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\title
%[]
{Upper motives of outer algebraic groups}


\keywords
{Algebraic groups,
projective homogeneous varieties,
Chow groups.\\
{\em 2000 Mathematical Subject Classifications:}
14L17; 14C25}


\author
%[N. Karpenko]
{Nikita A. Karpenko}



\address
{UPMC Univ Paris 06\\
%UMR 7586\\
Institut de Math\'ematiques de Jussieu\\
%4 place Jussieu\\
F-75252 Paris\\
%CEDEX 05\\
FRANCE}


\address
{{\it Web page:}
{\tt www.math.jussieu.fr/\~{ }karpenko}}

\email {karpenko {\it at} math.jussieu.fr}

\date{June 2009}


\thanks
{Supported by the Collaborative Research Centre 701 of the Bielefeld University}


\begin{document}

\begin{abstract}
Let $G$ be a semisimple affine algebraic group over a field $F$.
Let $E/F$ be a minimal field extension such that the group $G_E$ is of inner type.
Assuming that the degree of $E/F$ is a power of a prime $p$,
we determine the structure of the
Chow motives with coefficients in a finite field of characteristic $p$ of the projective
$G$-homogeneous varieties.
More precisely, it is known that
the motive of any such variety decomposes (in a unique way) into a sum of
indecomposable motives, and
we describe the indecomposable summands
which appear in the decompositions.
This description is already known for the groups $G$ of inner type and
is new for $G$ of outer type.
%The condition $(*)$ is satisfied, for instance, for an abelian $p$-primary extension, where $p$ is the characteristic
%of the coefficient field.
\end{abstract}



\maketitle

%\tableofcontents

\section
{Introduction}


We fix an arbitrary base field $F$.
Besides of that, we fix a finite field $\F$ and we consider the
Grothendieck Chow motives over $F$
with coefficients in $\F$.
These are the objects of the category $\CM(F,\F)$, defined as in \cite{EKM}.

%For any finite separable field extension $L/F$, there is an additive shift-respecting functor
%$\CM(L,\F)\to\CM(F,\F)$ associating to a motive $(X,p)\in\CM(L,\F)$, given by a smooth complete irreducible $L$-variety $X$
%and a projector $p\in \Ch_{\dim X}(X\times X)$, the motive $(Y,q)\in\CM(F,\F)$,
%where $Y$ is $X$ considered as an $F$-variety via the composition $X\to\Spec L\to\Spec F$ and where
%$q$ is the image of $p$ under the push-forward homomorphism $\Ch_{\dim X}(X\times X)\to \Ch_{\dim Y}(Y\times Y)$
%with respect to the morphism $X\times X=X\times_L X\to X\times_F X=Y\times Y$ induced by $\Spec L\to \Spec F$.


Let $G$ be a semisimple affine algebraic group over $F$.
According to \cite[Corollary 35(4)]{MR2264459}
(see also Corollary \ref{cor22} here), the motive of any projective $G$-homogeneous
variety decomposes (and in a unique way) into a finite direct sum of indecomposable motives.
One would like to describe the indecomposable motives which appear this way.
In this paper we do it under certain assumption on $G$ formulated in terms of a minimal field extension
$E/F$ such that the group $G_E$ is of inner type:
the degree of $E/F$ is assumed to be a power of $p$, where $p=\Char\F$.
(Note that in the case when $E=F$, that is, when $G$ is of inner type, the problem has been solved in \cite{upper}.)

Let us introduce a minimum of terminology and notation needed to formulate the answer.
For any intermediate field $L$ of the extension $E/F$ and any projective $G_L$-homogeneous variety $Y$,
we consider the upper (see \cite[Definition 2.10]{upper}) indecomposable summand $M_Y$ of the motive
$M(Y)\in\CM(F,\F)$ of $Y$ (considered as an $F$-variety at this point).
Note that $L$ is the {\em constant field} (the algebraic closure of the base field in the function field)
of the $F$-variety $Y$.
The set of the isomorphism classes of the motives $M_Y$ for all $L$ and
$Y$ is called the set of {\em upper motives}
of the algebraic group $G$.
%We note that for a given $G$ and $L$, the isomorphism class of $M_Y$ depends only on the Tits index of the %group $G$ over the function field of $Y$ by \cite{upper}.

We are going to claim that
the complete motivic decomposition of any projective $G$-homogeneous variety $X$
consists of shifts of upper motives of $G$.
In fact, the information we have is a bit more precise:

\begin{thm}
\label{main}
For $F$, $G$, $E$, and $X$ as above, the complete motivic decomposition of $X$ consists of shifts of
upper motives of the algebraic group $G$.
More precisely, any indecomposable summand of the motive of $X$ is isomorphic
the upper motive $M_Y$ of a variety $Y$
such that the Tits index of $G$ over the function field of $Y$ contains the Tits index of $G$ over the function field
of $X_L$, where $L$ is the constant field of $Y$.
\end{thm}

The proof of Theorem \ref{main} is given in \S\ref{Proof of Theorem main}.
Before this, we get some preparation results which are also of independent interest.
In \S\ref{Nilpotence principle for quasi-homogeneous varieties}, we prove the nilpotence principle
for the quasi-homogeneous varieties.
In \S\ref{Motives of $0$-dimensional varieties}, we establish some properties of a motivic
corestriction functor.

By {\em sum} of motives we always mean the {\em direct} sum;
a {\em summand} is a {\em direct} summand.

