\bar e_i$ is the image of $e_i$ in $T_s^*$.
\end{proof}

Here is a complement to Theorem \ref{t5}:

\begin{lemma}\label{l16} Let $r\ge 2$, and consider the embedding $\iota:\SL_{r+1}\inj
\SL_{r+2}$ given by
$u\mapsto \left(\begin{smallmatrix} u&0\\0&1\end{smallmatrix}\right)$. Then the induced
morphism
$\iota^*:c^i(\SL_{r+2})\to c^i(\SL_{r+1})$ is a quasi-isomorphism for $i=2,3$. 
\end{lemma}

\begin{proof} Let $T_{r+1},T_{r+2}$ be the diagonal tori of $\SL_{r+1}$ and $\SL_{r+2}$
respectively. It suffices to check that $\bfS^i(T_{r+2}^*)^{\fS_{r+2}}\iso
\bfS^i(T_{r+1}^*)^{\fS_{r+1}}$ for $i=2,3$. This follows from the computations in the proof of
Lemma \ref{l10}.
\end{proof}

\begin{rk}  For $G$ of type $C_r$, $CH^i(G_s)=0$ for all $i>0$, and for general $G$,
$CH^3(G_s)$ is a $2$-torsion group (see Lemma \ref{lgroth2}). Marlin computed $CH^*(G_s)$ for
$G$ of type $B_r,D_r,G_2$ or $F_4$ in
\cite{marlin}: he finds $CH^3(G_s)=\Z/2$ in each case. I don't know the value of $CH^3(G_s)$
for $G$ of type $E_6,E_7,E_8$: is it also $\Z/2$?
\end{rk}

\section{The generic element}\label{generic}

In this section, we prove Theorem \ref{tSL}, see \eqref{eq:xgen}, \eqref{eq:gen} and Theorem
\ref{t11.2}.


\subsection{The cohomological generic element}\label{ss:generic} Let $G$ be an
absolutely simple simply connected group of type
$A_r$.  From Theorem \ref{t5} and Diagram \eqref{eq6}, we first deduce:

\begin{cor} If $r=1$ or if $G$ is of outer type, we have 
$H^2(G,\sK_3^M)=\tilde H^5(G,\Z(3))=0$; the group $H^0(G,\sH^4(3))$ is at
most $\Z/2$ and is isomorphic to the kernel of the \'etale motivic cycle
map $CH^3(G)\to H^6(G,\Z(3))$.
\end{cor}

\begin{proof} All claims follow from the diagram and the fact that we have
$H^{-1}(F,c^3(G_s))=0$ in these cases, except for the second one (note
that obviously 
\[\Ker(CH^3(G)\to H^6(G,\Z(3))=\Ker(CH^3(G)\to \tilde
H^6(G,\Z(3))).\] 

The latter follows from Theorem \ref{t5}.\end{proof}


\begin{prop}\label{p4} If $G$ is inner, the map $\alpha$ in Diagram
\eqref{eq6} is $0$.
\end{prop}

\begin{proof} We have $G=\SL_1(A)$ for some central simple algebra $A$. If $CH^3(G)=0$, there is
nothing to prove; by Merkurjev \cite[Prop. 4.3]{merk2}, this happens if and only if $\ind(A)$
is odd. Suppose now
$\ind(A)$ even. If $A$ is a quaternion algebra, we have $\tilde H^0(G,\sH^4(3))=0$ by \cite[Lemma
5.1]{merk2}. In general, we proceed as in \cite[proof of Prop. 4.3]{merk2}. Note that $\alpha$
really comes from a map $\alpha':H^0(G,\sH^4(3))\to CH^3(G)$ and that $\alpha=0$ if and only if
$\alpha'=0$. Let $K/F$ be a field extension such that $\ind(A_K)=2$, so that $A_K=M_n(Q)$ for some
quaternion division algebra $Q$ over $K$ and $G_K=\SL_n(Q)$.  Set $H=\SL_1(Q)$ and $X= G_K/H$. By
loc. cit., the generic fibre of the projection $G_K\to X$ is $H_E$, with $E=K(X)$. We then have a
commutative diagram
\[\begin{CD}
H^0(G,\sH^4(3))\>>> H^0(G_K,\sH^4(3))\>>>H^0(H_E,\sH^4(3))\\
\V{\alpha'}VV \V{\alpha'}VV \V{\alpha'=0}VV\\
CH^3(G)\>>> CH^3(G_K)\>>>CH^3(H_E)
\end{CD}\]
and the bottom horizontal maps are isomorphisms by loc. cit. (see \cite[Rk
4.4]{merk2}).\end{proof}


From now on, we suppose $G$ inner and
$r\ge 2$, \ie $\deg(A)>2$ if $G=\SL_1(A)$. Then $H^{-1}(F,c^3(G_s))$ is
canonically isomorphic to
$\Z$, $H^2(F,\bG_m\otimes \bfS^2(T_s^*)^W)\simeq Br(F)$ and $H^{0}(F,c^3(G_s))=0$. For the
reader's  convenience, let us redraw Diagram \eqref{eq6} in this case, taking 
Proposition \ref{p4} into account: 
\begin{equation}\label{eq7}
\begin{CD}
&&&&0\\
&&&&\V{}VV\\
0&\to& H^2(G,\sK_3^M)\>>> \tilde H^5(G,\Z(3))&\to&
\tilde H^0(G,\sH^4(3))&\to& 0\\
&&&&\V{}VV\\ 
&&&&\Z\\
&&&&\V{\tilde d_2^{2,3}(G,3)}VV\\
&&&& Br(F) \\
&&&&\V{}VV\\
&&&&\tilde H^6(G,\Z(3))\\
&&&&\V{}VV\\
&&&&0
\end{CD}
\end{equation}

