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\newcommand{\M}{\mathcal{M}}     % Chow motives
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\DeclareMathOperator{\Br}{\mathrm{Br}}     % Brauer group
\DeclareMathOperator{\Gal}{\mathrm{Gal}}   % Galois group
\DeclareMathOperator{\CH}{\mathrm{CH}}      % Chow group
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\DeclareMathOperator{\Coker}{\mathrm{Coker}}    % Cokernel
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\DeclareMathOperator{\SB}{\mathrm{SB}}      % Severi-Brauer variety
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\title{Chow motives of generically split varieties}

\author{V.~Petrov, N.~Semenov, K.~Zainoulline
\footnote{Supported partially by CNRS, DAAD A/04/00348, INTAS 03-51-3251,  
RTN-Network HPRN-CT-2002-00287, SFB701}}

%\date{July 17, 2006}

%%%%%%%%%%%%%%%%%%%% DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\maketitle

\begin{abstract}
Let $G$ be an anisotropic linear algebraic group 
over a field $F$ which splits
by a field extension of a prime degree.
Let $X$ be a projective homogeneous $G$-variety such that
$G$ splits over the function field of $X$.
We prove that under certain conditions
the Chow motive of $X$ is isomorphic to a direct sum
of twisted copies of an indecomposable motive $\RR_X$.
This covers all known examples of motivic decompositions
of generically split projective homogeneous varieties
(Severi-Brauer varieties, Pfister quadrics, maximal orthogonal Grassmannians)
as well as provides new ones (exceptional varieties of types
$\E_6$ and $\E_8$).
\end{abstract}


\section{Introduction}
The history of the subject of the present paper starts
with a celebrated result of M.~Rost about motivic decomposition
of a Pfister quadric which became the crucial point
in the proof of Milnor conjecture by V.~Voevodsky.
Briefly speaking, it says (see \cite{Ro98})
that the Chow motive of a Pfister quadric $X$
decomposes as a direct sum of (twisted) copies of a certain indecomposable
motive $\RR_X$. Hence, the motivic (and, hence, cohomological)
behavior of $X$ depends on a rather
small object $\RR_X$. Indeed, over the algebraic closure $\RR_X$
splits as a direct sum $\zz\oplus\zz(r)$
of just two twisted copies of the motive of a point.
As a consequence, it allows to compute and estimate several important
cohomological invariants of $X$.

One of the crucial properties of a Pfister quadric is that
it becomes hyperbolic (totally isotropic) over its function field.
In the present paper we deal only with varieties satisfying
the analogous property which we call {\em generically split}.
Namely, $X$ is generically split if
its Chow motive splits over the generic point of $X$
as a direct sum of (twisted) copies of the motive of a point.


Note that the discussion about Pfister quadrics and Milnor conjecture
essentially involves the prime $p=2$ 
which is a minimal degree of a field extension that splits $X$.
One may ask what happens for other primes. Or more precisely,
is there some natural 
analog of a Pfister quadric which has the similar 
motivic decomposition. Observe that 
this question naturally arises in the context
of Bloch-Kato conjecture 
(generalization of Milnor conjecture to other primes).

The goal of the paper is to show that any
generically split projective homogeneous $G$-variety $X$
provides such an analog, where $G$
is an anisotropic simple linear algebraic group of inner type
which splits by a field extension of a prime degree $p$. 
Observe that the property
of being generically split depends on
Tits indices of the group $G$ (see \cite{Ti66}).
We prove (see Theorem~\ref{mainthm}) that
the motive of such variety splits as a direct sum
of (twisted) copies of a certain indecomposable motive $\RR_X$.
The prime $p$ here is a torsion prime of $G$.
In this way we obtain motivic analogs of a Pfister quadric
for $p=3$ ($\E_6/P_6$) and $p=5$ ($\E_8/P_8$).

Our work was mostly motivated by the paper of N.~Karpenko \cite{Ka01},
where he provided an elementary construction of the motive $\RR_X$ for
a Pfister quadric $X$. In fact, Theorem~\ref{mainthm} can be viewed 
as a further generalization
of Karpenko's and Rost's ideas and is based on Rost Nilpotence Theorem.
The key idea is to reduce the problem of decomposing the motive of $X$
to the problem of providing a certain family of algebraic cycles on $X$. 
The latter 
turns out to be a purely combinatorial problem related with properties
of characteristic map studied by Grothendieck,
Demazure, Karpenko and Merkurjev (see \cite{Gr58}, \cite{De74}, \cite{KM05}).

   
The paper is organized as follows.
In section~2 we provide general arguments concerning lifting of idempotents. 
In section~3 we study Chow motives of generically split varieties
and prove the main result Theorem~\ref{mainthm}. 
In the next section~4 we provide properties of projective
homogeneous varieties which will be extensively used in
the applications. 
In particular, we relate the question of indecomposability of $\RR_X$
with the question of existence of zero-cycles of degree one on $X$.
In section~5 we provide applications of Theorem~\ref{mainthm}
to various examples of projective homogeneous varieties.
Namely, to Severi-Brauer varieties, Pfister quadrics and
their neighbors, maximal orthogonal Grassmannians, varieties
of types $\G$, $\F$ and strongly inner $\E_6$.
In section~6 we decompose the motive of a variety of type $\E_8$
which splits by an extension of degree $p=5$.
In the last section we reduce the computations of motives
of general flag varieties to the examples of sections 5 and 6.







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5


\section{Decomposition of the diagonal}\label{method}

Let $p$ be an integer (not necessary a prime) and $\zz \to \zz/p$ be a specialization map.
The goal of the present section can be stated as follows.
Given a ring $R$ satisfying certain conditions, 
its  specialization $\bar R$
modulo $p$ and a family of pair-wise orthogonal idempotents
$q_i'$ in $\bar R$ such that $\sum_i q_i'=1_{\bar R}$, produce
a family of pair-wise orthogonal 
idempotents $q_i$ in $R$ such that $\bar q_i=q_i'$
and $\sum_i q_i=1_R$.

\begin{ntt}
Let $A^*$ be a graded commutative ring
which is a free $\zz$-module of rank $N$. 
By $A^n$ we denote the $n$-th graded component of $A$
(component of codimension $n$).
Fix a homogeneous $\zz$-basis $\{e_k\}_{k=1\ldots N}$ of $A$ which will
be called standard.
Assume there is a linear form $\deg\colon A\to \zz$ together
with a (dual) homogeneous $\zz$-basis $\{e_l^\vee\}_{l=1\ldots N}$ of $A$
such that
$$
\deg(e_ke_l^\vee)=\delta_{k,l}\text{ for all $k$ and $l$.}
$$ 
From now on fix an integer $p$ and 
denote by $\bar A$ the reduction of $A$ modulo $p$.
\end{ntt}

\begin{dfn}\label{rbalan}
Given an element $\rho\in \bar A^r$ of codimension $r$
a family $\{\alpha_i\}_{i\in \I}$ of homogeneous elements 
 in $\bar A$
is called {\em $\rho$-balanced} 
if it satisfies the following conditions
\begin{enumerate}
\item[(a)] the set of indices $\I$ 
is equipped with an involution $i\mapsto i^{\bullet}$, $p\cdot\#\I=N$;
\item[(b)] for all $m=0\ldots 2(p-1)$ and $i, j\in\I$
$$
\deg(\rho^{m}\alpha_i\alpha_{j^\bullet})
\begin{cases}
\in (\zz/p)^\times,& \text{if $m=p-1$ and $i=j$}\\
=0,& \textrm{otherwise}.
\end{cases}$$
To simplify the notation denote $C_i=\deg(\rho^{p-1}\alpha_i\alpha_{i^\bullet})\in (\zz/p)^\times$.
\end{enumerate}
\end{dfn}

\begin{lem}\label{rhobasis} Let
$\{\alpha_i\}_{i\in\I}$ be a $\rho$-balanced family of elements in $\bar A$.
The elements $\rho^m\alpha_i$, $m= 0\ldots p-1$, $i\in\I$ 
form a homogeneous basis of $\bar A$ over $\zz/p$ called a $\rho$-basis.
\end{lem}

\begin{proof}
By \ref{rbalan}.(b) the elements $\rho^m\alpha_i$ 
are linear independent.
Indeed, if $x=\sum_{m,i} c_{m,i}\rho^m\alpha_i$ 
then $c_{m,i}=
\tfrac{1}{C_i}\deg(x\rho^{p-1-m}\alpha_{i^\bullet})$.
Therefore, if $x=0$ then $c_{m,i}=0$ for all $i\in\I$ and $m=0\ldots p-1$.
Let $M$ be a quotient module of $\bar A$ by a submodule
generated by $\{\rho^m\alpha_i\}$.  
Since $p\cdot\#\I=N$, $M/dM=0$ for all prime divisors $d$ of $p$.
Therefore $M=0$ and the lemma is proved.  
\end{proof}