\section
{Nilpotence principle for quasi-homogeneous varieties}
\label
{Nilpotence principle for quasi-homogeneous varieties}

Let us consider the category
$\CM(F,\Lambda)$ of Grothendieck Chow motives over a field $F$ with coefficients in an
{\em arbitrary} associative commutative unital ring $\Lambda$.

We say that a smooth complete $F$-variety $X$ satisfies the nilpotence principle,
if for any field extension $K/F$ the kernel of the change of field homomorphism
$$
\End\big(M(X)\big)\to\End\big(M(X_K)\big)
$$
consists of nilpotents, where
$M(X)$ stands for the motive of $X$ in $\CM(F,\Lambda)$.

We say that an $F$-variety $X$ is {\em quasi-homogeneous}, if each connected component $X^0$ of $X$
has the following property: there exists a finite separable field extension $L/F$,
a semisimple affine algebraic group $G$ over $L$, and a projective $G_L$-homogeneous variety $Y$ such
that $Y$, considered as an $F$-variety via the composition
$Y\to\Spec L\to\Spec F$, is isomorphic to $X^0$.
(Note that the algebraic group $G$ needs not to be defined over $F$ in this definition.)

We note that any variety which is {\em projective quasi-homogeneous} in the sense of
\cite[\S4]{MR2178658} is also quasi-homogeneous in the above sense.
The following statement generalizes \cite[Theorem 8.2]{MR2110630} (see also
\cite[Theorem 5.1]{MR2178658}) and \cite[Theorem 25]{MR2264459}:

\begin{thm}
\label{nilp}
%Let $L/F$ be a finite separable field extension,
%let $G$ be a semisimple affine algebraic group over $L$,
%and let $X$ be a projective $G$-homogeneous variety.
%The scheme $X$ considered as an $F$-variety via the composition
%$X\to\Spec L\to\Spec F$
Any quasi-homogeneous variety
satisfies the nilpotence principle.
\end{thm}


\begin{proof}
%[Proof of Theorem \ref{nilp}]
By \cite[Theorem 92.4]{EKM} it suffices to show that the quasi-homogeneous varieties form
a {\em tractable class}.
We first recall the definition of a tractable class $\mathcal{C}$ (over $F$).
This is a disjoint union of classes $\mathcal{C}_K$ of smooth complete $K$-varieties,
where $K$ runs over {\em all} field extensions of $F$,
having the following properties:
\begin{enumerate}
\item
if $Y_1$ and $Y_2$ are in $\mathcal{C}_K$ for some $K$, then the disjoint union
$Y_1\coprod Y_2$ is also in $\mathcal{C}_K$;
\item
if $Y$ is in $\mathcal{C}_K$ for some $K$, then each component of $Y$
is also in $\mathcal{C}_K$;
\item
if $Y$ is in $\mathcal{C}_K$ for some $K$, then for any field extension $K'/K$
the $K'$-variety $Y_{K'}$ is in $\mathcal{C}_{K'}$;
\item
if $Y$ is in $\mathcal{C}_K$ for some $K$, $Y$ is irreducible, $\dim Y>0$, and $Y(K)\ne\emptyset$,
then $\mathcal{C}_K$ contains a (not necessarily connected) variety $Y_0$ such that $\dim Y_0<\dim Y$ and $M(Y)\simeq M(Y_0)$ in  $\CM(K,\Lambda)$.
\end{enumerate}
Let us define a class $\mathcal{C}$ as follows.
For any field extension $K/F$, $\mathcal{C}_K$ is the class of all
quasi-homogeneous $K$-varieties.
%finite disjoint unions
%of $K$-varieties $Y$ each of which has the following property:
%there exists a finite separable field extension $L/K$, a semisimple affine algebraic group $G$ over $L$,
%and a projective $G$-homogeneous variety $Y'$ such that $Y'$, considered as a $K$-variety, is isomorphic to %$Y$.
%(Note that the algebraic groups need not to be defined over $K$ in this definition.)

%Clearly, the $F$-variety $X$ is contained in $\mathcal{C}$ (that is, in $\mathcal{C}_F$.
We claim that the class $\mathcal{C}$ is tractable.
Indeed, the properties $(1)$--$(3)$ are trivial and
the property $(4)$ is \cite[Theorem 7.2]{MR2110630}.
\end{proof}

We turn back to the case where the coefficient ring $\Lambda$ is a finite field $\F$.

\begin{cor}
\label{cor22}
Let $M\in\CM(F,\F)$ be a summand of the motive of a quasi-homogeneous variety.
Then $M$ decomposes in a finite direct sum of indecomposable motives;
moreover, such a decomposition is unique (up to a permutation of the summands).
\end{cor}

\begin{proof}
Any quasi-homogeneous variety is {\em geometrically cellular}.
In particular, it is {\em geometrically split} in the sense of \cite[\S2a]{upper}.
Finally, by Theorem \ref{nilp}, it satisfies the nilpotence principle.
The statement under proof follows now by \cite[Corollary 2.6]{upper}.
\end{proof}


\section
{Corestriction of scalars for motives}
\label{Motives of $0$-dimensional varieties}

As in the previous section, we fix an arbitrary coefficient ring $\Lambda$.
We write $\Ch$ for the Chow group with coefficients in $\Lambda$.
Let $\C(F,\Lambda)$ be the category whose objects are pairs $(X,i)$, where
$X$ is a smooth complete equidimensional $F$-variety and $i$ is an integer.
A morphism $(X,i)\to(Y,j)$ in this category is an element of the Chow group
$\Ch_{\dim X+i-j}(X\times Y)$ (and the composition is the usual composition of correspondences).
The category $\C(F,\Lambda)$ is preadditive.
Taking first the additive completion of it, and taking then the idempotent completion of the
resulting category, one gets the category of motives $\CM(F,\Lambda)$, cf. \cite[\S63 and \S64]{EKM}.

