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\title[Essential dimension of finite groups] % colontitle
{Essential dimension of finite groups}

\keywords
{Algebraic groups, finite $p$-groups, essential dimension,
representations, $K$-theory.
{\em 2000 Mathematical Subject Classifications:}
20G15; 14C35}



\date
%{\today}
{June, 2007}

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

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


\address
{Institut de Math\'ematiques de Jussieu\\
Universit\'e Pierre et Marie Curie - Paris 6\\
4 place Jussieu\\
F-75252 Paris CEDEX 05\\
FRANCE}




\email {karpenko@math.jussieu.fr}



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

\email
{merkurev@math.ucla.edu}

\thanks
{The work of the first author has been partially supported by the
Collaborative Research Centre 701 ``Spectral Structures and
Topological Methods in Mathematics'' of the Bielefeld University}


\thanks{The work of the second author has been supported by the
NSF grant DMS \#0652316.}

\begin{document}

\begin{abstract}
We prove that the essential dimension of a $p$-group $G$ over a
field $F$ containing a primitive $p$-th root of unity is equal to
the least dimension of a faithful representation of $G$ over $F$.
\end{abstract}

\maketitle


The notion of the essential dimension $\ed_F(G)$ of a finite group
$G$ over a field $F$ was introduced in \cite{BuhlerReichstein97}.
The integer $\ed_F(G)$ is equal to the smallest number of
algebraically independent parameters needed to classify all Galois
$G$-algebras over any field extension of $F$. If $V$ is a faithful
linear representation of $G$ over $F$, then $\ed_F(G)\leq\dim(V)$
(cf. \cite[Prop. 4.11]{BerhuyFavi03}). The essential dimension of
$G$ can be smaller than $\dim(V)$ for every faithful representation
$V$ of $G$ over $F$. For example, we have $\ed_F(\Z/3\Z)=1$ over
$F=\Q$ or any field $F$ of characteristic $3$ (cf. \cite{Ledet02},
\cite[Example 2.3]{BerhuyFavi03}) and $\ed_{\C}(S_5)=2$ (cf.
\cite[Th. 6.5]{BuhlerReichstein97}).

In this paper we prove that if $G$ is a $p$-group and $F$ is a field
of characteristic different from $p$ containing a primitive $p$-th
root of unity, then $\ed_F(G)$ coincides with the least dimension of a
faithful representation of $G$ over $F$.

In the paper the word ``scheme" means a separated scheme of finite
type over a field and ``variety" an integral scheme.

\medskip

\noindent {\sl Acknowledgment:} We are grateful to Z.~Reichstein for
useful conversations.

\section{Preliminaries}

\subsection{Severi-Brauer varieties}(cf. \cite{ArtSB})
Let $A$ be a central simple algebra of degree $n$ over a field $F$.
The \emph{Severi-Brauer variety $P=\SB(A)$} of $A$ is the variety of
right ideals in $A$ of dimension $n$. For a field extension $L/F$,
the algebra $A$ is split over $L$ if and only if $P(L)\neq\emptyset$
if and only if $P_L\simeq\P_L^{n-1}$.

The change of field map $\deg:\Pic(P)\to \Pic(P_L)=\Z$ for a
splitting field extension $L/F$ identifies $\Pic(P)$ with $e\Z$,
where $e$ is the exponent (period) of $A$. In particular, $P$ has
divisors of degree $e$. The algebra $A$ is split over $L$ if and
only if $P_L$ has a divisor of degree $1$ (a hyperplane).



\subsection{Groupoids and gerbes}\label{groupgerbes}
Let $\cX$ be a groupoid over $F$ in the sense of \cite{Vistoli89}.
We assume that for any field extension $L/F$, isomorphism classes of
objects in the category $\cX(L)$ form a set which we denote by
$\widehat\cX(L)$. We can view $\widehat\cX$ as a functor from the
category $\cat{Fields}/F$ of field extensions of $F$ to
$\cat{Sets}$.

\begin{example1}
If $G$ is an algebraic group over $F$, then the groupoid $BG$ is
defined as the category of $G$-torsors over some base over $F$.
Hence the functor $\widehat{BG}$ takes a field extension $L/F$ to
the set of isomorphism classes of $G$-torsors over $L$.
\end{example1}


Special examples of groupoids are \emph{gerbes banded by a
commutative group scheme $C$} over $F$. There is a bijection between
the set of isomorphism classes of gerbes banded by $C$ and the
Galois cohomology group $H^2(F,C)$ (cf. \cite[Ch. 4]{Giraud71} and
\cite[Ch. 4, \S 2]{Milne80}). The gerbe $BC$ corresponds to the
trivial element of $H^2(F,C)$.



\begin{example1}(Gerbes banded by $\mu_n$)\label{gerbesbandedmu}
\ Let $A$ be a central simple $F$-algebra and $n$ an integer with
$[A]\in \Br_n(F)=H^2(F,\mu_n)$. Let $P$ be the Severi-Brauer variety
of $A$ and $S$ a divisor on $P$ of degree $n$.
 Denote
by $\cX_A$ the gerbe banded by $\mu_n$ corresponding to $[A]$. For a
field extension $L/F$, the set $\widehat\cX_A(L)$ has the following
direct description (cf. \cite{BRV07}): $\widehat\cX_A(L)$ is
nonempty if and only if $P$ is split over $L$. In this case
$\widehat\cX_A(L)$ is the set of equivalence classes of the set
\[
\{f\in L(P)^\times:\ \ \div(f)=nH-S_L, \ \ \text{where $H$ is a
hyperplane in $P_L$}\},
\]
and two functions $f$ and $f'$ are equivalent if $f'=f h^n$ for some
$h\in L(P)^\times$.
\end{example1}

\subsection{Essential dimension}
Let $T:\cat{Fields}/F\to\cat{Sets}$ be a functor. For a field
extension $L/F$ and an element $t\in T(L)$, the \emph{essential
dimension of $t$}, denoted $\ed(t)$, is the least $\tr_F(L')$ over
all subfields $L'\subset L$ over $F$ such that $t$ belongs to the
image of the map $T(L')\to T(L)$. The \emph{essential dimension
$\ed(T)$ of the functor $T$} is the supremum of $\ed(t)$ over all
$t\in T(L)$ and field extensions $L/F$.

