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\begin{document}


\title[On the essential dimension of infinitesimal group schemes]{On the essential dimension\\of infinitesimal group schemes}

\author{Dajano Tossici}
\author[Angelo Vistoli]{Angelo Vistoli}

\address{Scuola Normale Superiore\\Piazza dei Cavalieri 7\\
56126 Pisa\\ Italy}
\email[Vistoli]{angelo.vistoli@sns.it}
\email[Tossici]{dajano.tossici@sns.it}

\date{January 21, 2010}

\maketitle


\begin{abstract}
We discuss essential dimension of group schemes, with particular attention to infinitesimal group schemes. We prove that the essential dimension of a group scheme of finite type over a field $k$ is at least equal to the difference between the dimension of its Lie algebra and its dimension. Furthermore, we show that the essential dimension of a trigonalizable group scheme of length $p^{n}$ over a field of characteristic~$p>0$ is at most~$n$. We give several examples.

\end{abstract}


\section{Introduction}

The notion of essential dimension of a finite group over a field $k$ was introduced by Buhler and Reichstein (\cite{BR}). It was
later extended to various contexts. First Reichstein
generalized it to linear algebraic groups (\cite{Re}) in
characteristic zero; afterwards Merkurjev gave a general definition for covariant functors from the
category of extension fields of the base field $k$ to the category of sets
(\cite{BF}). Brosnan, Reichstein and Vistoli (\cite{BRV})
studied the essential dimension of an algebraic stack, a general class which
includes almost all the examples of interest.

Important results  on the essential dimension of finite groups in
characteristic $0$ have been proved by Florence (\cite{Flo}) and
Karpenko and Merkurjev (\cite{KM}). The essential dimension of
finite groups in positive characteristic has been studied by Ledet
\cite{Le}. As to higher dimensional groups, there are works of
Reichstein and Youssin (\cite{Rei-You}), Chernousov and Serre
(\cite{Che-Ser}), Gille and Reichstein \cite{Gil-Rei}, Brosnan,
Reichstein and Vistoli \cite{BRV2} on algebraic groups, Brosnan
\cite{Bro} on abelian varieties over $\mathbb{C}$, and Brosnan and
Shreekantan \cite{brosnan-shreekantan08} on abelian varieties over
number fields.

The case of non-smooth group schemes (necessarily in positive
characteristic) has not been investigated. It is certainly known
to experts that the essential dimensions of $\aalpha_{p^m}$ and
$\mmu_{p^m}$ over a field of characteristic~$p > 0$ are $1$, and that
the essential dimension of $\mmu_{p^{m}}^n$ is $n$.  To the
authors' knowledge nothing else was known. The purpose of this
paper is to throw some light on this subject. In particular we will
focus on the essential dimension of infinitesimal (i.e. connected
and finite) group schemes. This could give, for instance, some information
about the essential dimension of supersingular Abelian varieties
in characteristic $p > 0$ (an Abelian varieity is \emph{supersingular} if its $p$-torsion subgroup scheme is infinitesimal). If an Abelian variety $A$ is not supersingular, its essential dimension is conjecturally infinite, since for any $n > 0$ the group scheme $A$ contains, over an algebraic closure of $k$, the group scheme $\ZZ/p^n \ZZ$, whose essential dimension is conjecturally equal to $n$.

 If $G$ is a group scheme of
finite type over a field $k$, its essential dimension $\ed_{k}G$
is the essential dimension of the stack $\mathcal{B}_kG$. Let us
recall the definition. Let $k$ be a field, and $G$ be a
group scheme of finite type over $k$. If $X$ is a $k$-scheme, a
$G$-torsor is a $k$-scheme $P$ with a right action of $G$, with a
$G$-invariant morphism $P \arr X$, such that fppf locally on $X$
the scheme $P$ is $G$-equivariantly isomorphic to $X\times_{\spec
k}G$. Isomorphism classes of $G$-torsors on $X$ form a pointed set
$\H^{1}(X, G)$; if $G$ is commutative, then $\H^{1}(X, G)$ is a
group, and coincides with the cohomology group of $G$ in the fppf
topology.

\begin{definition}
Let $G$ be a group scheme of finite type over a field $k$. Let $k\In K$ be an
extension field and $[\xi]\in \H^1(X,\spec(K))$ the class of a
$G$-torsor $\xi$. Then the essential dimension of $\xi$ over $k$,
which we denote by $\ed_k(\xi)$, is the smallest nonnegative
integer $n$ such that there exists a subfield $K'$ of $K$
containing $k$, with $\trd(K'/k) \leq n$  such that $[\xi]$ is in the
image of the morphism $\H^1(X,\spec(K'))\arr \H^1(X,\spec(K))$.

The essential dimension of $G$ over $k$, which we denote by
$\ed_k(G)$, is the supremum  of $\ed_k(\xi)$, where $K/k$ ranges
through all the extension of $K$, and $\xi$ ranges through all the
$G$-torsors over $\spec(K)$.
\end{definition}

If $G$ is smooth, then $G$-torsors are locally trivial in the étale topology, and our definition coincides with that of Berhuy and Favi (\cite{BF}); in the general case, to get meaningful results one needs to use the fppf topology. For example, if $G$ is an infinitesimal group scheme, the $G$-torsors over a reduced scheme that are locally trivial in the étale topology are in fact trivial.