\begin{ntt}
Let $\{\bar e_k\}$ be the reduction modulo $p$ of the standard basis $\{e_k\}$.
Consider a $\rho$-basis $\{\rho^m\alpha_i\}$ of $\bar A$.
Let $\{\rho^m\alpha_i\}^{(n)}\subset \bar A^n$ 
be its component of codimension $n$. 
Let $D^{(n)}\in (\zz/p)^\times$ be the determinant of the 
matrix which expresses $\{\rho^m\alpha_i\}^{(n)}$
in terms of the components of the basis $\{\bar e_k\}$ of codimension $n$.
\end{ntt}

\begin{dfn}
We say a $\rho$-balanced family 
$\{\alpha_i\}_{i\in\I}$
is {\em unimodular in codimension $n$} if $D^{(n)}\pm 1$.
We say a $\rho$-balanced family is {\em unimodular} if it is
unimodular in all codimensions $n$.
\end{dfn}

\begin{lem}[Basis Lifting Property] 
Assume there is a basis
$\{f_k'\}$ of $\bar A^n$ such that
the transition matrix from $\{\bar e_k\}^{(n)}$ to $\{f_k'\}$ 
has determinant $\pm 1$.
Then there exists a $\zz$-basis $\{f_k\}$ of $A^n$ such that
$\bar f_k = f_k'$, where $k=1\ldots l$.
\end{lem}

\begin{proof}
Since $\zz/p$ is a semi-local ring, the group 
$\SL_l(\zz/p)$ is generated by elementary matrices.
Hence, the reduction map $\SL_l(\zz)\to \SL_l(\zz/p)$
is surjective and the lemma follows.
\end{proof}

\begin{cor}\label{liftbas} Assume we are given an unimodular $\rho$-balanced
family of elements in $\bar A$. Then the respective $\rho$-basis of $\bar A$
can be lifted to a homogeneous basis of $A$.
\end{cor}

\begin{proof}
Apply the lemma to all graded components
of the $\rho$-basis.
\end{proof}

\begin{rem} Observe that for $p=2$, $3$, $4$, $6$ any $\rho$-balanced
family of elements is unimodular. Therefore any $\rho$-basis can be lifted
to a homogeneous basis over $\zz$. 
\end{rem}

\begin{ntt}\label{compprod}
We endow the abelian group $A\otimes A$ with a (graded) ring structure
by means of the composition product 
$(\alpha\otimes \beta)\circ(\alpha'\otimes\beta') \mapsto
\deg(\alpha\beta')\cdot
(\alpha'\otimes \beta)$ and denote the resulting ring by $R$.
The identity element of $R$ we call a {\it diagonal} 
and denote it by $\Delta_R$.
Recall that an element $q\in R$ is called an {\em idempotent}
if $q\circ q=q$. We say two idempotents $q$ and $q'$ are {\em orthogonal}
if $q\circ q'=q'\circ q=0$.
\end{ntt}

\begin{ntt}
Assume we are given an unimodular $\rho$-balanced family in $\bar A$.
Let $\{u_{m,i}\}$ be a homogeneous $\zz$-basis of $A$ obtained
by lifting the $\rho$-basis 
$\{\bar u_{m,i}=\rho^m\alpha_i\}$ according to Corollary~\ref{liftbas}.
Let $\{u_{m,i}^\vee\}$ be the respective dual basis, i.e., defined by
\begin{equation}\label{dualbas}
\deg(u_{m,i}u_{m',i'}^\vee)=\delta_{m,m'}\delta_{i,i'}.
\end{equation} 
By \ref{rbalan}.(b) we obtain
that 
\begin{equation}\label{dualel}
\bar u_{m,i}^\vee=\tfrac{1}{C_i}\rho^{p-1-m}\alpha_{i^\bullet} \in \bar A.
\end{equation}
\end{ntt}

\begin{prop}\label{idlift} 
Homogeneous elements $q_i=\sum_{m=0}^{p-1} u_{m,i}\otimes u_{m,i}^\vee$,
$i\in\I$,
are pair-wise orthogonal idempotents in $R$ 
such that
$\sum_{i\in\I} q_i=\Delta_R$. Moreover, 
\begin{equation}\label{projbar}
\bar q_i= \tfrac{1}{C_i}
(\sum_{m=0}^{p-1} \rho^m\otimes \rho^{p-1-m})\cdot 
(\alpha_i\otimes\alpha_{i^\bullet})
\end{equation}
\end{prop}

\begin{proof} The fact that they are pair-wise orthogonal idempotents follows
from the definition of the composition product and (\ref{dualbas}).
On the other hand, since $\Delta_R$ is the identity of $R$, 
$\Delta_R=\sum_{m,i} u_{m,i}\otimes u_{m,i}^\vee$.
Formula \eqref{projbar} then
follows from the definition of $u_{m,i}$ and (\ref{dualel}).
\end{proof}

\begin{rem}\label{lifti} Consider an arbitrary $\rho$-basis. 
In each codimension $n$ such that $D^{(n)}\neq \pm 1$
choose an element $\rho^{m_n}\alpha_{i_n}$ and replace
it by $\tfrac{1}{D^{(n)}}\rho^{m_n}\alpha_{i_n}$. So obtained family
of elements will form a unimodular homogeneous basis of $\bar A$.
Let $\{v_{m,i}\}$ and $\{v_{m,i}^\vee\}$ be the respective
liftings to $A$. By definition $\bar v_{m_n,i_n}=\tfrac{1}{D^{(n)}} \bar u_{m_n,i_n}$
and $\bar v_{m_n,i_n}^\vee=D^{(n)} \bar u_{m_n,i_n}^\vee$.
Observe that by means of this procedure the idempotents 
$\bar q_i$ of \eqref{projbar} will not change.
  
\end{rem}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5


\section{Motives of generically split varieties}

In the present section we apply the results of the
previous section to Chow motives of generically split varieties.
For the definition and properties of Chow motives we refer
to the papers \cite{Ma68} and \cite[sect.2]{Ka01}. 

\begin{ntt} 
Recall that a category of Chow motives is defined as follows.
First, we define 
the category of \emph{correspondences} $Corr(F)$ over a field $F$.
Its objects are smooth projective varieties over $F$. 
For morphisms, called correspondences,
we set $\Mor(X,Y):=\CH_{\dim X}(X\times Y)$.
For any two correspondences $\alpha\in \CH(X\times Y)$ and 
$\beta\in \CH(Y\times Z)$ we define the composition product
$\beta\circ\alpha\in \CH(X\times Z)$ as
\begin{equation}
\beta\circ\alpha ={\pr_{13}}_*(\pr_{12}^*(\alpha)\cdot \pr_{23}^*(\beta)),
\end{equation}
where $\pr_{ij}$ denotes the projection
on the $i$-th and $j$-th factors of $X\times Y\times Z$ 
respectively and ${\pr_{ij}}_*$, ${\pr_{ij}^*}$ denote
the induced push-forwards and pull-backs for Chow groups. 
Taking the pseudo-abelian completion of $Corr(F)$  
we obtain the category of {\em effective} Chow motives $Chow^{eff}(F)$.
Its objects
are pairs $(X,p)$, where $X$ is a smooth projective variety
and $p\in\Mor(X,X)$ is an idempotent, that is, $p\circ p=p$. 
The morphisms between two objects $(X,p)$ and $(Y,q)$ 
are the compositions 
$q\circ \Mor(X,Y) \circ p$. Denote $\M(X)=(X,\id)$.
Following this definition the motive of a projective line
splits as a direct sum $\M(\pl^1)=\M(pt)\oplus L$ of the motive of a point
and some motive $L$ called {\em Lefschetz} motive.
Formally inverting $L$ one obtains the category of Chow motives
$Chow(F)$.