Let $L/F$ be a finite separable field extension.
We define a functor
$$
\cores_{L/F}:\C(L,\Lambda)\to\C(F,\Lambda)
$$
as follows:
on the objects $\cores_{L/F}(X,i)=(X,i)$, where on the right-hand side $X$ is considered as an $F$-variety
via the composition $X\to\Spec L\to\Spec F$;
on the morphisms, the map
$$
\Hom_{\C(L,\Lambda)}\big((X,i),(Y,j)\big)\to\Hom_{\C(F,\Lambda)}\big((X,i),(Y,j)\big)
$$
is the push-forward homomorphism
$\Ch_{\dim X+i-j}(X\times_L Y)\to\Ch_{\dim X+i-j}(X\times_F Y)$
with respect to the closed imbedding
$X\times_LY\hookrightarrow X\times_FY$.
Passing to the additive completion and then to the idempotent completion, we get
an additive and commuting with the Tate shift functor
$\CM(L,\Lambda)\to\CM(F,\Lambda)$, which we also denote by $\cores_{L/F}$.

The functor $\cores_{L/F}:\C(L,\Lambda)\to\C(F,\Lambda)$
is left-adjoint and right-adjoint to the change of field functor
$\res_{L/F}:\C(F,\Lambda)\to\C(L,\Lambda)$, associating to $(X,i)$
the object $(X_L,i)$.
Therefore the functor $\cores_{L/F}:\CM(L,\Lambda)\to\CM(F,\Lambda)$
is also left-adjoint and right-adjoint to the change of field functor
$\res_{L/F}:\CM(F,\Lambda)\to\CM(L,\Lambda)$.
(This makes a funny difference with the category of varieties, where the functor $\cores_{L/F}$
is only left-adjoint to $\res_{L/F}$, while the right-adjoint to $\res_{L/F}$ functor is the
Weil restriction.)
It follows that for any $M\in\CM(L,\Lambda)$ and any $i\in\Z$, the Chow groups
$\Ch^i(M)$ and $\Ch^i(\cores_{L/F}M)$ are canonically isomorphic as well as the
Chow groups
$\Ch_i(M)$ and $\Ch_i(\cores_{L/F}M)$ are.
Indeed, since $\res_{L/F}\Lambda=\Lambda\in\CM(L,\Lambda)$, we have
\begin{align*}
&\Ch^i(M):=\Hom\big(M,\Lambda(i)\big)=
%\Hom(M,\res_{L/F}\Lambda(i))=
\Hom\big(\cores_{L/F}M,\Lambda(i)\big)
=:\Ch^i(\cores_{L/F}M)
&\text{ and}
\\
&\Ch_i(M):=\Hom\big(\Lambda(i),M\big)
%=\Hom(\res_{L/F}\Lambda(i),M)
=\Hom\big(\Lambda(i),\cores_{L/F}M\big)
=:\Ch_i(\cores_{L/F}M).
&
\end{align*}

In particular, if the ring $\Lambda$ is connected and $M\in\CM(L,\Lambda)$
is an {\em upper} (see \cite[Definition 2.10]{upper})
motivic summand of an irreducible smooth complete $L$-variety
$X$, then $\cores_{L/F}M$ is an upper motivic summand of the $F$-variety $X$.


\begin{comment}
There is an additive and commuting with the Tate shift functor
$\cores_{L/F}:\CM(L,\F)\to\CM(F,\F)$ determined by the following two rules:

%\noindent
(1) for a smooth complete $L$-variety $X$, the image of the $L$-motive $M_L(X)\in\CM(L,\F)$ of $X$ is the $F$-motive
$M_F(X)\in\CM(F,\F)$ of $X$, considered as an $F$-variety
via the composition $X\to\Spec L\to\Spec F$;


%\noindent
(2) for smooth complete irreducible $L$-varieties $X$ and $Y$ , the homomorphism
$$
\Hom_{\CM(L,\F)}\big(M_L(X),M_L(Y)\big)\to\Hom_{\CM(F,\F)}\big(M_F(X),M_F(Y)\big)
$$
is the push-forward homomorphism
$\Ch_{\dim X}(X\times_L Y)\to\Ch_{\dim X}(X\times_F Y)$
with respect to the closed imbedding
$X\times_LY\hookrightarrow X\times_FY$.
\end{comment}

Now we turn back to the situation where $\Lambda$ is a finite field $\F$:

\begin{prop}
\label{three}
The following three conditions on a finite galois field extension $E/F$ are equivalent:
\begin{enumerate}
\item
for any intermediate field $F\subset K\subset E$, the $K$-motive of $Spec E$ is indecomposable;
\item
for any intermediate fields $F\subset K\subset L\subset E$ and any $L$-motive $M$, the $K$-motive
$\cores_{L/K}(M)$ is indecomposable;
\item
the degree of $E/F$ is a power of $p$ (where $p$ is the characteristic of the coefficient field $\F$).
\end{enumerate}
\end{prop}