Our first result is the determination  of a general lower bound for
the essential dimension.




\begin{theorem}\label{thm1}
Let $G$ be a group scheme of finite type over a field $k$. Then
   \[
   \ed_{k} G \geq \dim_{k} \lie G - \dim G\,.
   \]
\end{theorem}

Considering how hard it is in general to prove lower bounds for the essential dimension, it is a little surprising that the proof of Theorem~\ref{thm1} is little more than an observation.

We also have a general upper bound. We firstly recall the
definition of a trigonalizable group scheme.

\begin{definition}
Let $G$ be an affine group scheme of finite type over a field $k$.
$G$ is called trigonalizable if it has a normal unipotent subgroup
$U$ scheme such that $G/U$ is diagonalizable.
\end{definition}

The name is justified by the fact that, if $G$ is trigonalizable
and of finite type over $k$, then there exists a monomorphism of
$G$ in some trigonal group. Any affine commutative group scheme
over an algebraically closed field is trigonalizable (see
\cite[IV, \S 3, 1.1]{DG}).

\begin{theorem}\label{thm2}
Let $G$ be a finite trigonalizable group scheme, over a field of
characteristic $p > 0$, of order $p^{n}$. Then $\ed_k G \leq n$.
\end{theorem}

For constant $p$-group schemes (which are unipotent) the above result has already been proved by Ledet (\cite{Le}).

The second and the third sections are devoted to the proofs of these
two theorems. In the last section, combining the lower and upper
bounds above, we calculate the essential dimension of some classes
of infinitesimal group schemes. In particular we prove that the
essential dimension of a trigonalizable group scheme of height $\le
1$, i.e. such that the Frobenius $\fr$ is trivial on it, is equal to
the dimension of its Lie algebra (Corollary~\ref{Cor1}). We do not
have  examples of infinitesimal group schemes with essential
dimension strictly bigger than the dimension of its Lie algebra. For
not trigonalizable group schemes such an example should probably
exist: for example, there should be twisted forms of $\mmu_{p}$
whose essential dimension is larger than $1$. Over an algebraically
closed field, the issue is less clear. In
Example~\ref{ex:test-example} we propose a class of a commutative
unipotent group scheme whose essential dimensions we are unable to
determine, which should be an important test case to determine
whether it is reasonable to conjecture that equality hold for
trigonalizable group schemes.


\section{The proof of Theorem~\ref{thm1}}
We first state a well known lemma.
\begin{lemma}\label{lem1}
If $G$ is a subgroup scheme of a  group scheme $H$, then $\ed_{k}
G +\dim G\leq \ed_k H + \dim H$.
\end{lemma}
\begin{proof} This follows from \cite[3.2]{BRV} applied to the
stack morphism $\cB_k G\arr \cB_k H$.
\end{proof}

We now prove the Theorem. If the characteristic of $k$ is $0$,
then $G$ is smooth and there is nothing to prove. Suppose that the
characteristic of $k$ is $p
> 0$. Since the essential dimension does not increase after a base
change (\cite[1.5]{BF}), we may assume that $k$ is  algebraically
closed.

Let $G_{1}$ be the kernel of the Frobenius map $G \arr G$ defined at
the level of algebras by $a \arrto a^{p}$; then $G_{1}$ is an
infinitesimal group scheme, and $\lie G_{1} = \lie G$. Since by
\ref{lem1} we have $\ed_k G_{1} \leq \ed_k G - \dim G$, it is
sufficient to show that $\ed_k G_{1} \geq \dim_k \lie G_{1}$; in
other words, we may assume that $G = G_{1}$, i.e., by definition,
$G$ has height at most~$1$. This implies that $k[G]$ is isomorphic
to $k[x_{1}, \dots,~x_{n}]/(x_{1}^{p}, \dots,~x_{n}^{p})$ as a
$k$-algebra (\cite[II  \S 7, 4.2]{DG}).

Let $G$ act freely on an open subscheme $X$ of a representation of
$G$. If $K$ is the function field of the quotient $X/G$ and $E$ is
the function field of $X$, then we have a generic $G$-torsor
$\spec E \arr \spec K$. Set $n \eqdef \dim_{k} \lie G$. Suppose
that the $G$-torsor  is defined over an extension $L$ of $k$
contained in $K$; we need to show that the transcendence degree of
$L$ over $k$ is at least~$n$. Let $\spec R \arr \spec L$ the $G$-torsor yielding $\spec E \arr \spec K$ by base change. Clearly $R$ is a field. We claim that $R \subseteq L^{1/p}$; in other words, the $p\th$-power homomorphism $R \arr R$
 has its image contained in $L \subseteq R$.

