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\title
[Canonical $p$-dimension of algebraic groups]
{Canonical $p$-dimension of algebraic groups}



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


\date
%{\today}
{December, 2004}

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

\author
[A. Merkurjev]
{Alexander S. Merkurjev}


\address
{Laboratoire de Math{\'e}matiques de Lens\\
Facult{\'e} des Sciences Jean Perrin\\
Universit{\'e} d'Artois\\
rue Jean Souvraz SP 18\\
62307 Lens Cedex\\
FRANCE}

%\address
%{currently:
%Institute for Advanced Study\\
%1 Einstein Dr.\\
%Princeton, NJ 08540\\
%USA}

%\address
%{{\it Web page:}
%{\tt www.math.uni-bielefeld.de/\~{ }karpenko}}

\email
{karpenko@euler.univ-artois.fr}



\address
{Department of Mathematics\\
University of California\\
Los Angeles, CA 90095-1555\\
 USA}

\email
{merkurev@math.ucla.edu}

\thanks
{The James D. Wolfensohn Fund and The Ellentuck Fund support is
acknowledged by the first author. The first author was partially
supported by the European Community's Human Potential Programme
under contract HPRN-CT-2002-00287, KTAGS}

\thanks{The work of the second author has been supported by the
NSF grant DMS \#0355166.}


\begin{document}

\begin{abstract}
We describe a way to compute the $p$-relative version of the
Berhuy-Reichstein canonical dimension for an arbitrary split
semisimple algebraic group over an arbitrary field of an arbitrary
characteristic ($p$ is any prime integer). The canonical
$p$-dimension is computed for all split simple groups of classical
types.
\end{abstract}



\maketitle

\tableofcontents

The notion of the canonical dimension of an algebraic structure
was introduced by Berhuy and Reichstein in
\cite{Berhuy-Reichstein}. The canonical dimension measures the
size of generic splitting fields of the structure. The formal
definition is given in Section \ref{Canonical dimension of
determination functions}. Here we present two basic examples:

\smallskip

$\bullet$ Let $X$ be a scheme over a field $F$. A field extension
$L/F$ is called a \emph{splitting field of $X$}, if $X$ has a point
over $L$. A splitting field $L$ is called \emph{generic}, if for
any splitting field $K$ of $X$ there exists an $F$-place $L\to K$.
The \emph{canonical dimension of $X$} is the minimum of the
transcendence degree (over $F$) of all generic splitting fields of
$X$.

$\bullet$ Let $G$ be an algebraic group over $F$. The
\emph{canonical dimension of $G$} is the maximum of the canonical
dimension of all principal homogeneous varieties ($G$-torsors),
defined over field extensions of $F$.

\smallskip

When dealing with a given algebraic structure, we usually have
finitely many ``significant" prime integers involved. For example,
such primes associated with an algebraic group $G$ are the
\emph{torsion prime integers of $G$} (see Remark
\ref{grothendieck}). In order to locate contribution of a prime
integer $p$ to the canonical dimension, we define \emph{canonical
$p$-dimension} in a similar fashion.

It turns out that canonical dimension and $p$-dimension of an
arbitrary regular complete variety $X$ is closely related to the
algebraic cycles on $X$ (see Corollaries \ref{eshcho i takaja cd}
and \ref{eshcho i takaja cdp}). We express canonical $p$-dimension
of a generically cellular variety in terms of its Chow group (see
Theorem \ref{main-Nikita}).

The main result of the paper is Theorem \ref{compute}, giving a
recipe to compute canonical $p$-dimension of an arbitrary split
semisimple algebraic group over an arbitrary field (of arbitrary
characteristic). The
%precise
values of the canonical $p$-dimension
are given for all split simple groups of classical type (see
\S\ref{Split simple groups of classical types}).





\medskip
\noindent {\sl Acknowledgments:} Authors thank V.~Chernousov, I.~
Panin, and M.~Rost for helpful remarks.
The first author thanks the Princeton Institute for Advanced
Study: the results were obtained and the paper written during his
stay at the Institute;
he was also inspired by the lecture course of A. A. Suslin on the
Rost norm varieties.

\section
{Agreements and preliminaries}

\subsection
{Varieties}
\label{Varieties}

We refer as {\em schemes}  to separated schemes of finite type
over a field (there is no restrictions on the field, its
characteristic is arbitrary).

{\em Variety} in the paper is an integral scheme.

For a scheme $X$, the integer $d(X)$ is defined as the g.c.d. of
degree of all closed points on $X$; for a prime integer $p$,
$d_p(X)$ is the $p$-primary part of $d(X)$.

\subsection
{Chow groups}

Let $X$ be a scheme over a field $F$.
We write $\CH(X)$ for the integral Chow group of $X$ (see \cite{Fulton}).
Fixing a prime $p$, we write
$\Ch(X)$ for the modulo $p$ Chow group:
$$\Ch(X)=\CH(X)/p\cdot\CH(X)\;.$$
Furthermore, we write $\Ch(\overline X)$ (resp. $\CH(\overline X)$)
for the colimit of
$\Ch(X_L)$ (resp. $\CH(X_L)$) with $L$ running  over all field extensions $L/F$, and
we write $\BCh(X)$
(resp. $\BCH(X)$)
for the image of the restriction homomorphism
$\res\!:\Ch(X)\to\Ch(\overline X)$
(resp. $\CH(X)\to\CH(\overline X)$).
The group $\BCH(X)$ is called the
{\em reduced} Chow group of $X$;
the group $\BCh(X)$ is called the
{\em modulo $p$  reduced} Chow group of $X$.
Note that
$$
\BCh(X)=\BCH(X)/\big(\BCH(X)\cap p\CH(\overline X)\big)
$$
is not the same as $\BCH(X)/p\BCH(X)$.

%We say that an element of $\Ch(\overline X)$ is {\em rational}, if it actually
%lies in the subgroup $\BCh(X)\subset\Ch(\overline X)$.

\subsection
{Places}
\label{Places}

Let $K$ be a field. A {\em valuation ring} $R$ of $K$ is a subring
$R\subset K$, satisfying $K=R\cup(R\setminus\{0\})^{-1}$. Any
valuation ring is local; $R=K$ is a trivial example of a valuation
ring.


Given two fields $K$ and $L$, a {\em place} $K\to L$ is a local
ring homomorphism $\pi\!:R\to L$ of a valuation ring $R\subset K$
(an embedding of fields is a trivial example of a place).

If $K$ and $L$ are extensions of a field $F$, an \emph{$F$-place}
(or {\em place over $F$}) is a place $K\to L$ with $\pi$ defined
and identical on $F$.

Places are composable: if $K\to L$ is a place, given by a ring
homomorphism $\pi$, and $L\to E$ a place to a third field $E$,
given by a homomorphism $\rho$ of a ring $S\subset L$, then the
composition is the place $K\to E$, given the homomorphism
$\rho\compose\pi\!: \pi^{-1}(S)\to E$, defined on the valuation
ring $\pi^{-1}(S)$. In particular, any place $L\to E$ can be
restricted to any subfield $K\subset L$.

In this paper, a place $K\to L$ is said to be {\em geometric}, if it can be
represented as a composition of places with valuation rings being
discrete valuation rings.


\subsection
{Places and rational morphisms}

Let $X$ and $Y$ be varieties over a field $F$.
If $X$ is complete, then for any valuation ring $R$ of $F(X)$
there exists an $F$-morphism $\Spec R\to X$ \cite[ch. II th. 4.7]{Hartshorne};
therefore an
$F$-place $F(X)\to F(Y)$ produces a
rational morphism $Y \to X$.

Vice versa, if there is a rational morphism $Y \to X$ and $X$ is
regular, then there exists a {\em geometric} $F$-place $F(X)\to
F(Y)$. Indeed, since $X$ is regular at the image $x$ of the
generic point of $Y$, there exists a system of local parameters
around $x$, which produces a geometric place $F(X)\to F(x)$;
composing with the embedding $F(x)\subset F(Y)$, we get a required
place $F(X)\to F(Y)$.





\section
{Canonical dimension of determination functions}
\label{Canonical dimension of determination functions}

Let $F$ be a field, $\Fields_F$ the category of all field
extensions of $F$. Let $\Bin$ be the category of the subsets of a
$1$-elemental set $0$. A {\em determination function} $D$ over $F$
is a continuous functor $\Fields_F\to\Bin$, where by continuity we
mean that $D$ commutes with the filtered colimits. In other words,
$D$ is a rule assigning to each $E\in\Fields_F$ a value
$D(E)\in\{\emptyset,0\}$ such that
\begin{itemize}
\item
if $D(E)=0$ for some $E$, then $D(E')=0$ for any $E'$ admitting an
$F$-embedding $E\to E'$;
\item
(continuity property) if $D(E)=0$ for some field $E$ covered by a
(possibly infinite) filtered family of subfields $E_i$, then
$D(E_i)=0$ for some $E_i$.
\end{itemize}


A field $E\in\Fields_F$ is called a {\em splitting field} of a
determination function $D$, if $D(E)=0$. A splitting field $E$ of
$D$ is called {\em generic}, if for any splitting field $L$ there
exists an $F$-place $E\to L$. If $D$ has at least one generic
splitting field, {\em canonical dimension} $\cd(D)$ of $D$ is
defined as the minimum of the transcendence degree (over $F$) of
all generic splitting fields of $D$; if $D$ does not admit a
generic splitting field, we set $\cd(D)=\infty$.


\begin{lemma}
\label{generic splitting subfield}
For a given determination function $D$,
any its splitting field, which is a subfield of a generic splitting field, is also
generic.
Besides, any splitting field contains a finitely generated
splitting field and $\cd(D)=\infty$ only if $D$ does not admit generic
splitting.
\end{lemma}

\begin{proof}
If $E$ is a generic splitting field and $E'$ a splitting field contained
in $E$,
then for any splitting field $L$, restricting a place $E\to L$ to
$E'$, we get a place $E'\to L$; therefore $E'$ is also generic.

Any splitting field contains a finitely generated splitting field
by the continuity of the determination function.

If $D$ has a generic splitting field,
then, taking a finitely generated splitting subfield,
we get a finitely generated generic splitting field, showing that
$\cd(D)$ is finite.
\end{proof}

A determination function $D$ over $F$ is {\em split}, if $D(F)=0$.
In this case, $F$ is a generic splitting field of $D$ and
$\cd(D)=0$.

Our basic example of a determination function is the determination function
associated with a scheme $X$ over $F$:
$$
L\mapsto
\begin{cases}
\emptyset, \text{ if $X(L)=\emptyset$;}\\
0, \text{ otherwise.}
\end{cases}
$$

\emph{Canonical dimension $\cd(X)$} of an $F$-scheme $X$ is
defined as the canonical dimension of the associated determination
function.


\begin{example}[{\cite[th. 4.3]{KM}}]
\label{cd(quadric)}
Let $F$ be a field of characteristic $\ne2$.
Let $X$ be an anisotropic smooth projective quadric over $F$.
Then
$\cd(X)=\dim X-\iw_1(X)+1$, where $\iw_1(X)$ is the first Witt index of $X$.
\end{example}

Let $\Sets$ be the category of pointed sets and let $k$ be a
field. A functor
$$\cF\!:\Fields_k\to\Sets$$ is called {\em continuous}, if
it commutes with filtered colimits. If $\cF$ is a continuous
functor, then for any $F\in\Fields_k$ and $\alpha\in\cF(F)$, we
get a determination function $D_\alpha$ over $F$ by setting
$$
D_{\alpha}(L)=
\begin{cases}
0, \text{ if $\alpha_L$ is the distinguished point of the set $\cF(L)$;}\\
\emptyset, \text{ otherwise.}
\end{cases}
$$
{\em Berhuy-Reichstein  canonical dimension} of a continuous
functor $\cF$ \cite[\S10]{Berhuy-Reichstein} is the supremum of
$\cd(D_\alpha)$ for all $F$ and $\alpha\in\cF(F)$. If $G$ is an
algebraic group over the field $k$, \emph{canonical dimension of
$G$} is defined as canonical dimension of the (continuous) functor
$\Tors_G$, taking a field $F$ to the set of isomorphism classes
$\Tors_G(F)$ of $G$-torsors over $F$. We note that canonical
dimension of an algebraic group $G/k$ is not the same as canonical
dimension of the underlying variety of $G$ (which is always $0$
because $G(k)\ne\emptyset$).