Since
$H^0(G,\sH^4(3))$ and $Br(F)$ are torsion, we recover Merkurjev's
result that $H^2(G,\sK_3^M)=H^2(G,\sK_3)$ is infinite cyclic
\cite[Lemma 5.7]{merk2}. We also find that $\tilde H^5(G,\Z(3))$ is infinite cyclic and that $\tilde H^0(G,\sH^4(3))$ is cyclic of order $(\tilde H^5(G,\Z(3)):H^2(G,\sK_3^M))$. 

\begin{defn} Let $G=\SL_1(A)$. We denote by $c_A$  the ``positive" generator of $\tilde H^5(G,\Z(3))\subset H^5(G,\Z(3))$, that is, the generator that maps to a positive multiple of $1\in\Z$, and by $\bar c_A$ its image in $\tilde H^0(G,\sH^4(3))\subset H^0(G,\sH^4(3))$ ($\bar c_A$ generates $\tilde H^0(G,\sH^4(3))$).
\end{defn}

\begin{lemma} \label{l12} Let still $G=\SL_1(A)$, and let $p_1,p_2,\mu:G\times_F G\to G$ be repectively the first projection, the second projection and the multiplication map. Then
\[\mu^*c_A=p_1^* c_A +p_2^* c_A.\]
\end{lemma}

\begin{proof} Since $\tilde H^5(G,\Z(3))\to H^{-1}(F,c^3(G_s))$ is injective for
any group $G$, it is sufficient to show that the maps $\mu^*$ and $p_1^*
+p_2^*$ from $c^3(G_s)$ to $c^3(G_s\times_{F_s} G_s)$ are
equal.\footnote{Note that morphisms  between reductive groups preserving the unit sections act on the spectral sequences \eqref{eq5} by
preserving the spectral sequences $\tilde E_r^{p,q}$. This applies to
$\mu$ and to the maps $\iota_1$ and $\iota_2$
further below.}

The K\"unneth formula \eqref{eq13.1}
gives an isomorphism
\[c^3(G_s)\oplus c^3(G_s)\iso c^3(G_s\times_{F_s} G_s)\]
induced by $p_1^*\oplus p_2^*$, since $c^1(G_s)=0$.

Let $C=c^3(G_s)$. The inclusion $\iota_1:G\times\{1\}\to G\times G$
induces a map $\iota_1^*:C\oplus C\to C$; since $p_1\circ \iota_1= Id$
and $p_1\circ \iota_1$ is the trivial map, $\iota_1^*$ is the first
projection. Similarly, $\iota_2:\{1\}\times G\to G\times G$  induces the
second projection. We conclude that $\mu^*: C\to C\oplus C$ is the
diagonal map, using the left and right unit formulas $\mu\circ
\iota_1=\mu\circ\iota_2=Id$.\end{proof}

Let $X$ be a smooth $F$-variety. To any morphism $f:X\to \SL_1(A)$, we
associate the pull-back of $c_A$:
\[c_A(f)=f^* c_A\in H^5(X,\Z(3)).\]

Lemma \ref{l12} shows that we have
\[c_A(fg)=c_A(f)+c_A(g)\]
for two maps $f,g$, where $fg:=\mu\circ(f,g)$.

Consider the embedding $\iota_n:\SL_1(A)\inj \SL_n(A)$ given by $u\mapsto \left(\begin{smallmatrix}
u&0\\0&1\end{smallmatrix}\right)$. Noting that $\SL_n(A)=\SL_1(M_n(A))$, Lemma \ref{l16} shows that
\[c_{M_n(A)}(\iota_n) = c_A.\]

In particular, $\iota_n^*:\tilde H^5(\SL_n(A),\Z(3))\to \tilde H^5(\SL_1(A),\Z(3))$ is an
isomorphism. So, if $f$ is a morphism from $X$ to
$\SL_n(A)$, we may define
$c_A(f)= (\iota_n^*)^{-1}c_{M_n(A)}(f)$, and this definition is ``stable".

In particular, suppose $X=\Spec R$ affine. Then $\Hom_F(X,\SL_n(A))=SL_n(A\otimes_F R)$. Define
$SL(A\otimes_F R)=\colim SL_n(A\otimes_F R)$ as usual, and
\[SK_1(X,A)= SL(A\otimes_F R)^\ab.\]

For $X$ smooth in general, we may similarly define
\[SL(X,A) = \colim\Hom_F(X,\SL_n(A)),\quad SK_1(X,A) = SL(X,A)^\ab.\]

The above discussion then yields a homomorphism
\begin{equation}\label{eq:xgen}
SK_1(X,A)\to H^5(X,\Z(3))
\end{equation}
which is contravariant in $X$.