By construction, $Chow(F)$ is a tensor additive category,
where the tensor product is given by the usual product
$(X,p)\otimes(Y,q)=(X\times Y, p\times q)$. 
For a given motive $M$ denote by $M(k)$ its twist, i.e.,
the tensor product $M\otimes L^{\otimes k}$.
For a cycle $\alpha$ denote by $\alpha^t$ the
corresponding transposed cycle.
\end{ntt}

\begin{dfn}
Let $X$ be a smooth irreducible projective variety over a field $F$.
We say $L$ is a {\em splitting field} of $X$ if
the Chow motive $\M(X_L)$ of the scalar extension $X_L=X\times_F L$
is isomorphic to a direct sum
of twisted Lefschetz motives and the natural
map $\res_{L/F}: \M(X)\to \M(X_L)$ is an isomorphism
after tensoring with $\mathbb{Q}$.
We say $X$ is {\em generically split} over $F$ if its function field
$K=F(X)$ is a splitting field of $X$.
\end{dfn}

\begin{rem}
According to Lemma~\ref{gensplem} any projective homogeneous $G$-variety
of a linear algebraic group $G$ of inner type over $F$ which 
splits over the generic point of $X$ 
provides an example of a generically split variety.
\end{rem}

Let $L$ be a splitting field of $X$.
In view of the notation of the previous section 
set $A=\CH(X_L)$ and denote by $A_{rat}$ the image of the restriction map
$\res_{L/F}\colon \CH(X)\to\CH(X_L)$. 
An element (cycle) of $A_{rat}$ will be called {\em rational}.

\begin{ntt}
According to \cite{Ma68} and \cite[Rem.~5.6]{KM05}
the ring $A=\CH(X_L)$ is a free $\zz$-module,
there is K\"unneth formula 
$\CH(X_L\times X_L)=A\otimes A$ and 
Poincare duality. The latter means that for a given basis $\{e_k\}$ of $A$
there is the dual basis $\{e_l^\vee\}$ with respect to
the degree map $\deg: \CH(X_L)\to\zz$.
Observe that the composition product introduced in \ref{compprod}
coincides with the composition of endomorphisms 
in $\End(\M(X_L))$, hence,
$\End(\M(X_L))=(A\otimes A)^{(\dim X)}$.
\end{ntt}



The following facts will be extensively used in the sequel

\begin{lem}\label{prat} Let $L$ be a splitting field of a variety $X$
of finite degree over $F$.
Then for any $\alpha\in A$ we have $[L:F]\cdot \alpha \in A_{rat}$.
\end{lem}

\begin{proof} Choose an element $\alpha$ in $A=\CH(X_L)$.
Since $\res_{L/F}\otimes \mathbb{Q}$ is surjective
there exists an element $\beta\in\CH(X)$ and a non-zero integer $m$
such that $\res_{L/F}(\beta)=m\alpha$.
By projection formula 
$$
m\cdot \cores(\alpha)=\cores(\res(\beta))=[L:F]\cdot \beta.
$$
Applying $\res$ to both sides we obtain
$m(\res(\cores(\alpha)))=m[L:F]\cdot\alpha$. Therefore
$\res_{L/F}(\cores(\alpha))=[L:F]\cdot\alpha$ and the lemma is proven.
\end{proof}

\begin{lem}\label{gensp}
Let $X$ be a generically split variety and 
$L$ be a splitting field of $X$. Let 
$\pr_1: (A\otimes A)^r \to A^r$ be the projection on the
first summand in the direct sum decomposition
$(A\otimes A)^r=(A^r\otimes A^0)\oplus\ldots\oplus (A^0\otimes A^r)$.
Then for any $\rho \in A^r$ there exists a rational preimage 
of $\rho$ by means of $\pr_1$.
\end{lem}

\begin{proof} Lemma follows from the commutative diagram
$$
\xymatrix{
\CH^r(X\times X) \ar[r]^{\res_{L/F}}\ar[d]_{p_1^*}& \CH^r(X_L\times X_L)\ar[d] 
\ar@{=}[r]^-{\text{K\"unneth}} & (A\otimes A)^r \ar[d]^{\pr_1}\\
\CH^r(X_K)\ar[r]^-\simeq & \CH^r(X_{LK}) \ar[r]^-\simeq & A^r
}
$$
where $K$ is the quotient field of $X$, the first
vertical arrow $p_1^*$ is taken from localization sequence for Chow groups
and, hence, is surjective and the isomorphisms come from the fact
that $L$ and $K$ are splitting fields of $X$.
\end{proof}

\begin{lem}[Rost Nilpotence] (see \cite{VZ06})
Let $X$ be a generically split variety and 
$E/F$ be a field extension. Then
the kernel of the natural map
$$
\End(\M(X)) \to \End(\M(X_E))
$$
consists of nilpotent elements.
\end{lem}

\begin{cor}\label{rostcor}
Let $X$ be a generically split variety and $E/F$ be a field extension.
Given a direct summand $M$ of $\M(X)$ and a family $q_i\in\End(M_E)$ 
of rational pair-wise orthogonal idempotents with
$\sum_i q_i=\Delta_E$ there exist a family of pair-wise orthogonal idempotents
$p_i\in\End(M)$ such that $\sum_i p_i=\Delta$ and $\res_{E/F}(p_i)=q_i$.
Moreover, if $M$, $N$ are direct summands of $\M(X)$ and
$\varphi$ is a rational isomorphism between the $N_E$ and $M(i)_E$ for
some $i$, then $N\simeq M(i)$.
\end{cor}




Now we are ready to state and prove the main result of this paper

\begin{thm}\label{mainthm}
Let $X$ be a generically split variety.
Assume there exists a splitting field $L$ of $X$ 
of a prime degree $p$ together with 
an element $\rho\in\bar A^r=\CH^r(X_L)/p$ for some $r$ such that
\begin{enumerate}
\item[(i)] for all $s<r$ we have $\bar A^s=\bar A_{rat}^s$ and
$\bar A^r=\langle \bar A_{rat}^r,\rho\rangle$;
\item[(ii)] there is a unimodular 
$\rho$-balanced family $\{\alpha_i\}_{i\in\I}$ in $\bar A_{rat}$.
\end{enumerate}
Then the Chow motive of $X$ with integral coefficients 
can be expressed as the direct sum
$$
\M(X)\simeq\bigoplus_{i\in\I} \RR(\codim\alpha_i),$$
where
the motive $\RR$ is indecomposable
if and only if there are no
zero-cycles of degree one on $X$. 
\end{thm}

\begin{proof} 
By Lemma~\ref{prat} all cycles divisible by $p$
are rational. Hence, to prove that a certain cycle $\alpha$
is rational in $R=A\otimes A$ 
is the same as to prove it in $\bar R=\bar A\otimes_{\zz/p} \bar A$.

Consider two cycles $\rho\otimes 1-\eps(1\otimes\rho)$, where $\eps=\pm 1$.
We claim that one of them is rational (cf. \cite[Lemma~5.1]{Ka01}).
Indeed, since $X$ is generically split
by Lemma~\ref{gensp}
there exists a rational preimage of $\rho\otimes 1$
$$
\rho\otimes 1+\delta(1\otimes\rho)+\sum_k \mu_k\otimes \nu_k +
1\otimes \gamma \in \pr_1^{-1}(\rho\otimes 1),$$
where $\delta\in\zz/p$, codimensions of $\mu_k$ and
$\nu_k$ are less than $r$ and $\gamma \in \bar A_{rat}^r$. 
By (i)
the cycles $\mu_k$ and $\nu_k$ are rational. Therefore
$\tau:=\rho\otimes 1+\delta(1\otimes\rho)$ is rational.
If $\delta=-1$ take $\eps=1$. Otherwise the cycle 
$\tau+\tau^t=(1+\delta)(\rho\otimes 1+1\otimes\rho)$
is rational and therefore 
we can take $\eps=-1$.
Hence, we obtain a rational cycle in $\bar R^{r(p-1)}$
\begin{equation}\label{rcycl}
\sigma=(\rho\otimes 1-\eps(1\otimes\rho))^{p-1}=\sum_{m=0}^{p-1} \varepsilon^m (\rho^m\otimes \rho^{p-1-m}).
\end{equation}



Applying Proposition~\ref{idlift} to the unimodular $\rho$-balanced family
$\{\alpha_i\}$ in $\bar A_{rat}$, we obtain the family of pair-wise
orthogonal idempotents $q_i$ in
$R$ with $\bar q_i$ given by formula 
\eqref{projbar}. Note that idempotents
$q_i$ are rational. Indeed,
$q_i=q_i'\circ q_i'$, where 
$q_i'=\sum_m \varepsilon^m(u_{m,i}\otimes u_{m,i}^\vee)$ and
the reduction
$\bar q_i'=\tfrac{1}{C_i}\sigma\cdot(\alpha_{i}\otimes\alpha_{i^\bullet})$
is rational.


Note that the motives $(X_L,q_i)$ and $(X_L,q_{i'})(\codim(\alpha_i)-\codim(\alpha_{i'}))$
are isomorphic by means of an isomorphism $\varphi_{i,i'}$
given by 
\begin{equation}\label{isoms}
\varphi_{i,i'}=\sum_m \varepsilon^m u_{m,i'}\otimes u_{m,i}^\vee.
\end{equation}
Observe that 
$\bar \varphi_{i,i'}= \tfrac{1}{C_i}
\sigma \cdot 
(\alpha_{i'}\otimes\alpha_{i^\bullet})$ and hence $\varphi_{i,i'}$ is rational.