\section
{Notion of canonical $p$-dimension}

Let us fix an arbitrary prime $p$ and refer to a splitting field $E$ of a determination
function
$D$ as {\em $p$-generic},\footnote{Our notion of $p$-generic splitting is based on
on the notion of {\em $p$-generic splitting varieties} of
symbols in a modulo $p$ Milnor's $K$-group of a field, introduced in \cite{Rost-NV}.}
if for any splitting field $L$ of $D$
there exists a finite field extension $L'/L$
of degree prime to $p$ admitting a place $E\to L'$.
Replacing generic splitting fields by the $p$-generic ones in the definitions of section
\ref{Canonical dimension of determination functions}, we get a modified
notion of canonical dimension which we call {\em canonical $p$-dimension}
and denote $\cd_p$.

We refer to a finite field extension as {\em $p$-coprime}, if its degree
is not divisible by $p$.

The following two Lemmas are useful when working with $\cd_p$.

\begin{lemma}[{cf. \cite[lemma 3.3]{KM}}]
\label{polia} Let $K$ be an arbitrary field, $p$ a prime,
$K'/K$ a $p$-coprime field extension, and $L/K$ an
arbitrary field extension.
Then there exists a field $L'$,
containing $K'$ and $L$, such that the extension $L'/L$ is also
$p$-coprime.
\end{lemma}

\begin{proof}
We almost repeat the proof of \cite[lemma 3.3]{KM}, where the case
of $p=2$ was treated. We may assume that $K'$ is generated over
$K$ by one element; let $f(t)\in F[t]$ be its minimal polynomial.
Since the degree of $f$ is coprime with $p$, there exists an
irreducible divisor $g\in L[t]$ of $f$ over $L$ such that
$\deg(g)$ is coprime with $p$ as well. We set $L'=L[t]/(g)$.
\end{proof}


\begin{lemma}
\label{places} Let $K$ be a field extension of $F$ of finite
transcendence degree over $F$; let $K\to L$ be a geometric
$F$-place and let $K'$ be a $p$-coprime field extension of $K$.
Then there exists a $p$-coprime field extension $L'/L$ such that
the place $K\to L$ extends to a place $K'\to L'$.
\end{lemma}

\begin{proof}
By Lemma \ref{polia} it suffices to prove Lemma
\ref{places} in the case where the place $K\to L$ is surjective
and its valuation ring $R$ is a discrete valuation ring.
Also it is clear, that is suffices to consider only two cases:
(1) $K'/K$ is purely inseparable and
(2) $K'/K$ is separable.

In the first case, the degree $[K':K]$ is a power of a prime $q\ne
p$. We take an arbitrary valuation ring $R'$ of $K'$, {\em lying
over} $R$, i.e., such that $R'\cap K=R$ and the embedding $R\to
R'$ is local (such $R'$ exists in the case of an arbitrary field
extension $K'/K$, \cite[ch. VI th. $5'$]{Samuel-Zariski}). Let
$L'$ be the residue field of $R'$ so that we have a surjective
place $K'\to L'$. We show that $L'$ is also purely inseparable
over $L$ (and therefore $[L':L]$, being a power of the same $q$,
is coprime to $p$). For this, we take an element $l\in L'$ and
show that $l^{q^n}\in L$ for some $n$: let $k\in R'$ be a preimage
of $l$; then $k^{q^n}\in K$ for some $n$ and consequently
$l^{q^n}\in L$ for the same $n$.

In the second case we consider
%a maximal number of extensions to $K'$ of
%the given place of $K$, having pairwise different valuation rings.
all valuation rings $R_1,\dots,R_r$ of $K'$, lying over $R$
(the number of such valuation rings is finite by
\cite[ch. VI th. 12 cor. 4]{Samuel-Zariski}).
The residue field of each $R_i$ is a finite extension of $L$.
Moreover, $\Sum_{i=1}^re_in_i=[K':K]$ \cite[ch. VI th. 20 and p. 63]{Samuel-Zariski}
(the discrete valuation
ring assumption and the separability assumption are
needed for this equality), where $n_i$ is the degree
over $L$ of the residue field of $R_i$, and $e_i$ is the {\em reduced ramification index}
of $R_i$ over $R$, \cite[def. on pp. 52--53]{Samuel-Zariski}.
It follows that at least one of $n_i$ is not divisible by $p$.
\end{proof}


Let us make some first general observations on $\cd_p$. Clearly, a
generic splitting field of a determination function is also
$p$-generic; therefore we always have $\cd\geq\cd_p$.

Also it is clear, that $\cd_p$ is not interesting, if the
determination function in question has a $p$-coprime splitting
field. More precisely, one has a simple

\begin{comment}
\begin{prop}
\label{razvlekuha}
Let $D$ be a determination function over $F$, $F'/F$ a finite $p$-coprime field
extension, $D'$ the restriction of $D$
to $\Fields_{F'}$.
Then $\cd_p(D')=\cd_p(D)$.
\end{prop}

\begin{proof}
Let $E$ be a $p$-generic splitting field of $B$.
Let $E'$ be a field containing $E$ and $F'$ and finite $p$-coprime over $E$
(existing by Lemma \ref{polia}).
The transcendence degree of $E'/F$ is the same as that of $E/F$.

Any splitting field $L$ of $B'$ is also a splitting field of $B$;
therefore there exists a place $E\to L'$ with some finite $p$-coprime $L'/L$.
The place $E\to L'$ extends to a place $E'\to L''$ with some finite $p$-coprime
$L''/L'$.
Since the degree $[L'':L]=[L'':L']\cdot[L':L]$ is finite $p$-coprime,
we see that $E'$ is a $p$-generic splitting field of $B'$;
therefore $\cd_p(B')\leq \cd_p(B)$.

Now let $E'$ be an arbitrary $p$-generic splitting field of $B'$.
If $L$ is a splitting field of $B$, let us take a field $L'$
containing $L$ and $F'$ and finite  $p$-coprime over $L$.
Since $L'\supset L$, the field $L'$ is also a splitting field of $B'$;
consequently, there exists a
place $E'\to L''$ with some finite $p$-coprime $L''/L'$,
showing that $E'$ is a $p$-generic splitting
field also for $B$;
therefore $\cd_p(B)\leq\cd(B')$.
\end{proof}

\end{comment}

\begin{lemma}
If a determination function $D$ is split by an algebraic field extension $E/F$ of degree
non-divisible by $p$,
then $\cd_p(D)=0$.
\end{lemma}

\begin{proof}
Taking a finitely generated splitting subfield of
$E$, we get a $p$-coprime splitting field.
It follows by Lemma \ref{polia}, that $F$ is a $p$-generic
splitting field of $D$.
\end{proof}

\begin{example}
The computation of $\cd(X)$ for an anisotropic smooth projective quadric $X$
(over a field of characteristic $\ne2$), given in
\cite{KM} (see Example \ref{cd(quadric)}), shows in fact also that $\cd_2(X)=\cd(X)$.
\end{example}


\begin{example}\label{severi}
Let $X$ be a Severi-Brauer variety.
% of a central simple
%$F$-algebra $A$.
If
%$A$ has a $p$-primary degree $p^r$, then
$d(X)=d_p(X)$ (that is, $d(X)$ is a power of the prime $p$), then
$\cd(X)=\cd_p(X)=d_p(X)-1$, \cite[th. 11.4]{Berhuy-Reichstein}.
Now if
%the degree of $A$ is non-primary,
$d(X)$ is not a power of a prime,
the value of $\cd(X)$ is not
known, while $\cd_p(X)$ is still $d_p(X)-1$
%equal to $p^r-1$, where $p^r=d_p(\deg A)$.
%the canonical $p$-dimension of the Severi-Brauer variety of
%the $p$-primary component of $A$
%(see (\ref{an})).
(see Example \ref{cd_p(SB)}).
\end{example}

\begin{example}\label{so}
The computation of $\cd(\gSO_n)$, given in \cite{cd(SO)}, also
shows that $\cd_2(\gSO_n)=\cd(\gSO_n)$ (see also Example \ref{cd_2(Grassm)}
as well as (\ref{bn}) and
(\ref{dn})).
\end{example}


\begin{rem}\label{inject}
Let $\cF$ and $\cF'$ be continuous functors $\Fields_k\to\Sets$
with a morphism $f\!:\cF\to\cF'$. If the kernel of $f$ is trivial,
then for any $F\in\Fields$ and any $\alpha\in\cF(F)$ the
determination function of $\alpha$  coincides with the
determination function of $f(\alpha)$ (cf. \cite[lemma
10.2(a)]{Berhuy-Reichstein}); therefore $\cd(\cF)\leq\cd(\cF')$
(cf. \cite[lemma 10.2(b)]{Berhuy-Reichstein}) and
$\cd_p(\cF)\leq\cd_p(\cF')$ (for any $p$). If moreover $f$ is
surjective (but may not be necessarily injective), then
$\cd(\cF)=\cd(\cF')$ (cf. \cite[lemma 10.2(c)]{Berhuy-Reichstein})
and $\cd_p(\cF)=\cd_p(\cF')$.
\end{rem}

\section
{Canonical ($p$-)dimension of regular complete varieties}

\begin{lemma}
\label{X}
The function field of a regular variety $X$ is a generic splitting field of
$X$;
in particular, $\cd(X)\leq\dim X$ for regular $X$.
\end{lemma}

\begin{proof}
The function field $F(X)$ is a splitting field of $X$ (even in the
non-regular case). To check that it is generic, we take an
arbitrary splitting field $L$ of $X$, find a finitely generated
splitting subfield of $L$, take a model $Y/F$ of it, and consider
a rational morphism $Y\to X$, that exists because $X$ has a point
over the field $F(Y)$.
%Let $Y\subset X$ be the closure of the image of this rational morphism.
Since $X$ is regular, there exists a geometric place $F(X)\to
F(Y)$; composing it with the embedding $F(Y)\subset L$, we get the
required place $F(X)\to L$.
\end{proof}

\begin{rem}
\label{otleglo}
The place $F(X)\to L$, constructed in the end of proof of Lemma \ref{X},
is geometric (as defined in \S\ref{Places}).
\end{rem}

We have the following generalization of Lemma \ref{X}:



\begin{lemma}
\label{1}
If $Y$ is a closed subvariety of a regular variety $X$, admitting a dominant
rational morphism $X\to Y$,
then the function field of $Y$ is a generic splitting field of $X$.
In particular,
$\cd(X)\leq\dim Y$.
\end{lemma}

\begin{proof}
Clearly, $F(Y)$ is a splitting field of $X$.
The dominant rational morphism produces an embedding of $F(Y)$
into the field $F(X)$, which by Lemma \ref{X} is a generic splitting field
of $X$.
It follows by Lemma \ref{generic splitting subfield} that $F(Y)$
is a generic splitting field too.
\end{proof}


\begin{lemma}
\label{polnaja fignia}
Let $Y$ be a scheme over a field $F$, $X$ a variety over $F$.
\begin{enumerate}
\item
If $Y$ admits a dominant rational morphism $X\to Y$,
then the $F(X)$-scheme $Y_{F(X)}$ has a closed rational point.
\item
If the $F(X)$-scheme $Y_{F(X)}$ has a closed rational point, then there
exists a closed subvariety $Y'\subset Y$, admitting a
dominant rational morphism $X\to Y'$.
\end{enumerate}
\end{lemma}

\begin{proof}
Existence of a rational morphism $X\to Y$ is equivalent to existence of a
closed rational point on $Y_{F(X)}$.
To prove the second statement, we take as $Y'$ the closure of the image
of the rational morphism $X\to Y$.
\end{proof}

\begin{prop}
\label{2}
Any regular complete variety $X$ has a closed subvariety $Y\subset
X$ of dimension $\dim Y=\cd(X)$,
admitting a dominant rational morphism $X\to Y$.
\end{prop}

\begin{proof}
Let us take a generic splitting field of $X$, having the
transcendence degree $\cd(X)$ over $F$, replace it by a finitely generated splitting subfield
(Lemma \ref{generic splitting subfield}),
and consider a projective model $T/F$ of it.
Since $X$ is regular, $F(X)$ is a generic splitting field of $X$ (Lemma \ref{X}) and
there exists a place $F(X)\to F(T)$.
Since the variety $X$ is complete, such a place produces a rational morphism $T\to X$.
Replacing $T$ by the closure in $T\times X$ of the graph of such a
rational
morphism, we get another model of the same
field, but now we have a regular morphism $T\to X$ instead of the rational one
we had before.