\begin{proof}
We start by showing that $(3)\Rightarrow(2)$.
So, we assume that $[E:F]$ is a power of $p$ and we prove $(2)$.
The extension $L/K$ decomposes in a finite chain of galois degree $p$ extensions.
Therefore we may assume that $L/K$ itself is a galois degree $p$ extension.
Let $R=\End(M)$.
This is an associative, unital, but not necessarily commutative $\F$-algebra.
Moreover, since $M$ is indecomposable, the ring $R$ has no non-trivial idempotents.
We have $\End\big(\cores_{L/K}(M)\big)=R\otimes_\F\End\big(M_K(\Spec L)\big)$.
According to \cite[\S7]{MR2264459}, the ring $\End\big(M_K(\Spec L)\big)$ is isomorphic
to the group ring $\F\Gamma$, where $\Gamma$ is the Galois group of $L/K$.
Since the group $\Gamma$ is (cyclic) of order $p$, we have $\F\Gamma\simeq\F[t]/(t^p-1)$.
Since $p=\Char \F$, $\F[t]/(t^p-1)\simeq\F[t]/(t^p)$.
It follows that the ring $\End\big(\cores_{L/K}(M)\big)$ is isomorphic to the ring $R[t]/(t^p)$.
We prove $(2)$ by showing that the latter ring does not contain non-trivial idempotents.
An arbitrary element of $R[t]/(t^p)$ can be (and in a unique way) written in the form $a+b$,
where $a\in R$ and $b$ is divisible by $t$.
Note that $b$ is nilpotent.
%Note that $a+bt$ is an invertible element of $R[t]/(t^p)$ provided that $a$ is invertible in $R$.
Let us take an idempotent of $R[t]/(t^p)$ and write it in the above form $a+b$.
Then $a$ is an idempotent of $R$.
Therefore $a=1$ or $a=0$.
If $a=1$, then $a+b$ is invertible and therefore $a+b=1$.
If $a=0$, then $a+b$ is nilpotent and therefore $a+b=0$.

We have proved the implication $(3)\Rightarrow(2)$.
The implication $(2)\Rightarrow(1)$ is trivial.
We finish by proving that $(1)\Rightarrow(3)$.

We assume that $[E:F]$ is divisible by a different from $p$ prime $q$ and we show that $(1)$ does not hold.
Indeed, the galois group of $E/F$ contains an element $\sigma$ of order $q$.
Let $K$ be the subfield of $E$ consisting of the elements of $E$ fixed by $\sigma$.
We have $F\subset K\subset E$ and $E/K$ is galois of degree $q$.
The endomorphisms ring of $M_K(\Spec E)$ is isomorphic to $\F[t]/(t^q-1)$.
Since $q\ne\Char\F$, the factors of the decomposition
$t^q-1=(t-1)\cdot(t^{q-1}+t^{q-2}+\dots+1)\in\F[t]$ are coprime.
Therefore the ring $F[t]/(t^q-1)$ is the direct product of the rings $\F[t]/(t-1)=\F$ and $\F[t]/(t^{q-1}+\dots+1)$,
and it follows that  the motive $M_K(\Spec E)$ is not indecomposable.
\end{proof}

\begin{comment}

Let $S$ be a finite set.
We write $\F S$ for the abelian group of the formal linear combinations of the elements of $S$ with
coefficients in $S$.

Let $\Gamma$ be a finite abstract group.
The abelian group $\F\Gamma$ has than a natural structure of an associative commutative ring with $1$.
For a (not necessarily normal) subgroup $\Gamma'$ of $\Gamma$, we write $\Gamma/\Gamma'$
for the set of the left cosets of $\Gamma'$ in $\Gamma$.
The abelian group $\F(\Gamma/\Gamma')$ has a natural structure of a (permutation) module over the ring $\F\Gamma$.

\begin{dfn}
\label{def (*)}
We say that $\Gamma$ satisfied condition $(*)$, if for any subgroups $\Gamma''\subset\Gamma'\subset\Gamma$
the endomorphisms ring of the $\F\Gamma'$-module $\F(\Gamma'/\Gamma'')$ is connected.
We say that a given finite galois field extension satisfied $(*)$, if its galois group satisfies $(*)$.
\end{dfn}

\begin{lemma}
A finite galois field extension $E/F$ satisfies $(*)$ if and only if
for any intermediate fields $F\subset K\subset L\subset E$, the motive of the $K$-variety $\Spec L$
is indecomposable.
\end{lemma}

\begin{proof}
Let $\Gamma_K$ be the subgroup of $\Gamma$ consisting of the identical on $K$ elements.
%Let $\Gamma_L$ be the subgroup of $\Gamma$ consisting of the identical on $L$ elements.
Since $K\subset L$, we have $\Gamma_L\subset\Gamma_K$.
By \cite[\S7]{MR2264459}, the endomorphisms ring of the $K$-motive of $\Spec L$ is isomorphic to the endomorphism
ring of the $\F\Gamma_K$-module $\F(\Gamma_K/\Gamma_L)$.
A motive is indecomposable if and only if its endomorphisms ring is connected.
\end{proof}

\begin{lemma}
Any $p$-primary abelian group $\Gamma$, where $p=\Char\F$, satisfies $(*)$.
\end{lemma}

\begin{proof}
For any $\Gamma''\subset\Gamma'\subset\Gamma$, the endomorphisms ring of the $\F\Gamma'$-module
$\F(\Gamma'/\Gamma'')$ is the ring $\F(\Gamma'/\Gamma'')$.
The $p$-primary abelian group $\Gamma'/\Gamma''$ is isomorphic to a direct product of $n$ cyclic
groups of orders $p^{r_1},\dots,p^{r_n}$ for some $n$ and some $r_1,\dots,r_n$.
Therefore the ring $\F(\Gamma'/\Gamma'')$ is isomorphic to the polynomial ring
$\F[t_1,\dots,t_n]$ modulo the ideal generated by the polynomials $t_1^{p^{r_1}}-1,\dots,t_n^{p^{r_n}}-1$.
Killing the nil-radical (remember that  $\Char\F=p$), we get the connected ring $\F$.
\end{proof}