Let $G$ be an algebraic group over $F$. The \emph{essential
dimension $\ed(G)$ of $G$} is the essential dimension of the functor
taking a field extension $L/F$ to the set of isomorphism classes of
$G$-torsors over $\Spec L$. If $G$ is a finite group, we view $G$ as
a constant group over a field $F$ and write $\ed_F(G)$ for the
essential dimension of $G$ over $F$.

\begin{example1}\label{edstack}
Let $\cX$ be a groupoid over $F$. The \emph{essential dimension of
$\cX$}, denoted by $\ed(\cX)$, is the essential dimension
$\ed(\widehat \cX)$ of the functor $\widehat\cX$ defined in $\S
\ref{groupgerbes}$. In particular, $\ed(BG)=\ed(G)$ for an algebraic
group $G$ over $F$.
\end{example1}


\subsection{Canonical dimension} (cf. \cite{BerhuyReichstein05},
\cite{KM06}) Let $F$ be a field and $\cC$ a class of field
extensions of $F$. A field $E\in\cC$ is called \emph{generic} if for
any $L\in\cC$ there is an $F$-place $E\rightsquigarrow L$.

The \emph{canonical dimension} $\cdim(\cC)$ of the class $\cC$ is
the minimum of the $\tr_F E$ over all generic fields $E\in\cC$.

Let $T:\cat{Fields}/F\to\cat{Sets}$ be a functor. Denote by $\cC_T$
the class of \emph{splitting fields of $T$}, i.e., the class of
field extensions $L/F$ such that $T(L)\neq\emptyset$. The
\emph{canonical dimension of $T$}, denoted $\cdim(T)$, is the
canonical dimension of the class $\cC_T$.

\begin{comment}
\begin{example1}\label{am:candimvariety}
Let $X$ be a scheme over $F$ and $\cC$ the class of field extensions
$L$ of $F$ with $X(L)\neq\emptyset$. If $X$ is a regular scheme,
by \cite[Lemma 4.4]{KM06}, the field $F(X)$ is generic in
$\cC$.
\end{example1}
\end{comment}



If $X$ is a scheme over $F$, we write $\cdim(X)$ for the canonical
dimension of $X$ viewed as a functor $L\mapsto X(L)=\Mor_F(\Spec L,
X)$.





\begin{example1}\label{stack}
Let $\cX$ be a groupoid over $F$. We define the \emph{canonical
dimension $\cdim(\cX)$ of $\cX$} as the canonical dimension of the
functor $\widehat\cX$. In particular, if $G$ is an algebraic group
over $F$, then $\cdim(BG)$ coincides with $\cdim(G)$ as defined in
\cite{BerhuyReichstein05}.
\end{example1}

\begin{example1}\label{kuku} (A relation between essential and canonical
dimension) Let $T:\cat{Fields}/F\to\cat{Sets}$ be a functor. We
define the ``contraction" functor $T^c:\cat{Fields}/F\to\cat{Sets}$
as follows. For a field extension $L/F$, we have $T^c(L)=\emptyset$
if $T(L)$ is empty and $T^c(L)$ is a one element set otherwise. If
$X$ is a regular and complete variety over $F$ viewed as a functor
then $\ed(X^c)=\cdim(X)$ and this integer is equal to the smallest
dimension of a closed subvariety $Z\subset X$ such that there is a
rational morphism $X\dashrightarrow Z$. (cf. \cite[Cor. 4.6]{KM06}).
\end{example1}

\subsection{Valuations}\label{valuation}
Let $K/F$ be a regular field extension, i.e., for any field
extension $L/F$, the ring $K\tens_F L$ is a domain. We write $KL$
for the quotient field of $K\tens_F L$.

Let $v$ be a valuation on $L$ over $F$ with residue field $R$. Let
$O$ be the valuation ring and $M$ its maximal ideal. As $K\tens_F R$
is a domain, the ideal $\widetilde M:=K\tens_F M$ in the ring
$\widetilde O:=K\tens_F O$ is prime. The localization ring
$\widetilde O_{\widetilde M}$ is a valuation ring in $KL$ with
residue field $KR$. The corresponding valuation $\tilde v$ on $KL$
is called the \emph{canonical extension of $v$ on $KL$}. Note that
the groups of values of $v$ and $\tilde v$ coincide.




\section{Essential and canonical dimension of gerbes banded by $(\gmu_n)^s$}

In this section we relate the essential and canonical dimension of
gerbes banded by $(\gmu_n)^s$. The following statement is a
generalization of \cite[Th. 7.1]{BRV07}.

\begin{theorem}\label{esdimgerbe}
Let $\cX$ be a gerbe banded by  $(\gmu_n)^s$ over an arbitrary field
$F$ with $n>1$ and $s\geq 0$. Then $\ed(\cX)=\cdim(\cX)+s$.
\end{theorem}

\begin{proof}
The gerbe $\cX$ is given by an element in
$H^2(F,\mu_n)^s=\Br_n(F)^s$, i.e., by an $s$-tuple of central simple
algebras $A_1,A_2,\dots, A_s$ with $[A_i]\in\Br_n(F)$. Let $P$ be
the product of the Severi-Brauer varieties $P_i:=\SB(A_i)$,
$i=1,\dots, s$. As the classes of splitting fields for $\cX$ and $P$
coincide, we have $\cdim(\cX)=\cdim(P)$. We shall prove that
$\ed(\cX)=\cdim(P)+s$.

Let $S_i$ be a divisor on $P_i$ of degree $n$. Let $L/F$ be a field
extension and $f_i\in L(P_i)^\times$ with $\div(f_i)=nH_i-(S_i)_L$
for $i=1,\dots, s$. Write $\langle f_i\rangle$ for the corresponding
element in $\widehat\cX(L)$ (cf. \S\ref{groupgerbes}).

By Example \ref{kuku}, there is a closed subvariety $Z\subset P$ and
a rational dominant morphism $P\dashrightarrow Z$ with
$\dim(Z)=\cdim(P)$. We view $F(Z)$ as a subfield of $F(P)$. As
$P(L)\neq\emptyset$ and $P$ is regular, there is an $F$-place
$\gamma:F(P)\rightsquigarrow L$ (cf. \cite[\S 4.1]{KM06}). Since $Z$
is complete, the valuation ring of the restriction
$\gamma|_{F(Z)}:F(Z)\rightsquigarrow L$ dominates a point in $Z$. It
follows that $Z(L)\neq\emptyset$. Choose a point $y\in Z$ such that
$F':=F(y)\subset L$.

Since $P(F')\neq\emptyset$, there are $g_i\in F'(P_i)^\times$ with
$\div(g_i)=nH'_i-(S_i)_{F'}$ for $i=1,\dots, s$. As
$\Pic(P_i)_L=\Z$, there are functions $h_i\in L(P_i)^\times$ with
$\div(h_i)=(H_i')_L-H_i$. We have
\[
\div(g_i)_L=\div(f_i)+\div(h_i^{n}),
\]
hence
\[
a_ig_i=f_i h_i^n
\]
for some $a_i\in L^\times$. Hence $\langle f_i\rangle=\langle
a_ig_i\rangle$ in $\cX(L)$, therefore $\langle f_i\rangle$ is
defined over the field $F'(a_1,a_2,\dots,a_s)$. It follows that
\[
\ed\langle f_i\rangle\leq\tr_F(F')+s\leq\dim(Z)+s=\cdim(P)+s,
\]
hence $\ed(\cX)\leq\cdim(P)+s$.