Since $F(T)$ is a generic splitting field of $X$, there exists a
place $F(T)\to F(X)$. By completeness of $T$, it produces a
rational morphism $X\to T$. Composing it with the regular morphism
$T\to X$, we get a rational morphism $X\to X$; let $Y$ be the
closure of its image. Clearly, $\dim Y\leq\dim T=\cd(X)$. On the
other hand, by Lemma \ref{1}, $\dim Y\geq\cd(X)$. Therefore, $\dim
Y=\cd(X)$.
\end{proof}

Combining Lemma \ref{1} and Proposition \ref{2}, we get

\begin{cor}
\label{takaja cd}
Canonical dimension of a regular complete variety $X$ is the
minimum of dimension of closed subvarieties $Y\subset X$,
admitting a dominant rational morphism $X\to Y$.
\qed
\end{cor}

Taking into account Lemma \ref{polnaja fignia}, we get the
following variation of Corollary \ref{takaja cd}:

\begin{cor}
\label{eshcho i takaja cd}
Canonical dimension of a regular complete variety $X$ is the
minimum of dimension of closed subvarieties $Y\subset X$,
satisfying $Y(F(X))\ne\emptyset$.
\qed
\end{cor}



Now we establish variations of Lemma \ref{1}, Proposition \ref{2},
and Corollaries \ref{takaja cd} and \ref{eshcho i takaja cd},
related to the canonical $p$-dimension.

Let $Y$ be a scheme over a field $F$ and let $X$ be a variety over
$F$. We say that $Y$ satisfies condition $(\ast)$ (with respect to
$X$), if there exists an $F$-variety $X'$, a dominant rational
morphism $X'\to Y$, and a dominant rational morphism $X'\to X$
such that the field extension $F(X')/F(X)$ is $p$-coprime.


\begin{lemma}
\label{1p}
If $Y$ is a closed subvariety of a regular variety $X$, satisfying condition
$(\ast)$,
then the function field of $Y$ is a $p$-generic splitting field of $X$.
In particular,
$$\cd_p(X)\leq\dim Y\;.$$
\end{lemma}

\begin{proof}
Clearly, $F(Y)$ is a splitting field of $X$.
Let now $L/F$ be any splitting field of $X$.
Then we can find a geometric place $F(X)\to L$
(see Lemma \ref{X} with Remark \ref{otleglo}).
Applying Lemma \ref{places} to this place
and the field extension
$F(X')/F(X)$,
we get a place $F(X')\to L'$ for some $p$-coprime field extension $L'/L$.
Restricting the latter place to the subfield $F(Y)\subset F(X')$,
we get a place $F(Y)\to L'$;
therefore, the splitting field $F(Y)$ is $p$-generic.
\end{proof}

\begin{lemma}
\label{p-polnaja fignia}
Let $Y$ be a scheme over a field $F$, $X$ a variety over $F$.
\begin{enumerate}
\item
If $Y$ satisfies condition $(\ast)$,
then $d_p(Y_{F(X)})=1$ (see \S\ref{Varieties} for definition of $d_p$).
\item
If $d_p(Y_{F(X)})=1$, then there exists a closed subvariety
$Y'\subset Y\subset X$, satisfying condition $(\ast)$. \qed
\end{enumerate}
\end{lemma}


\begin{prop}
\label{2p}
Any regular complete variety $X$ has a closed subvariety $Y\subset
X$ of $\dim Y=\cd_p(X)$, satisfying condition $(\ast)$.
\end{prop}

\begin{proof}
Let us take a finitely generated $p$-generic splitting field of
$X$, having the transcendence degree $\cd_p(X)$ over $F$, and
consider a projective model $T/F$ of it. Since $F(X)$ is a generic
splitting field of $X$, while $F(T)$ is a splitting field, there
exists a place $F(X)\to F(T)$. Since the variety $X$ is complete,
such a place produces a rational morphism $T\to X$. Replacing $T$
by the closure in $T\times X$ of the graph of this morphism, we
get another model of the same field, but now we have a regular
morphism $T\to X$ instead of the rational one.

Since $F(T)$ is a $p$-generic splitting field of $X$, there exists
a place $F(T)\to L'$, where $L'\supset F(X)$ is a field,
$p$-coprime over $F(X)$. Let $X'/F$ be a model of the field $L'$.
By completeness of $T$, the place $F(T)\to F(X')$ produces a
rational morphism $X'\to T$. Composing it with the regular
morphism $T\to X$, we get a rational morphism $X'\to X$; let $Y$
be the closure of its image. Clearly, $\dim Y\leq\dim T=\cd_p(X)$.
On the other hand, by Lemma \ref{1p}, $\dim Y\geq\cd_p(X)$.
Therefore, $\dim Y=\cd_p(X)$.
\end{proof}

Lemma \ref{1p} and Proposition \ref{2p} together produce

\begin{cor}
\label{takaja cdp} Canonical $p$-dimension of a regular complete
variety $X$ is the minimum of dimension of the closed subvarieties
$Y\subset X$, satisfying $(\ast)$. \qed
\end{cor}

By Lemma \ref{p-polnaja fignia}, the following variation of
Corollary \ref{takaja cdp} also holds:

\begin{cor}
\label{eshcho i takaja cdp} Canonical $p$-dimension of a regular
complete variety $X$ is the minimum of dimension of the closed
subvarieties $Y\subset X$ with $d_p(Y_{F(X)})=1$. \qed
\end{cor}



\section
{Generically $p$-split varieties}

In this section $X$ stands for a smooth complete absolutely
irreducible variety over a field $F$.

\begin{lemma}
\label{dim=1} Degree homomorphism $\deg\!:\Ch_0(\overline
X)\to\BBF_p$ is an isomorphism if and only if $\dim_{\BBF_p}\Ch_0(\overline
X)=1$.
\end{lemma}

\begin{proof}
The degree homomorphism is non-zero and therefore surjective.
\end{proof}

\begin{lemma}
\label{rombik} Assume that $\dim_{\BBF_p}\Ch_0(\overline X)=1$.
Let $T$ be an arbitrary $F$-scheme, let $E_1$ and $E_2$ be field
extensions of $F$, and let $f_1\!:\Spec E_1\to X$ and $f_2\!:\Spec
E_2\to X$ be $F$-morphisms. Then the diagram
$$
\begin{diagram}
\node{}\node{\BCh(T\times X)}\arrow{sw,t}{(\id_T\times f_1)^*}
\arrow{se,t}{(\id_T\times f_2)^*}\node{}\\
\node{\BCh(T_{E_1})}\arrow{se,J}\node{}\node{\BCh(T_{E_2})}
\arrow{sw,L}\\
\node{}\node{\Ch(\overline T)}\node{}
\end{diagram}
$$
is commutative.
\end{lemma}

\begin{proof}
Let $E$ be a field extension of $F$, containing $E_1$ and $E_2$.
Replacing $T$ and $X$ by $T_E$ and $X_E$, we come to the following
situation:
$E=E_1=E_2=F$ and for some closed rational points $x_1,x_2\in X$,
$f_i$ is the embedding $T=T\times\{x_i\}\hookrightarrow T\times
X$.
We want to show that $f_1^*=f_2^*\!:\BCh(T\times X)\to\BCh(T)$.
Since $\pr_*\compose f_{i*}=\id$ for $i=1,2$, where
$\pr$ is the projection $T\times X\to T$, it suffices to show that
$$
f_{1*}\compose f_1^*=f_{2*}\compose f_2^*\!:\BCh(T\times X)\to\BCh(T\times X)\;.
$$
The composition $f_{i*}\compose f_i^*$ coincides with the
multiplication by $[T\times x_i]$. Since by the assumption on
$\dim_{\BBF_p}\Ch_0(\overline X)$ and Lemma \ref{dim=1}, the
degree homomorphism $\deg\!:\Ch(\overline X)\to\BBF_p$ is an
isomorphism, we have $[x_1]=[x_2]\in\BCh(X)$, and therefore
$[T\times x_1]=[T\times x_2]$ in $\BCh(T\times X)$. The required
assertion follows.
\end{proof}

Let $i\!:Y\hookrightarrow X$ be a closed subvariety of $X$.
The closed embedding
$$
(\id_Y\times i\times\id_X)\!:Y\times X\to Y\times X\times X
$$
is regular, and
we define a paring
$$
\BCh(Y)\otimes\BCh(X\times X)\to\BCh(Y\times X)\;,
$$
by the formula $\alpha\otimes\beta\mapsto
(\id_Y\times i\times\id_X)^*(\alpha\times\beta)$.


\begin{prop}
\label{ASMprop} Let $Y$ be a closed subvariety of $X$. Assume that
$\BCh(X_{F(X)})=\Ch(\overline X)$ and
$\dim_{\BBF_p}\Ch_0(\overline X)=1$. Then the above paring is
surjective.
\end{prop}

\begin{proof}
We proceed by induction on $\dim Y$. We have a commutative diagram
$$
\begin{CD}
\Oplus_{Y'}\BCh(Y')\otimes\BCh(X\times X)@>>>
\BCh(Y)\otimes\BCh(X\times X)@>>>
\BCh(X\times X)@>>>0\\
@VVV @VVV @VVV\\
\Oplus_{Y'}\BCh(Y'\times X)@>>>\BCh(Y\times
X)@>>>\BCh(X_{F(Y)})@>>>0
\end{CD}
$$
where $Y'$ runs over closed subvarieties of codimension $1$ in $Y$
(the upper row is the obvious exact sequence
$\Oplus_{Y'}\BCh(Y')\to\BCh(Y)\to\Z\to0$, tensored by
$\BCh(X\times X)$ over $\BBF_p$). The rows are exact and the left
vertical map is surjective by the induction hypothesis. The right
vertical map is surjective because the rhombus
$$
\begin{diagram}
\node{}\node{\BCh(X\times X)}\arrow{sw}\arrow{se,A}\node{}\\
\node{\BCh(X_{F(Y)})}\arrow{se,J}\node{}\node{\BCh(X_{F(X)})}\arrow{sw,=}\\
\node{}\node{\Ch(\overline X)}\node{}
\end{diagram}
$$
is commutative (as guaranteed by the assumption on
$\Ch_0(\overline X)$ and Lemma \ref{rombik} applied to $T=X$).
\end{proof}

\begin{cor}
\label{ASMcor} Under condition of Proposition \ref{ASMprop}, if
the push-forward $$(i\times\id_X)_*\!:\BCh(Y\times
X)\to\BCh(X\times X)$$ is non-zero, then the push-forward
$i_*\!:\BCh(Y)\to\BCh(X)$ is also non-zero and, in particular,
$\BCh_i(X)\ne0$ for at least one $i\leq\dim Y$.
\end{cor}

\begin{proof}
The square
$$
\begin{diagram}
\node{\BCh(Y)\otimes\BCh(X\times X)}\arrow{e,A}\arrow{s,r}{i_*\otimes\id}\node{\BCh(Y\times
X)}\arrow{s,r}{(i\times\id)_*}\\
\node{\BCh(X)\otimes\BCh(X\times X)}\arrow{e}\node{\BCh(X\times
X)}
\end{diagram}
$$
is commutative.
\end{proof}


\begin{dfn}
\label{def-pbalanced}
We say that a (complete smooth absolutely irreducible) variety
$X$ over $F$ as {\em $p$-balanced}, if
the symmetric bilinear form
$$
\Ch(\overline X)\times\Ch(\overline X)\to\BBF_p\;,\;\;
(\alpha,\beta)\mapsto\deg(\alpha\cdot\beta)
$$
is non-degenerate (in the sense that its radical is trivial; note that
$\dim_{\BBF_p}\Ch(\overline X)$ can be infinite).