In particular, for $X=\Spec F$, we get a homomorphism
\begin{equation}\label{eq:gen}
c_A:SK_1(A)\to H^5(F,\Z(3))\osi H^4(F,\Q/\Z(3)).
\end{equation}

The following theorem was (embarrassingly) pointed out by Philippe Gille, whom I thank here.

\begin{thm}\label{t11.2} \eqref{eq:gen} defines the universal invariant of $\SL_1(A)$ of
degree
$4$ with values in $H^4(3)$, in the sense of Merkurjev \cite[Def. 2.1]{merk2}.
\end{thm}

\begin{proof} Let $G=\SL_1(A)$. In view of \cite[Th. 2.3]{merk2}, the only thing which remains
to be proven is that $\tilde H^0(G,\sH^4(3))= A^0(G,H^4(3))_{mult}$ (notation as in
\cite[1.3]{merk2}): this follows from Gersten's conjecture (to identify $H^0(G,\sH^4(3))$ and
$A^0(G,H^4(3))$) and Lemma \ref{l12}.
\end{proof}

\begin{rk}\label{r11.1} Let $r$ be a divisor of $d=\deg(A)$. Let us write $H^4(3)/r[A]$ for the
degree
$4$ part of the cycle module given by
\begin{multline*}
K\mapsto H^n(K,n-1)/r[A]\\:=\Coker(H^{n-2}(K,\mu_r^{\otimes{n-2}})\by{\cdot
r[A]}H^n(K,\Q/\Z(n-1))).
\end{multline*}

It is tempting to conjecture that the map
\[A^0(\SL_1(A),H^4(3))_{mult}\to A^0(\SL_1(A),H^4(3)/r[A])_{mult}\]
is surjective, which would provide a relationship between the invariant $c_A$ and the
invariant $\sigma^1_r$ of Corollary \ref{c8.1}. However, since $A^0(-)_{mult}$ is left exact
rather than right exact, this does not look straightforward at all. A description of the kernel
of cup-product with $r[A]$ seems a major issue to solve.
\end{rk}


\subsection{The $K$-theoretic generic element}
In the universal case $X=\SL_1(A)$, we may write $SK_1(\SL_1(A),A)=SK_1(A)\oplus
\widetilde{SK}_1(\SL_1(A),A)$ using the unit section of $\SL_1(A)$. The induced morphism
\[\widetilde{SK}_1(\SL_1(A),A)\to \tilde H^5(\SL_1(A),\Z(3))\]
is surjective, hence split surjective since $\tilde H^5(\SL_1(A),\Z(3))=\Z$. An explicit
splitting sends $c_A$ to the class of the inclusions $\iota_n:\SL_1(A)\inj \SL_n(A)$.

\begin{lemma}\label{l13} a) For any smooth $F$-variety $Y$, the map
\[H^0(Y,SK_1(\sO_Y\otimes_F A))\to SK_1(F(Y)\otimes_F A)\]
is surjective; the image of $c_{F(Y)\otimes_F A}$ is contained in
$H^0(Y,\sH^4(3))$.\\ 
b) For $Y=\SL_1(A)$ and $K=F(Y)$, the map
$c_{A_K}$ induces a surjection
\[SK_1(A_K)/SK_1(A)\Surj \tilde H^0(\SL_1(A),\sH^4(3)).\]
\end{lemma}

\begin{proof} The first assertion of a) is classical (Rost, \cf
\cite[p. 38]{ctazumaya}), and the second one follows from this and the
construction of
$e_A$. For b), let
$\eta=\Spec K$ be the generic point of $\SL_1(A)$. It defines an element
$\bar
\eta\in SK_1(A_K)$: the \emph{generic element}. By construction, we have
\[c_{A_K}(\bar\eta)=\bar c_A\] 
from which b) follows.
\end{proof}

I don't know if the surjection of Lemma \ref{l13} b) is split. In fact, 

\begin{conj} The surjection of Lemma \ref{l13} b) is an isomorphism.
\end{conj}

\begin{rk} The homomorphism 
\[c_A:\Hom(\SL_1(A),\SL_1(A))\to H^5(\SL_1(A),\Z(3))\] 
also behaves well with respect to composition: for $f\in \Hom(\SL_1(A),\SL_1(A))$, we have
$c_A(f)\in
\tilde H^5(\SL_1(A),\Z(3))$ if and only if $f(1)=1$. If this is the case, set
$c_A(f) = n(f)c_A$. Then, clearly, $n(g\circ f) = n(g)n(f)$. Can one describe this ``degree"
map in a more na\"\i ve fashion?
\end{rk}

\section{Some computations}\label{comp} We now  try and evaluate the groups
$SK_1(A_K)/SK_1(A)$, where
$K$ is the function field of $\SL_1(A)$, and $\tilde H^0(\SL_1(A),\sH^4(3))$: our main results
in this direction are Theorem \ref{t11.1} and Corollary \ref{c11.1}. By \cite[Th. 5.4]{merk2},
the latter group is cyclic of order $2$ when $A$ is a (division) biquaternion algebra, thanks to
Rost's Theorem \ref{tr}. Unfortunately we are not able to prove its nontriviality when
$\ind(A)$ is odd by the present methods.

We assume that $n=\deg(A)$ is of the form $l^m$, $l$ prime.

\subsection{Comparing some quotients}  First
we have

\begin{lemma} \label{l14}
$|\tilde H^0(\SL_1(A),\sH^4(3))|\le \ind(A)/l$.
\end{lemma}

\begin{proof} This follows from Lemma \ref{l13} b) and the fact that
$SK_1(A_K)$  has exponent $\le \ind(A)/l$.\end{proof}

Let $G=\SL_1(A)$. We note the isomorphisms
\begin{align*}
H^2(G,K_3^M)&\iso H^2(G,K_3)\\
K_2(F)&\iso H^0(G,K_2).
\end{align*}