In fact we are mainly interested in the case of $\Gamma=\Z/2$ by the following reason:

\begin{rem}
Assume that our algebraic group $G$ is absolutely simple, outer,
and not of the type $\cat{D}_4$.
Let $E$ be a {\em minimal} field extension of $F$ such that $G_E$ is of inner type.
Then the extension $E/F$ is galois and quadratic and therefore satisfies $(*)$ provided that $\Char\F=2$.
\end{rem}

\begin{example}
Assume that $\Char\F\ne2$.
Then the group $\Z/2$ does not satisfy $(*)$.
Indeed, the ring $\F(\Z/2)$ is isomorphic to $\F[t]/(t^2-1)$ and is not connected.
It follows for any quadratic galois field extension $E/F$ that the $F$-motive of $\Spec E$ is decomposable.
Since $\Ch^*(\Spec E)=\Ch^0(\Spec E)=\F$, there is a non-zero motivic summand $M$ of the $F$-variety
$\Spec E$ such that $\Ch^*(M)=0$.
This gives a counter-example for Theorem \ref{main} with condition $(*)$ removed (because by definition of an upper
motive, its  Chow group
$\Ch^0$  is non-zero).
\end{example}

\end{comment}


\begin{cor}
\label{upper v upper}
Let $E/F$ be a finite $p$-primary galois field extension and let $L$ be an intermediate field:
$F\subset L\subset E$.
Let $M\in\CM(L,\F)$ be an upper indecomposable motivic summand of an irreducible smooth complete $L$-variety
$X$.
Then $\cores_{L/F}M$ is an upper indecomposable summand of the $F$-variety $X$.
\qed
\end{cor}


\section
{Proof of Theorem \ref{main}}
\label{Proof of Theorem main}


Before starting the proof of Theorem \ref{main},
let us recall some classical facts and introduce some notation.

We write
$\D$
(or $\D_G$) for the set
of vertices of the Dynkin diagram of $G$.
The galois group $\Gamma$ of the field extension $E/F$ acts on $\D$.

Let $L$ be an intermediate field of the extension $E/F$.
Any subset $\tau$ in $\D$ which is stable under the action of the galois group of $E/L$, determines a projective $G_L$-homogeneous
variety $X_{\tau, G_L}$ in the way described in \cite[\S3]{upper}.
This is the variety corresponding to the set $\D\setminus\tau$ in the sense of \cite{MR0224710}.
For instance, $X_{\D,G_L}$ is the variety of the Borel subgroups of $G_L$, and $X_{\emptyset,G_L}=\Spec L$.
Any projective $G_L$-homogeneous variety is $G_L$-isomorphic to $X_{\tau,G_L}$ for some $\tau\subset \D$
stable under the action of the galois group of $E/L$.

For any intermediate field $L$ of $E/F$ and any $\Gal(E/L)$-stable subset $\tau\subset \D$,
%stable under the action of the galois group of the extension $E/L$,
we write $M_{\tau,G_L}$ for the upper indecomposable motivic summand of the $F$-variety $X_{\tau,G_L}$.

For any field extension $L/F$, we consider the Tits index (see \cite{MR0224710}) of $G$ over $L$ and
we write $\tau_L$ (or $\tau_{L,G}$) for the subset in $\D$ consisting of the circled vertices of the Dynkin diagram.


\begin{proof}[Proof of Theorem \ref{main}]
%We rewrite (with appropriate modifications)
This is a recast of
\cite[proof of Theorem 3.5]{upper}.

We proof  Theorem \ref{main}
simultaneously for all $F,G,X$ using an induction on $n=\dim X$.
The base of the induction is $n=0$ where $X=\Spec F$ and the statement is trivial.

From now on we are assuming that $n\geq1$ and that Theorem
\ref{main} is already proven for all varieties of dimension $<n$.

Let $M$ be an indecomposable summand of $M(X)$.
We have to show that $M$ is isomorphic to a shift of $M_{\tau,G_L}$ for some
intermediate field $L$ of $E/F$ and some $\Gal(E/L)$-stable subset
$\tau\subset \D_G$ containing $\tau_{L(X)}$.

%Since $\dim X>0$, the set $\tau_{F(X),G}$ is non-empty.
Let $G'/F(X)$ be the semisimple anisotropic kernel of the group $G_{F(X)}$.
The set $\D_{G'}$ is identified with $\D_G\setminus\tau_{F(X),G}$.
%In particular, $\rk G'<n$.

We note that the group $G'_{E(X)}$ is of inner type.
The field extension $E(X)/F(X)$ is galois with the galois group $\Gal(E/F)$.
In particular, its degree is a power of $p$ and
any its intermediate field is of the form $L(X)$ for some intermediate field
$L$ of the extension $E/F$.