A variety $X$ over $F$ is called \emph{cellular}, if there is a
filtration
\[
\emptyset=X_0\subset X_1\subset\dots\subset X_{n-1}\subset X_n=X
\]
by closed subschemes such that for every $i=0,1,\dots, n-1$ the
scheme $X_{i+1}\setminus X_i$ is isomorphic to an affine space
over $F$.
\end{dfn}

\begin{rem}
Let $X$ be a geometrically cellular variety, that is, $X_E$ is
cellular for some field extension $E/F$. We claim that $X$ is
$p$-balanced (for any $p$). Indeed, the Chow-motive of the
cellular variety $X_E$ decomposes into a finite direct sum of
twists of the motive of the point (see, e.g., \cite[th.
6.5]{flag}). Therefore $\CH(X_E)=\CH(\overline X)$. Moreover, the
mutually inverse isomorphisms of the motive of $X_E$ with the
above direct sum are given by certain sequences $e_0,\dots,e_n$
and $e'_0,\dots,e'_n$ of homogeneous elements in $\CH(X_E)$, that
are bases of $\CH(X_E)$ mutually dual with respect to the
$\Z$-bilinear form $(\alpha,\beta)\mapsto\deg(\alpha\cdot\beta)$
(simply because they define mutually inverse isomorphisms of
motives).
\end{rem}

Note that for any $p$-balanced $X$ and any integer $i$, one has
$\dim_{\BBF_p}\Ch^i(\overline X)=\dim_{\BBF_p}\Ch_i(\overline X)$,
if at least one of these dimensions is finite. Since
$\dim_{\BBF_p}\Ch^0(\overline X)$=1, the above relation, taken with
$i=0$, implies that $\dim_{\BBF_p}\Ch_0(\overline X)=1$ for a
$p$-balanced $X$.

\begin{dfn}
\label{def-psplit}
A $p$-balanced variety $X$ over $F$ is called {\em $p$-split}, if
$\BCh(X)=\Ch(\overline X)$.
\end{dfn}

A cellular variety is $p$-split.

We say that a variety $X$ has a property {\em generically}, if $X$
over its own function field has this property.
%For example, a
%variety $X$ is {\em generically cellular}, if $X_{F(X)}$ is a cellular
%variety over $F(X)$.
This way we get a notion of {\em generically $p$-split} variety.
According to above remarks, a {\em generically cellular} variety is
generically $p$-split.


We are ready to prove the main result of the first half of the
paper, interpreting the canonical $p$-dimension of a generically
$p$-split variety in terms of its modulo $p$ reduced Chow group:


\begin{thm}
\label{main-Nikita}
If $X$ is a generically $p$-split variety
(see Definitions \ref{def-psplit} and \ref{def-pbalanced}),
then
$$
\cd_p(X)=\min\{i\;|\;\;\BCh_i(X)\ne0\}\;.
$$
In particular, the formula holds
for a generically cellular $X$.
\end{thm}

\begin{proof}
Two inequalities are proved separately.

\noindent
\fbox{$\leq$}
Let $i$ be an integer such that the group $\BCh_i(X)$ is non-zero.
Then $[Y]\ne0$
for a closed $i$-dimensional subvariety $Y\subset X$.
We are going to show that $d_p(Y_{F(X)})=1$ for such $Y$, what is
enough for our aim by Corollary \ref{eshcho i takaja cdp}.

Since the variety $X_{F(X)}$ is $p$-split,
there exists a prime cycle $Z\subset X_{F(X)}$ such that $\deg([Y_{F(X)}]\cdot[Z])\ne0$.
Since the product $[Y_{F(X)}]\cdot[Z]$ can be represented by a cycle on the intersection
$Y_{F(X)}\cap Z$ (see \cite[\S8.1]{Fulton}),
the scheme $Y_{F(X)}$ has a closed $p$-coprime point, meaning
that $d_p(Y_{F(X)})=1$.

\noindent
\fbox{$\geq$}
Let now $Y$ be a closed subvariety of $X$,
satisfying condition $(\ast)$
We will show that $\BCh_i(X)\ne0$ for some $i\leq \dim Y$.
The desired inequality will then follow by Proposition \ref{2p}.

Let $Z$ be the closure of the image of the rational morphism
$X'\to Y\times X$ (the variety $X'$ with the rational morphisms $X'\to
Y$ and $X'\to X$ comes from condition $(\ast)$).
The cycle $(i\times\id_X)_*([Z])\in\BCh(X^2)$ is non-zero, because
for the second projection
$\pr\!:X^2\to X$, we have
$$\pr_*(i\times\id_X)_*[Z]=[F(X'):F(X)]\cdot[X]\ne0\in\BCh(X)\;.$$
It follows by Corollary \ref{ASMcor} that
$\BCh_i(X)\ne0$ for some $i\leq \dim Y$.
\end{proof}

\begin{rem}
If we take $Y$ with $\dim Y=\cd_p(X)$ in the beginning of the
($\geq$)-part of the proof of Theorem \ref{main-Nikita},
then, since we have already proved the ($\leq$)-part of the theorem,
we come to the conclusion that the ($\dim Y$)-dimensional component of
the homomorphism $i_*\!:\BCh(Y)\to\BCh(X)$ is non-zero.
Since the ($\dim Y$)-dimensional component of the image of
$i_*$ is generated by $[Y]\in\BCh(X)$, we see that in fact the class in $\BCh(X)$ of
$Y$ itself is non-zero.
\end{rem}

\begin{example}
\label{cd_p(SB)}
Let $X$ be the Severi-Brauer variety
of a central simple $F$-algebra $A$.
%Let $p^k$ be the $p$-primary part of the index of $A$
%(in other words, $p^k=d_p(X)$).
Since by Theorem \ref{main-Nikita},
$\cd_p(X)=\cd_p(X_L)$ for any $p$-coprime field extension $L/F$,
$\cd_p(X)=\cd_p(Y)$, where $Y$ is the Severi-Brauer variety of a
division algebra, Brauer-equivalent to the $p$-primary part of $A$.
Furthermore, $\BCh(Y)=\BCh^0(Y)$ by \cite[prop. 2.1.1]{SB-motives}.
Therefore, by Theorem \ref{main-Nikita},
$\cd_p(Y)=\dim Y$, so that we get
$$\cd_p(X)=\cd_p(Y)=\dim Y=d_p(X)-1\;.$$
\end{example}

\begin{example}
\label{cd_2(Grassm)} In this example $p=2$.
% and $\Char F\ne2$.
Let $X/F$ be the
orthogonal grassmannian of $n$-dimensional totally isotropic
subspaces of a ($2n+1$)-dimensional quadratic form.
If
$d_2(X)=2^n$, then $\BCh(X)=\BCh^0(X)$ by \cite[prop. 1.4]{cd(SO)}
and therefore $$\cd_2(X)=\dim(X)=n(n+1)/2\;.$$
%Note that the proof
%of \cite[prop. 1.4]{cd(SO)} makes use only of the ring structure
%of $\CH(\overline X)$, which does not depend on characteristic of the
%base field (see Remark \ref{demazurovo}); therefore the
%characteristic $\ne2$ assumption, made in \cite{cd(SO)}, can be
%neglected.
Without any restriction on $d_2(X)$, canonical $2$-dimension of
$X$ can be expressed as the sum of all $i$ such that the $i$-th special Schubert
class $e_i\in\CH^i(\overline X)$ is {\em non-rational}, i.e, does not lie in $\BCH(X)$
(cf. \cite[conj. 6.6]{Vishik-QG}):
indeed, by \cite[main th. 5.7]{Vishik-QG},
the product of all {\em rational} $e_i$, reduced modulo $2$, is a non-zero element
of $\BCh(X)$ of the smallest possible dimension.
\end{example}

\section{Canonical $p$-dimension of algebraic groups}


If $P$ is an algebraic group over a field $F$, we write $\CH(BP)$
for the $P$-equivariant Chow ring $\CH_P(\Spec F)$ of the point
$\Spec F$ (see \cite{Edidin98}).

Let $G$ be a connected algebraic group over $F$ and let $P$ be a
subgroup of $G$. Consider the homomorphism
\[
\varphi_G=\varphi_{G,P}:\CH(BP)=\CH_P(\Spec F)\xra{q^*} \CH_P(G)=\CH(G/P),
\]
where $q:G\ra \Spec F$ is the structure morphism.

\begin{rem}
\label{vlozhenie grupp}
If $G$ is a
subgroup of a group $G'$, then $\varphi_{G}=i^*\circ
\varphi_{G'}$, where $i:G/P\ra G'/P$ is the morphism, induced by embedding
of the groups.
\end{rem}

\begin{prop}\label{GL}
Let $G=\gGL_n$. Then the map $\varphi_{G}$ is surjective and the
left $G$-action on $G/P$ induces the trivial action on $\CH(G/P)$.
\end{prop}

\begin{proof}
The group $G$ is embedded into the affine space of $\End(F^n)$ as a
$G$-equivariant open subset. The map $q^*$ factors as the
composite
\[
\CH_P(\Spec F)\ra \CH_P(\End(F^n))\ra \CH_P(G),
\]
where the first pull-back map is an isomorphisms by the homotopy
invariance property and the second restriction map is surjective
by the localization. Hence $\varphi_{G}$ is surjective.