Applying Corollary~\ref{rostcor} to the idempotents $q_i$
and isomorphisms $\varphi_{i,i'}$
we obtain the desired decomposition with $\RR=(X,p_{i_0})$, where 
$i_0$ is a unique index such that
$\codim(\alpha_{i_0})=0$. 

Finally observe that the ring of endomorphisms
$\End(\RR_L)$ is additively generated by $u_{m,i_0}\otimes u_{m,i_0}^\vee$,
$m=0\ldots p-1$. Assume there is a rational idempotent 
$q=\sum_m a_m(u_{m,i_0}\otimes u_{m,i_0}^\vee)$. A straightforward computation
shows that $a_m=0$, $1$ for all $m$. Consider the rational cycle 
$q\cdot q^t=(\sum_m a_m)\cdot (pt\otimes pt)$, where the class $pt=u_{p-1,i_0}
\in\CH_0(X_L)$
is represented by a zero-cycle of degree one. The push-forward induced
by the first projection sends this cycle to $(\sum_m a_m)\cdot pt$.
Denote $a=\sum_m a_m$.
Note that $q$ is non-trivial if and only if $a \neq 0\mod p$.
Therefore, the motive $\RR$ is decomposable if and only if the cycle 
$a\cdot pt$ is rational, where $a\neq 0\mod p$.
But the cycle $p\cdot pt$ is rational by Lemma~\ref{prat} and
since $a$ and $p$ are coprime, $pt$ must be rational as well.
Conversely, if $pt$ is rational we can take $q=1\otimes pt\in\End(\RR_L)$.
The latter implies that the motive of a point $\zz$ 
is a direct summand in $\RR$, i.e.,
$\RR$ is decomposable. 
\end{proof}

\begin{rem}\label{primecond} The fact that $p$ is a prime integer 
was used in the proof two times.
First, to prove that the cycle $\sigma$ from \eqref{rcycl}
is rational. Indeed, to obtain the formula \eqref{rcycl} 
we used the congruence $\binom{p-1}{i}=(-1)^i \mod p$
which holds only if $p$ is prime.
Second, to prove that the cycle 
$pt$ is rational. In fact, we proved that the cycle
$\mathop{g.c.d.}(a,p)\cdot pt$ is
rational. Then using the fact that $p$ is prime and $a<p$
we concluded that $\mathop{g.c.d.}(a,p)=1$.
\end{rem}

\begin{rem}\label{assunim} If one drops the assumption of unimodularity for
the $\rho$-balanced family the proof still works by Remark~\ref{lifti},
excluding the lifting
of isomorphisms \eqref{isoms}.
So one produces a motivic decomposition $\M(X)\simeq\oplus_{i\in\I}\RR_i$,
where each $\RR_i$ has a property that modulo $p$ (but not integrally)
it can be identified with
$\RR_i\otimes\zz/p\simeq \RR_0(\codim\alpha_i)\otimes \zz/p$. 
\end{rem}

\begin{prop}\label{compar} 
Let $X$ and $Y$ satisfy the hypothesis of Theorem~\ref{mainthm}
with the same splitting field $L$ of a prime degree $p$ and 
the same codimension $r$.
Assume that one of two cycles
$\rho_X\otimes 1-\eps (1\otimes \rho_Y)\in \CH^r(X_L\times Y_L)$,  
$\eps=\pm 1$
is rational.
Then the motives $\RR_X$ and
$\RR_Y$ appearing in the decomposition of $\M(X)$ and $\M(Y)$ respectively
are isomorphic.
\end{prop}

\begin{proof} Let $\{\alpha_i\}_{i\in\I}$ and $\{\alpha_j\}_{j\in\mathcal{J}}$
denote the respective $\rho_X$- and $\rho_Y$-balanced families.
The explicit isomorphism between $\RR_X=(X,p_{i_0})$ and
$\RR_Y=(Y,p_{j_0})$ is given by the cycle
$$
\varphi_{i_0,j_0}=\sum_m \varepsilon^m u_{m,i_0}\otimes u_{m,j_0}^\vee,
$$
where $u_{m,i}$ and $u_{m,j}$ are liftings of the respective $\rho_X$-
and $\rho_Y$-basis. Note that modulo $p$ it coincides with the rational cycle
$$
\bar \varphi_{i_0,j_0}= \tfrac{1}{C_{j_0}}
(\rho_X\otimes 1-\eps (1\otimes \rho_Y))^{p-1} \cdot 
(\alpha_{i_0}\otimes\alpha_{{j_0}^\bullet})
$$
and so is rational. It remains to apply Corollary~\ref{rostcor} to the
isomorphism $\varphi_{i_0,j_0}$.
\end{proof}

\begin{lem}\label{p23}  
Let $X$ and $Y$ satisfy the hypothesis of Theorem~\ref{mainthm}
with the same splitting field $L$ of a prime degree $p$ and 
the same codimension $r$.
If $F(Y)$ is a splitting field of $X$ and $F(X)$
is a splitting field of $Y$
then the cycle $\rho_X\otimes 1-\eps (1\otimes \rho_Y)$ is rational
for some $\eps\in (\zz/p)^\times$. In particular, for $p=2$ and $3$
one of two cycles $\rho_X\otimes 1 -\eps(1\otimes\rho_Y)$,
$\eps=\pm 1$ is rational.
\end{lem}

\begin{proof} The proof is the same as in the beginning
of the proof of Theorem~\ref{mainthm} using a slightly modified
version of Lemma~\ref{gensp}, where instead of the product 
$A\otimes A=\CH(X_L)\otimes\CH(X_L)$ one considers $\CH(X_L)\otimes\CH(Y_L)$.
\end{proof}








%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Projective homogeneous varieties}\label{examples}

In the present section we list several important properties
of projective homogeneous varieties needed to apply Theorem~\ref{mainthm}.
From now on we assume $G$ is a simple linear algebraic group over
a field $F$.

\begin{lem}\label{surj} 
Let $X$ be a projective homogeneous variety of a group of inner type
over a field $F$ then
for any field extension $E/F$ the natural map
$\res_{E/F}: \M(X)\to\M(X_E)$ 
induces an isomorphism after tensoring with $\mathbb{Q}$.
\end{lem}

\begin{proof}
Consider an isomorphism induced by Chern character
$\CH(X)\otimes\qq\simeq\K_0(X)\otimes\qq$.
According to the results of paper \cite{Pa94} the map
$\res_{E/F}: K_0(X)\otimes \qq\to K_0(X_E)\otimes \qq$ is an isomorphism.
 To finish the proof apply Manin's identity principle.
\end{proof}
 
\begin{lem}\label{gensplem} Let $X$ be a projective homogeneous 
$G$-variety where $G$ is a group of inner type over $F$ which splits
over the function  field of $X$ 
Then $X$ is generically split.
\end{lem}

\begin{proof} Follows from \cite[Corollary~7.6]{CGM} and Lemma~\ref{surj}.
\end{proof}

\begin{lem}\label{ratpic} 
Let $G$ be a group of strongly inner type over $F$ and $X$
be a projective homogeneous $G$-variety. Then for any splitting
field $L$ of $X$ the natural map $\res_{L/F}: \Pic(X) \to \Pic(X_L)$
is an isomorphism. 
\end{lem}

\begin{proof} 
Since for any field extension $E/L$ the induced map
$\res_{E/L}$ is an isomorphism we may assume $L=F_s$ is
the separable closure of $F$.
Consider an exact sequence from \cite{MT95}
\begin{equation}\label{seqpic}
0\to \Pic(X)\xra{\res_{L/F}}\Pic(X_L)^\Gamma \xra{\alpha}\Br(F),
\end{equation}
where $\Gamma$ denotes the absolute Galois group.
For all groups of strongly inner type 
the image of $\alpha$ is trivial by \cite{Ti71} and
\cite[Prop.~2.5]{MT95}.
Since $G$ is of inner type, 
$\Gamma$ acts trivially on $\Pic(X_L)$ and, hence, 
$\Pic(X_L)\simeq\Pic(X_L)^\Gamma$ and the lemma is proven.
\end{proof}