We shall prove the opposite inequality. As $P\(F(Z)\)\neq\emptyset$,
there are $f_i\in F(Z)(P_i)^\times$ with
$\div(f_i)=nH_i-(S_i)_{F(Z)}$. Consider the field
$L:=F(Z)(t_1,t_2,\dots,t_s)$, where $t_i$ are variables, and the
point $\langle t_if_i\rangle\in\widehat\cX(L)$.

We claim that $\ed\langle t_if_i\rangle\geq\cdim(P)+s$. Suppose
$\langle t_if_i\rangle$ is defined over a subfield $L'\subset L$,
i.e., there are $g_i\in L'(P_i)^\times$ and $h_i\in L(P_i)^\times$
with $t_if_i=g_ih_i^n$. We shall show that
$\tr_F(L')\geq\cdim(P)+s$.

The sequence of variables $t_1,t_2,\dots,t_s$ yields a valuation $v$
on $L$ over $F$ with residue field $F(Z)$ and the group of values
$\Z^s$ such that $v(t_i)=e_i$, where the $e_i$ denote the standard
basis elements of $\Z^s$. Write $w$ for the restriction of $v$ on
$L'$.

Let $K=F(P)$. We extend canonically the valuations $v$ and $w$ to
valuations $\tilde v$ and $\tilde w$ on $KL$ and $KL'$ respectively
(cf. \S \ref{valuation}). Note that $f_i\in K(Z)^\times$, $g_i\in
(KL')^\times$ and $h_i\in (KL)^\times$. We have
\[
e_i=\tilde v(t_if_i)\equiv \tilde w(g_i)\ \ (mod \ \ n).
\]
Since $n>1$, the elements $w(g_i)$ generate a subgroup of $\Z^s$ of
finite index. It follows that the value group of $\tilde w$ is of
rank $s$, hence $\rank(w)=\rank(\tilde w)=s$.

Let $R$ be the residue field of $w$. By \cite[Ch. VI, Th. 3, Cor.
1]{Zariski75/2}, we have
\begin{equation}\label{inequality}
\tr_F(L')\geq \tr_F(R)+\rank(w)=\tr_F(R)+s.
\end{equation}
As $P(L')\neq\emptyset$, there is an $F$-place $F(P)\rightsquigarrow
L'$. Composing it with the place $L'\rightsquigarrow R$ given by
$w$, we get an $F$-place $F(P)\rightsquigarrow R$. As $P$ is
complete, we have $P(R)\neq\emptyset$, i.e., $R$ is a splitting
field of $P$. On the other hand, $R$ is a subfield of $F(Z)$ that is
the residue field of $v$ and a generic splitting field of $P$. Hence
$R$ is also a generic splitting field of $P$. By definition of the
canonical dimension,
\[
\cdim(P)=\tr_F F(Z)\geq \tr_F (R)\geq\cdim(P).
\]
It follows that $\tr_F(R)=\cdim(P)$ and therefore, we have
$\tr_F(L')\geq \cdim(P)+s$ by (\ref{inequality}). The claim is
proved.

It follows from the claim that $\ed(\cX)\geq \cdim(P)+s$.
\end{proof}

\section{Canonical dimension of a subgroup of $\Br(F)$}

Let $p$ be a prime integer and $D$ a finite subgroup of $\Br_p F$ of
dimension $r$ over $\Z/p\Z$. In this section we determine the
canonical dimension $\cdim D$ of the class of common splitting
fields of all elements of $D$. We say that a basis $\{a_1,
a_2,\dots,a_r\}$ of $D$ is \emph{minimal} if for any $i=1,\dots,r$
and any element $d\in D$ outside of the subgroup generated by
$a_1,\dots,a_{i-1}$, we have $\ind d\geq\ind a_i$.

One can construct a minimal basis of $D$ by induction as follows.
Let $a_1$ be a nonzero element of $D$ of minimal index. If the
elements $a_1,\dots,a_{i-1}$ are already chosen for some $i\leq r$,
then we take for $a_i$ an element of $D$ of the minimal index among
the elements outside of the subgroup generated by
$a_1,\dots,a_{i-1}$.

In this section we prove the following

\begin{theorem}\label{vottakaiafignia}
\label{nk:cdim algebr} Let $p$ be a prime integer, $D\subset \Br_pF$
a subgroup of dimension $r$ and $\{a_1, a_2,\dots,a_r\}$ a minimal basis
of $D$. Then
\[
\cdim(D)=\(\sum_{i=1}^r\ind a_i\)-r\;.
\]
\end{theorem}

\begin{comment}
\begin{remark}
It is not obvious that the sum $\sum_{i=1}^r\ind a_i$ is the
smallest for a minimal basis $\{a_1, a_2,\dots,a_r\}$. However, this
fact is a consequence of Theorem \ref{nk:cdim algebr}.
\end{remark}
\end{comment}


We prove Theorem \ref{nk:cdim algebr} in several steps.

Let $\{a_1, a_2,\dots,a_r\}$ be a minimal basis of $D$. For every
$i=1,2,\dots,r$, let $P_i$ be the Severi-Brauer variety of a central
division $F$-algebra $A_i$ representing the element $a_i\in\Br_p F$.
We write $P$ for the product $P_1\times P_2\times\dots\times P_r$.
We have
\[
\dim P=\sum_{i=1}^r\dim P_i=\(\sum_{i=1}^r\ind a_i\)-r.
\]
Moreover, the classes of splitting fields of $P$ and $D$ coincide,
hence $\cdim(D)=\cdim(P)$. Thus, the statement of Theorem
\ref{nk:cdim algebr} is equivalent to the equality
$\cdim(P)=\dim(P)$.