For a rational point $g$ of $G$, let $\lambda_g:G\ra G$ is the
morphism of the left multiplication by $g$. It follows from
$q\circ \lambda_g=q$ that $\lambda_g^*\circ q^*=q^*$. Since $q^*$
is surjective, $\lambda_g^*$ is the identity, i.e., $G$ acts
trivially on $\CH(G/P)$.
\end{proof}


Recall that we write $\CH(\overline{G/P})$ for the colimit of
$\CH(G_L/P_L)$ over all field extensions $L/F$.
We define a homomorphism $\overline\varphi_G$ as
the composite
\[
\overline\varphi_G: \CH(BP)\xra{\varphi_{G}}\CH(G/P)\xra{\res}
\CH(\overline{G/P})\;.
\]

Let $E$ be a (right) $G$-torsor over a field extension $F'$ of
$F$. Set $K=F'(E)$.
Let $\psi_E\!:\CH(E/P)\to\CH(G_K/P_K)$ be pull-back map with
respect to the morphism $G_K/P_K\ra E/P$, induces by
the $G$-equivariant morphism $G_K\ra E$,
taking the identity of $G$ to the generic point of $E$.
We define a homomorphism $\overline\psi_E$ as
the composite
\[
\overline\psi_E: \CH(E/P)\xra{\psi_E} \CH(G_K/P_K)\xra{\res_K}
\CH(\overline{G/P})\;.
\]

We identify $G$ with a subgroup of $S=\gGL_n$ for some $n$.

\begin{lemma}\label{freedom}
Suppose that there is a $G$-equivariant morphism $E\ra S$ over $F$
and let $h:E/P\ra S/P$ be the induced morphism. Then
$\overline\varphi_G=\overline\psi_E\circ h^*\circ\varphi_{S}$.
\end{lemma}

\begin{proof}
The composition $G_K\ra E_K\ra S_K$ of the morphisms induced by
$G_K\ra E$ and $E\ra S$, differs from the inclusion
$G_K\hookrightarrow S_K$ by a left multiplication by an element of
$S(K)$. By Proposition \ref{GL}, the induced pull-back
homomorphisms $\CH(S_K/P_K)\ra\CH(G_K/P_K)$ coincide. Composing with
the restriction homomorphism $\CH(S/P)\ra\CH(S_K/P_K)$, we get
$\psi_E\circ h^*=\res_{K/F}\circ \,i^*$, where
%$r:G_K/P_K\ra G/P$ and
$i:G/P\ra S/P$ is the morphisms, induced by the embedding $G\hookrightarrow S$.
We have:
$$
%\theta= &\ \lambda\circ\varphi_{G/P}\\
%      =  &\ \lambda_K\circ r^*\circ i^*\circ \varphi_{S/P}\\
%      =  &\ \lambda_K\circ f^*\circ h^*\circ \varphi_{S/P}\\
%      =  &\ \psi_E\circ h^*\circ\varphi_{S/P}.
\overline\varphi_G= \res \compose\, \varphi_{G}
      =  \res_K \compose \,\res_{K/F} \compose\, i^* \compose \varphi_{S}
      =  \res_K \compose \psi_E \compose h^* \compose \varphi_{S}
      =  \overline\psi_E \compose h^* \compose \varphi_{S}
$$
(for the second equality, see Remark \ref{vlozhenie grupp}).
\end{proof}



\begin{thm}\label{main}
1. For any $G$-torsor $E$ (over any field extension of $F$) we
have
$$\Im(\overline\varphi_G)\subset \Im(\overline\psi_E)\;.$$
2. There exists a $G$-torsor $E$ (over a field extension of $F$)
such that $\Im(\overline\varphi_G)= \Im(\overline\psi_E)$.
\end{thm}

\begin{proof}
1. We may assume that $E$ is a $G$-torsor over $F$. By the Hilbert
theorem 90, the $S$-torsor $(E\times S)/G$ is trivial
(where $(E\times S)/G$ stands for the quotient of $E\times S$ by the action
$(e,s)\cdot g=(eg,g^{-1}s)$ of $G$; the action of $G$ on this quotient
is defined by the formula $(e,s)\cdot g=(e,sg)$, so that the embedding
$E=E\times1\hookrightarrow E\times S$ induces a $G$-equivariant morphism
$E\to(E\times S)/G$).
In
particular, there is a $G$-equivariant morphism $E\ra S$. By Lemma
\ref{freedom},
%$\Im(\theta)\subset \Im(\psi_E)$.
$\Im(\overline\varphi_G)\subset \Im(\overline\psi_E)$.

%\medskip

2. Let $X=S/G$ and $K=F(X)$. Denote by $E\ra\Spec K$ the generic
fiber of the projection $S\ra X$. Clearly, $E$ is a $G$-torsor
over $K$. Denote by
$
h:E/P_K\ra S/P
$
the morphism induced by the canonical $G$-equivariant morphism
$E\ra S$. Since $E/P_K$ is a localization of $S/P$, the pull-back
homomorphism $h^*$ is surjective. By Proposition \ref{GL},
$\varphi_{S}$ is also surjective. It follows from Lemma
\ref{freedom} that
%$\Im(\theta)= \Im(\psi_E)$.
$\Im(\overline\varphi_G)=\Im(\overline\psi_E)$.
\end{proof}




Let $G$ be an algebraic group over a field $F$ and let $\Tors_G$
be the functor $\Fields_F\to \Sets$, taking a field $K$ to the set of isomorphism
classes $\Tors_G(K)$ of $G$-torsors over $K$. For a $G$-torsor
$E/K$, the determination function of $E\in\Tors_G(K)$ coincides with the
determination function of the $K$-variety $E$.

Let $P\subset G$ be a subgroup. We assume that $P$ is a special
group, that is, the functor $\Tors_P$ is trivial.

\begin{lemma}\label{binary}
The determination functions of the varieties $E$ and $E/P$ coincide.
\end{lemma}
\begin{proof}
Suppose $E/P$ has a point over $K$. We need to show that
$E(K)\neq\emptyset$. The fiber of the natural morphism $E\ra E/P$
over the point is a $P$-torsor. Since $P$ is special, this torsor is
trivial, i.e., the fiber has a point over $K$.
\end{proof}

\begin{rem}\label{splitsellular}
Let $G$ be a split semisimple algebraic group and let $P$ be a
parabolic subgroup. The variety $G/P$ is cellular
(see, e.g., \cite{Chernousov-Gille-Merkurjev}), therefore
%its
%Chow group is a free group generated by closures of the cells and
%hence it does not change under field extensions. Therefore,
$\CH(\overline{G/P})=\CH(G/P)$.
\end{rem}

\begin{rem}\label{grothendieck}
Suppose further that $P$ is a Borel subgroup of $G$. The image of
the composite
\[
\CH(BP)\xra{\varphi_{G}}\CH(G/P)\xra{\deg}\Z
\]
is a subgroup $t_G\Z$ with a positive integer $t_G$ known as the
\emph{torsion index of $G$} (see \cite{Grothendieck58}). It
follows from Theorem \ref{main} and Lemma \ref{binary} that $t_G$
is the l.c.m. of the numbers $d(E)$ over all $G$-torsors over all
field extension. This statement is known as Grothendieck's theorem
\cite[th. 2]{Grothendieck58}. The prime divisors of the torsion
index $t_G$ are called the \emph{torsion primes of $G$}.
\end{rem}

It follows that the canonical dimension of $G$ (resp. $\cd_p(G)$)
is the supremum of the canonical dimension of $E/P$ (resp.
$\cd_p(E/P)$) over all $G$-torsors over all field extensions of
$F$.

Let $G$ be a split semisimple group and let $P\subset G$ be a
parabolic subgroup. Suppose $P$ is a special group (for example,
$P$ is a Borel subgroup of $G$). Let $E$ be a $G$-torsor. Note
that the variety $E/P$ is projective. In order to apply Theorem
\ref{main-Nikita} to the variety $E/P$, we need the following

\begin{cor}\label{cellular}
The variety $E/P$ is generically cellular.
\end{cor}
\begin{proof}
By Lemma \ref{binary}, the torsor $E$ is split over the function
field $L=F(E/P)$, hence $E_L\simeq G_L$ and therefore
$(E/P)_L\simeq (G/P)_L$. The latter variety is cellular.
\end{proof}

%Set $X=E/P$ where $E$ is a $G$-torsor. Since over a splitting
%field of $E$ the variety $X$ is isomorphic to $G/P$, the group $A$
%defined above coincides with $\CH(G/P)$ (see Remark
%\ref{splitsellular}). Denote by $\widetilde \CH(G/P)$ the image of
%the graded ring homomorphism
%\[
%\varphi_{G/P}:\CH(BP)\ra \CH(G/P).
%\]

Theorems \ref{main-Nikita} and \ref{main} yield

\begin{thm}\label{compute}
Let $G$ be a split semisimple group and let $P\subset G$ be a
special parabolic subgroup (for example, a Borel subgroup).
Denote by $\widetilde \CH(G/P)$ the image of
the graded ring homomorphism
$
\varphi_{G}:\CH(BP)\ra \CH(G/P)
$.
Then
$\cd_p(G)$ for a prime $p$, is equal to the smallest integer $i$
such that $\widetilde \CH_i(G/P)$ is not contained in
$p\CH_i(G/P)$.
\qed
\end{thm}

\begin{rem}
The canonical dimension $\cd_p(G)$ is positive if and only if $p$
is a torsion prime of $G$ (see Remark \ref{grothendieck}). Indeed,
by Theorem \ref{compute}, $\cd_p(G)=0$ if and only if $\widetilde
\CH_0(G/P)$ is not divisible by $p$ in $\CH_0(G/P)$, where $P$ is
a Borel subgroup of $G$. Since $\CH_0(G/P)$ is an infinite cyclic
group generated by the class of a rational point, the latter is
equivalent to the condition that $p$ does not divide $t_G$, i.e.,
$p$ is not a torsion prime of $G$.
\end{rem}





\section
{Remarks on $\widetilde\CH(G/P)$}
\label{Explicit description}


Let $P$ be an arbitrary subgroup of an algebraic group $G$. Let
$P\ra\gGL(V)$ be a finite-dimensional representation. The group
$P$ acts (on the right) on the product $G\times V$ by $(g,v)\cdot
p=(g\cdot p,\;p\inv\cdot v)$. The factor variety $(G\times V)/P$ is a
vector bundle over $G/P$, we denote it by $\Bun(V)$.

We can view $V$ as a $P$-equivariant vector bundle over the point
$\Spec F$. For any $n\geq0$, the $n$-th
$P$-equivariant Chern class $c_n(V)$ is an element
of $\CH^n(BP)$ (see \cite{Edidin98}).

Let $T$ be a split torus. There is a canonical isomorphism
\[
\cat S(\widehat T)\iso \CH(BT),
\]
(where $\widehat T$ is the character group of $T$,
$\cat S$ stands for the symmetric algebra) defined by the
property that the image of a character $\chi$ is the first Chern
class $c_1(\chi)$ where $\chi$ is considered as a 1-dimensional
representation of $T$ \cite[3.2]{Edidin98}.

Let $P$ be a special parabolic subgroup of a split semisimple algebraic group $G$.
Let $T$ be a maximal split torus contained in $P$ and let $W_P$ be
the Weyl group of $P$. Since $P$ is special, the canonical
homomorphism
\[
\CH(BP)\ra \CH(BT)^{W_P}=\cat S(\widehat T)^{W_P}
\]
is an isomorphism \cite[prop. 6]{Edidin98}. Identifying $\CH(BP)$
with $\cat S(\widehat T)^{W_P}$, we get a homomorphism
\[
\varphi_{G}: \cat S(\widehat T)^{W_P}\ra \CH(G/P)
\]
with the image the subring $\widetilde \CH(G/P)$.

\begin{lemma}\label{technical}
Let $\chi_1,\chi_2,\dots,\chi_m\in\widehat T$ be all characters
(with multiplicities) of a representation $P\ra\gGL(V)$. Let
$s_n\in \cat S^n(\widehat T)^{W_P}$ be the standard symmetric
functions on the characters $\chi_i$. Then $
\varphi_{G}(s_n)=c_n\(\Bun(V)\)$.
\end{lemma}
\begin{proof}
By naturality of the Chern classes, we have $
\varphi_{G}\(c_n(V)\)=c_n\(\Bun(V)\)$. On the other hand,
$c_n(V)$ is the $n$-th standard symmetric functions on the
characters of $V$.
\end{proof}


\begin{rem}
\label{demazurovo}
Let $G$ be a split semisimple group over an arbitrary field (of an arbitrary
characteristic), $B\subset G$ a Borel
subgroup, $T\subset B$ a split maximal torus, $W$ the Weyl group of $G$.
The closures $X_w$ of the cells $BwB/B$ of the cellular variety $G/B$ are indexed by the
elements $w\in W$ and called {\em generalized Schubert varieties} of $G/B$;
moreover, $\dim X_w=l(w)$, where $l\!:W\to\Z_{\geq0}$ is the
length function.
% (with respect to the set of simple roots, given by $B$).
Taking the unique maximal length element $w_0\in W$ and setting
$X^{w}=X_{w_0w}$, we get a different (preferable for us) indexation of the same
varieties, for which $\codim X^w=l(w)$.
The group $\CH(G/B)$ is free and the classes $[X^w]$, called {\em generalized Schubert
classes}, form its basis.