 In order to
 prove this we may base change to an algebraic closure $\overline{K}$ of $K$; then $\overline{K}\otimes_{K}E$
 is isomorphic as a $\overline{K}$-algebra to the Hopf algebra $\overline{K}[G]$.
 Since $\overline{K}[G]$ is isomorphic to $\overline{K}[x_{1}, \dots,~x_{n}]/(x_{1}^{p}, \dots,~x_{n}^{p})$
 as a $\overline{K}$-algebra, it is clearly true that the Frobenius map carries $\overline{K}\otimes_{K}E$ into $\overline{K}$; hence $R \subseteq L^{1/p}$, as claimed.
Since the degree $[R:L]$ is $p^{n}$, the result follows from the following well known fact.

\begin{lemma}
Let $L$ be a finitely generated extension of transcendence degree $d$ of a perfect field $k$ of characteristic $p > 0$. Then $[L^{1/p}:L] = p^{d}$.
\end{lemma}



\section{The proof of Theorem~\ref{thm2}}

Let us start with stating a few Lemmas that we will use in the
proof. The first two are well known.


\begin{lemma}\cite[IV \S 2, 2.5]{DG}
\label{lem3} Let $G$ be a commutative unipotent group scheme over
a field $K$. Then there exists  a central decomposition series
   \[
   1 = G_{0} \subseteq G_{1} \subseteq \dots \subseteq G_{r} = G
   \]
of $G$, such that each successive quotient $G_{i}/G_{i-1}$ is a subgroup scheme of $\ga$.
\end{lemma}

\begin{lemma}\cite[IV \S 2, 2.6]{DG}\label{lem:unichange}
If $G$ is a group scheme over a field $K$ and $E$ is an extension
of $K$, then $\spec E\times_{\spec K}G \arr \spec E$ is unipotent
if and only if $G$ is unipotent. In particular, any twisted form
of a unipotent group scheme is unipotent.
\end{lemma}

\begin{lemma}\label{lem2}
Let $G$ be a commutative unipotent group scheme over a field $K$.
Then $\H^{i}(K, G) = 0$, for $i\ge 2$.
\end{lemma}


\begin{proof}
By Lemma~\ref{lem3}, we may assume that $G$ is a subgroup of
$\ga$. The quotient $\ga/G$ is isomorphic to $\ga$ (\cite[IV \S 2,
1.1]{DG}); then the result follows from the fact that $\H^{i}(K,
\ga) =  0$ for $i\ge 1$.
\end{proof}

We here prove the key Lemma.
\begin{lemma}\label{lem:main}
Suppose that we have an extension
   \[
   1 \arr G_1 \arr G \arr G_2 \arr 1
   \]
of group schemes over $k$, where $G_1$ is a commutative unipotent
normal group subscheme of a group scheme $G$. Let $P \arr \spec K$
be a $G$-torsor over an extension $K$ of $k$. Then there exists an
intermediate extension $k \subseteq E \subseteq K$ and a twisted
form $\widetilde{G_1} \arr \spec E$ of $G_1$ over $E$, such that
$P$ is defined over an intermediate extension of transcendence
degree at most $\ed_{k} G_2 + \ed_{E} \widetilde{G_1}$ over $k$.

Furthermore, if $G_1$ is central in $G$, then $\widetilde{G_1} =
\spec E \times_{\spec k}G_1$.
\end{lemma}

\begin{proof}
Consider the induced $G_2$-torsor $Q \eqdef P/G_1 \arr \spec K$.
There exists an intermediate extension $k \subseteq E \subseteq K$
with $\trdeg_{k}E \leq \ed G_2$, such that $Q$ comes by base
change from a $G_2$-torsor $Q_{E} \arr \spec E$. I claim that
$Q_{E}$ lifts to a $G$-torsor $P_{E} \arr \spec E$.

To see this, consider the fppf gerbe of liftings $\cL \arr
\catsch{E}$. It is a fibered category over the category $\catsch{E}$ of $E$-schemes, whose objects over an
$E$-scheme $T \arr \spec E$ are $G$-torsors $P_{T} \arr T$, together
with isomorphisms of $G_2$-torsors $P_{T}/G_1 \simeq T \times_{\spec
E} Q_{E}$, or, equivalently, $G$-equivariant morphisms of
$T$-schemes $P_{T} \arr T \times_{\spec E} Q_{E}$. The arrows from
$P_{T} \arr T \times_{\spec E} Q_{E}$ to $P'_{T'} \arr T'
\times_{\spec E} Q_{E}$ are defined in the obvious way, as diagrams
   \[
   \xymatrix{
   P_{T}\ar[d]^{F}\ar[r] &
   T\times_{\spec E}Q_{E} \ar[r]^-{\pr_{1}} \ar[d]^{f\times\id} &
   T \ar[d]^{f}\\
   P_{T'}\ar[r] &
   T'\times_{\spec E}Q_{E} \ar[r]^-{\pr_{1}} &
   T'
   }
   \]
in which $F$ is $G$-equivariant. We need to show that $\cL$ has a global section over $\spec E$.