The first one is trivial and the second  one is \cite[Cor. B.3]{eklv}. 
By the second one, the BGQ spectral sequence yields an injection 
\begin{equation}\label{eq16}
K_1(G)^{(2/3)}\Inj H^2(G,K_3).
\end{equation}



\begin{prop}\label{p3} If $G$ is split, with $r\ge 2$, the maps $\tilde
H^5(G,\Z(3))\to\Z$ and $H^2(G,\sK_3^M)\to \tilde H^5(G,\Z(3))$ from 
\eqref{eq7} are both bijective. The same is true of the map \eqref{eq16}.
\end{prop}

\begin{proof} Mixing the coniveau spectral sequence for \emph{Nisnevich} motivic
cohomology with the slice spectral sequence \eqref{eq5Nis} (also for Nisnevich motivic
cohomology) yields a diagram similar to \eqref{eq7} and mapping to it:
\begin{equation}\label{eq6Zar}
\begin{CD}
&&&& H^2(G,\sK_3^M)\>\sim>> \tilde H^5_\Zar(G,\Z(3))\\
&&&&&&\V{\wr}VV\\ 
&&&&&&\Z\\
\end{CD}
\end{equation}
 
This proves the first two claims of Proposition \ref{p3} at once. For 
the last one, we notice that if $G$ is split then all its Chow groups 
are $0$, hence all differentials leaving from $H^2(G,\sK_3)$ in the 
BGQ spectral sequence vanish.\end{proof}

Note that the horizontal map in \eqref{eq6Zar} is an isomorphism for any
$G$, whether split or not.

\begin{cor} \label{c1} In Diagram \eqref{eq7} for $G=\SL_1(A)$, we have 
\[\tilde d_2^{2,3}(G,3)(1)= t[A]\]
for some integer $t$, where $[A]$ is the class of $A$ in $Br(F)$. In
particular, $(\Z: \tilde H^5(G,\Z(3)))$ divides the exponent of $[A]$.
\end{cor}

\begin{proof} Let $K$ be the function field of the Severi-Brauer variety of $A$.
Then $A$ splits over $K$. The first statement now follows from
Proposition \ref{p3} and Amitsur's theorem \cite{amitsur} that
$\Ker(Br(F)\to Br(K))=\langle [A]\rangle$. The second statement is
obvious.\end{proof}

\begin{cor} \label{l11} In general, 
\begin{multline*}
(\Z:H^2(G,K_3^M))=(H^2(G_s,K_3^M):H^2(G,K_3^M))\\
\mid (K_1(G_s)^{(2/3)}:K_1(G)^{(2/3)}).
\end{multline*}
\end{cor}

\begin{proof} This follows immediately from Proposition \ref{p3}.
\end{proof}

The following diagram is a little more precise and may be helpful to the
reader ($G=\SL_1(A)$):
\[\begin{CD}
\displaystyle\frac{SK_1(A_K)}{SK_1(A)}&\Surj& \tilde H^0(G,\sH^4(3))&&
\displaystyle\frac{K_1(G_s)^{(2/3)}}{K_1(G)^{(2/3)}}\\ 
&& \A{\wr}AA \V{\text{onto}}VV\\
0 &\to& \displaystyle\frac{\tilde
H^5(G,\Z(3))}{H^2(G,\sK_3^M)}&\to&\Z/H^2(G,\sK_3^M)&\to&
\Z/t\Z &\to& 0
\end{CD}\]
where $t$ is as in Corollary \ref{c1}.

\subsection{The map $Br(F)\to \tilde H^6(G,\Z(3))$}In order to better understand
the differential
$\tilde d_2^{2,3}(G,3)$, we note:

\begin{prop} Let $G=\SL_1(A)$.\\
a) We have an exact sequence
\[0\to H^1(G,\sH^4(3))\to \tilde H^6(G,\Z(3))/CH^3(G)\to \tilde
H^0(G,\sH^5(3)).\]
b) The composition 
\[Br(F)\to \tilde H^6(G,\Z(3))\to \tilde H^6(G,\Z(3))/CH^3(G)\to \tilde
H^0(G,\sH^5(3))\]
from Diagram \eqref{eq7} is $0$, and so is the map $\tilde H^6(G,\Z(3))/CH^3(G)\to \tilde
H^0(G,\sH^5(3))$. Hence we have in fact an exact sequence
\[0\to CH^3(G)\to \tilde H^6(G,\Z(3))\to H^1(G,\sH^4(3))\to 0.\]
\end{prop}

\begin{proof} a) follows from the coniveau spectral sequence for the
\'etale motivic cohomology of $G$. b) The second vanishing follows from the first, since $Br(F)\to
\tilde H^6(G,\Z(3))$ is surjective. For the first vanishing, given the definition of the
homomorphism
$Br(F)\to
\tilde H^6(G,\Z(3))$, it suffices to show that the map $\alpha^*c_i(V)\to
\alpha^*c_i(G)$ induces $0$ on homology sheaves for $i=1,2,3$ if $V$ is a
suitable open subset $V$ of $G$.

Let $B$ be a Borel subgroup containing $T_s\subset G_s$. Consider the big
cell $\bar U_0\subset G_s/B$: it is an affine space, hence all its Chow
groups are $0$. Observe that $U_0$ is defined over a finite extension of
$F$, hence it has only a finite number of Galois conjugates: then their
intersection $\bar U$ is defined over $F$, and its geometric Chow groups
are still $0$. Let $U$ be the inverse image of $\bar U$ in $Y_s$: then $U$
is defined over $F$ and all its geometric Chow groups are $0$. Hence, for
all $p>0$, the \'etale complex $\alpha^* c_p(U)$ is concentrated in
degrees $<0$.