The following formula for the product of a
$1$-codimension Schubert class with an arbitrary Schubert class is given in
\cite[\S4.4 cor.2]{Demazure2}:
$$
[X^{s_\alpha}]\cdot[X^w]=\Sum_\beta \langle\beta^\vee,\omega_\alpha\rangle\cdot[X^{w\cdot
s_\beta}]\;,
$$
where $\alpha$ is a simple root,
$\omega_\alpha$ its fundamental weight,
$s_\alpha\in W$ the reflection with respect to $\alpha$;
$\beta$ runs over the set of positive roots such that $l(w\cdot
s_\beta)=l(w)+1$, and $\beta^\vee$ is the dual to $\beta$ root.
Note that the coefficients of this formula depend only on the root system; in
particular, they do not depend on the base field and its
characteristic.
Moreover, this formula completely determines the multiplication
table of the basis $[X^w]$, $w\in W$, because the $\mathbb{Q}$-algebra
$\CH(G/B)\otimes\mathbb{Q}$
is generated by $\CH^1(G/B)$ \cite{Demazure2}.

\begin{comment}
Let now $P\supset B$ be a parabolic subgroup of $G$, $W_P\subset W$ its Weyl group.
The closures $Y^\sigma$ of the cells $BwP/P$, $w\in W$
of the cellular variety $G/P$ are indexed by
the cosets $\sigma=wW_P\in W/W_P$ and called
{\em generalized Schubert varieties} of $G/P$;
moreover,
$\codim Y^\sigma$ is the minimum of lengths of
elements in the coset $\sigma$.
The group $\CH(G/P)$ is free and the classes $[Y^\sigma]$,
called {\em generalized Schubert
classes}, form its basis.


The projection $G/B\to G/P$ produces an injective  homogeneous ring homomorphism
$\CH^*(G/P)\to\CH^*(G/B)$, mapping $[Y^{\sigma}]$ to $[X^w]$, where
$w\in\sigma$ is the (unique) minimal length element of $\sigma$
(see \cite[lemma 1.2(b)]{Koeck} and \cite[\S3.3]{Demazure2}).
Therefore
the above formula for multiplication in $\CH(G/B)$ also determines
multiplication in $\CH(G/P)$.
\end{comment}
\end{rem}


\begin{rem}\label{borel}
Let $P=B$ be a Borel subgroup of $G$.
We have $W_B=1$ and therefore
the subring $\widetilde \CH(G/B)$ is generated  by
%the image of
%$(\phi_G)|_{\widehat T}:\widehat T\ra \CH^1(G/B)$.
$\phi_G(\widehat T)$.
In the case of simply connected $G$,
for the weight $\omega_\alpha$ of a simple root $\alpha$,
one has the formula
$$
\phi_G(\omega_\alpha)=
%-\Sum_{\beta}\langle\beta^\vee,\alpha\rangle[X^{s_\beta}]\;,
-[X^{s_\alpha}]
\;,\;\;\;\;\;\;\;\text{\cite[\S4 formula (7)]{Demazure2}}\;,
$$
which determines $\phi_G$ also in the non simply connected case.
% because $\widehat T$ is
%rationally  generated by the simple roots.
This formula also shows that if the group $G$ is
simply connected, then $\phi_G(\widehat T)=\CH^1(G/B)$, and
therefore
$\widetilde \CH(G/B)$ is the subring of $\CH(G/B)$, generated  by $\CH^1(G/B)$.
\end{rem}

%In fact, the ring $\widetilde \CH(G/P)$ is generated by the
%Chern classes of all vector bundles over $G/P$ of the form
%$(G\times V)/P$ for all representations $P\ra S$ of the
%group $P$.

\begin{rem}
%%% FROM PROBLEM:
%>From Theorem \ref{compute} and Remark \ref{borel}, we see that:
From Theorem \ref{compute} and Remark \ref{borel}, we see that:
\begin{enumerate}
\item
if $G_1$ and $G_2$ are split semisimple groups, then
$\cd_p(G_1\times G_2)=\cd_p(G_1)+\cd_p(G_2)$;
\item
if $G'\to G$ is a central isogeny of split semisimple groups, then
$\cd_p(G')\leq\cd_p(G)$.
\end{enumerate}
\end{rem}

\begin{rem}
Let us consider pairs $(\Phi, A)$, consisting of a root system
$\Phi$ and a subgroup of the quotient of the weight lattice of
$\Phi$ by its root lattice.
An isomorphism of pairs $(\Phi,A)\to(\Phi',A')$ is an isomorphism
of the root systems $\Phi\to\Phi'$
such that the induced isomorphism of the
lattice quotients maps $A$ to $A'$.
%a subgroup
%conjugated with $A'$ under an automorphism of the Dynkin diagram
%of $\Phi'$.
To any split semisimple algebraic group $G$ one attaches an
isomorphism class of above pairs, to which we refer as
{\em extended type} of $G$.
Theorem \ref{compute} with Remarks \ref{demazurovo} and
\ref{borel} shows that $\cd_p(G)$ (for any $p$) depends only on the
extended type of $G$;
in particular, it does not depend on the base field and especially
on its characteristic (so that computing $\cd_p(G)$
one may always assume that $G$ is defined over $\mathbb{C}$).
\end{rem}

\section{Canonical $p$-dimension of split simple groups of classical types}
\label{Split simple groups of classical types}

In this section we compute  canonical $p$-dimension of all split simple
groups of classical types.
We will need the following

\begin{lemma}\label{invariant}
Let $R$ be a commutative ring, $r\in R$, and let $A$ be the factor
ring of the polynomial ring $R[x_1,x_2,\dots,x_n]$ modulo the
ideal generated by the polynomial $x_1+x_2+\dots +x_n-r$. The
symmetric group $W=S_n$ acts on $A$ by permuting the $x_i$. If $R$
has trivial $\Z$-torsion, then $A^W=R[s_2,s_3,\dots,s_n]$, where
$s_i$ are the standard symmetric functions.
\end{lemma}

\begin{proof}
Consider the natural $W$-action on the ring
$R[x]=R[x_1,x_2,\dots,x_n]$. We have the exact sequence
$
0\ra R[x]\xra{f} R[x]\ra A\ra 0
$,
where the first map is the multiplication by $f=x_1+x_2+\dots
+x_n-r$. Applying the $W$-invariants we get an exact sequence
\[
0\ra R[x]^W\xra{f} R[x]^W\ra A^W\ra H^1(W,R[x]).
\]
The ring $R[x]^W$ coincides with $R[s]=R[s_1,s_2,\dots,s_n]$. The
monomials in the variables $x_i$ form a permutation basis of the
$R$-module $R[x]$. By the Faddeev-Shapiro lemma, the group
$H^1(W,R[x])$ is a direct sum of the groups $H^1(W',R)=\Hom(W',R)$
for certain subgroups $W'\subset W$. Since $R$ has trivial
$\Z$-torsion, the latter group is trivial. Therefore,
\[
A^W=R[s]/(f)=R[s_2,s_3,\dots,s_n].\qedhere
\]
\end{proof}

\subsection{Type $A_{n-1}$}\label{an}

A split simple group of type $A_{n-1}$ is isomorphic to $G=\gSL(n)/\gmu_l$,
where $l$ is a divisor of $n$.
Let $P\subset \gSL(n)$ be the stabilizer of the line
$U=[1:0:\ldots:0]\in\P^{n-1}$ with respect to the natural action
of $\gSL(n)$ on $\P^{n-1}$. The semisimple part of $P$ is
$\gSL(n-1)$ and it intersects $\gmu_l$ trivially. Hence the
parabolic subgroup $P_l=P/\gmu_l$ of $G$ is special. We have
$G/P_l=\P^{n-1}$.


The intersection $T$ of the group of diagonal matrices $\gD(n)$ of
$\gGL(n)$ with $\gSL(n)$ is a maximal torus of $\gSL(n)$. The
character group $\widehat T$ is identified with the factor group
of $\Z^n=\widehat{\gD(n)}$ with the standard basis $x_1,x_2,\dots,
x_n$ by the subgroup generated by $x_1+x_2+\dots +x_n$. The
character group of the maximal torus $T_l=T/\gmu_l$ of $G$ is the
subgroup of $\widehat T$ consisting of all sums $\sum a_ix_i$ such
that $\sum a_i$ is divisible by $l$. Hence, $\widehat T_l$ is
generated by $lx_1$ and $x_i-x_1$ for all $i=2,\dots, n$ with the
relation $\sum_{i\geq 2}(x_i-x_1)=-nx_1.$

The Weyl group $W=W_{P_l}$ is the symmetric group $S_{n-1}$,
permuting $x_2,\dots, x_n$. Applying Lemma \ref{invariant} to the
ring $R=\Z[lx_1]$, the element $r=-nx_1$, the variables $x_i-x_1$
and the group $W$, we get
$
\cat S(\widehat T_l)^W=\Z[lx_1,s_2,s_3,\dots, s_{n-1}]
$,
where the $s_i$ are the standard symmetric functions on the
$x_i-x_1$, $i\geq 2$.

The group $P$ acts naturally on the space $V=F^n$. The characters
of this representation are $x_1,x_2,\dots, x_n$. The corresponding
vector bundle $\Bun(V)$ over $\P^{n-1}=\gSL(n)/P=G/P_l$ is the
trivial vector bundle of rank $n$. The line $U$ can be viewed as a
1-dimensional representation of $P$ given by the character $x_1$.
We have $\Bun(U)=L^\vee$, where $L$ is the canonical line bundle
on $\P^{n-1}$ (with the sheaf of sections $\cO(1)$). Consider the
representation $M=(V/U)\tens U^\vee$ of the group $P$ with the
characters $x_i-x_1$ for all $i=2,\dots, n$. Note that the group
$\gmu_l$ is contained in the kernel of the representation, hence
$M$ is a representation of $P_l$.

By Lemma \ref{technical}, we have
$\varphi_{G}(lx_1)=lc_1(L^\vee)=-lh$, where
$h\in\CH_1(\P^{n-1})$ is the class of a hyperplane, and also
$\varphi_{G}(s_i)=c_i\(\Bun(M)\)$ for all $i$. Hence the
subring $\widetilde \CH(\P^{n-1})$ of $\CH(\P^{n-1})=\Z[h]/(h^n)$
is generated by $lh$ and the Chern classes $c_i\(\Bun(M)\)$. Since
\[
\Bun(M)=\(\Bun(V)/\Bun(U)\)\tens \Bun(U^\vee)=(\Bun(V)/L^\vee)\tens
L,
\]
the class $[\Bun(M)]$ is equal to $n[L]-1$ in
$K_0(\P^{n-1})$. Hence,
$
c_\bullet(\Bun(M))=c_\bullet(L)^n=(1+h)^n
$.
Thus the subring $\widetilde \CH(\P^{n-1})$ is generated by $lh$
and $\binom{n}{i}h^i$ for $i=2,\dots,n-1$.


Let $p$ a be prime integer and let $p^k$ be the largest power of $p$
dividing $n$. Note that the binomial coefficient $\binom{n}{i}$ is
divisible by $p$ unless $i$ is divisible by $p^k$. The largest
value of $i<n$ such that $\binom{n}{i}$ is not divisible by $p$ is
$n-p^k$. By Theorem \ref{compute},
\[
\cd_p \(\gSL(n)/\gmu_l\)=
\left\{%
\begin{array}{ll}
    p^k-1, & \hbox{if $p$ divides $l$;} \\
    1, & \hbox{otherwise.} \\
\end{array}%
\right.
\]

Denote by $\CSA_{n,l}(K)$ the set of isomorphism classes of
central simple $K$-algebras of degree $n$ and exponent dividing
$l$. The exact sequence $1\ra \gmu_l\ra \gSL(n)\ra
\gSL(n)/\gmu_l\ra 1$ yields a surjective map
$
\Tors_{\gSL(n)/\gmu_l}(K)\ra \CSA_{n,l}(K)
$
with trivial kernel. By Remark \ref{inject},
\[
\cd_p \(\CSA_{n,l}\)=
\left\{%
\begin{array}{ll}
    p^k-1, & \hbox{if $p$ divides $l$;} \\
    1, & \hbox{otherwise.} \\
\end{array}%
\right.
\]


\subsection{Type $B_n$}\label{bn}

The only torsion prime is $p=2$.