The action of $G$ on $G_1$ by conjugation descends to an action of
$G_2$ on $G_1$, since $G_1$ is commutative. Denote by
$\widetilde{G_1}$ the twisted form of the group scheme $\spec
E\times_{\spec{k}} G_1$ coming from the $G_2$-torsor $Q_{E} \arr
\spec E$; in other words, $\widetilde{G_1}$ is the quotient
$(Q_{E}\times_{\spec k}G_1)/G_2$, where $G_2$ acts on the right on
$Q_{E}$, and on $G_1$ by right conjugation. We claim that the
gerbe $\cL$ is banded by $\widetilde{G_1}$; that is, if $P_{T}
\arr T\times_{\spec E}Q_{E}$ is an object of $\cL(T)$, the
automorphism group $\aut(P_{T})$ is isomorphic to
$\widetilde{G_1}(T)$, and this isomorphism is functorial in $T$.
In fact, the twisted form $\widetilde{G}$ of $G$ obtained as the
quotient $P\times_{\spec k}G$ by the action of $G$ on conjugation
on $G$ is the automorphism group scheme of the $G$-torsor $P_{T}
\arr T$, and it contains $\widetilde{G_1}$ as the subgroup scheme
of automorphisms inducing the identity on $T\times_{\spec
E}Q_{E}$. Hence $\widetilde{G_1}$ is the automorphism group scheme
of the object $P_{T} \arr T\times_{\spec E}Q_{E}$, and this proves
the claim.

By \cite[IV 3.4]{Gir}, the equivalence classes of gerbes banded by
$\widetilde{G_1}$ are para\-metrized by the group $\H^{2}(K,
\widetilde{G_1})$; a gerbe corresponds to $0$, i.e., it is
equivalent to the classifying stack $\cB_{E}\widetilde{G_1}$, if and
only if it has a section. Now, by Lemma~\ref{lem:unichange},
$\widetilde{G_1}$ is unipotent; hence by Lemma~\ref{lem2} $\H^{2}(E,
\widetilde{G_1}) = 0$, so $\cL$ has a section, and the $G_2$-torsor
$Q_{E} \arr \spec E$ lifts to a $G$-torsor $P_{E} \arr Q_{E} \arr
\spec E$.

There is no reason why $\spec K \times_{\spec E}P_{E}$ should be
isomorphic to $P \arr \spec K$ as a $G$-torsor. However, by
construction, we have $P/G_1 \simeq \spec K\times_{\spec E}Q_{E}$.
Since as we just saw $\cL$ is a trivial gerbe banded by
$\widetilde{G_1}$, we have that $\cL$ is equivalent to the
classifying stack $\cB_{E}\widetilde{G_1}$, in such a way that the
lifting $P_{E} \arr Q_{E}$ corresponds to the trivial torsor
$\widetilde{G_1} \arr \spec E$. Then $P \arr Q$ gives an object of
$\cL(\spec K)$; this will be defined over a intermediate extension
$E \subseteq F \subseteq K$ of transcendence degree at most
$\ed_{E} \widetilde{G_1}$ over $E$. So over $F$ there will exist
an object $P_{F} \arr \spec L\times_{\spec E}Q_{E}$ which is
isomorphic to $P \arr Q$ when pulled back to $K$. Hence $P$ is
defined over $F$, and, since the transcendence degree of $F$ is at
most equal to $\ed_{k}G_2 + \ed_{E} \widetilde{G_1}$, the result
follows.
\end{proof}


Now we are ready to prove the theorem.

First of all, suppose that $G$ is a diagonalizable group of order $p^{n}$; then $G$ is a product $\mmu_{p^{d_{1}}} \times \dots \times\mmu_{p^{d_{r}}}$
 for certain positive integers $d_{1}$, \dots,~$d_{r}$ with $d_{1} + \dots + d_{r} = n$. Then $G$ is a subgroup scheme of $\gm^{r}$,
 hence by the Lemma \ref{lem1} we have
$\ed G \leq r\leq n$.

Now, assume that $G$ is commutative unipotent of order $p^{n}$,
with $n
> 0$. Suppose that $G$ is a subgroup of $\ga$; then again by
Lemma~\ref{lem1} we have $\ed_{k}G \leq \dim \ga = 1 \leq n$, and
we are done.

If it is not a subgroup scheme of $\ga$, we proceed by induction on
$n$. Assume that the result holds for all commutative unipotent
subgroup schemes of order $p^{m}$ with $m < n$. Let $G_1$ be a
nontrivial subgroup scheme that is a subgroup scheme of $\ga$ and
call $p^{m}$ its order. The group scheme $G_1$ exists by \ref{lem3}.
Then by Lemma~\ref{lem:main} we have
   \[
   \ed_k G \leq \ed_k (G/G_1) + \ed_{E}{G_1}_{E} \leq (n-m) + m = n;
   \] so the result holds for $G$.