According to \cite[Theorem 4.2]{MR2178658},
%(see also \cite{MR2110630}),
the motive of $X_{F(X)}$ decomposes into a sum of shifts of motives
of projective $G'_{L(X)}$-homogeneous (where $L$ runs over intermediate fields of
the extension $E/F$) varieties $Y$,
satisfying $\dim Y<\dim X=n$ (we are using the assumption that $n>0$ here).
It follows by the induction hypothesis and
%Proposition \ref{three},
Corollary \ref{upper v upper},
that each summand of the complete motivic decomposition of $X_{F(X)}$ is a shift of
$M_{\tau',G'_{L(X)}}$ for some $L$ and some $\tau'\subset\D_{G'}$.
The complete decomposition of $M_{F(X)}$ is a part of the above decomposition.

%Let us fix an algebraic closure $\bar{F}$ of $F$ containing $E$.
%Each summand of the complete decomposition of $M_{F(X)}$
%decomposes over $\bar{F}(X)$ into a sum of
%Tate motives.
%This gives a decomposition of $\bar{M}=M_{\bar{F}(X)}$ into a sum of
%Tate motives.
%Let us choose a Tate summand $\F(i)$ with the smallest $i$ in the decomposition of $\bar{M}$.
Let us choose a summand $M_{\tau',G'_{L(X)}}(i)$ with minimal $i$ in the complete decomposition of
$M_{F(X)}$.
%This summand comes from the decomposition of the $i$th shift of
%some $\bar{M}_{\tau',G'_{L(X)}}$ for some $L$ and some $\tau'\subset\D_{G'}$.
We set $\tau=\tau'\cup\tau_{L(X)}\subset\cT_G$.
We shall show that $M\simeq M_{\tau,G_L}(i)$
%(note that we have $G$, not $G'$ in this formula!)
for these $L$, $\tau$, and $i$.

We write $Y$ for the $F$-variety $X_{\tau,G_L}$ and we write
$Y'$ for the $F(X)$-variety $X_{\tau',G'_{L(X)}}$.
We write $N$ for the $F$-motive $M_{\tau,G_L}$ and we write
$N'$ for the $F(X)$-motive $M_{\tau',G'_{L(X)}}$.

By \cite[Lemma 2.14]{upper} and since $M$ is indecomposable,
it suffices to construct morphisms
$$
\alpha:M(Y)(i)\to M\;\;\text{and}\;\;
\beta:M\to M(Y)(i)
$$
satisfying $\mult(\beta\compose\alpha)=1$.
%Indeed, a power of $\alpha\compose\beta$ is then a non-zero
%projector, therefore an identity (by the indecomposability of
%$M_{l,D}$); consequently, $M_{l,D}\simeq M$ by the
%indecomposability of $M$.

%The fixed above summand $\F(i)$ of $\bar{M}$ is produced
%by two elements
%$$
%a\in\Ch_i(\bar{X})=\Hom\big(\F(i),M(\bar{X})\big)
%\text{ and }
%b\in\Ch^i(\bar{X})=\Hom\big(M(\bar{X}),\F(i)\big)
%$$
%such that the degree of the $0$-cycle class $(a\cdot b)\in\Ch_0(\bar{X})$ is $1\in\F$
%(the condition on the degree means that the composition
%$$
%\begin{CD}
%\F(i)@>a>> M(\bar{X})@>b>>\F(i)
%\end{CD}
%$$
%is the identity).

%Let us fix an algebraic closure $\bar{F}$ of $F$ containing $E$.

We construct $\alpha$ first.
%We recall that the function field $F(Y)$
Since $\tau'\subset\tau$, the $F(X)(Y)$-variety $Y'\times_{L(X)}\Spec F(X)(Y)$
has a rational point.
Let $\alpha_1\in\Ch_0\big(Y'\times_{L(X)}\Spec F(X)(Y)\big)$ be the class of a rational point.
Let $\alpha_2\in\Ch_i(X_{F(X)(Y)})$ be the image of $\alpha_1$ under the composition
$$
\Ch_0\big(Y'\times_{L(X)}\Spec F(X)(Y)\big)\to
\Ch_0(Y'_{F(X)(Y)})\to\Ch_0(N'_{F(X)(Y)})\hookrightarrow\Ch_i(X_{F(X)(Y)}),
$$
where the first map is the push-forward with respect to the closed imbedding
$$
Y'\times_{L(X)}\Spec F(X)(Y)\hookrightarrow Y'_{F(X)(Y)}:=Y'\times_{F(X)}\Spec F(X)(Y).
$$
%The element $a$ is the image under the embedding
%$\Ch_0(\bar{N}')\hookrightarrow\Ch_i(\bar{X})$
%of an element in $\Ch_0(\bar{Y}')=\Ch_0(\bar{N}')$.
%%(see Lemma \ref{upper and 0-cycles}).
%Note that the variety $\bar{Y}':=Y'\times_{F(X)}\Spec\bar{F}(X)$
%is a disjoint union of $[L:F]$ copies of the irreducible variety
%$Y'\times_{L(X)}\Spec\bar{F}(X)$.
%Since $\tau'\subset\tau$, the $L(X)$-variety $Y'$ has an $L(X)(Y)$-point, and it follows that
%the element $a_{\bar{F}(X)(Y)}$ is $L(X)(Y)$-rational
%(that is, lies in the image of the change of field homomorphism of the Chow groups).
Since $\tau_{L(X)}\subset\tau$, the variety $X$ has an $F(Y)$-point
and therefore the field extension $F(X)(Y)/F(Y)$ is purely transcendental.
Consequently, the element $\alpha_2$ is $F(Y)$-rational and
lifts to an element $\alpha_3\in\Ch_{\dim Y+i}(Y\times X)$.
We mean here a lifting with respect to the composition
$$
\begin{CD}
\Ch_{\dim Y+i}(Y\times X) \onto \Ch_i(X_{F(Y)}) @>{\res_{F(X)(Y)/F(Y)}}>> \Ch_i(X_{F(X)(Y)})
\end{CD}
$$
where the first map is the epimorphism given by the pull-back with respect to the morphism
$X_{F(Y)}\to Y\times X$ induced by the generic point of the variety $Y$.