Let $0\leq n_1\leq n_2\leq\dots\leq n_r$ be integers and
$K=K(n_1,\dots,n_r)$ the subgroup of the polynomial ring $\Z[x]$ in
$r$ variables $x=(x_1,x_2,\dots,x_r)$ generated by the monomials
$p^{e(j_1,\dots,j_r)}x_1^{j_1}\dots x_k^{j_r}$ for all
$j_1,\dots,j_r\geq0$, where the exponent $e(j_1,\dots,j_r)$ is $0$
if all the $j_1,\dots,j_r$ are divisible by $p$, otherwise
$e(j_1,\dots,j_r)=n_k$ with the maximum $k$ such that $j_k$ is not
divisible by $p$. In fact, $K$ is a subring of $\Z[x]$.

\begin{remark}
Let $A_1,\dots,A_r$ be central division algebras over some field
such that for any non-negative integers $j_1,\dots,j_r$, the index
of the tensor product $A_1^{\otimes j_1}\otimes\dots\otimes
A_r^{\otimes j_r}$ is equal to $p^{e(j_1,\dots,j_r)}$. The group $K$
can be interpreted as
%a subgroup of
the colimit of the Grothendieck
groups of the product over $i=1,\dots,r$ of the Severi-Brauer
varieties of the matrix algebras $M_{l_i}(A_i)$ over all positive
integers $l_1,\dots,l_r$.
\end{remark}

We set $h=(h_1,\dots,h_r)$ with $h_i=1-x_i\in\Z[x]$.

\begin{proposition}
\label{nk:vot} Let $bh_1^{i_1}\dots h_r^{i_r}$ be a monomial of the
lowest total degree of a polynomial $f$ in the variables $h$ lying
in $K$. Assume that the integer $b$ is not divisible by $p$. Then
$p^{n_1}|i_1,\dots,p^{n_r}|i_r$.
\end{proposition}

\begin{proof}
We recast the proof for $r=1$ given in \cite[Lemma 2.1.2]{Karpenko95}
to the case of arbitrary $r$.

We proceed by induction on $m=r+n_1+\dots +n_r$. The case $m=1$ is
trivial. If $m>1$ and $n_1=0$, then $K=K(n_2,\dots,n_r)[x_1]$ and we are done
by induction applied to $K(n_2,\dots,n_r)$. In what follows we
assume that $n_1\geq1$.

Since $K(n_1,n_2,\dots,n_r)\subset K(n_1-1,n_2,\dots,n_r)$, by the
induction hypothesis $p^{n_1-1}|i_1,p^{n_2}|i_2,\dots,p^{n_r}|i_r$.
It remains to show that $i_1$ is divisible by $p^{n_1}$.

Consider the additive operation $\varphi\!:\Z[x]\to\Q[x]$ which
takes a polynomial $g\in\Z[x]$ to the polynomial $p\inv x_1\cdot
g'$, where $g'$ is the partial derivative of $g$ with respect to
$x_1$. We have
\[
\varphi(K)\subset K(n_1-1,n_2-1,\dots,n_r-1)\subset
K(n_1-1)[x_2,\dots,x_r]
\]
and
\[
\varphi(h_1^{j_1}h_2^{j_2}\cdots h_r^{j_r})= -p\inv j_1
h_1^{j_1-1}h_2^{j_2}\cdots h_r^{j_r} +p\inv
j_1h_1^{j_1}h_2^{j_2}\cdots h_r^{j_r}.
\]
Since $bh^{i_1}\cdots h^{i_r}$ is a monomial of the lowest total
degree of the polynomial $f$, it follows that $-bp\inv i_1
h_1^{i_1-1}h_2^{i_2}\cdots h_r^{i_r}$ is a monomial of $\varphi(f)$
considered as a polynomial in $h$. As
$$
\varphi(f)\in K(n_1-1)[x_2,\dots,x_r]\;,
$$
we see that $-bp\inv  i_1h_1^{i_1-1}$ is a monomial of a polynomial
from $K(n_1-1)$. It follows that $p\inv i_1$ is an integer and by
Lemma \ref{nk:razmernost 0} below, this integer is divisible by
$p^{n_1-1}$. Therefore $p^{n_1}|i_1$.
\end{proof}

\begin{lemma}
\label{nk:razmernost 0} Let $g$ be a polynomial in $h_1$ lying in
$K(m)$ for some $m\geq0$. Let $bh_1^{i-1}$ be a monomial of $g$ such
that $i$ is divisible by $p^m$. Then $b$ is divisible by $p^m$.
\end{lemma}

\begin{proof}
We write $h$ for $h_1$ and $x$ for $x_1$. Note that $h^i\in K(m)$
since $i$ is divisible by $p^m$. Moreover, the quotient ring
$K(m)/(h^i)$ is additively generated by $p^{e(j)}x^j$ with $j<i$.
Indeed, the polynomial $x^i-(-h)^i=x^i-(x-1)^i$ is a linear
combination with integer coefficients of $p^{e(j)}x^j$ with $j<i$.
Consequently,  for any $k\geq0$, multiplying by $p^{e(k)}x^{k}$, we
see that the polynomial $p^{e(i+k)}x^{i+k}=p^{e(k)}x^{i+k}$ modulo
the ideal $(h^i)$ is a linear combination with integer coefficients
of the $p^{e(j)}x^j$ with $j<i+k$.

Thus, $K(m)/(h^i)$ is additively generated by $p^{e(j)}(1-h)^j$ with
$j<i$. The only one generator $p^{e(i-1)}(1-h)^{i-1}=p^m(1-h)^{i-1}$
has nonzero $h^{i-1}$-coefficient and that coefficient is divisible
by $p^m$.
\end{proof}


Let $Y$ be a scheme over the field $F$. We write $\CH(Y)$ for the
Chow group of $Y$ and set $\Ch(Y)=\CH(Y)/p\CH(Y)$. We define
$\Ch(\overline{Y})$ as the colimit of $\Ch(Y_L)$ where $L$ runs over
all field extensions of $F$. Thus for any field extension $L/F$, we
have a canonical homomorphism $\Ch(Y_L)\to\Ch(\overline{Y})$. This
homomorphism is an isomorphism if $Y=P$, the variety defined above,
and $L$ is a splitting field of $P$.

We define $\BCh(Y)$ as the image of the homomorphism
$\Ch(Y)\to\Ch(\overline{Y})$.

\begin{proposition}\label{vsenol}
\label{nk:BCh(P)} We have $\BCh^j(P)=0$ for any $j>0$.
\end{proposition}

\begin{proof}
Let $K_0(P)$ be the Grothendieck group of $P$. We write
$K_0(\overline{P})$ for the colimit of $K_0(P_L)$ taken over all
field extensions $L/F$. The group $K(\overline{P})$ is canonically
isomorphic to $K_0(P_E)$ for any splitting field $E$ of $P$. Each of
the groups $K_0(P)$ and $K_0(\overline{P})$ is endowed with the
topological filtration. The subsequent factor groups $G^jK_0(P)$ and
$G^jK_0(\overline{P})$ of these filtrations fit into the commutative
square
$$
\begin{CD}
\CH^j(\overline{P})@>>> G^jK_0(\overline{P})\\
@AAA @AAA\\
\CH^j(P)@>>> G^jK_0(P)
\end{CD}
$$
where the top map is an isomorphism. Therefore it suffices to show
that the image of the homomorphism $G^jK_0(P)\to
G^jK_0(\overline{P})$ is divisible by $p$ for any $j>0$.