Let $G$ be a trigonalizable infinitesimal group scheme of order
$p^{n}$. Once again, we proceed by induction on $n$. Let us
suppose that the result is true for all trigonalizable groups of
order $p^{m}$ with $m < n$. By definition, $G$ is an extension
   \[
   1 \arr G_{\rmu} \arr G \arr G_{\rmd}\arr 1\,,
   \]
where $G_{\rmu}$ is unipotent and $G_{\rmd}$ is diagonalizable. We
may assume that $G_{\rmu}$ is non-trivial, otherwise $G$ is
diagonalizable and we are done. If $G_1$ denotes the center of
$G_{\rmu}$, then $G_1$ is a nontrivial commutative unipotent
normal subgroup of $G$; set $G_2 \eqdef G/G_1$. Call $p^{m}$ the
order of $G_2$; by induction hypothesis we have $\ed_{k}G_2
\leq m$. Once again using Lemma~\ref{lem:main}, we have that
$\ed_{k}G \leq \ed_{k}G_2 + \ed_{E}\widetilde{G_1}$ for some
twisted form of $G_1$; but by Lemma~\ref{lem:unichange} the group
scheme $\widetilde{G_1}$ is still commutative unipotent, hence by
the previous case $\ed_{E}\widetilde{G_1} \leq n-m$, and we are
done.




\section{The essential dimension of some group schemes}

The two Theorems above give rise to two natural questions.

\begin{itemize}
\item For which group schemes $G$ of finite type over a field $k$ does the
equality
\begin{equation}\label{eq:equality}
\ed_k(G)=\dim_k (\Lie(G))-\dim G
\end{equation}
hold?

\item Is it true that for  a finite group scheme of order $p^n$ over a field of characteristic $p>0$ we have
$\ed_k(G)\le n$?
\end{itemize}

It is clear that if $G$ is a finite group scheme of order $p^{n}$ over a field of characteristic $p> 0$ that satisfies \eqref{eq:equality}, then $\ed_{k}G \leq n$.


The class of smooth group schemes satisfying the first equality is well known, it is that of the so-called \emph{special groups}. This were introduced by J.P.~Serre in \cite{Ser} and studied by Grothendieck in \cite{grothendieck-torsion}, and by many other authors since then.

For singular groups the question becomes substantially harder.
The observation above about smooth group schemes motivates  the
following definition.

\begin{definition}\label{def: almost special} Let $G$ be a group scheme of finite
type over a field $k$. If
\begin{equation*}
\ed_k(G)=\dim_k (\Lie(G))-\dim G
\end{equation*}
then $G$ is called \textsl{almost special}.
\end{definition}

In the following we give several examples of almost special group schemes. Most of them are connected and trigonalizable. We do not
know whether all connected trigonalizable group scheme are almost
special. For diagonalizable group scheme this is true (see
Example~\ref{ex: ed of some group schemes}). In the unipotent case this is open: in \ref{ex4} we discuss what we consider to be a key example to analyze to clarify this question.


It is easy to give examples of infinitesimal almost special group schemes that are not trigonalizable: for example, if $\cG$ is a non-trigonalizable smooth group scheme (e.g. $\cG=\Gl_n$), the kernel of its Frobenius ${}_{\fr} \cG$ is almost special, as it follows from Theorem~\ref{thm1} and Lemma~\ref{lem1}.

Finally we remark that if
$G$ and $H$ are almost special
 then the product $G\times H$ is also almost special, and
 $\ed_k(G\times H)=\ed_k(G)+\ed_k(H)$.

About the second question above, our guess is that the upper bound
does not hold in general. An example could be given by some twisted form of
$\mmu_p$; however, we are unable to prove this, as, unlike what happens for smooth group schemes, a group scheme of essential dimension $1$ is not necessarily contained in $\PGL_2$.

The following simple strategy lets us to find some almost special
group schemes. If an infinitesimal group scheme $G$ can be
embedded in a special smooth algebraic group scheme of dimension
$n\eqdef\dim_k \Lie(G)$ then, using Theorem \ref{thm1} and Lemma
\ref{lem1}, we can conclude that the essential dimension is
exactly $n$, i.e. $G$ is almost special. Here is an example when
this happens. Later in \ref{ex:test-example} we will see another
example in which this argument can not be applied.


\begin{example}\label{ex: ed of some group schemes}
If $n$ is a positive integer, we denote by $W_{n}$ the group scheme of truncated Witt vectors of length $n$ (see \cite[Chapter~2, \S~6]{Ser-local}, \cite[Chapter~3, \S~1]{Haz} and \cite[Chapter~5, \S~1]{DG}). Let $G$ be the group scheme
\begin{equation*}
\prod_{j=1}^{t} {}_{\fr^{m_j}} W_{n_j}\times \prod_{i=1}^s
\mmu_{p^{l_i}} ,
\end{equation*}
where ${}_{\fr^{m}} W_{n}$ is the kernel of the iterated Frobenius
$\fr^m:W_n\arr W_n$.
 Then $G$ is almost special, i.e.
\begin{equation}\label{eq: 1}
\ed_k G=s+\sum_{j=1}^{t}n_j=\dim_{k}\Lie G. \end{equation} Indeed,
by the remark above, it is enough to remark that $G$ is a
closed subgroup of $\prod_{j=1}^{t} W_{n_j} \times \GG_m^s$, which is special.
\end{example}