Taking a ($2n+1$)-dimensional vector space, endowed with
a completely split quadratic form, let
a vector $g$ together with vectors
$e_i,f_i$, $i=1,2,\dots, n$ form a basis such that $\{e_i,f_i\}$ are pairwise
orthogonal hyperbolic pairs, while $g$ is orthogonal to all $e_i,f_i$.
%The vectors $e_i$, $f_i$ together
%with one more vector $g$ form a basis of the $(2n+1)$-dimensional
%underlying vector space.
Let $G=\gSO(2n+1)$ be the corresponding special
orthogonal group.  The inclusion of $\gD(n)$ into
$\gSO(2n+1)$ given by $t(e_i)=t_ie_i$, $t(f_i)=t_i\inv f_i$ and
$t(g)=g$, where $t=\operatorname{diag}(t_1,\dots, t_n)$,
identifies $\gD(n)$ with a maximal torus $T$ of $\gSO(2n+1)$. In
particular, the group $\widehat T$ is identified with
$\Z^n=\widehat{\gD(n)}$. We write
 $x_1,x_2,\dots, x_n$ for the standard basis of $\Z^n$.

Let $V$ be the totally isotropic subspace of dimension $n$
generated by all the $e_i$. Denote by $P$ the stabilizer of $V$ in
$G$, so that $X=G/P$ is the variety of all dimension $n$ totally
isotropic subspaces. The characters of the natural representation
$P\ra \gGL(V)$ are $x_1,x_2,\dots, x_n$. The vector bundle
$\Bun(V)$ over $X$ is the tautological vector bundle.

The group $W=W_P$ is the symmetric group $S_n$ permuting the
$x_i$. The semisimple part of $P$ is $\gSL(n)$, so that $P$ is
special.

We have $\cat S(\widehat T)^W=\Z[s_1,s_2,\dots, s_n]$, where $s_k$
are the standard symmetric functions on the $x_i$. By Lemma
\ref{technical}, the subring $\widetilde \CH(X)$ of $\CH(X)$ is
generated by the Chern classes of $\Bun(V)$. These Chern classes
are divisible by 2 in $\CH(X)$ \cite[ch. III th. 6.11]{Mimura91}.
Thus, $\widetilde {\Ch^j}(X)=0$ if $j>0$. We conclude by Theorem
\ref{compute} that
\[
\cd_2 \gSO(2n+1)=\frac{n(n+1)}{2}
\]
(see also Examples \ref{so} and \ref{cd_2(Grassm)}).
The set $\Tors_{\gSO(2n+1)}(K)$ is
identified with the set of similarity classes $\Q_{2n+1}(K)$ of
non-degenerate quadratic forms of dimension $2n+1$ over $K$. Thus,
\[
\cd_2 \Q_{2n+1}=\frac{n(n+1)}{2}.
\]



Let $G=\gSpin (2n+1)$ be the spinor group. There is an exact
sequence
\[
1\ra \gmu_2\ra T'\ra T\ra 1,
\]
where $T'$ is a maximal torus of $\gSpin (2n+1)$. We have
$\widehat T'=\widehat T+\Z y=\Z^n+\Z y$, where
$y=(x_1+\dots+x_n)/2$. By Lemma \ref{invariant} applied to the
ring $R=\Z[y]$, the element $r=2y$ and the group $W$, the ring
$\cat S(\widehat T)^W$ is the polynomial ring
$\Z[y,s_2,s_3,\dots,s_{n}]$.

By Lemma \ref{technical}, $\varphi_{G}(s_1)=c_1\(\Bun(V)\)$. The
latter class coincides with $2e$ where $e$ is a generator of
$\CH^1(X)$ \cite[ch. III th. 6.11]{Mimura91}. Since $s_1=2y$ and
$\CH^1(X)$ is torsion free, we have $\varphi_{G}(y)=e$.



As noted above, the images of the $s_i$ in $\CH(X)$ are divisible
by 2. Hence the image of $\widetilde \CH(X)$ in $\Ch(X)=\CH(X)/2$
is the subring generated by $e$ mod 2. Let $m$ be the smallest
integer such that $2^m>n$. Then $e^{2^m}=0$ and $e^{2^m-1}\neq 0$
in $\Ch(X)$ \cite[ch. III th. 6.11]{Mimura91}. Thus,
\[
\cd_2 \gSpin (2n+1)=\frac{n(n+1)}{2}-2^m+1.
\]

Let $\overline \Q_{2n+1}(K)$ be the subset of $\Q_{2n+1}(K)$
consisting of all classes of forms with trivial even Clifford
invariant. The exact sequence $1\ra \gmu_2\ra \gSpin(2n+1)\ra
\gSO(2n+1)\ra 1$  yields a surjective map
$
\Tors_{\gSpin (2n+1)}(K)\ra \overline \Q_{2n+1}(K)
$
with trivial kernel. In particular,
\[
\cd_2 \overline \Q_{2n+1}=\frac{n(n+1)}{2}-2^m+1.
\]

\subsection{Type $C_n$}

The group $\gSp(2n)$ is special, so that $\cd_p\gSp(2n)=0$ for all
$p$.

Let $G=\gPGSp(2n)$ be the projective symplectic group. The number
$p=2$ is the only torsion prime of $G$. Instead of applying the
general method, we proceed as follows.

The set
$\Tors_{\gPGSp(2n)}(K)$ is identified with the set of isomorphism
classes $\ASI_{2n}(K)$ of central simple $K$-algebras $A$ of
degree $2n$ with a symplectic involution \cite[\S29.22]{Book}.
%The pair
%$(A,\sigma)$ is split if and only if the algebra $A$ is split.
The forgetful functor $\ASI_{2n}\to\CSA_{2n,2}$ has trivial kernel
and is surjective. Therefore, by Remark \ref{inject} and
(\ref{an}),
\[
\cd_2 \gPGL(2n)=\cd_2 \ASI_{2n}=\cd_2\CSA_{2n,2}=2^k-1,
\]
where $2^k$ is the largest power of $2$ dividing $2n$.



\subsection{Type $D_n$}\label{dn}
\label{Type $D_n$}

Let $\{e_i,f_i\}$, $i=1,2,\dots, n$ be pairwise orthogonal
hyperbolic pairs of a
%non-degenerate
hyperbolic
quadratic form of dimension
$2n$. The inclusion of $\gD(n)$ into $\gSO(2n)$ given by
$t(e_i)=t_ie_i$ and $t(f_i)=t_i\inv f_i$, where
$t=\operatorname{diag}(t_1,\dots, t_n)$, identifies $\gD(n)$ with
a maximal torus $T'$ of $\gSO(2n)$. In particular, the group
$\widehat T'$ is identified with $\Z^n=\widehat{\gD(n)}$. We write
 $x_1,x_2,\dots, x_n$ for the standard basis of $\Z^n$.

 Let $V$ be the totally isotropic subspace of dimension $n$
generated by all the $e_i$ and let $U$ be the line $Fe_1$. Denote
by $P$ the stabilizer of the flag $U\subset V$ in $G=\gSpin(2n)$
and set $X=G/P$. The semisimple part of $P$ is isomorphic to
$\gSL(n-1)$ and intersects trivially the center of $G$. Hence the
image of $P$ in any simple group of type $D_n$ (under a central isogeny
of $G$) is a special group.

Let $Y$ be the connected component of the scheme of maximal
($n$-dimensional) totally isotropic subspaces such that $V$ is a
point of $Y$. The natural morphism $f:X\ra Y$ is the projective
bundle associated with the tautological vector bundle $E$ over $Y$
of rank $n$. In particular,
\[
\dim X=\dim Y+(n-1)=\frac{n(n-1)}{2}+(n-1).
\]
Note that $Y$ is isomorphic to the projective homogeneous variety of the
group $\gSpin(2n-1)$ considered in the type $B_{n-1}$. The Chern
classes of $E$ in $\CH(Y)$ are divisible by 2 (see the type
$B_n$), hence $\Ch(X)=\Ch(Y)[h]/(h^n)$, where $h=c_1(L)$ for the
canonical line bundle $L$ over $X$.

Similar to the case $B_n$, the character group of the maximal
torus $T$ of $\gSpin(2n)$ is equal to $\Z^n+\Z y$, where
$y=(x_1+x_2+\dots +x_n)/2$. Set $x'_i=x_i-x_1$ for $i=2,\dots,n$,
so that $x'_2+\dots+x'_n=2y-nx_1$. The symmetric group $W=W_P$
permutes the $x'_i$ and acts trivially on $y$ and $x_1$. Applying
Lemma \ref{invariant} to the variables $x'_i$, the ring
$R=\Z[y,x_1]$ and the element $r=2y-nx_1$ we see that
$
\cat S(\widehat T)^W=\Z[y,x_1,s_2,\dots s_{n-1}]
$,
where the $s_i$ are the standard symmetric functions on the
%$x_i-x_1$.
$x'_i$.


%The characters of the natural representation $P\ra \gGL(V)$ are
%$x_1,x_2,\dots, x_n$. The vector bundle $\Bun(V)$ over $X$ is the
%tautological vector bundle.



Consider the homomorphism (reduced modulo 2)
\[
\varphi_{G}: \Z[y,x_1,s_2,\dots s_{n-1}]\ra
\Ch(X)=\Ch(Y)[h]/(h^n).
\]
As in the case $A_{n-1}$, we have $\Bun(U)=L^\vee$ and therefore
$\varphi_{G}(x_1)=c_1(L^\vee)=-h$. Similar to the case $B_n$,
the class $e=\varphi_{G}(y)$ is a generator of $\Ch^1(Y)$.
Recall that $e^{2^m-1}\neq 0$ and $e^{2^m}=0$ where $m$ is the
smallest integer such that $2^m\geq n$.

Similar to the case $A_{n-1}$, we observe by Lemma \ref{technical}
that the images of the $s_i$ in $\Ch(X)$ are the Chern classes of
the vector bundle $(f^*(E)/L^\vee)\tens L$. The class of this
bundle in $K_0(X)$ is equal to $[f^*(E)\tens L]-1$. Since the
Chern classes of $E$ are divisible by 2, we can replace $E$ by the
trivial bundle of rank $n$ and replace $[f^*(E)\tens L]$ by
$n[L]$. As in the case $A_{n-1}$, we see that
$\varphi_{G}(s_i)=\binom{n}{i}h^i$.

The subring $\widetilde \Ch(X)=\Im(\varphi_{G})$ is generated by
$h$ and $e$. The largest degree nontrivial monomial in $h$ and $e$
is $h^{n-1}e^{2^m-1}$. By Theorem \ref{compute},
\[
\cd_2 \gSpin (2n)=\dim X-(n-1)-(2^m-1)=\frac{n(n-1)}{2}-2^m+1.
\]


% The torsors of $\gSpin(2n)$ over $K$ are
%identified with the set $\overline \Q_{2n}(K)$ of all isomorphism
%classes of non-degenerate quadratic forms with trivial
%discriminant and Clifford invariant. The canonical dimensions of
%$\overline \Q_{2n}$ and $\overline \Q_{2n-1}$ coincide. Hence,
%\[
%\cd_2 \gSpin (2n)=\cd_2 \overline \Q_{2n}=\frac{n(n-1)}{2}-2^m+1,
%\]
%where $m$ is the smallest integer such that $2^m\geq n$.