The ring $K_0(\overline{P})$ is identified with the quotient of the
polynomial ring $\Z[h]$ by the ideal generated by $h_1^{\ind
a_1},\dots,h_r^{\ind a_r}$. Under this identification, the element
$h_i$ is the pull-back to $P$ of the class of a hyperplane in $P_i$
over a splitting field and the $j$-th term $K_0(\overline{P})^{(j)}$
of the filtration is generated by the classes of monomials of degree
at least $j$. The group $G^jK_0(\overline{P})$ is identified with
the group of all homogeneous polynomials of degree $j$.

The group $K_0(P)$ is isomorphic to the direct sum of $K_0(B)$,
where $B=A_1^{\otimes j_1}\otimes\dots\otimes A_r^{\otimes j_r}$,
over all $j_i$ with $0\leq j_i< \ind  a_i$ (cf. \cite{Quillen73}).
The image of the natural map $K_0(B)\to K_0(B_L)=\Z$, where $L$ is a
splitting field of $B$, is equal to $\ind (a_1^{j_1}\cdots
a_r^{j_r})\Z$. The image of the homomorphism $K_0(P)\to
K_0(\overline{P})$ (which is in fact an injection) is generated by
\[
\ind (a_1^{j_1}\cdots a_r^{j_r})(1-h_1)^{j_1}\cdots(1-h_r)^{j_r}
\]
over all $j_1,\dots,j_r\geq0$.

We embed $K_0(\overline{P})$ into the polynomial ring
$\Z[x]=\Z[x_1,\dots,x_r]$ as a subgroup by identifying a monomial
$h_1^{j_1}\cdots h_r^{j_r}$ where $0\leq j_i<\ind a_i$ with the
polynomial $(1-x_1)^{j_1}\cdots(1-x_r)^{j_r}$. As the elements
$a_1,\dots,a_r$ form a minimal basis of $D$, the index $\ind
(a_1^{j_1}\cdots a_r^{j_r})$ is a power of $p$ with the exponent at
least $e(\ind a_1,\dots,\ind a_r)$. Therefore,
\[
K_0(P)\subset K(\ind a_1,\dots,\ind a_r)\subset\Z[x].
\]
An element of $K_0(P)^{(j)}$ with $j>0$ is a polynomial $f$ in $h$
of degree at least $j$. The image of $f$ in $G^jK_0(\overline{P})$
is the $j$-th homogeneous part $f_j$ of $f$. As the degree of $f$
with respect to $h_i$ is less than $\ind a_i$, it follows from
Proposition \ref{nk:vot} that all the coefficients of $f_j$ are
divisible by $p$.
\end{proof}

Let $d=\dim P$ and $\alpha\in\CH^d(P\times P)$. The \emph{first
multiplicity $\mult_1(\alpha)$ of $\alpha$} is the image of $\alpha$
under the push-forward map $\CH^d(P\times P)\to\CH^0(P)=\Z$ given by
the first projection $P\times P\to P$ (cf. \cite{Merkurjev03c}).
Similarly, we define the \emph{second multiplicity
$\mult_2(\alpha)$}.

\begin{corollary}
\label{nk:mult} For any element $\alpha\in\CH^d(P\times P)$, we have
$$\mult_1(\alpha)\equiv \mult_2(\alpha)\;\;\text{ modulo $p$}.$$
\end{corollary}

\begin{proof}
We follow the proof of \cite[Th. 2.1]{Karpenko00}. The homomorphism
\[
f\!:\CH^d(P\times P)\to(\Z/p\Z)^2,
\]
taking an $\alpha\in\CH^d(P\times P)$ to
$\(\mult_1(\alpha),\mult_2(\alpha)\)$ modulo $p$, factors through
the group $\BCh^d(P\times P)$. Since for any $i$, any projection
$P_i\times P_i\to P_i$ is a projective bundle, the Chow group
$\BCh^d(P\times P)$ is a direct some of several copies of
$\BCh^i(P)$ for some $i$'s and the value $i=0$ appears once. By
Proposition \ref{vsenol}, the dimension over $\Z/p\Z$ of the vector
space $\BCh^d(P\times P)$ is equal to $1$ and consequently the
dimension of the image of $f$ is at most $1$. Since the image of the
diagonal class under $f$ is $(1,1)$, the image of $f$ is generated
by $(1,1)$.
\end{proof}

\begin{corollary}
\label{nk:dominant}
Any rational map $P\dashrightarrow P$ is dominant.
\end{corollary}

\begin{proof}
Let $\alpha\in\CH^d(P\times P)$ be the class of the closure of the
graph of a rational map $P\dashrightarrow P$. We have
$\mult_1(\alpha)=1$. Therefore, by Corollary \ref{nk:mult},
$\mult_2(\alpha)\ne0$, and it follows that the rational map is
dominant.
\end{proof}

Example \ref{kuku} then yields

\begin{corollary}
\label{nk:cdimP} $\cdim P=\dim P$. \qed
\end{corollary}

The corollary completes the proof of Theorem \ref{vottakaiafignia}.

\begin{remark}
Theorem \ref{nk:cdim algebr} can be generalized to the case of any
finite subgroup $D\subset\Br(F)$ consisting of elements of
$p$-primary orders. Let $\{a_1,a_2,\dots,a_r\}$ be elements of $D$
such that their images $\{a'_1,a'_2,\dots,a'_r\}$ in $D/D^p$ form a
minimal basis, i.e., for any $i=1,\dots r$ and any element $d\in D$
with the class in $D/D^p$ outside of the subgroup generated by
$a'_1,\dots,a'_{i-1}$, the inequality $\ind d\geq\ind a_i$ holds.
Then, as in Theorem \ref{nk:cdim algebr}, we have
\[
\cdim(D)=\(\sum_{i=1}^r\ind a_i\)-r\;.
\]
Indeed, the quotient $D/D^p$ has the same splitting fields as the
group $D$ itself. Therefore, $\cdim(D)=\cdim(D/D^p)=\cdim(P)$ for
$P=P_1\times\dots\times P_r$, where $P_i$ for every $i=1,\dots,r$ is
the Severi-Brauer variety of a central division algebra
representing the element $a_i$. Corollaries
\ref{nk:mult}, \ref{nk:dominant} and \ref{nk:cdimP} hold for $P$
since $K_0(P)\subset K(\ind a_1,\dots,\ind a_r)$.
\end{remark}


\begin{comment}
\begin{remark}
We have actually computed $\cdim_pP$.
\end{remark}

\begin{remark}
The variety $P$ belongs to the class of the so-called {\em
generically split} projective homogenous varieties (cf. \cite{???}).
For such a variety $Y$, it is shown in \cite{???} that
$$
\cdim_p Y=\min\{i\in\Z\;|\;\;\Ch_i(Y)\ne0\}\;.
$$
This result produces Corollary \ref{nk:cdimP} directly from
Proposition \ref{nk:BCh(P)}.
\end{remark}
\end{comment}



\section{Main theorem}

The main result of the paper is the following

\begin{theorem}\label{main2}
Let $G$ be a $p$-group and $F$ a field of characteristic different
from $p$ containing a primitive $p$-th root of unity. Then
$\ed_F(G)$ coincides with the least dimension of a faithful
representation of $G$ over $F$.
\end{theorem}

The rest of the paper is devoted to the proof of the theorem. As was
mentioned in the introduction, we have $\ed_F(G)\leq\dim(V)$ for any
faithful representation of $G$ over $F$. We shall construct a
faithful representation of $G$ over $F$ with $\ed_F(G)\geq\dim(V)$.