Another example in which the equality \eqref{eq:equality} holds is
given by the following Corollary.
\begin{corollary}\label{Cor1}
 Any trigonalizable group scheme $G$ of height
$\le 1$ and of finite type over $k$ is almost special.
\end{corollary}
\begin{proof}
If $G$ has order $p^n$ then, since $G$ has height $\le 1$, the
dimension of its Lie algebra is $n$.
 Therefore the result follows from Theorems \ref{thm1} and \ref{thm2}.
\end{proof}

As a consequence of  Theorem \ref{thm2} and  Lemma \ref{lem:main}
we prove the following result.

\begin{corollary}\label{cor2}
Let
   \[
   1 \arr G_1 \arr G \arr G_2 \arr 1
   \]
be an extension of group schemes over a field $k$ of
characteristic $p>0$, with $G_1$ unipotent commutative of order
$p^n$, then
$$
\ed_k(G_1)\le n +\ed_k(G_2).
$$
\end{corollary}
\begin{remark}
In particular one can apply the corollary under the further
hypothesis $\ed_k(G_1)=n$. In this situation we have the interesting
upper bound
$$
\ed_k(G)\le \ed_k(G_1)+\ed_k(G_2).
$$
Later we will see some cases in which equality holds.

 By Corollary \ref{Cor1}, the
hypothesis $\ed_k(G_1)=n$ is satisfied if $G_1$ is
commutative unipotent of height at most $1$. If $k$ is a perfect
field, any unipotent commutative $k$-group scheme of height at most
$1$ is isomorphic to a finite direct product of $ _{\fr} W_n$, $n\in
\NN$, where ${}_{\fr} W_{n}$ is the kernel of the Frobenius
$\fr:W_n\arr W_n$ (see \cite[IV, \S 2, 2.14]{DG}).

This hypothesis is also satisfied when $G_1$ is a twisted form of $\ZZ/p^n\ZZ$ with $n\le 2$ and conjecturally for any $n$ (see Example \ref{ex4}).

\end{remark}
\begin{proof}
Let $P \arr \spec K$ be a $G$-torsor over an extension $K$ of $k$.
Then, from Lemma \ref{lem:main}, there exists an intermediate
extension $k \subseteq E \subseteq K$ and a twisted form
$\widetilde{G_1}
\arr \spec E$ of $G_1$ over $E$, such that
$$\ed_k P\le
\ed_E (\widetilde{G_1})+ \ed_k (G_2).$$ By Lemma
\ref{lem:unichange}, $\widetilde{G_1}$ is unipotent. Hence  by
Theorem \ref{thm2} we have $\ed_E (\widetilde{G_1})\le n$,
 and  the result follows.
\end{proof}

\begin{example}\label{ex 1}
In addition to the hypotheses of the Corollary \ref{cor2}, let us
suppose that $G_1$ and $G_2$ are almost special and
   \[
   \dim_k\Lie(G)=\dim_k \Lie(G_1)+\dim_k \Lie(G_2).
   \]
In particular $G_1$
must be of height $\le 1$. From Corollary \ref{cor2} and Theorem
\ref{thm1} it follows that
$$\ed_k (G)=\ed_k (G_1)+\ed_k (G_2).$$ The hypothesis on the
dimension of the Lie algebra of $G$ is satisfied, for instance, if
the extension is split.

So, for example, if $G_1$is a commutative unipotent group scheme
of height $\le 1$ (hence almost special) and $G_2$ is almost
special then the group scheme $G=G_1\rtimes G_2$ is almost special and its
 essential dimension is $\ed_k(G_1)+\ed_k(G_2)$. This could be applied for example when $G_1$ is a commutative unipotent group scheme of height $\le 1$
and $G_2$ is  one of the finite group schemes of the Example
\ref{ex: ed of some group schemes}.
\end{example}
\begin{example}
The above example lets us to compute, in particular, the essential
dimension of $\aalpha_p\rtimes \mmu_{p^n}$, which is $2$. But the
argument does not apply to $\aalpha_{p^m}\rtimes \mmu_{p^n}$, for
$m>1$. However in this case the essential dimension is also $2$.
From the Theorem \ref{thm1} it is at least $2$ and  we now prove it
is exactly $2$. One observes (quite easily) that for any field $K$
of positive characteristic the automorphism group scheme of
$\aalpha_{p^n}$ is smooth and trigonalizable. In particular this is
true for the field $\FF_p$. Hence we have that
$\Aut_{\FF_p-gr}(\aalpha_{p^n})$ is special. Indeed it is an
extension of a smooth unipotent group scheme by a smooth
diagonalizable group scheme and both are special. For a smooth
diagonalizable group scheme it is immediate to see that it is
special while for an unipotent smooth group scheme it is enough to
observe that, since $\FF_p$ is perfect, it has a central
decomposition series with quotients isomorphic to $\ga$ (see
\cite[IV, \S2 3.9]{DG}). A fortiori $\Aut_{K-gr}(\aalpha_{p^n})$ is
almost special  for any field $K$ of characteristic $p>0$, since the
essential dimension does not increase by base change (see
(\cite[1.5]{BF})). This implies that there are no non trivial
twisted forms of $\aalpha_{p^n}$. The claim now follows from
 the Lemma \ref{lem:main}, since $\ed_k(\aalpha_{p^n})=1$. With the same argument
one obtains, if $G$ is an almost special group scheme $G$ acting
on $\aalpha_{p^r}$, that $\aalpha_{p^r}\rtimes G$ is almost
special, i.e.
$$
\ed_k (\aalpha_{p^r}\rtimes G)=\ed_k G+1.
$$
\end{example}

\begin{example}\label{ex:test-example}
We do not have examples of infinitesimal group
schemes that are not  almost special. Some such group schemes  could be maybe found among the twisted forms of diagonalizable group schemes, i.e. group schemes of multiplicative type. If  such an example exists among
trigonalizable group schemes, a candidate should be
the following.

