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\title[Essential dimension of central simple algebras] % colontitle
{Essential dimension of central simple algebras}

\keywords { Essential dimension, Brauer group, algebraic tori {\em
2000 Mathematical Subject Classifications: primary 16K50, secondary 14L30,
20G15  }}


\date
%{\today}
{}

\author
[S. Baek] {Sanghoon Baek}



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



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


\email {shbaek@math.ucla.edu}


\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 supported by the
Beckenbach Dissertation Fellowship at the University of California at Los Angeles.}

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




\begin{document}

\begin{abstract}
Let $p$ be a prime integer, $1\leq s \leq r$ integers and $F$ a field of characteristic different from $p$. We find upper and lower bounds for the essential $p$-dimension $\ed_{p}(\cat{Alg}_{p^r,p^s})$ of the class $\cat{Alg}_{p^r,p^s}$ of central simple algebras of degree $p^r$ and exponent dividing $p^s$. In particular, we show that $\ed_{2}(\cat{Alg}_{8,2})=8$ and $\ed_{p}(\cat{Alg}_{p^2,p})=p^{2}+p$ for $p$ odd.
\end{abstract}


\maketitle



\section{Introduction}

Let $\cF:\cat{Fields}/F\to\cat{Sets}$ be a functor from the category $\cat{Fields}/F$ of field extensions over $F$ to the category $\cat{Sets}$ of sets. Let $E\in\cat{Fields}/F$ and $K\subset E$ a subfield over $F$. An
element $\alpha\in \cF(E)$ is said to be \emph{defined over $K$} (and
$K$ is called a \emph{field of definition of $\alpha$}) if there
exists an element $\beta\in \cF(K)$ such that $\alpha$ is the image of
$\beta$ under the map $\cF(K)\to \cF(E)$. The \emph{essential dimension
of $\alpha$}, denoted $\ed^{\cF}(\alpha)$, is the least transcendence
degree $\td_F(K)$ over all fields of definition $K$ of $\alpha$. The
\emph{essential dimension of the functor $\cF$} is
\[
\ed(\cF)=\sup \{\ed^{\cF}(\alpha)\},
\]
where the supremum is taken over all fields $E\in\cat{Fields}/F$ and all
$\alpha\in \cF(E)$ (see \cite[Def. 1.2]{BerhuyFavi03} or \cite[Sec.1]{Merkurjev09}). Informally, the essential dimension of $\cF$ is the smallest number of algebraically independent parameters required to define $\cF$ and may be thought of as a measure of complexity of $\cF$.

Let $p$ be a prime integer. The \emph{essential $p$-dimension of  $\alpha$}, denoted $\ed_p^{\cF}(\alpha)$, is
defined as the minimum of $\ed^{\cF}(\alpha_{E'})$, where $E'$ ranges over all field extensions of $E$ of degree prime to $p$. The \emph{essential $p$-dimension of $\cF$} is
\[
\ed_p(\cF)=\sup \{\ed_p^{\cF}(\alpha)\},
\]
where the supremum ranges over all fields $E\in\cat{Fields}/F$ and all $\alpha\in \cF(E)$.
By definition, $\ed(\cF)\geq\ed_p(\cF)$ for all $p$.

For every integer $n\geq 1$, a divisor $m$ of $n$ and any field extension $E/F$, let $\cat{Alg}_{n,m}(E)$ denote the set of isomorphism classes of central simple $E$-algebras of degree $n$
and exponent dividing $m$. Equivalently, $\cat{Alg}_{n,m}(E)$ is the subset of the $m$-torsion part $\Br_m(E)$ of the Brauer group of $E$ consisting of all elements $a$ such that $\ind(a)$ divides $n$. In particular, if $n=m$, then $\cat{Alg}_{n}(E):=\cat{Alg}_{n,n}(E)$ is the set of isomorphism classes of central simple $E$-algebras of degree $n$. We view $\cat{Alg}_{n,m}$ and $\cat{Alg}_{n}$ as functors $\cat{Fields}/F\to\cat{Sets}$.

In the present paper we give upper and lower bounds for $\ed_p(\cat{Alg}_{n,m})$ for a prime integer $p$. Let $p^r$ (respectively, $p^s$) be the largest power of $p$ dividing $n$ (respectively, $m$). Then $\ed_p(\cat{Alg}_{n,m})=\ed_p(\cat{Alg}_{p^{r},p^{s}})$ and $\ed_p(\cat{Alg}_{n})=\ed_p(\cat{Alg}_{p^{r}})$ (see Section \ref{mainpf}). Thus, we may assume that $n$ and $m$ are the $p$-powers $p^r$ and $p^s$ respectively.