Denote by $C$ the subgroup of all central elements of $G$ of
exponent $p$ and set $H=G/C$, so we have an exact sequence
\begin{equation}\label{exactseq}
1\to C\to G\to H\to 1.
\end{equation}

Let $E\to\Spec F$ be an $H$-torsor and $\Spec F\to BH$ be the
corresponding morphism. Set $\cX^E:=BG\times _{BH}\Spec F$. Then
$\cX^E$ is a gerbe over $F$ banded by $C$ and its class in
$H^2(F,C)$ coincides with the image of the class of $E$ under the
connecting map $H^1(F,H)\to H^2(F,C)$ (cf. \cite[Ch. 4, \S
2]{Milne80}). An object of $\cX^E$ over a field extension $L/F$ is a
pair $(E',\alpha)$, where $E'$ is a $G$-torsor over $L$ and
$\alpha:E'/C\iso E_L$ is an isomorphism of $H$-torsors over $L$.

Alternatively. $\cX^E=[E/G]$ (cf. \cite{Vistoli89}) with objects
(over $L$) $G$-equivariant morphisms $E'\to E_L$, where $E'$ is a
$G$-torsor over $L$.



The following lower bound for $\ed(G)$ was established in
\cite[Prop. 2.20]{BRV07}. We give a short proof here for
completeness.

\begin{theorem}\label{gerbesed}
For any $H$-torsor $E$ over $F$, we have $\ed_F(G)\geq \ed(\cX^E)$.
\end{theorem}

\begin{proof}
Let $L/F$ be a field extension and $x=(E',\alpha)$ an object of
$\cX^E(L)$. Choose a subfield $L_0\subset L$ over $F$ such that
$\tr(L_0)=\ed(E')$ and there is a $G$-torsor $E'_0$ over $L_0$ with
$(E'_0)_L\simeq E'$.


Let $Z$ be the (zero-dimensional) scheme of isomorphisms
$\Iso_{L_0}(E'_0/C,E_{L_0})$ of $H$-torsors over $L_0$. The image of
the morphism $\Spec L\to Z$ over $L_0$ representing the isomorphism
$\alpha$ is a one point set $\{z\}$ of $Z$. The field extension
$L_0(z)/L_0$ is algebraic since $\dim Z=0$.

The isomorphism $\alpha$ descents to an isomorphism of the
$H$-torsors $E'/C$ and $E$ over $L_0(z)$. Hence the isomorphism
class of $x$ belongs to the image of the map
$\widehat\cX^E\(L_0(z)\)\to \widehat\cX^E(L)$. Therefore,
\[
\ed_F(G)\geq \ed(E')=\tr(L_0)=\tr(L_0(z)\)\geq \ed(x).
\]
It follows that $\ed_F(G)\geq \ed(\cX^E)$.
\end{proof}

Let $C^*:=\Hom(C,\gm)$ denote the character group of $C$. An
$H$-torsor $E\to \Spec F$ yields a homomorphism
\[
\beta^E:C^*\to\Br(F)
\]
taking a character $\chi:C\to\gm$ to the image of the class of $E$
under the composition
\[
H^1(F,H)\xra{\partial} H^2(F,C)\xra{\chi^*} H^2(F,\gm)=\Br(F),
\]
where $\partial$ is the connecting map for the exact sequence
(\ref{exactseq}). Note that as $\mu_p\subset F^\times$, the
intersection of $\Ker(\chi^*)$ over all characters $\chi\in C^*$ is
trivial. It follows that the classes of splitting fields of the
gerbe $\cX^E$ and the subgroup $\Im(\beta^E)$ coincide. It follows
that
\begin{equation}\label{canequal}
\cdim(\cX^E)=\cdim \(\Im(\beta^E)\).
\end{equation}

Let $\chi_1,\chi_2,\dots,\chi_s$ be a basis of $C^*$ over $\Z/p\Z$
such that $\{\beta^E(\chi_1),\dots, \beta^E(\chi_r)\}$ is a minimal
basis of $\Im(\beta^E)$ for some $r$ and $\beta^E(\chi_i)=1$ for
$i>r$. By Theorem \ref{vottakaiafignia}, we have
\begin{equation}\label{canbeta}
\cdim \(\Im(\beta^E)\)=\(\sum_{i=1}^r
\ind\beta^E(\chi_i)\)-r=\(\sum_{i=1}^s \ind\beta^E(\chi_i)\)-s.
\end{equation}

In view of Theorem \ref{gerbesed} and (\ref{canequal}), we shall
find an $H$-torsor $E$ (over a field extension of $F$) so that the
integer in (\ref{canbeta}) is as large as possible. We embed $H$
into $X:=\gGL_n$ as a subgroup for some $n$ and set $Y:=X/H$. Let
$E$ be the generic fiber of the $H$-torsor $\pi:X\to Y$. It is a
``generic" $H$-torsor over the function field $L:=F(Y)$.

Let $\chi:C\to\gm$ be a character. Let $\Rep^{(\chi)}(G)$ be the
category of all finite dimensional representations $\rho$ of $G$
such that $\rho(c)$ is multiplication by $\chi(c)$ for any $c\in C$.
Choose a representations $\rho:G\to\gGL(V)$ in $\Rep^{(\chi)}(G)$.
The conjugation action of $G$ on $B:=\End(V)$ factors through an
$H$-action. By descent (cf. \cite[Ch. 1, \S 2]{Milne80}), there is
(a unique up to canonical isomorphism) Azumaya algebra $\cA$ over
$Y$ and an $H$-equivariant algebra isomorphism $\pi^*(\cA)\simeq
B_X$. Let $A$ be the generic fiber of $\cA$; it is a central simple
algebra over $L=F(Y)$.