Let us consider the kernel $G_{r,m,n}$ of the morphism $\fr^m
+V^r:\W_n\arr \W_n$, where $V$ is the Verschiebung (called ``shift''
in \cite{Ser-local} and ``decalage'' in \cite{DG}), $m>0$ and
$0<r<n$. This is a group scheme of order $p^{mn}$. The dimension of
its Lie algebra is $r$, and it it is embedded in a special group
scheme of dimension $n$, therefore $r\le \ed_k(G_{r,m,n})\le n$.

Now consider the case $r=1$.  Then we will prove that $G$ can not be
embedded in any special algebraic group scheme $\cG$ of dimension
$1$, so we can not use an argument as in the Example \ref{ex: ed of
some group schemes} to conclude that $\ed_k(G_{1,m,n})=1$.  To prove
this fact we can clearly suppose that $k$ is algebraically closed.
Now let us suppose that such a group $\cG$ exists. First of all we
can suppose it is connected. Secondly we observe that it should be
unipotent. Indeed, since it is special, it should be affine
(\cite[Théorème 1]{Ser}) and smooth (see Theorem \ref{thm1});
therefore, if it was not unipotent, from \cite[IV \S 3.11]{DG} we
conclude that it contains a subgroup scheme isomorphic to $\GG_{m}$.
This implies that the kernel of Frobenius, $ _{\fr} \cG$, is not
unipotent, since it contains a subgroup scheme isomorphic to
$\mmu_{p}$. But since the Lie algebra of $G_{1,m,n}$ is equal to the
Lie algebra of $\cG$ we have that $_{\fr}\cG=  { _{\fr}} G_{1,m,n}$,
which is a contradiction since $G_{1,m,n}$ is unipotent. But any
unipotent smooth and connected group scheme of dimension $1$ over a
perfect field is isomorphic to $\GG_a$ (\cite[IV, \S 2, 2.10]{DG}).
And $G_{1,m,n}$ is not a subgroup scheme of $\GG_a$.

This example is also interesting since,  if $k$ is perfect, the
group scheme $G_{1,1,2}$ is the $p$-torsion group scheme of a
supersingular elliptic curve. The $p^n$-torsion group
scheme of the same curve has a decomposition series with quotients isomorphic to $G_{1,1,2}$

\end{example}


\begin{example}\label{ex4}It is conjectured that the essential dimension of
$\ZZ/p^n\ZZ$ over a field $k$ of characteristic $p$ is $n$. The
conjecture has been proved (easily) for $n\le 2$. Since $\ZZ/p^n\ZZ$
is contained in the group scheme of Witt vectors of dimension $n$,
$\W_n$, and this group is special then $\ed_k(\ZZ/p^n\ZZ)\le n$. The
same result also follows from Theorem \ref{thm2}, which for constant
$p$-group schemes was already known. The open problem (for $n>2$) is
to prove the opposite inequality. We remark that if this conjecture
is true over an algebraic closure $\bar{k}$ of $k$ then $\ed_k(G)$
is equal to $n$ for any twisted form $G$ of $\ZZ/p^n\ZZ$. Indeed
such a group scheme would be unipotent (see Lemma \ref{lem3}) so, by
the Theorem \ref{thm2}, its essential dimension is smaller or equal
to $n$. On the other hand it is at least $n$, since the essential
dimension does not increase by base change (see \cite[1.5]{BF}).

Even if the essential dimension of a cyclic constant $p$-group is
not known it is possible to determine the essential dimension of
group schemes with $\ZZ/p^n\ZZ$ as direct summand. Let $G$ be an
infinitesimal group scheme with essential dimension $n$ and
contained in $\W_n$. Then
$$
\ed_k(G\times \ZZ/p^m\ZZ)=n
$$
if $m\le n$. First $\ZZ/p^m\ZZ$ is a subgroup of $\ZZ/p^n\ZZ$ and
this one is the kernel of $\fr-1:\W_n\arr \W_n$. Moreover since
$G$ is infinitesimal and $\ZZ/p^n\ZZ$ is étale the morphism,
induced by multiplication inside $\W_n$,
$$
G\times\ZZ/p^m\ZZ\arr
 \W_n
$$
is injective. Therefore
$$
n=\ed_k(G)\le \ed_k(G\times\ZZ/p^m\ZZ)\le n
$$
For instance, if $m,n,r\in \NN$ and $m\le n$ then  we have
$$
\ed_k(_{\fr^r}\W_n\times \ZZ/p^m\ZZ)=n.
$$
\end{example}



\bibliographystyle{amsalpha}
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\providecommand{\MRhref}[2]{%
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\begin{thebibliography}{BRV09b}

\bibitem[BF03]{BF}
Gr{\'e}gory Berhuy and Giordano Favi, \emph{Essential dimension: a functorial
  point of view (after {A}. {M}erkurjev)}, Doc. Math. \textbf{8} (2003),
  279--330 (electronic).