In the case of projective homogeneous varieties we can say
more about the motive $\RR$ appearing in Theorem~\ref{mainthm}.
Namely,

\begin{prop}\label{zerocycl} 
Let $X$ be a projective homogeneous 
$G$-variety such that
over its function  field the group $G$ splits. 
Assume we are under the hypothesis of Theorem~\ref{mainthm}.
If the motive $\RR$ appearing in \ref{mainthm} is decomposable then
it decomposes as the direct sum
$$
\RR\simeq\bigoplus_{m=0}^{p-1}\zz(rm).
$$
\end{prop}

\begin{proof}
Indeed, if $X$ possess a zero-cycle of degree one, then there exists
a field extension $E/F$ of degree $l$ coprime to $p$ such that $X_E$
has a rational point. 
By \cite[Thm.~3.10 and Cor.~3.9]{KR94}
since $X$ is generically split, $E$ is a splitting field of $X$
and, therefore, the cycles 
$l\alpha$ are rational for all $\alpha\in\CH(X_E)$ by Lemma~\ref{prat}.
Since $l$ and $p$ are coprime, 
all cycles in $\CH(X_E)$ are rational. By Rost Nilpotence
$\RR\simeq\bigoplus_{m=0}^{p-1}\zz(rm)$.
\end{proof}

\begin{rem} We refer to \cite[Q.~0.2 and Section~9]{To04} for
the discussion about existence of zero cycles of degree one
on anisotropic projective homogeneous varieties.
\end{rem}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Applications and Examples}

As the first application of Theorem~\ref{mainthm} we obtain
the prime degree case of the main result of paper
\cite{Ka96}
concerning motives of Severi-Brauer varieties

\begin{cor}[Karpenko]
Let $D$ be a division algebra of a prime degree $p$ over a field $F$
and $\SB(M_n(D))$ be the Severi-Brauer variety of
a central simple algebra $M_n(D)$. 
Then
$$
\M(\SB(M_n(D)))\simeq \bigoplus_{i=0}^{n-1} \M(\SB(D))(ip),
$$
where $\M(\SB(D))$ is indecomposable.
\end{cor}

\begin{proof}
The variety $X$ is generically split and there is a splitting
field $L$ of $X$ of degree $p$. 
Set $A=\CH(\pl^{np-1})$. Set $\rho=h$ to be the
class of a hyperplane section of $\pl^{np-1}$,
$\I=\{0,1,\ldots,n-1\}$, $i^\bullet=n-1-i$ and $\alpha_i=h^{pi}$.
Note that $h^p$ is rational since $h^p=\res(j_*(1))$,
where $j\colon\SB(M_{n-1}(D))\hookrightarrow \SB(M_n(D))$
is the closed embedding and $j_*$ is the induced push-forward.

Hence, $\{\alpha_i\}_{i\in\I}$ form a $\rho$-balanced family in $\bar A$.
Note that the respective $\rho$-basis coincides with
the standard basis of $A$ (which consists of powers of $h$)
and, hence, is unimodular.
Applying Theorem~\ref{mainthm} to the unimodular
$\rho$-balanced family $\{\alpha_i\}$ we obtain the decomposition
$$
\M(\SB(M_n(D)))\simeq \bigoplus_{i=0}^{n-1} \RR(ip)
$$
for some indecomposable motive $\RR$.

To show that the motive $\RR$ is isomorphic 
to the motive $\M(\SB(D))$ we apply Proposition~\ref{compar} to 
$X=\SB(M_n(D))$, $Y=\SB(D)$ and the rational cycle
$h_X\otimes 1 - 1\otimes h_Y$ which generates the kernel
of the map $\alpha$ of the exact sequence \eqref{seqpic},
where $X$ is taken to be the product $\SB(M_n(D))\times\SB(D)$.
\end{proof}

\begin{rem}
In view of Remark~\ref{primecond}
to prove an arbitrary degree case
it is enough to show that the cycle $\sigma$ from \eqref{rcycl}
is rational. 
In \cite[Claim~7.8]{Ka00} it is shown
that $\sigma$ lies
in the subring generated by rational cycles $(h_X\otimes 1-1\otimes h_Y)$,
$c_i(T_X)\otimes 1$ and $1\otimes c_i(T_Y)$, where
$c_i(T)$ is the $i$-th Chern class
of the respective tangent bundle, and, therefore, is rational.
\end{rem}

As the next application we obtain one of the results
of paper \cite{Ro98}
about motives of Pfister quadrics

\begin{cor}[Rost]\label{pfist}
Let $X$ be a $n$-fold 
Pfister quadric or its maximal neighbor.
Then
$$\M(X)\simeq\bigoplus_{i=0}^{\dim X-r}\RR(i),$$
where the motive $\RR$ is decomposable
if and only if $X$ is isotropic, and in this case
$\RR\simeq\zz\oplus\zz(r)$, where $r=2^{n-1}-1$. The motive $\RR$
is called Rost motive.
\end{cor}

\begin{proof}
By \cite[Cor.~9.10]{EKM} and Lemma~\ref{gensplem}
the variety $X$ is generically split.
Moreover, it is split by a quadratic field extension $L/F$, i.e.,
in this case $p=2$.
Let $A=\CH(X_L)$ and $\bar A$ be its specialization modulo $2$. 
Set $\rho\in \bar A^r$ to be the class of a maximal totally isotropic
subspace of the respective quadratic space.
Set $\I=\{0,1,\ldots,\dim X -r\}$ and the involution 
$i \mapsto i^\bullet=\dim X -r-i$.
By Lemma~\ref{ratpic} all powers $h^i$, $i=0\ldots\dim X$ 
of a class of a hyperplane section $h\in\bar A^1$ are rational cycles in 
$\bar A$.
Set $\alpha_i=h^i$ for $i\in \I$.
Knowing the multiplicative structure of $\bar A$ (see \cite{Ka01})
one immediately
obtains that $\{\alpha_i\}_{i\in \I}$ form a unimodular $\rho$-balance family
of elements. Applying Theorem~\ref{mainthm} we finish
the proof.
\end{proof}

The next example deals with maximal orthogonal Grassmannians

\begin{cor} 
Let $(V,q)$ be a quadratic space corresponding to a maximal $n$-fold
Pfister neighbor $q$.
Let $X$ be the respective maximal orthogonal Grassmannian, 
i.e., the variety
of maximal totally isotropic subspaces in $(V,q)$.
Then                       
$$\M(X)\simeq\bigoplus_{i=0}^{\frac{r(r-1)}2}\RR(i)^{\oplus a_i},$$
where the motive $\RR$ is the Rost motive
of the maximal $n$-fold Pfister neighbor $q$. In particular, $\RR$
is decomposable
if and only if $q$ is isotropic, and in this case
$\RR\simeq\zz\oplus\zz(r)$, where $r=2^{n-1}-1$.
The integers $a_i$ are the coefficients at $t^i$ in 
$\prod_{k=1}^{r-1}(1+t^k)$.
\end{cor}

\begin{proof}
According to Tits diagrams $X$ is generically split and, moreover,
splits by a quadratic field extension $L$ which splits $q$.
The Chow ring $A$ of $X_L$ has the
following presentation (see \cite{EKM})
$$
A=\zz[e_1,\ldots,e_r]/\langle e_i^2=e_{2i},\,2i\le r;
e_i^2=0,\,2i> r\rangle,$$
where $\codim e_i=i$.
For a subset $I=\{i_1,\ldots,i_k\}\subset\{1,\ldots,r\}$ we denote
$e_I=e_{i_1}\ldots e_{i_k}$. The elements $e_I$ form an
additive basis of $A$. Moreover, for two cycles
$e_I$, $e_J$ with $\codim e_I+\codim e_J=\dim X=\frac{r(r+1)}2$ we have
$$
e_Ie_J=\begin{cases}
pt, &J=\{1,\ldots,r\}\setminus I,\\
0, &\text{otherwise.}
\end{cases}$$    
According to \cite[Corollary~66.7]{EKM}
the cycles $e_1,\ldots,e_{r-1}$ are rational.
Define $\rho=e_r$.
Let $\I$ be the set of all subsets of $\{1,\ldots,r-1\}$
with involution being set-theoretical complement.
Set $\alpha_I=e_I$, $I\in\I$.
Observe that $\{\alpha_I\}_{I\in\I}$ form a $\rho$-balanced
family of elements in $\bar A$. Applying Theorem~\ref{mainthm}
we obtain the desired motivic decomposition.