We now take for $V$ the inverse image of $U$ (viewed as an open subset of
$Y$) in $G$. As in \cite[Prop. 9.3]{hk}, we have for all $N\ge 0$ a
spectral sequence
\[E_1^{p,q}(V_s)=H^q(c_{N-p}(U_s))\otimes
\Lambda^p(T_s^*)\Rightarrow H^{p+q}(c_N(V_s))\]
which maps to the corresponding spectral sequence $E_r^{p,q}(G_s)$ for
$G_s$ (that yields the complexes \eqref{eq2}). For $N>0$, we have
$E_1^{p,q}(G_s)=0$ for $q\ne 0$ and $E_1^{p,q}(V_s)=0$ for $q=0$, hence
all maps
$H^i(c_N(V_s))\to H^i(c_N(G_s))$ are $0$. This completes the proof of b).
\end{proof}

\subsection{A Chern class computation}

\begin{comment}
\begin{rk} There are two conventions in the literature
to index higher Chern classes: that of Soul\'e \protect\cite{soule} where
in $c_{i,j}$ $i$ refers to the weight and $j$ to the cohomology degree,
and that of Gillet \protect\cite{gillet} where in $c_{i,j}$ $i$ still
refers to the weight but $j$ refers to the
$K$-theory degree. Soul\'e's convention is also used by Suslin. However,
here we shall have to compare higher Chern classes with values in
various cohomology theories. If we used the Soul\'e-Suslin convention, we
would have to change indices when we pass from $K$-cohomology to \'etale
or motivic cohomology, as the cohomology degree doubles. Hence we choose
to use the Gillet convention here, as we won't have to change indices
then. This means that Suslin's \'etale Chern classes denoted above by
$c_{2,2}$ and
$c_{3,4}$ will be respectively denoted by $c_{2,2}$ (no change) and
$c_{3,2}$ from now on.
\end{rk}
\end{comment}

We use Gillet's convention for higher Chern classes \cite{gillet}.

\begin{lemma}\label{l7} For a smooth variety $X$, consider the higher
Chern class
\[c_{3,1}:K_1(X)\to H^2(X,\sK_3).\]
Then $2d_2^{0,-2}=0$ and the diagram
\[\begin{CD}
K_1(X)^{(2)}\>c_{3,1}>> H^2(X,\sK_3)\<2<< H^2(X,\sK_3)/d_2^{0,-2}
H^0(X,\sK_2)\\
\V{}VV&&\A{}AA\\
K_1(X)^{(2/3)}&\>\sim >> & E_\infty^{2,-3}\\
\end{CD}\]
commutes, where $d_2^{0,-2}$ and $E_\infty^{2,-3}$ are relative to the BGQ
spectral sequence for $X$.
\end{lemma}

\begin{proof} The BGQ spectral sequence for $X$ may be considered as the coniveau
spectral sequence for $X$ relative to algebraic $K$-theory. For a
given $i\ge 0$, consider the corresponding coniveau spectral sequence
$'E_r^{p,q}$ relative to
$U\mapsto H^*(U,\sK_i)$ (for $U$ running through open subsets of $X$). By
\cite[pp. 239--240]{gillet}, the $i$-th Chern class $C_i$
defines a morphism of spectral sequences $E_r^{p,q}\to\ {}'E_r^{p,q}$ ($r\ge
1$) converging to the higher Chern classes $c_{i,-p-q}:K_{-p-q}(X)\to
H^{p+q+i}(X,\sK_i)$.

The group $'E_1^{p,q}$ is $0$ for $q\ne -i$ and
$'E_1^{p,-i}=\bigoplus_{x\in X^{(p)}}K_{i-p}(F(x))$. Hence
$'E_2^{p,q}=0$ for $q\ne -i$ and
$'E_2^{p,-i}=H^p(X,\sK_i)=\ {}'E_\infty^{p,-i}$. By \cite[Th. 3.9]{gillet},
the map from $E_1^{p,-i}$ to $'E_1^{p,-i}$ induced by $C_i$ equals
$\frac{(-1)^p(i-1)!}{(i-p-1)!}c_{i-p,i-p}$ on each summand
$K_{i-p}(F(x))$. In particular, for $i=3$, $c_{1,1}$ is the identity for
fields and we get a commutative diagram
\[\begin{CD}
E_2^{0,-2}\>d_2^{0,-2}>> E_2^{2,-3}\\
\V{}VV \V{2}VV\\
0\>>> 'E_2^{2,-3}=E_2^{2,-3}
\end{CD}\]
which proves the first claim of the lemma; the second one follows
from the morphism of spectral sequences.
\end{proof}



\subsection{Some computations, continued} The group $H^1(G,\sH^4(3))$ is
mysterious and would require a further analysis: we shall refrain from
starting it in this paper and will concentrate on computing the index
$(K_1(G_s)^{(2/3)}:K_1(G)^{(2/3)})$, which can be done in some
interesting cases.

For this, we may try  and look at
the map
$K_1(G)\to K_1(G_s)$ and use the results of Levine
\cite{levine3} and Suslin \cite{suslin3}. In particular, we have an
isomorphism \cite[Th. 4.3]{levine3}
\[K_1(G)\simeq K_1(F)\oplus \bigoplus_{i=1}^r K_0(A^{\otimes i})\]
where $r=\rank G=\deg A-1$. If $G$ (equivalently $A$) is split, the 
summand
$K_0(A^{\otimes i})\simeq\Z$ is generated by the class of 
$\Lambda^i(\rho_r)$, where $\rho_r$ is the standard representation of 
$G=\SL_{r+1}$ into $\GL_{r+1}$. While Levine thinks of $\rho_r$ as a
representation, Suslin thinks of it as the generic matrix and denotes it
by $\alpha_{r+1}$: the two viewpoints are of course the same.