We define the morphism $\alpha$ as the composition
$$
\begin{CD}
M(Y)(i)@>{\alpha_3}>> M(X)@>>> M
\end{CD}
$$
where the second map is the projection of $M(X)$ onto its summand $M$.

We proceed by constructing $\beta$.
%Let $b'\in\Ch^0(Y')$ be the class of $Y'$.
%Let $b\in\Ch^i(X_{\bar{F}(X)})$ be the image of $b'$ under the composition
%$$
%\Ch^0(Y')\to
%\Ch^0(Y'\times_{L(X)}\Spec \bar{F}(X))\to
%\Ch^0(\bar{Y}')=\Ch^0(\bar{N}')\hookrightarrow\Ch^i(X_{\bar{F}(X)}).
%$$
%Note that the degree of the $0$-cycle class $(a\cdot b)$ is equal to $1\in\F$
Let $\beta_1\in\Ch_{\dim Y'}(Y'\times_{F(X)} Y_{F(X)})$ be the class of
the closure of the graph of a rational map of $L(X)$-varieties
$Y'\RatM Y_{F(X)}$ (which exists because $\tau\subset\tau_{L(X)}\cup \tau'$).
Note that this graph is a subset of $Y'\times_{L(X)}Y_{F(X)}$, which we consider as a subset
of $Y'\times_{F(X)}Y_{F(X)}$ via the closed imbedding
$Y'\times_{L(X)}Y_{F(X)}\hookrightarrow Y'\times_{F(X)}Y_{F(X)}$.
%Also note that $Y_{L(X)}$, defined as $Y\times_L\Spec L(X)$coincides with
%$Y_{F(X)}:=Y\times_F F(X)$.
Let $\beta_2$ be the image of $\beta_1$ under the composition
%of the homomorphisms
\begin{multline*}
\Ch^{\dim Y}(Y'\times_{F(X)} Y_{F(X)})=
\\
\Ch^{\dim Y}\big(M(Y')\otimes M(Y_{F(X)})\big)\to
\Ch^{\dim Y}\big(N'\otimes M(Y_{F(X)})\big)\to \\
\Ch^{\dim Y+i}\big(M(X_{F(X)})\otimes M(Y_{F(X)})\big)=
\Ch^{\dim Y+i}\big((X\times Y)_{F(X)}\big)
\end{multline*}
where the first arrow is induced by the projection $M(Y')\to N'$ and the second arrow is induced
by the imbedding $N'(i)\to M(X_{F(X)})$.
%Since the field extension $L(X)/L$ is purely transcendental,
%the element $(\beta_2)_{\bar{L}(X)}$ is $L$-rational.
The element $\beta_2$ lifts to an element
$$
\beta_3\in\Ch^{\dim Y+i}(X\times X\times Y).
$$
%(by means of the generic point of the second factor in this triple direct product).
We mean here a lifting with respect to the epimorphism
$$
\begin{CD}
\Ch^{\dim Y+i}(X\times X\times Y) \onto \Ch^{\dim Y+i}\big((X\times Y)_{F(X)}\big)
%@>{\res_{\bar{L}(X)/L}}>> \Ch^{\dim Y+i}\big((X\times Y)_{\bar{L}(X)}\big)
\end{CD}
$$
given by the pull-back with respect to the morphism
$X\times X\times Y\to (X\times Y)_{F(X)}$ induced by the generic point of the second factor in this
triple direct product.

Let $\pi\in\Ch_{\dim X}(X\times X)$ be the projector defining the
summand $M$ of $M(X)$.
Considering $\beta_3$ as a correspondence from $X$ to $X\times Y$, we define
$$
\beta_4\in\Ch^{\dim Y+i}(X\times X\times Y)
$$
as the
composition $\beta_3\compose\pi$.
We get
$$
\beta_5\in\Ch^{\dim Y+i}(X\times Y)=
\Ch_{\dim X-i}(X\times Y)
$$
as the image
of $\beta_4$ under the pull-back with respect to the diagonal
of $X$.
Finally, we define the morphism $\beta$ as the
composition
$$
\begin{CD}
M@>>>M(X)@>{\beta_5}>> M(Y)(i).
\end{CD}
$$

%We finish the proof by checking that
The verification of the relation
$\mult(\beta\compose\alpha)=1$, finishing the proof,
is similar to that of \cite[proof of Theorem 3.5]{upper}.
\begin{comment}
Since the multiplicity is not changed under extension of scalars, we may do the computation over the field
$\bar{F}(X)$.
Decompositions of the motives of
the varieties $\bar{Y}:=Y\times_F\Spec\bar{F}(X)$ and $\bar{X}:=X\times_F\Spec\bar{F}(X)$
into sums of Tate motives give certain homogeneous $\F$-bases of their Chow groups with coefficients
in $\F$ (a Tate summand $\F(j)$ of, say, $M(\bar{X})$ gives the basis element
$\F(i)\to M(\bar{X})\in\Ch_i(\bar{X})$).
We use here a motivic decomposition of $\bar{Y}$ which is a refinement of the decomposition of
$\bar{Y}$ into the sum of its ($[L:F]$) connected components.
Let us use a motivic decomposition of $\bar{X}$
which is obtained as follows: we decompose (over $\bar{F}(X)$) the complementary to $N'$ motivic summand
of $X_{\bar{F}(X)}$ and we take a refinement of the decomposition of $\bar{N'}$
induced by the decomposition of the variety $\bar{Y}'$ into the union of its ($[L:F]$)
connected components.