To show that $\RR$ is the Rost motive
apply Proposition~\ref{compar} and Lemma~\ref{p23}
to the variety $X$ and the Pfister quadric $Y$.
\end{proof}

\begin{rem}
In the case $r=3$ the proof goes through for the  maximal orthogonal
Grassmannian corresponding to a quadratic form $q$ with
trivial even Clifford algebra $C_0(q)$.
\end{rem}



Next we provide the proof
of the main result of paper \cite{Bo03} concerning
motives of $\G$-varieties.

\begin{cor}[Bonnet]
Let $G$ be a twisted form of a group of type $\G$.
Consider a projective homogeneous $G$-variety $X$
corresponding to a maximal parabolic subgroup of $G$
(there are two such varieties).
Then
$$
\M(X)\simeq\bigoplus_{i=0}^2 \RR(i),
$$
where $\RR$ is the Rost motive of the maximal $3$-fold
Pfister neighbor. In particular, $\RR$ is decomposable
iff $X$ is isotropic, and in this case
$\RR\simeq\zz\oplus\zz(3).$
\end{cor}


\begin{proof}
Again by Tits diagrams $X$ is generically split and there is
a quadratic field extension $L/F$ which splits $X$, i.e., $p=2$.
Using Pieri formula \ref{Pieri} it can be shown
that
the Chow ring $\bar A$ of $X_L$ modulo 2 is isomorphic 
to the Chow ring of the maximal neighbor of the
3-fold Pfister quadric given by the norm of the underlying octonion
algebra.
Moreover, since $G$ is of strongly inner type over $F$,
by Lemma~\ref{ratpic} the subgroup of rational cycles of $\bar A$
is the same as for the Pfister neighbor.
The proof then follows from the proof of Corollary~\ref{pfist}.
To show that the corresponding motives $\RR$
are the same for both varieties of type $\G$
use Proposition~\ref{compar} and
Lemma~\ref{p23} together with 
the fact that $G$ splits over the function fields 
of both.
\end{proof}

Now we give a short proof of the main result of paper \cite{NSZ}
about the motives of $\F$-varieties.

\begin{cor}\label{f4case} 
Let $G$ be a twisted form of a split group of type $\F$
which splits by a cubic field extension. Let $X$ be a projective
homogeneous $G$-variety
corresponding to a maximal parabolic subgroup of $G$ given by
the first or the last vertex
of the Dynkin diagram. Then
$$
\M(X)\simeq\bigoplus_{i=0}^7\RR(i),
$$
where the motive $\RR$ is the same for both varieties.
$\RR$ is decomposable
if and only if $X$ is isotropic, and in this case
$\RR\simeq\zz\oplus\zz(4)\oplus\zz(8).$
\end{cor}

\begin{proof}
By Tits diagrams $X$ is generically split. Indeed,
if $G_{F(X)}$ is not split, then $G_{F(X)}$ must have Tits index  
$\mathrm{F}_{4,1}^{21}$ with anisotropic kernel $\mathrm{Spin}(q)$,
where $q$ is a quadratic form of dimension $7$.
By the hypothesis $\mathrm{Spin}(q)$ must split by a cubic field extension,
contradiction. 

Let $L$ be a cubic field extension which splits $G$ ($p=3$).
Since $G$ is of strongly inner type over $F$ by Lemma~\ref{ratpic}, the class
of a hyperplane section $h$ generating the Picard group $\Pic(X_L)$
is rational.
Consider the Chow ring $\bar A$ modulo 3.
Using Pieri formula \ref{Pieri} we show that $\langle h^i\rangle =\bar A^i$
for $i<4$ and $h^4=g_4^{(1)}+g_4^{(2)}$
is the sum of standard generators of $\bar A^4$. Set $\rho=g_4^{(1)}$.
Define $\I=\{0,\ldots,7\}$ and the involution $i^\bullet=7-i$, $i\in \I$.
Set $\alpha_i=h^i$.
Direct computations show that the elements $\{\alpha_i\}_{i\in\I}$
form a unimodular $\rho$-balanced family.
Applying Theorem~\ref{mainthm} we obtain the desired motivic decomposition.

As in the case of $\G$ observe that the group $G$ splits over the function
fields of both varieties.
\end{proof}

Following the arguments of $\F$-case one immediately
obtains $\E_6$-case, where $\E_6$ is of strongly inner type. Namely,

\begin{cor}\label{e6case}
Let $G$ be a strongly inner form of a split group of type $\E_6$
which splits by a cubic field extension.
Let $X$ be a projective homogeneous variety corresponding to
a maximal parabolic subgroup of $G$ given by the first or the last
vertex of the Dynkin diagram. Then
$$
\M(X)\simeq\bigoplus_{i=0}^8\RR(i),
$$
where the motive $\RR$ is the same as in $\F$-case. In particular,
$\RR$ is decomposable if and only if $X$ is isotropic, and in this case
$\RR\simeq\zz\oplus\zz(4)\oplus\zz(8).$
\end{cor}

\begin{proof}
By Tits diagrams $X$ is generically split. Indeed,
if $G_{F(X)}$ is not split, then $G_{F(X)}$ must have Tits index  
$^1{\E_{6,2}^{28}}$ with anisotropic kernel $\mathrm{Spin}(q)$,
where $q$ is a quadratic form of dimension $8$.
By the hypothesis $\mathrm{Spin}(q)$ must split by a cubic field extension,
contradiction. 

Let $L$ be a cubic field extension which splits $G$ ($p=3$).
Since $G$ is of strongly inner type over $F$ by Lemma~\ref{ratpic}, the class
of a hyperplane section $h$ generating the Picard group $\Pic(X_L)$
is rational.
Consider the Chow ring $\bar A$ modulo 3.
Using Pieri formula \ref{Pieri} we show that $\langle h^i\rangle =\bar A^i$
for $i<4$ and $h^4=g_4^{(1)}+g_4^{(2)}$
is the sum of standard generators of $\bar A^4$. Set $\rho=g_4^{(1)}$.
Define $\I=\{0,\ldots,8\}$ and the involution $i^\bullet=8-i$, $i\in \I$.
Set $\alpha_i=h^i$.
Direct computations show that the elements $\{\alpha_i\}_{i\in\I}$
form a unimodular $\rho$-balanced family.
It remains to apply Theorem~\ref{mainthm}.

The motive $\RR$ is the same as in the $\F$-case, since
$X$ splits over the function field of the $\F$-variety $Y$
and vice versa. Indeed, the underlying Jordan algebra splits
over the function fields of $X$ and $Y$.
\end{proof}

\section{The case of $\E_8$}

Let $G$ be a group of type $\E_8$,
which splits by a field extension of degree $5$. Such a group
exists according to \cite[A.6.]{Ga06}.
Let $X$ be a projective $G$-homogeneous variety of 
parabolic subgroups of type $P_8$. The variety
$X$ has dimension $57$ and is the minimal possible $\E_8$-variety
in the sense of dimension. In the present section we prove

\begin{thm}\label{e8}
The Chow motive of $X$ is isomorphic to
$$
\M(X)\simeq \bigoplus_{i=0\ldots 1, j=0\ldots 23} \RR_{i,j},
$$
where the motive $\RR_{i,j}$ is decomposable if and only if
$X$ has a zero-cycle of degree one, and in this case $\RR_{i,j}$
is isomorphic to the direct sum
$$
\RR_{i,j}\simeq (\zz\oplus\zz(6)\oplus\zz(12)\oplus\zz(18)\oplus\zz(24))(10i+j).
$$
Moreover, considered with $\zz/5$-coefficients it is isomorphic to
$$
\RR_{i,j}\otimes \zz/5\simeq \RR_{0,0}(10i+j)\otimes \zz/5.
$$ 
\end{thm}



\begin{proof}
Let $L$ be a splitting field of $X$ of degree $5$, i.e., $p=5$.
Since $G$ is of strongly inner type over $F$, by Lemma~\ref{ratpic} 
the Picard group of $X_L$ is generated by a rational cycle $h$.
Consider the Chow group $\bar A=\CH(X_L)/5$.
Pieri formula \ref{Pieri}
shows that $\bar A^i=\langle h^i\rangle$
for $i\le 5$ and in codimension $6$
it is generated by two elements. One of them is $h^6$ and, hence,
is rational. According to \ref{mainthm}.(i) we can take any other generator
as a cycle $\rho$. 
In terms of the standard basis of $\bar A^6$ (see Appendix), 
$h^6=g_6^{(1)}+g_6^{(2)}$ and
we choose $\rho=g_6^{(1)}$.