If we pass to the separable closure, we get a commutative diagram
\[\begin{CD}
K_1(G)^{(2/3)}\>\gamma_3>> H^2(G,\sK_3)\\
\V{}VV \V{}VV\\
K_1(G_s)^{(2/3)}\>\gamma_3>> H^2(G_s,\sK_3)\simeq \Z.
\end{CD}\]

\begin{lemma} \label{l15} Suppose $G=\SL_n$, with $n=r+1$.\\
a) All $[\Lambda^i(\rho_r)]$ belong to $K_1(G)^{(1)}$ and the image of 
$[\Lambda^i(\rho_r)]$ in $H^1(G,\sK_2)\allowbreak=\Z$ is
$\binom{n-2}{i-1}$.\\ 
b) For all $i$, $[\Lambda^i(\rho_r)] -\binom{n-2}{i-1}[\rho_r]\in 
K_1(G)^{(2)}$ and its image in
$H^2(G,\sK_3)=\Z$ is $\binom{n-3}{i-2}$.
\end{lemma}

\begin{proof} (It may not be the most direct, but it works.)  For the first
assertion of a), we need to show that
$[\Lambda^i(\rho_r)]_{| F(SL_n))}\allowbreak=0$ or, which amounts to the
same, that
$\Lambda^i(\alpha_n)$ is a product of commutators, where $\alpha_n$ is the
generic matrix with determinant $1$. For this, it suffices to see that
$\det \Lambda^i(\alpha_n)=1$. But, for any matrix $u$, $\det\Lambda^i(u)$
is a certain power of $\det(u)$, hence the claim.

For the second assertion of a) and for b), we first do a Chern class
computation. Let $\bar\gamma_j=\gamma_j([\rho_r])=\gamma_j([\alpha_n])$,
where $\gamma_j$ is the $j$-th gamma operation in $K$-theory.
Note the formula (\cf \cite[p. 65]{suslin3})
\[\sum[\Lambda^i(\alpha_n)]u^i=\sum \bar\gamma_i u^i(1+u)^{n-i}.
\]

Also, from \cite[1.3.4 a) p.277 and Remark p. 297]{soule2} (see also
\cite[IV.6]{soule}), we find
\[c_{2,1}(\bar\gamma_j)=
\begin{cases}
0&\text{for $j>2$}\\
-c_{2,1}(\alpha_n)&\text{for $j=2$}\\
c_{2,1}(\alpha_n)&\text{for $j=1$}
\end{cases}
\]
and
\[c_{3,1}(\bar\gamma_j)=
\begin{cases}
0&\text{for $j>3$}\\
2c_{3,1}(\alpha_n)&\text{for $j=3$}\\
-3c_{3,1}(\alpha_n)&\text{for $j=2$}\\
c_{3,1}(\alpha_n)&\text{for $j=1$}
\end{cases}
\]
from which we deduce
\begin{multline}\label{eq11.1}
\sum
c_{2,1}([\Lambda^i(\alpha_n)])u^i=
c_{2,1}(\alpha_n)(u(1+u)^{n-1}-u^2(1+u)^{n-2})\\ =
c_{2,1}(\alpha_n)u(1+u)^{n-2}=c_{2,1}(\alpha_n)\sum
\binom{n-2}{i-1}u^i=:c_{2,1}(\alpha_n)\phi(u)
\end{multline}
and
\begin{multline*}
\sum
c_{3,1}([\Lambda^i(\alpha_n)])u^i\\=
c_{3,1}(\alpha_n)(u(1+u)^{n-1}-3u^2(1+u)^{n-2}+2u^3(1+u)^{n-3})\\ =
c_{3,1}(\alpha_n)u(1+u)^{n-3}(1+u)
\end{multline*}
hence
\begin{multline}\label{eq11.2}
\sum
c_{3,1}([\Lambda^i(\alpha_n)])u^i-c_{3,1}(\alpha_n)\phi(u)=
-2c_{3,1}(\alpha_n)u^2(1+u)^{n-3}\\
= -2c_{3,1}(\alpha_n)\sum
\binom{n-3}{i-2}u^i.
\end{multline}

We now use the fact that, for $i\ge 1$, $H^i(\SL_n,\sK_{i+1})$ is generated
by $c_{i+1,1}([\alpha_n])$ \cite[Th. 2.9]{suslin3}. By an analogue
of Lemma \ref{l7}, the edge homomorphism $K_1(X)^{(1)}\to H^1(X,\sK_2)$
of the BGQ spectral sequence coincides with $-c_{2,1}$ for any smooth
variety $X$. With \eqref{eq11.1}, this proves the second part of a) and
the first part of b). Then the second part of b) follows from Lemma
\ref{l7} and \eqref{eq11.2}.\end{proof}

Let $G$ not be necessarily split anymore. Let $e_i$ be the positive 
generator of the summand $K_0(A^{\otimes i})$: $e_i\mapsto
\ind(A^{\otimes i})[\Lambda^i(\alpha_n)]$. Lemma
\ref{l15} shows that 
$\frac{\ind(A)}{\ind(A^{\otimes i})} e_i -\binom{n-2}{i-1} e_1\in 
K_1(G)^{(2)}$ and that its image in $H^2(G_s,\sK_3)=\Z$ is 
$\ind(A)\binom{n-3}{i-2}$.

\begin{lemma}\label{l11.1} $v_l(\binom{n-2}{i-1})=v_l(i)$.
\end{lemma}

\begin{proof} For an integer $m$, let $s_l(m)$ be the sum of the digits
of $m$ written in base $l$. It is well-known that
\[v_l\binom{a}{b}=\frac{s_l(b)+s_l(a-b)-s_l(a)}{p-1}.\]