Consider the homomorphism $\beta^E:C^*\to\Br(L)$.

\begin{lemma}\label{calc}
The class of $A$ in $\Br(L)$ coincides with $\beta^E(\chi)$.
\end{lemma}

\begin{proof}
Consider the commutative diagram
\[
\begin{CD}
1  @>>> C  @>>> G  @>>> H @>>> 1  \\
  @.  @V{\chi}VV    @V{\rho}VV   @V{\alpha}VV    \\
1  @>>> \gm  @>>> \gGL(V)  @>>> \gPGL(V) @>>> 1
\end{CD}
\]
The image of the $H$-torsor $\pi:X\to Y$ under $\alpha$ is the
$\gPGL(V)$-torsor
\[
E':=\gPGL(V)_X/H\to Y
\]
where $\gPGL(V)_X:=\gPGL(V)\times X$ and $H$ acts on $\gPGL(V)_X$ by
$h(a,x)=(ah\inv,hx)$. The conjugation action of $\gPGL(V)$ on $B$
gives rise to an isomorphism between $\gPGL(V)_X$ and the $H$-torsor
$\Iso_X\(B_X,\End(V)_X\)$ of isomorphisms between the (split)
Azumaya $\mathcal{O}_X$-algebras $B_X$ and $\End(V)_X$. Note that this
isomorphism is $H$-equivariant if $H$ acts by conjugation on $B_X$
and trivially on $\End(V)_X$. By descent,
\[
E'\simeq \Iso_Y\(\cA,\End(V)_Y\).
\]
Therefore, the image of the class of the torsor $E'\to Y$ under the
connecting map for the bottom row of the diagram coincides with the
class of the Azumaya algebra $\cA$. Restricting to the generic fiber
yields $[A]=\beta^E(\chi)$.
\end{proof}

\begin{theorem}\label{index}
For any character $\chi\in C^*$, we have
$\ind\beta^E(\chi)=\min\dim(V)$ over all representations $V$ in
$\Rep^{(\chi)}(G)$.
\end{theorem}
\begin{proof}
We follow the approach given in \cite{Merkurjev96}. Let $G$ act on a
scheme $Z$ over $F$. Denote by $\cM(G,Z)$ the (abelian) category of
left $G$-modules on $Z$ that are coherent $\mathcal{O}_Z$-modules (cf.
\cite[\S 1.2]{Thomason83}). In particular, $\cM(G,\Spec F)=\Rep(G)$,
the category off all finite dimensional representations of $G$.

For a character $\chi:C\to \gm$, let $\cM^{(\chi)}(G,Z)$ be the full
subcategory of $\cM(G,Z)$ consisting of $G$-modules on which $C$
acts via $\chi$. For example, $\cM^{(\chi)}(G,\Spec
F)=\Rep^{(\chi)}(G)$.



We write $K_0(G,Z)$ and $K_0^{(\chi)}(G,Z)$ for the Grothendieck
groups of $\cM(G,Z)$ and $\cM^{(\chi)}(G,Z)$ respectively.


\begin{comment}
The exact sequence (\ref{exactseq}) extends to a commutative diagram
\[
\begin{CD}
1  @>>> C  @>>> G  @>>> H @>>> 1  \\
  @.  @VVV    @VVV   @|    \\
1  @>>> \gm  @>>> G'  @>>> H @>>> 1
\end{CD}
\]
with exact rows. Note that $\Rep^{(\chi)}(G)=\Rep^{(\rho)}(G')$
where $\rho$ is the identity character of $\gm$. Hence replacing the
exact sequence (\ref{exactseq}) by the bottom row of the diagram, we
may assume that $C$ is a diagonalizable group (even $C=\gm$).
\end{comment}

By representation theory of diagonalizable group schemes, every $M$
in $\cM(G,Z)$ is a direct sum of unique submodules $M^{(\chi)}$ of
$M$ in $\cM^{(\chi)}(G,Z)$ over all characters $\chi$ of $C$. It
follows that
\[
K_0(G,Z)=\coprod K_0^{(\chi)}(G,Z).
\]

By Lemma \ref{calc}, it suffices to show that $\ind(A)=\gcd\dim(V)$
over all representations $V$ in $\Rep^{(\chi)}(G)$.

The image of the map $\dim:K_0(A)\to\Z$ given by the dimension over
$L$ is equal to $\ind(A)\cdot\dim(V)\cdot\Z$. To finish the proof of
the theorem is suffices to construct a surjective homomorphism
\begin{equation}\label{mainiso}
K_0\(\Rep^{(\chi)}(G)\)\to K_0(A)
\end{equation}
so that the composition $K_0\(\Rep^{(\chi)}(G)\)\to
K_0(A)\xra{\dim}\Z$ is given by the dimension times $\dim(V)$.

First of all we have
\begin{equation}\label{first}
K_0\(\Rep^{(\chi)}(G)\)\simeq K_0^{(\chi)}(G,\Spec F).
\end{equation}


Recall that $H$ is embedded into $X=\gGL_n$. Next, we embed $X$ into
$\A^m$, where $m=n^2$, as an open set in the standard way. The
$G$-action on $X$ extends to a linear $G$-action on $\A^m$. By
homotopy invariance in the equivariant $K$-theory \cite[Cor.
4.2]{Thomason83},
\[
K_0(G,\Spec F)\simeq K_0(G,\A^m).
\]
It follows that
\begin{equation}\label{second}
K_0^{(\chi)}(G,\Spec F)\simeq K_0^{(\chi)}(G,\A^m).
\end{equation}


By localization \cite[Th. 2.7]{Thomason83}, the restriction
homomorphism
\begin{equation}\label{third}
K_0^{(\chi)}(G,\A^m)\to K_0^{(\chi)}(G,X).
\end{equation}
is surjective.

Denote by $\cM^{(1)}(G,X,B_X)$ the category of left $G$-modules $M$
on $X$ that are coherent $\mathcal{O}_X$-modules and right $B_X$-modules such
that $C$ acts trivially on $M$ and the $G$-action on $M$ and the
conjugation $G$-action on $B_X$ agree. The corresponding
Grothendieck group is denoted by $K_0^{(1)}(G,X,B_X)$. For any
object $L$ in $\cM^{(\chi)}(G,X)$, the group $C$ acts trivially on
$L\tens_F V^*$ and $B$ acts on the right on $L\tens_F V^*$. We have
Morita equivalence
\[
\cM^{(\chi)}(G,X)\iso \cM^{(1)}(G,X,B_X)
\]
given by $L\mapsto L\tens_F V^*$ (with the inverse functor $M\mapsto M\tens_B V$). Hence
\begin{equation}\label{forth}
K_0^{(\chi)}(G,X)\simeq K_0^{(1)}(G,X,B_X).
\end{equation}


Now, as $C$ acts trivially on $X$ and $B_X$, the category
$\cM^{(1)}(G,X,B_X)$ is equivalent to the category $\cM(H,X,B_X)$.
Hence
\begin{equation}\label{forthplus}
K_0^{(1)}(G,X,B_X)\simeq K_0(H,X,B_X).
\end{equation}