We set $\I=\{0,1\}\times\{0,\ldots,23\}$ and the involution
$(i,j)^\bullet=(1-i,23-j)$.
Define a $\rho$-balanced family $\{\alpha_{i,j}\}_{(i,j)\in \I}$ as
$\alpha_{i,j}=c_{10}^ih^j$, where $c_{10}$ is the 10-th Chern class
of the tangent bundle over $X_L$. 
The choice
of the 10-th Chern class is not accidental. Indeed, this is the next
codimension (after 6) where the number of generators jumps by one.
Direct computations (using formulae from Appendix)
show that $\{\alpha_{(i,j)}\}_{(i,j)\in \I}$ is a $\rho$-balanced family.
The respective $\rho$-basis is non-unimodular, since there are determinants
which are equal to $\pm 2 \mod 5$. Applying Theorem~\ref{mainthm}
and Remark~\ref{assunim} we finish the proof of the Theorem.
\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Chow motives of fibered spaces}

The main result of  paper \cite{CPSZ} says that 
under certain restrictions
the Chow motive
of a twisted flag variety $X$ can be expressed
in terms of the motive of a 'minimal' flag, i.e., the one which
corresponds to a maximal parabolic subgroup.
These restrictions cover almost all twisted flag varieties corresponding 
to groups of types $\A_n$ and $\B_n$ together with
some examples of types $\C_n$, $\G$
and $\F$.
In the present section using the result of Edidin and Graham \cite{EG97}
about cellular fibrations
we extend this set of examples for groups of inner types $\C_n$, $\D_n$ and
exceptional groups. We also provide a shortened and uniform proof
of the results obtained in \cite{CPSZ}.
More precisely, we prove the following 

\begin{thm}\label{mainthmf} Let $G_0$ be a split simple linear algebraic group 
over a field $F$
and $P\subset P'$ parabolic subgroups of $G_0$. 
For a cocycle $\xi\in Z^1(F,G_0)$ 
let $G={}_\xi G_0$ denote the twisted form of $G_0$ and
$X={}_\xi (G_0/P)$ and $X'={}_\xi (G_0/P')$ the respective twisted flag varieties.
If $G$ splits over the generic point of $X'$, then
the Chow motive $\M(X)$ of $X$ is isomorphic (non-standardly)
to a direct sum
of twisted copies of the motive $\M(X')$, i.e.,
$$
\M(X) \simeq \bigoplus_{i\in\mathcal{I}} \M(X')(a_i).
$$
\end{thm}

\begin{proof}
Consider the standard map 
$\pi: X\to X'$. 
Since $G$ splits over $F(X')$,
the fiber of $\pi$ over $F(X')$ splits, i.e.,
is isomorphic to $P'/P$. Moreover, there is an open subset of $X'$
over which $G$ splits. 
Since $G$ acts transitively on $X'$,
the map $\pi$ is a locally trivial fibration
whose fiber is isomorphic to $P'/P$, i.e., has a decomposition
into affine cells. To finish the proof apply the proposition below.
\end{proof}

\begin{prop} Let $f: Y\to X$ be a smooth projective locally trivial
fibration with $X$ smooth and projective whose 
fiber $\mathcal{F}$ has a decomposition into affine cells. 
Then $\M(Y)$ is (non-standardly) isomorphic to 
$\M(X)\otimes \M(\mathcal{F})$. 
\end{prop}

\begin{proof}
We follow the proof of \cite[Prop.~1]{EG97}. Define the morphism
$$
\varphi: \bigoplus_{i\in\mathcal{I}} \M(X)(a_i) \to \M(Y)
$$
to be the direct sum $\varphi=\bigoplus_{i\in\mathcal{I}}\varphi_i$,
where each $\varphi_i$ is given by the cycle 
$[\pr_Y^*(B_i)\cdot \Gamma_f]\in\CH(X\times Y)$ 
produced from  the graph cycle $\Gamma_f$ and the chosen (non-standard)
basis $\{B_i\}_{i \in \mathcal{I}}$
of $\CH(Y)$ over $\CH(X)$.
The realization of $\varphi$ coincides exactly with
an isomorphism of abelian groups $\CH(X)\otimes\CH(\mathcal{F})\to 
\CH(Y)$ constructed in \cite[Prop.~1]{EG97}.
By Manin's identity principle \cite{Ma68} $\varphi$ is an isomorphism 
and we are done.
\end{proof}
 
Let $P=P_\Theta$ be a standard parabolic subgroup corresponding
to a subset $\Theta$ of the Dynkin diagram $\mathcal{D}$ and
$P_k$, $k\in\mathcal{D}$, stands for the maximal parabolic subgroup 
$P_{\mathcal{D}\setminus \{k\}}$. Enumeration of roots follows Bourbaki.
To simplify the notation
we will write $X_\Theta$ for the form of $G_0/P_\Theta$
twisted by $\xi\in Z^1(F,G_0)$ and 
$X_k$ for the form of $G_0/P_k$ respectively. 
By definition all forms $G={}_\xi G_0$ of $G_0$
are of inner type over $F$. Since we are interested in twisted flag
varieties we may assume $G_0$ is adjoint.
As an immediate consequence of the theorem we obtain
that


\begin{cor} For a group $G$ of inner type $\mathcal{D}$ over $F$
and twisted flag varieties $X_\Theta$ and $X_k$
there is an isomorphism of Chow motives
$$
\M(X_\Theta) \simeq \bigoplus_{i\in\mathcal{I}}\M(X_k)(a_i),
$$
where the number $k\in\mathcal{D}$ corresponds to a parabolic
subgroup $P_k$ containing $P_\Theta$ 
(i.e., $k\in\mathcal{D}\setminus \Theta$) 
and satisfies the following conditions depending on $\mathcal{D}$
\begin{itemize}
\item[$\A_n$:] $k$ is prime to the degree of a
splitting field of $G$ (cf. \cite[Thm.~2.1]{CPSZ});
\item[$\B_n$:] $k=n$ (cf. \cite[Thm.~2.9]{CPSZ}); 
if $G=\PGO^+(q)$, where $q$ is a maximal Pfister neighbor, $k$ is arbitrary;
\item[$\C_n$:] $k$ is odd (cf. \cite[Thm.~2.11]{CPSZ});
\item[$\D_{2m+1}$:] $k=2m$, $2m+1$;
\item[$\D_{2m}$:] $k=2m-1$, $2m$ in the case 
$G=\PGO^+(q)$, where $q$ is a non-singular quadratic form over $F$;
moreover, if $q$ is a Pfister form, then $k$ is arbitrary; 
\item[$\G$:] $k$ is arbitrary (cf. \cite[Thm.~2.13]{CPSZ});
\item[$\F$:] $k\neq 4$ or $G$ splits by a cubic field extension 
(cf. \cite[Thm~2.14]{CPSZ});
\item[$\E_6$:] $k=\begin{cases} 3,\,5 & \\
2,\,4 & \text{ if $G$ is of strongly inner type}\\
1,\,6 & \text{ if $G$ is of strongly inner type and 
splits by a cubic field extension}.
\end{cases}$
\item[$\E_8$:] $k$ is arbitrary in the case $G$ splits by a field
extension of degree $5$.
\end{itemize}
\end{cor}

\begin{proof} 
In order to apply the theorem to the fibration $X_\Theta\to X_k$
we need to assume that the group $G$ splits over the generic point of $X_k$.
The latter condition can be extracted from the Tits diagrams of $G$
(see \cite{Ti66}). Namely, it holds if
the only Tits diagram of $G$ containing
a circled vertex numbered by $k$ is the split one, i.e., 
having all vertices circled.

We provide the proof for the most complicated cases
$G_0=\E_6$ and $\E_8$ only.
All the other cases can be treated similarly and are left to the reader.

Analyzing Tits diagrams for a general $\E_6$
we obtain immediately the cases 
$k=3$, $5$. For strongly inner forms of $\E_6$ using the diagrams provided
in \cite[5.2]{Ti90} we obtain the cases $k=2$, $4$.
In the cases $k=1$, $6$ the only possible non-split diagram
is ${}^1\E_{6,2}^{28}$ whose anisotropic kernel is $\Spin(q)$,
where $q$ is a 3-fold Pfister form. But according to Springer's Theorem
$q$ can not split by a cubic field extension.

In the case of $\E_8$ the anisotropic kernel of $G_{F(X_k)}$ is trivial
since it can't split by a field extension of degree 5 by \cite[Thm.~5.1]{To04}.
\end{proof}

Let $S$ be a coefficient ring 
such that Krull-Schmidt theorem holds 
in the category of Chow motives of projective homogeneous 
$G$-varieties. 
For instance, $S$ is a field or a discrete valuation ring 
(see \cite[Cor.~9.7]{CM04}). By $M\otimes S$ we denote a motive $M$
considered with $S$-coefficients. 