Clearly, we have $s_l(l^m-2)=m(l-1)-1$. Let $t=v_l(i)$ and write $i-1=\sum
a_j l^j$, with $0\le a_j\le l-1$, $a_j=l-1$ for $j<t$ and $a_t<l-1$.
Then $l^m-i-1=\sum b_jl^j$ with $b_j=l-1$ for $j<t$, $b_t=l-2-a_t$ and
$b_j=l-1-a_j$ for $t<j\le m$. Hence
\begin{multline*}
s_l(i-1)+s_l(l^m-i-1)-s_l(l^m-2)=\\
2t(l-1)+(m-t)(l-1)-1
-(m(l-1)-1)=t(l-1).
\end{multline*}
\end{proof}

\begin{thm} \label{t11.1} We have
\[(K_1(G_s)^{(2/3)}:K_1(G)^{(2/3)})=
\begin{cases}
\inf(l^{2t}\ind(A^{\otimes l^t}))&\text{if $l>2$}\\
\inf(l^{2t-1}\ind(A^{\otimes l^t}))&\text{if $l=2$.}
\end{cases}\]
\end{thm}

\begin{proof} Since the index $(K_1(G_s)^{(2/3)}:K_1(G)^{(2/3)})$ \emph{a
priori} divides $\ind(A)$ (transfer argument), to evaluate it we may
tensor both groups with
$\Z_l$, as well as $H^2(G,\sK_3)$ and
$H^2(G_s,\sK_3)$. Note also that, since $K_1(G)^{(1/2)}\allowbreak\inj
H^1(G,\sK_2)\simeq
\Z$ is torsion-free, $x\in K_1(G)^{(1)}\otimes\Z_l$ and $mx\in
K_1(G)^{(2)}\otimes\Z_l$ for some $m\in\Z_l-\{0\}$ implies $x\in
K_1(G)^{(2)}\otimes\Z_l$. This will allow us to divide freely by
$l$-units below.

With the notation just above Theorem \ref{t11.1}, setting
$n=l^m$, we have by Lemma
\ref{l15} under the composite map
$K_1(G)^{(2/3)}\otimes\Z_l\to K_1(G_s)^{(2/3)}\otimes\Z_l\iso
H^2(G_s,\sK_3)\otimes\Z_l$: 
\begin{equation}\label{eq11.3a}
\frac{l^m}{\ind(A^{\otimes
i})\binom{l^m-2}{i-1}}e_i-e_1\mapsto
\frac{l^m(i-1)}{l^m-2}
\end{equation}
(note that the coefficient of $e_i$ is an $l$-integer by Lemma
\ref{l11.1}).

Let $x=\sum \lambda_i e_i\in K_1(G)^{(2)}\otimes\Z_l$ (with
$\lambda_i\in\Z_l$). In $\Q_l$, write
\[\lambda_i=\mu_i\frac{l^m}{\ind(A^{\otimes i})\binom{l^m-2}{i-1}}\]
so that
\begin{equation}\label{eq11.4}
x=\sum\mu_i\left(\frac{l^m}{\ind(A^{\otimes
i})\binom{l^m-2}{i-1}}e_i-e_1\right)+\sum \mu_i e_1
\end{equation}
hence $x\in K_1(G)^{(2)}\otimes\Z_l$ if and only if $\sum\mu_i=0$. Note
that
\[x\mapsto \sum \mu_i\frac{l^m(i-1)}{l^m-2}=\sum
i\mu_i\frac{l^m}{l^m-2}.\]

Since $v_l(\mu_i)\ge v_l\left(\frac{l^m}{\ind(A^{\otimes
i})\binom{l^m-2}{i-1}}\right)$, we have
\[v_l(i\mu_i \frac{l^m}{l^m-2})
= \begin{cases}
2v_l(i)+v_l(\ind(A^{\otimes i}))&\text{if $l>2$}\\
2v_l(i)+v_l(\ind(A^{\otimes i}))-1&\text{if $l=2$}
\end{cases}\] 
(see Lemma \ref{l11.1}).

This proves the inequality $\ge$ in Theorem \ref{t11.1}. To
get equality, let $s=\inf\{t\mid l^{2t}\ind(A^{\otimes l^t})\text{ is
minimum}\}$. Suppose first that $l>2$. Choose
$\lambda_{l^s}=1$, $\mu_{2l^s}=-\mu_{l^s}$ and $\lambda_i=0$ otherwise,
and we are done.

Suppose now that $l=2$. We can then argue as above by taking
$\mu_{32^s}=-\mu_{2^s}$ provided
$32^s<2^m$, \ie $s\le r- 2$; $s=r$ is clearly impossible and $s=r-1$ may
occur only when $2^{2r-3}\ind(A^{\otimes 2^{r-1}})<2^m$, \ie when
$2^m\ind(A^{\otimes 2^{r-1}})\le 4$. This
means $r=1$ or $r=2$, $\exp(A)=2$. In the first case we clearly have
equality. In the second one we may compute directly
\begin{align*}
2e_2-e_1&\mapsto 2\\
e_3-e_1&\mapsto 4
\end{align*}
which shows that $(K_1(G_s)^{(2/3)}:K_1(G)^{(2/3)})= 2$. So equality
still holds in this case.\end{proof}

\begin{cor}\label{c11.1} 
a) If $\ind(A)=\exp(A)$, then
$(K_1(G_s)^{(2/3)}:K_1(G)^{(2/3)})\allowbreak=\ind(A)$.\\
b) Suppose $\exp(A)=l$. If $l>2$ we have
\[(K_1(G_s)^{(2/3)}:K_1(G)^{(2/3)})=
\begin{cases}
l&\text{if $\ind(A)=l$}\\
 l^2&\text{if $\ind(A)>l$}
\end{cases}\]
while if $l=2$ we always have
$(K_1(G_s)^{(2/3)}:K_1(G)^{(2/3)})\allowbreak=2$.
\end{cor}