\bibitem[BR97]{BR}
Joe Buhler and Zinovy Reichstein, \emph{On the essential dimension of a finite
  group}, Compositio Math. \textbf{106} (1997), no.~2, 159--179.

\bibitem[Bro07]{Bro}
Patrick Brosnan, \emph{The essential dimension of a g-dimensional complex
  abelian variety is 2g}, Transformation Groups \textbf{12} (2007), no.~3,
  437--441.

\bibitem[BRV09a]{BRV}
Patrick Brosnan, Zinovy Reichstein, and Angelo Vistoli, \emph{Essential
  dimension of moduli of curves and other algebraic stacks}, arXiv:0907.0924,
  to appear on Journal of European Mathematical Society, 2009.

\bibitem[BRV09b]{BRV2}
\bysame, \emph{Essential dimension, spinor groups and quadratic forms},
  arXiv:0907.0922, to appear on Annals of Mathematics, 2009.

\bibitem[BS08]{brosnan-shreekantan08}
Patrick Brosnan and Ramesh Sreekantan, \emph{Essential dimension of abelian
  varieties over number fields}, arXiv:0803.0016, 2008.

\bibitem[CS06]{Che-Ser}
Vladimir Chernousov and Jean-Pierre Serre, \emph{Lower bounds for essential
  dimensions via orthogonal representations}, J. Algebra \textbf{305} (2006),
  no.~2, 1055--1070.

\bibitem[DG70]{DG}
Michel Demazure and Pierre Gabriel, \emph{Groupes alg\'ebriques. {T}ome {I}:
  {G}\'eom\'etrie alg\'ebrique, g\'en\'eralit\'es, groupes commutatifs}, Masson
  \& Cie, \'Editeur, Paris, 1970, Avec un appendice {{\i}t Corps de classes
  local} par Michiel Hazewinkel.

\bibitem[Flo07]{Flo}
Mathieu Florence, \emph{{On the essential dimension of cyclic p-groups}},
  Inventiones mathematicae \textbf{171} (2007), 175--189.

\bibitem[Gir71]{Gir}
Jean Giraud, \emph{Cohomologie non ab\'elienne}, Springer-Verlag, Berlin, 1971,
  Die Grundlehren der mathematischen Wissenschaften, Band 179.

\bibitem[GR09]{Gil-Rei}
Philippe Gille and Zinovy Reichstein, \emph{A lower bound on the essential
  dimension of a connected linear group}, Comment. Math. Helv. \textbf{84}
  (2009), no.~1, 189--212.

\bibitem[Gro58]{grothendieck-torsion}
Alexander Grothendieck, \emph{Torsion homologique et sections rationnelles},
  Anneaux de Chow et Applications, S{\'e}minaire Claude Chevalley, 1958,
  expoe\'e n. 5.

\bibitem[Haz78]{Haz}
Michiel Hazewinkel, \emph{Formal groups and applications}, Pure and Applied
  Mathematics, vol.~78, Academic Press Inc. [Harcourt Brace Jovanovich
  Publishers], New York, 1978.

\bibitem[KM08]{KM}
Nikita~A. Karpenko and Alexander~S. Merkurjev, \emph{Essential dimension of
  finite p-groups}, Inventiones Mathematicae \textbf{172} (2008), no.~3,
  491--508.

\bibitem[Led04]{Le}
Arne Ledet, \emph{On the essential dimension of {$p$}-groups}, Galois Theory
  and Modular Forms, Dev. Math., vol.~11, Kluwer Acad. Publ., 2004,
  pp.~159--172.

\bibitem[Rei00]{Re}
Zinovy Reichstein, \emph{On the notion of essential dimension for algebraic
  groups}, Transformation Groups \textbf{5} (2000), no.~3, 265--304.

\bibitem[RY00]{Rei-You}
Zinovy Reichstein and Boris Youssin, \emph{Essential dimensions of algebraic
  groups and a resolution theorem for {$G$}-varieties}, Canad. J. Math.
  \textbf{52} (2000), no.~5, 1018--1056, With an appendix by J{\'a}nos
  Koll{\'a}r and Endre Szab{\'o}.

\bibitem[Ser58]{Ser}
Jean-Pierre Serre, \emph{Espaces fibres alg{\'e}briques}, Anneaux de Chow et
  Applications, S{\'e}minaire Claude Chevalley, 1958, expos{\'e} n. 1.

\bibitem[Ser79]{Ser-local}
\bysame, \emph{Local fields}, Graduate Texts in Mathematics, vol.~67,
  Springer-Verlag, New York, 1979.

\end{thebibliography}

\end{document}