The basis of $\Ch(\bar{Y})$ contains (a multiple with an invertible coefficient of)
the element $y$ which we define as the push-forward of $[Y\times_L\bar{F}(X)]$
with respect to the closed imbedding
$Y\times_L\bar{F}(X)\hookrightarrow Y\times_F\bar{F}(X)$
and (a multiple with an invertible coefficient of) the element $x$ which we define as
as the push-forward with respect to the same closed imbedding of the rational point class
on $[Y\times_L\bar{F}(X)]$.
The element $a$ is in the basis of $\Ch(\bar{X})$.
As to the element $b$, it is in the {\em dual} basis of $\Ch(\bar{X})$, where ``dual'' means dual with respect to the bilinear
form $\Ch(\bar{X})\times\Ch(\bar{X})\to\F$, $(x_1,x_2)\mapsto \deg(x_1\cdot x_2)$.
%Note that (non-zero multiples of) the elements $1,x\in\Ch\big(\bar{X}(p^l,D)\big)$ are also in the dual basis of
%$\Ch\big(\bar{X}(p^l,D)\big)$.

We consider the Chow group of the products
$\bar{Y}\times \bar{X}$ and $\bar{X}\times \bar{X}$
together with the bases given by the external products of the elements of the bases
of the Chow groups of the factors, where in the case of $\bar{X}\times \bar{X}$
we are using the {\em dual} basis for the first factor and we are using the
``original'' basis for the second factor.
We have $\bar{\alpha}_1=1\times a+\dots$, where ``$\dots$'' stands for a linear combination of only those basis elements
whose first factor has codimension $>\codim 1=0$.
The projector $\pi$ which determines the summand $M$ of the motive of $X$ looks over $\bar{L}$
as $\bar{\pi}=b\times a+\dots$, where ``$\dots$'' stands for a linear combination of only those different from $b\times a$
basis elements of $\Ch(\bar{X}\times\bar{X})$ whose first factor has codimension $\geq\codim b=i$.
Since $\alpha=\pi\compose\alpha_1$, it follows that
$\bar{\alpha}=1\times a+\dots$, where ``$\dots$'' stands
for a linear combination of only those basis elements whose first factor is of positive codimension.

Now let us go through the construction of $\beta$.
To describe $\bar{\beta}_1$, we fix a homogeneous basis of $\Ch(\bar{Y}')$ and use it to built up a basis of
$\Ch(\bar{Y}'\times \bar{Y})$.
Abusing notation we write $1$  also for the unit class in $\Ch(\bar{Y}')$.

We have $\bar{\beta}_1=1\times x+\dots$
%with the usual convention on the meaning of ``$\dots$''.
where ``$\dots$'' stands
for a linear combination of only those basis elements whose first factor is of positive codimension.
Then we have $\bar{\beta}_2=b\times x+\dots$,
where we are using the basis of
$\Ch(\bar{X}\times \bar{Y})$ obtained out of the {\em dual} basis of $\Ch(\bar{X})$, and
where ``$\dots$'' stands
for a linear combination of only those basis elements whose first factor is of codimension $>\codim b=i$.
For $\beta_3$ we have
$\bar{\beta}_3=b\times1\times x+\dots$, where $1$ is the unit of $\Ch(\bar{X})$ and ``$\dots$'' stands for a linear combination
of basis elements of $\Ch(\bar{X}\times \bar{X}\times\bar{Y})$
which have a second factor of positive codimension {\em or} the first factor of dimension $>\codim b=i$.
For this triple product, we are using the basis obtained out of the dual basis for the first factor and, say,
the original basis for the second factor (a choice of a basis for the second factor is not important).

The element $\bar{\beta}_4$ has the same shape with the additional property that the codimension of the
first factor in each basis element
appearing in the linear combination is $\geq i$.
By this reason, $\bar{\beta}_5$ and also $\bar{\beta}$ look as
$b\times x+\dots$, where ``$\dots$'' stands
for a linear combination of only those basis elements whose first factor is of codimension $>\codim b=i$.
Therefore $\bar{\beta}\compose\bar{\alpha}=1\times x+\dots$, where ``$\dots$'' stands
for a linear combination of only those basis elements whose first factor is of positive codimension.
It follows that
$\mult(\bar{\beta}\compose\bar{\alpha})=1$.
\end{comment}
\end{proof}


%\begin{example}
%\end{example}

\begin{rem}
Theorem \ref{main} can be also proved using a weaker result
in place of \cite[Theorem 4.2]{MR2178658}, namely, \cite[Theorem 7.5]{MR2110630}.
\end{rem}

%\bibliographystyle{acm}
%\bibliography{../nikita}

\def\cprime{$'$}
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{\sc Chernousov, V., Gille, S., and Merkurjev, A.}
\newblock Motivic decomposition of isotropic projective homogeneous varieties.
\newblock {\em Duke Math. J. 126}, 1 (2005), 137--159.

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\newblock {\em Transform. Groups 11}, 3 (2006), 371--386.

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{\sc Elman, R., Karpenko, N., and Merkurjev, A.}
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{\sc Karpenko, N.~A.}
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\end{thebibliography}



\end{document}