Let $\overline \Q_{2n}(K)$ be the subset of the set $\Q_{2n}(K)$
of isomorphism classes of non-degenerate quadratic forms of
dimension $2n$ consisting of all classes of forms with trivial
discriminant and Clifford invariant. The exact sequence $1\ra
\gmu_2\ra \gSpin(2n)\ra \gSO(2n)\ra 1$ yields a surjective map
$
\Tors_{\gSpin (2n)}(K)\ra \overline \Q_{2n}(K)
$
with trivial kernel. In particular,
\[
\cd_2 \overline \Q_{2n}=\frac{n(n-1)}{2}-2^m+1.
\]

Now let $G=\gSO(2n)$. Recall that the character group $\widehat
T'$ of the maximal torus $T'$ of $G$ is the subgroup of $\widehat
T$ generated by all the $x_i$. Thus we have
$
\cat S(\widehat T')=\Z[x_1,x_2,\dots, x_n]
$
and therefore,
$
\cat S(\widehat T')^W=\Z[x_1,s_1,\dots, s_{n-1}].
$
The subring $\widetilde \Ch(X)$ is then generated by $h$. The
largest degree nontrivial monomial in $h$ is $h^{n-1}$. By Theorem
\ref{compute},
\[
\cd_2 \gSO(2n)=\dim X-(n-1)=\frac{n(n-1)}{2}.
\]

Let $\Q'_{2n}(K)$ be the subset of the set $\Q_{2n}(K)$ consisting
of all classes of forms with trivial discriminant. There is a
canonical bijection
$
\Tors_{\gSO(2n)}(K)\iso \Q'_{2n}(K)
$.
Therefore,
\[
\cd_2 \Q'_{2n}=\frac{n(n-1)}{2}.
\]

%The torsors of $\gSO(2n)$ over $K$ are identified with the set
%$\Q_{2n}(K)$ of all isomorphism classes of non-degenerate
%quadratic forms with trivial discriminant. The canonical
%dimensions of $\Q_{2n}$ and $\Q_{2n-1}$ coincide. Hence,
%\[
%\cd_2 \gSO (2n)=\cd_2 \Q_{2n}=\frac{n(n-1)}{2}.
%\]

Let $G=\gPGO^+(2n)$ be the projective orthogonal group. Let
$\overline T$ be the image of the maximal torus $T$ under the
canonical isogeny $\gSpin(2n)\ra G$. The character group
$\overline T$ is the subgroup of $\widehat T$ generated by all the
simple roots. Thus we have
$
\cat S(\widehat {\overline T})=\Z[2x_1,x'_2,\dots, x'_n]
$
and therefore,
$
\cat S(\widehat {\overline T})^W=\Z[2x_1,s_1,\dots, s_{n-1}]
$.
The subring $\widetilde \Ch(X)$ is then generated by
$\binom{n}{i}h^i$. Let $2^k$ be the largest power of 2 dividing
$n$. Note that the binomial coefficient $\binom{n}{i}$ is even
unless $i$ is divisible by $2^k$. The largest value of $i<n$ such
that $\binom{n}{i}$ is odd is $n-2^k$. The largest degree
nontrivial monomial in $h$ is $\binom{n}{n-2^k}h^{n-2^k}$. By
Theorem \ref{compute},
\[
\cd_2 \gPGO^+(2n)=\dim X-(n-2^k)=\frac{n(n-1)}{2}+2^k-1.
\]

Let $\AQP_{2n}(K)$ be the set of isomorphism classes of central
simple algebras of degree $2n$ with a quadratic pair with trivial
discriminant \cite[\S29.F]{Book}. The exact sequence $1\ra
\gPGO^+(2n)\ra \gPGO(2n)\ra \Z/2\Z\ra 1$ yields a surjective map
$
\Tors_{\gPGO^+ (2n)}(K)\ra \AQP_{2n}(K)
$
with trivial kernel. In particular,
\[
\cd_2 \AQP_{2n}=\frac{n(n-1)}{2}+2^k-1.
\]


Suppose now that $n$ is even. There are two isomorphic semispinor
groups. We set $\gSpin^\sim(2n)=\gSpin(2n)/H$, where $H$ is the
intersection of $\Ker(y)$ with the center of $\gSpin(2n)$. Let
$T''$ be the image of the maximal torus $T$ under the canonical
isogeny $\gSpin(2n)\ra G$. The character group of $T''$ is the
subgroup of $\widehat T$ generated by all the simple roots and
$y$. Thus we have
$
\cat S(\widehat T'')=\Z[y,2x_1,x'_2,\dots, x'_n]
$.

Applying Lemma \ref{invariant} to the elements $x'_i$, the ring
$R=\Z[y,2x_1]$, and the element $r=2y-nx_1$, we see that
$
\cat S(\widehat T'')^W=\Z[y,2x_1,s_2,\dots s_{n-1}]
$.


\begin{comment}


The character group of a maximal torus $T$ of $\gSpin^\sim(2n)$
is the subgroup of $\Z^n+\Z y$ generated by the roots
$x_i-x_{i+1}$ and $y$ or, equivalently, by $2x_1$, $y$ and
$x_i-x_1$, $i=2,\dots n$.

Consider the homogeneous variety $X=G/P$ as above (used for
$\gPGO^+(2n)$). The Weyl group $W_P$ is the symmetric group
$S_{n-1}$ permuting $x_i-x_1$, $i=2,\dots n$. Hence
\[
\cat S(\widehat T'')^W=\Z[2x_1,y,s_2,\dots,s_{n-1}]
\]
where $s_i$ are the standard symmetric functions in $x_i-x_1$.
(Note that $s_1=2y-nx_1$.)

\end{comment}


The subring $\widetilde \Ch(X)$ is then generated by $e$ and
$\binom{n}{i}h^i$. The largest degree nontrivial monomial in $h$
and $e$ is $\binom{n}{n-2^k}h^{n-2^k}e^{2^m-1}$. By Theorem
\ref{compute},
\[
\cd_2 \gSpin^\sim(2n)=\dim
X-(n-2^k)-(2^m-1)=\frac{n(n-1)}{2}+2^k-2^m.
\]
Let $\AQP'_{2n}(K)$ be the set of isomorphism classes of central
simple algebras of degree $2n$ with a quadratic pair with trivial
discriminant and trivial component of the
%even
Clifford algebra.
The exact sequence $1\ra \gmu_2\ra \gSpin^\sim(2n)\ra
\gPGO^+(2n)\ra 1$ yields a surjective map
$
\Tors_{\gSpin^\sim(2n)}(K)\ra \AQP'_{2n}(K)
$
with trivial kernel. In particular,
\[
\cd_2 \AQP'_{2n}=\frac{n(n-1)}{2}+2^k-2^m.
\]

\subsection
{Appendix: Type $G_2$}

The only torsion prime is $2$.
Since $\Tors_G\simeq\overline{Q}_8$ for a split simple $G$ of type $G_2$,
we have
$
\cd_2(G)=\cd_2(\overline{Q}_8)=3
$ (see \S\ref{Type $D_n$}).


\begin{thebibliography}{99}



\bibitem{Berhuy-Reichstein}
G.~Berhuy and Z.~Reichstein. {\em On the notion of canonical
dimension for algebraic groups.} Adv. Math., to appear.

\bibitem{Chernousov-Gille-Merkurjev}
V. Chernousov, S. Gille, A. Merkurjev.
{\em Motivic decomposition of isotropic projective homogeneous varieties.}
Duke Math. J., to appear.


\bibitem{Demazure1}
M. Demazure.
{\em Invariants sym\'etriques des groupes de Weyl et torsion.}
Invent. Math. {\bf 21} (1973), 287--301.

\bibitem{Demazure2}
M. Demazure.
{\em D\'esingularisation des vari\'et\'es de Schubert g\'en\'eralis\'ees.}
Ann. Sci. Ecole Norm. Sup. (4) {\bf 7} (1974), 53--88.


\bibitem{Edidin98}
D.~Edidin and W.~Graham.
\emph{Equivariant intersection theory.}
Invent. Math.
\textbf{131} (1998), no.~3, 595--634.


\bibitem{Fulton}
W.~Fulton. {\em Intersection Theory.} Springer-Verlag, Berlin,
Heidelberg, New York, Tokyo, 1984.

\bibitem{Grothendieck58}
A.~Grothendieck.
\emph{Torsion homologique et sections
rationnelles.}
Exp. {\bf 5} in:
{\em Anneaux  de Chow et applications.}
S\'eminaire C. Chevalley  1958. Paris (1958).


\bibitem{Hartshorne}
R. Hartshorne.
{\em Algebraic Geometry.}
Springer-Verlag; New York, Heodelberg, Berlin, 1977.


\bibitem{SB-motives}
N. A. Karpenko.
{\em The Grothendieck Chow-motifs of Severi-Brauer varieties.}
Algebra i Analiz {\bf 7} (1995), no. 4, 196--213 (in Russian).
Engl. transl.:
St. Petersburg Math. J. {\bf 7} (1996), no. 4, 649--661.

\bibitem{flag}
N. A.~Karpenko. {\em Cohomology of relative cellular spaces and
isotropic flag varieties.} Algebra i Analiz {\bf 12} (2000), no.
1, 3--69 (in Russian). Engl. transl.: St. Petersburg Math. J. {\bf
12} (2001), no. 1, 1--50.

\bibitem{cd(SO)}
N. A.~Karpenko. {\em Canonical dimension of orthogonal groups.}
Transform. Groups, to appear.

\bibitem{KM}
N. A.~Karpenko and A. S.~Merkurjev. {\em Essential dimension of
quadrics.} Invent. Math. {\bf 153} (2003), 361--372.

\bibitem{Book} M.-A.~Knus, A.S.~Merkurjev, M.~Rost, J.-P.~Tignol.
{\em The Book of Involutions.}
American Mathematical Society
Colloquium Publications, {\bf 44}. American Mathematical Society,
Providence, RI, 1998, xxii+593 pp.

\bibitem{Koeck}
B. K\"ock.
{\em Chow motif and higher Chow theory of $G/P$.}
Manuscripta Math. {\bf 70} (1991), 363--372.

\bibitem{Mimura91}
M.~Mimura and H.~Toda.
\emph{Topology of {L}ie groups. {I} and {II}.}
  Translations of Mathematical Monographs, vol.~91, American Mathematical
  Society, Providence, RI, 1991, Translated from the 1978 Japanese edition by
  the authors.

%\bibitem{Rost-A}
%M. Rost.
%{\em Chow groups with coefficients.}
%Doc. Math. {\bf 1} (1996), 319--393.

%\bibitem{Popov74}
%V.~L. Popov, \emph{Picard groups of homogeneous spaces of linear
%algebraic
%  groups and one-dimensional homogeneous vector fiberings}, Izv. Akad. Nauk
%  SSSR Ser. Mat. \textbf{38} (1974), 294--322.

\bibitem{Rost-NV}
M.~Rost. {\em Norm varieties and algebraic cobordism.} Li, Ta
Tsien (ed.) et al., Proceedings of the International Congress of
mathematicians, ICM 2002, Beijing, China, August 20-28, 2002. Vol.
II: Invited lectures. Beijing: Higher Education Press. 77--85
(2002).

\bibitem{Samuel-Zariski}
P.~Samuel and O.~Zariski. {\em Commutative Algebra. Volume II.}
Princeton (New Jersey), London, New York, Toronto; D.Van Nostrand
Company, Inc, 1958.

%\bibitem{Suslin-Princeton}
%A. Suslin.
%{\em Norm varieties.}
%Institute for Advanced Study, Princeton, New Jersey;
%Fall 2004 Lectures.


\bibitem{Vishik-QG}
A. Vishik.
{\em On the Chow groups of quadratic grassmannians.}
Linear Algebraic Groups and Related Structures (Preprint Server) {\bf 168}
(Nov. 30, 2004), 23 p.

\end{thebibliography}


\end{document}