\begin{proof} a) is obvious, since in this case necessarily $\ind(A^{\otimes
l^t})=l^{r-t}$ for all $t\le r$. For b), we have
$(K_1(G_s)^{(2/3)}:K_1(G)^{(2/3)})=\inf(\ind(A),l^2)$ (for $l=2$) or
$\inf(\ind(A),2)$ (for $l=2$) and the result immediately follows.
\end{proof}

\begin{rk} An easier computation gives that \break
$(K_1(G_s)^{(1/2)}:K_1(G)^{(1/2)})=\ind(A)$.
\end{rk}

\begin{comment}
 and that $K_1(GL_n)^{(i)}$ is generated by the
$\bar\gamma_j$ for $i+1\le j\le n$, with
$\bar\gamma_{i+1}$ mapping to $(-1)^ic_i$ \cite[Th.
3.10]{suslin3}. From this, by the same argument as in
\cite[proof of Th. 2.9]{suslin3}, one easily deduces by double induction
on $i$ and $n$ that
\[K_1(SL_n)^{(i)}=
\Z[\bar\gamma_1+\dots+\bar\gamma_i,\bar\gamma_{i+1},\dots,
\bar\gamma_{n-1}]
\]
(use the formula $\bar\gamma_1+\dots+\bar\gamma_n=0$, which follows from
\cite[p. 65]{suslin3}).
\end{comment}

\appendix

\section{A cancellation theorem over imperfect fields}\label{cancel}


\begin{thm}\label{tA.1} Let $F$ be a field, $M\in DM_-^\eff(F)$ and $n\ge 0$. Then the map
$-\otimes
\Z(1)$ induces an isomorphism
\[\Hom_{DM}(M,\Z(n))\iso \Hom_{DM}(M(1),\Z(n+1)).\]
\end{thm}

\begin{proof} It is enough to prove this for $M=C_*(X)[i]$, $X$ a smooth variety and $i\in\Z$.
By \cite[Prop. 3.2.3]{voetri} and \cite{allagree}, the left hand side is functorially
isomorphic to Bloch's higher Chow group $CH^n(X,2n+i)$. By \cite[Th. 15.12]{mvw} (projective
bundle formula in $DM$), the right hand side is a direct summand of
$CH^{n+1}(X\times\P^1,2n+2+i)$. By the projective bundle formula for higher Chow groups
(\cite[Th. 7.1]{bloch}, \cite[Cor. 5.4]{levinerev}), the latter decomposes as a direct sum
\[CH^{n+1}(X\times\P^1,2n+2+i)\simeq CH^{n+1}(X,2n+2+i)\oplus CH^{n}(X,2n+i).\]

Moreover, the construction of the projective bundle isomorphisms in \cite{mvw} and
\cite{bloch,levinerev} show that the latter two are compatible via the isomorphism between
motivic cohomology and higher Chow groups in \cite{allagree}. This proves the
theorem.
\end{proof}

Theorem \ref{tA.1} is sufficient to extend to imperfect fields the construction of the slice
spectral sequences in the form of \eqref{eq5Nis}, \ie for motivic cohomology computed in the
Nisnevich topology (= Bloch's higher Chow groups). It is not sufficient, however, to obtain a
version of the \'etale spectral sequences of \eqref{eq5} which is interesting at $p$,
since $p$ is automatically inverted in $DM_{-,\et}^\eff(F)$ (see Remark \ref{r2.2}). In order
to achieve this, one may presumably proceed by working directly on Bloch's cycle
complexes, as follows:

By the work of Geisser-Levine \cite{gl2}, the \'etale hypercohomology of Bloch's cycle complexes
provides an interesting theory modulo $p$. The first thing to do is to find a version of the
slice filtration directly on the cycle complexes of a given smooth $F$-variety $X$: this can be
achieved by using the ``homotopy coniveau filtration" (which is at the basis of the
construction of the Bloch-Lichtenbaum spectral sequence), see \cite{levine4} and \cite[\S
4]{kl}.

This will give spectral sequences comparable to those of Theorem \ref{t4.1} and \eqref{eq5}. The
issue is then to identify the $E_2$-terms. This can presumably be done by a slightly tedious
imitation of the computations in \cite{hk} and \S \ref{s:red}, where the tediousness comes from
the fact that one is limited to work with smooth varieties rather than general motives.

In the course of the computation, the following ingredients will certainly appear: \'etale
versions of the localisation theorem for higher Chow groups (see e.g. the proof of
\cite[Prop. 4.11]{hk}) and of Bloch's projective bundle theorem. They should be obtained  much
as in \cite[Th. 4.2 and Th. 5.1]{w2}. Hopefully a partial purity statement similar to \cite[Th.
4.2]{w2} will be sufficient for the applications.

We leave this programme to the interested reader.


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\end{document}

\begin{comment}
To compute
$\Coker \gamma^3$, we use Lemma \ref{lgroth}: it yields a commutative diagram with exact rows and columns
\[\begin{CD} 
T_s^*\otimes \bfS^2(T_s^*)\>>> \bfS^3(T_s^*)\>>> 0\\
\V{1\otimes\gamma^2}VV \V{\gamma^3}VV\V{}VV\\
T_s^*\otimes CH^2(Y_s)\>>> CH^3(Y_s)\>>> CH^3(G_s)\>>> 0\\
\V{}VV \V{}VV\V{=}VV\\
T_s^*\otimes CH^2(G_s)\>>> \Coker \gamma^3\>>> CH^3(G_s)\>>> 0\\
\V{}VV \V{}VV\V{}VV\\
0 && 0 && 0
\end{CD}\]
\end{comment}