Recall that $Y=X/H$. By descent, the category $\cM(H,X,B_X)$ is
equivalent to the category $\cM(Y,\cA)$ of coherent $\mathcal{O}_Y$-modules
that are right $\cA$-modules. Hence
\begin{equation}\label{fifth}
K_0(H,X,B_X)\simeq K_0(Y,\cA).
\end{equation}


The restriction to the generic point of $Y$ gives a surjective
homomorphism
\begin{equation}\label{sixth}
K_0(Y,\cA)\to K_0(A).
\end{equation}

Finally, the homomorphism (\ref{mainiso}) is the composition of
(\ref{first}), (\ref{second}), (\ref{third}), (\ref{forth}),
(\ref{forthplus}), (\ref{fifth}) and (\ref{sixth}).
\end{proof}

We can now complete the proof of Theorem \ref{main2}. By Theorem
\ref{index}, there are representations $V_i$ in $\Rep^{(\chi_i)}(G)$
such that $\ind\beta^E(\chi_i)=\dim V_i$, $i=1, \dots, s$. Let $V$
be the direct sum of all
the $V_i$. By Theorem \ref{gerbesed}
(applied to the group $G$ over $L$ and the generic torsor $E$),
Theorem \ref{esdimgerbe}, (\ref{canequal}) and (\ref{canbeta}), we
have
\begin{multline*}
\ed_F(G)\geq\ed_L(G)\geq\ed(\cX^E)=\cdim(\cX^E)+s=\cdim\(\Im(\beta^E)\)+s\\
=\sum_{i=1}^s\ind\beta^E(\chi_i)=\sum_{i=1}^s \dim(V_i)=\dim(V).
\end{multline*}




Since $\chi_1,\chi_2,\dots,\chi_s$ generate $C^*$, the restriction
of $V$ on $C$ is faithful. As every nontrivial normal subgroup of
$G$ intersects $C$ nontrivially, the $G$-representation $V$ is
faithful. We have constructed a faithful representation $V$ of $G$
over $F$ with $\ed_F(G)\geq\dim(V)$. The theorem is proved.


\begin{remark}\label{construction}
The proof of Theorem \ref{main2} shows how to compute the essential
dimension of $G$ over $F$. For every character $\chi\in C^*$ choose
a representation $V_\chi\in\Rep^{(\chi)}(G)$ of the smallest
dimension. It appears as an irreducible component of the smallest
dimension of the induced representation $\Ind_C^G(\chi)$. We
construct a basis $\chi_1,\dots,\chi_s$ of $C^*$ by induction as
follows. Let $\chi_1$ be a nonzero character with the smallest
$\dim(V_{\chi_1})$. If the characters $\chi_1,\dots,\chi_{i-1}$ are
already constructed for some $i\leq s$, then we take for $\chi_i$ a
character with minimal $\dim(V_{\chi_i})$ among the characters
outside of the subgroup generated by $\chi_1,\dots,\chi_{i-1}$. Then
$\ed_F(G)=\sum_{i=1}^s\dim(V_{\chi_i})$.
\end{remark}

\section{An application}

\begin{theorem}\label{product}
Let $G_1$ and $G_2$ be two $p$-groups and $F$ a field of
characteristic different from $p$ containing a primitive $p$-th root
of unity. Then
\[
\ed_F(G_1\times G_2)=\ed_F(G_1)+\ed_F(G_2).
\]
\end{theorem}
\begin{proof}
The index $j$ in the proof takes two values $1$ and $2$. If $V_j$ is
a faithful representation of $G_j$ then $V_1\oplus V_2$ is a
faithful representation of $G_1\times G_2$. Hence $\ed_F(G_1\times
G_2)\leq\ed_F(G_1)+\ed_F(G_2)$
%(cf. \cite[Lemma 7.5]{BerhuyReichstein05}).
(cf. \cite[Lemma 4.1(b)]{BuhlerReichstein97}).

Denote by $C_j$ the subgroup of all central elements of $G_j$ of
exponent $p$. Set $C=C_1\times C_2$. We identify $C^*$ with
$C_1^*\oplus C_2^*$.

For every character $\chi\in C^*$ choose a representation
$\rho_\chi:G_1\times G_2\to \gGL(V_\chi)$ in
$\Rep^{(\chi)}(G_1\times G_2)$ of the smallest dimension. We
construct a basis $\{\chi_1,\chi_2,\dots,\chi_s\}$ of $C^*$
following Remark \ref{construction}. We claim that all the $\chi_i$
can be chosen in one of the $C_j^*$. Indeed, suppose the characters
$\chi_1,\dots,\chi_{i-1}$ are already constructed, and let $\chi_i$
be a character with minimal $\dim(V_{\chi_i})$ among the characters
outside of the subgroup generated by $\chi_1,\dots,\chi_{i-1}$. Let
$\chi_i=\chi^{(1)}_i+\chi^{(2)}_i$ with $\chi^{(j)}_i\in C_j^*$.
Denote by $\varepsilon_1$ and $\varepsilon_2$ the endomorphisms of
$G_1\times G_2$ taking $(g_1,g_2)$ to $(g_1,1)$ and $(1,g_2)$
respectively. The restriction of the representation
$\rho_{\chi_i}\circ\varepsilon_j$ on $C$ is given by the character
$\chi^{(j)}_i$. We replace $\chi_i$ by $\chi^{(j)}_i$ with $j$ such
that $\chi^{(j)}_i$ does not belong to the subgroup generated by
$\chi_1,\dots,\chi_{i-1}$. The claim is proved.


Let $W_j$ be the direct sum of all the $V_{\chi_i}$ with $\chi_i\in
C_j^*$. Then the restriction of $W_j$ on $C_j$ is faithful, hence so
is the restriction of $W_j$ on $G_j$. It follows that
$\ed_F(G_j)\leq\dim(W_j)$. As $W_1\oplus W_2=V$, we have
\[
\ed_F(G_1)+\ed_F(G_2)\leq
\dim(W_1)+\dim(W_2)=\dim(V)=\ed_F(G_1\times G_2).\qedhere
\]
\end{proof}

Denote by $\xi_{p^m}$ a primitive $p^m$-th root of unity (in a
separable closure of $F$).
\begin{corollary}\label{}

Let $F$ be a field as in Theorem \ref{product}. Then
\[
\ed_F\(\Z/p^{n_1}\Z\times \Z/p^{n_2}\Z\times \cdots \times
\Z/p^{n_s}\Z\)=\sum_{i=1}^s \big[F(\xi_{p^{n_i}}):F\big].
\]
\end{corollary}
\begin{proof}
By Theorem \ref{product}, it suffices to consider the case $s=1$.
This case has been done in \cite{Florence}. It is also covered by
Theorem \ref{main2} as the natural representation of the group
$\Z/p^n\Z$ in the $F$-space $F(\xi_{p^n})$ is faithful irreducible
of the smallest dimension.
\end{proof}



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