\begin{cor} For a group $G$ of inner type $\mathcal{D}$ over $F$
there is an isomorphism of Chow motives with $S$-coefficients
$$
\M(X_\Theta)\otimes S \simeq \bigoplus_{i\in\mathcal{I}}(\RR\otimes S)(a_i),
$$
where an indecomposable motive $\RR$ (depending only on $G$)
and the subsets $\Theta$ are taken from the following list
\begin{itemize}
\item[$\A_n$:] For $G=\PGL_{n/d}(D)$, where $D$ is a division algebra of index
$d$,
take $\RR=\M(\SB(D))$ and $\Theta$ such that 
$\mathop{gcd}(\Theta \cup \{d\})=1$ (cf. \cite[Prop.~2.4]{CPSZ}). 
\item[$\C_n$:] For $G=\PGU_{2n/d}(D,h)$, 
where $D$ is a division algebra of index
$d=2^r$, with an anti-hermitian form $h$,
take $\RR=\M(\SB(D))$ and $\Theta$ such that
$\mathcal{D}\setminus\Theta$ contains an odd number.
\item[$\B_n$] and $D_n$: Assume $G=\PGO^+(q)$, where $q$ is a 
Pfister form or its maximal neighbor. Take $\RR$ to be
the Rost motive corresponding to $q$ and arbitrary $\Theta$.
\item[$\G$:] For $G=\Aut(\mathbb{O})$, where $\mathbb{O}$ is an
octonion algebra take $\RR$ to be the Rost motive of a maximal
Pfister neighbor of $N_\mathbb{O}$ and arbitrary $\Theta$.
\item[$\F$:] Assume $G=\Aut(J)$, where a Jordan
algebra $J$ splits by a cubic field extension.
Take $\RR$ to be the integral motive
appearing in the decomposition of Cor.~\ref{f4case}
and arbitrary $\Theta$. 
\item[$\E_6$:] Assume $G=\Aut(N_J)_{ad}$ and $J$ splits 
by a cubic field extension. Take the same $\RR$ as in the $\F$-case
and arbitrary $\Theta$.
\item[$\E_8$:] Assume $G$ splits by a field
extension of degree $5$. Take $\RR$ to be an integral
motive appearing in the decomposition of
Theorem~\ref{e8} and arbitrary $\Theta$.
\end{itemize}
\end{cor}

\begin{proof}
Consider the variety $X_k$, where $k=8$ if $G$ has a type $\E_8$ and 
$k=1$ otherwise.  
The motive $M(X_k)$ with $S$-coefficients
splits as a direct sum of copies of $\RR\otimes S$
by the results of the previous section.
Our conditions on $G$ and $\Theta$ imply that
$G$ splits over $F(X_\Theta)$ and over $F(X_k)$. 
If $k\in \mathcal{D}\setminus\Theta$ we apply
the previous corollary. Otherwise consider the flag 
$X_{\Theta\setminus \{k\}}$. According to the theorem
the motive $\M(X_{\Theta\setminus \{k\}})\otimes S$ has two different decompositions
$$
\bigoplus_{i\in\mathcal{I}} \M(X_\Theta)(b_i)\otimes S \simeq \M(X_{\Theta\setminus \{k\}})\otimes S
\simeq \bigoplus_{j\in\mathcal{J}} \M(X_k)(b_j')\otimes S\simeq 
\bigoplus_{l\in\mathcal{L}} \RR(b_l'')\otimes S.
$$ 
By Krull-Schmidt Theorem for motives with $S$-coefficients 
\cite[Cor.~9.7]{CM04} the motive $\M(X_\Theta)\otimes S$ has the desired
decomposition.
\end{proof}





\section{Appendix}
\begin{ntt}
Let $G_0$ be a split simple algebraic group of rank 
$n$ defined over a field $F$.
We fix a maximal split torus $T$ in $G_0$ and a Borel subgroup $B$ of $G_0$
containing $T$ and defined over $F$. 
We denote by $\Phi$ the root system of $G_0$,
by $\Pi=\{\alpha_1,\ldots,\alpha_n\}$ the set of simple roots of $\Phi$
corresponding to $B$, by $W$ the Weyl group, and by $S=\{s_1,\ldots,s_n\}$
the corresponding set of fundamental reflections. 
Enumeration of roots follows Bourbaki.

Let $P=P_\Theta$ be a (standard) parabolic subgroup corresponding
to a subset $\Theta\subset\Pi$, i.e., $P=BW_\Theta B$, where
$W_\Theta=\langle s_\theta, \theta\in\Theta\rangle$.
As $P_i$ we denote the maximal parabolic subgroup 
$P_{\Pi\setminus\{\alpha_i\}}$.


Denote $W^\Theta=\{w\in W\mid\forall\, s\in\Theta\quad l(ws)=l(w)+1\},$
where $l$ is the length function.
The pairing 
$W^\Theta\times W_\Theta\to W$ with $(w,v)\mapsto wv$
is a bijection and $l(wv)=l(w)+l(v)$.
It is easy to see that $W^\Theta$ consists of all representatives
in the left cosets $W/W_\Theta$ which have minimal length.
 


\end{ntt}

\begin{ntt}[Standard basis]
Now consider the Chow ring of a projective
homogeneous variety $G_0/P_\Theta$. 
It is well known that $\CH(G_0/P_\Theta)$ is a free abelian group
with a basis given by classes of varieties $X_w=\overline{PwP/P}$ 
that correspond 
to the elements $w\in W^{\Theta}$. We call such a basis standard.
The codimension of the basis element $[X_w]$ equals $l(w_{\theta})-l(w)$, where $w_\theta$ is the longest element of $W_\Theta$.
Standard generators of codimension $n$ will be denoted by $g_n^{(i)}$.

There exists a natural injective pull-back homomorphism
$$\CH(G_0/P)\to\CH(G_0/B)$$
$$[X_w]\mapsto[X_{ww_\theta}]$$
\end{ntt}

The following results provide tools to perform computations in the Chow
ring $\CH(G_0/P_\Theta)$.
\begin{ntt}[Poincar\'e duality]
In order to multiply two basis elements $h$ and $g$ of 
$\CH(G_0/P_\Theta)$ such that 
$\codim h+\codim g=\dim G_0/P_\Theta$ we use the following formula 
(see \cite[1.4]{Ko91}):
$$
[X_w]\cdot [X_{w'}]= \delta_{w,w_0w'w_\theta}\cdot [X_1].
$$
\end{ntt}

\begin{ntt}[Pieri formula]\label{Pieri}
In order to multiply two basis elements of $\CH(G_0/B)$ one of which
is of codimension $1$ we use the following formula 
(see \cite[Cor.~2 of 4.4]{De74}):
$$
[X_{w_0s_\alpha}][X_w]=
\sum_{\beta\in\Phi^+,\, l(ws_\beta)=l(w)-1}
\langle\beta^\vee,\overline{\omega}_\alpha\rangle[X_{ws_\beta}],
$$
where $\alpha$ is a simple root and the sum runs through
the set of positive roots $\beta\in\Phi^+$, $s_\alpha$ denotes
the simple reflection corresponding to $\alpha$
and $\w_\alpha$ is the fundamental weight corresponding
to $\alpha$. Here $[X_{w_0s_\alpha}]$ is the element of codimension $1$.
\end{ntt}

\begin{ntt}[Characteristic map]
Let $\PP=\PP(\Phi)$ be the weight space. 
We denote as $\w_1,\ldots\w_l$ the basis of $\PP$
consisting of fundamental weights. 
The symmetric algebra $S^*(\PP)$ is isomorphic to
$\mathbb{Z}[\w_1,\ldots\w_l]$. 
The Weyl group $W$ acts on $\PP$, hence, on $S^*(\PP)$. 
Namely, for a simple root $\alpha_i$,
$$w_{\alpha_i}(\w_j)=
\begin{cases}
\w_i-\alpha_i, & i=j, \\
\w_j, & \text{otherwise}.
\end{cases}$$
We define a linear map $c\colon S^*(\PP)^{W_{P_\Theta}}\to\CH^*(G_0/P_\Theta)$
as follows.
For a homogeneous $W_{P_\Theta}$-invariant $u\in\mathbb{Z}[\w_1,\ldots,\w_l]$ 
$$
c(u)=\sum_{w\in W^\Theta,\, l(w)=\deg(u)}\Delta_w(u)[X_{w_0ww_\theta}],
$$ 
where for $w=s_{i_1}\ldots s_{i_k}$ we denote by $\Delta_w$
the composition of derivations
$\Delta_{s_{i_1}}\circ\ldots\circ\Delta_{s_{i_k}}$ and
the derivation $\Delta_{s_i}\colon S^*(\PP)\to S^{*-1}(\PP)$ is 
defined by $\Delta_{s_i}(u)=\frac{u-s_i(u)}{\alpha_i}$.
\end{ntt}

\begin{ntt}[Tangent bundle]\label{chern}
Consider the tangent vector bundle $T$ over $G_0/P_\Theta$ and
observe that $$c(T)=c(\prod_{\gamma\in\Sigma_u(P_\Theta)}(1+t\gamma)),$$
where $\Sigma_u(P_\Theta)$ is the set of (positive) roots lying in the
unipotent radical of the parabolic subgroup $P_\Theta$
and $c(T)$ denotes the total Chern class.
Since the tangent vector bundle is rational, the
cycles $c_i(T)\in\CH^i(G_0/P_\Theta)$ are rational.
\end{ntt}

The formulae above were implemented in Maple package
\cite{NS}. Most of the computations in the paper were performed and checked
using this package.

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\end{document}