Using structure theorems on central simple algebras, we can compute the essential ($p$)-dimension of $\cat{Alg}_{p^{r},p^{s}}$ for certain small values of $r$, $s$ or $p$ as follows.
As every central simple algebra $A$ of degree $p$ is cyclic over a finite field extension of degree prime to $p$, $A$ can be given by two parameters (see Section \ref{chandbr}). In fact, $\ed_{p}(\cat{Alg}_{p})=2$ by \cite[Lemma 8.5.7]{Reichstein00a}. 

By Albert's theorem, every algebra in $\cat{Alg_{4,2}}$ is biquaternion and hence can be given by $4$ parameters. In fact, $\ed(\cat{Alg_{4,2}})=\ed_{2}(\cat{Alg_{4,2}})=4$ (see Remark \ref{lastst}).







The upper and lower bounds for $\ed_{p}(\cat{Alg}_{p^{r}})$ can be found in \cite{MR09a} and \cite{Merkurjev09b} respectively.
In this paper (see Sections \ref{mainpf} and \ref{upup}), we establish the following upper and lower bounds for $\ed_{p}(\cat{Alg_{p^r,p^s}})$:

\smallskip

\noindent{\bf Theorem.} \emph{Let $F$ be a field and $p$ a prime
integer different from $\ch(F)$. Then, for any integers $r\geq 2$ and $s$
with $1\leq s\leq r$,}
\[
2p^{2r-2}-p^{r}+p^{r-s}\ \geq\ \ed_{p}(\cat{Alg}_{p^{r},p^{s}})\ \geq\ \begin{cases}
(r-1)2^{r-1} & \text{if $p=2$ and $s=1$},\\
(r-1)p^r+p^{r-s} & \text{otherwise}.
\end{cases}
\]




\smallskip

\noindent{\bf Corollary.} (cf. \cite{Merkurjev09a})
\emph{Let $p$ be a prime integer and $F$ a field of characteristic different from $p$. Then}
\[\ed_{p}(\cat{Alg_{p^2}})=p^{2}+1.\]


\smallskip

\noindent{\bf Corollary.}
\emph{Let $p$ be an odd prime integer and $F$ a field of characteristic different from $p$. Then}
\[\ed_{p}(\cat{Alg_{{p^2},p}})=p^{2}+p.\]

 The corollary recovers a result in \cite{Tignol87} that for $p$ odd, there exists a central simple algebra of degree $p^2$ and exponent $p$ which is not decomposable as a tensor product of two algebras of degree $p$. Indeed, if every central simple algebra of degree $p^2$ and exponent $p$ is decomposable, then the essential $p$-dimension of $\cat{Alg_{{p^2},p}}$ would be at most $4$.


\smallskip

\noindent{\bf Corollary.}
\emph{Let $F$ be a field of characteristic different from $2$. Then}
\[\ed_{2}(\cat{Alg}_{8,2})=\ed(\cat{Alg}_{8,2})=8.\]

The proof is given in Section \ref{verylast}. The corollary recovers a result in \cite{ART79} that there is a central simple algebra of degree $8$ and exponent $2$ which is not decomposable as a tensor product of three quaternion algebras. Indeed, if every central simple algebra of degree $8$ and exponent $2$ is decomposable, then the essential $2$-dimension of $\cat{Alg_{8,2}}$ would be at most $6$.










\section{Character, Brauer group and algebraic tori}\label{brtori}

\subsection{Character and Brauer group}\label{chandbr}

Let $F$ be a field, $F_{\sep}$ a separable closure of $F$, $\Gamma_F=\Gal(F_\sep/F)$. For a (discrete) $\Gamma_F$-
module $M$, we write $H^n(F,M)$ for the Galois cohomology group $H^n(\Gamma_F,M)$.

If $S$ is an algebraic group over $F$, we let $H^1(F,S)$ denote
the set $H^1\(\Gamma_F,S(F_\sep)\)$ (see \cite{Serre97}).

The \emph{character group} of $F$ is defined by
\[
\Ch(F):=\Hom_{\cont}(\Gamma_F, \Q/\Z)=H^{1}(F, \Q/\Z)\simeq H^{2}(F, \Z).
 \]
The \emph{$n$-torsion character group} $\cat{\Ch}_{n}(F)$ is identified with $H^{1}(F, \Z/n\Z)$.
For a character $\chi\in\Ch(F)$, set
$F(\chi)=(F_{\sep})^{\Ker(\chi)}$. The field extension $F(\chi)/F$ is cyclic of degree $\ord(\chi)$.
If $\Psi\subset \Ch(F)$ is a finite subgroup, we set 
\[
F(\Psi):=(F_{sep})^{\cap Ker(\chi)},
\]
where the intersection is taken over all $\chi\in \Psi$.
The Galois group $G=\Gal\(F(\Psi)/F\)$ is abelian and $\Psi$ is canonically
isomorphic to the character group $\Hom(G,\Q/\Z)$ of $G$. Note that a character $\eta\in\Ch(F)$ is trivial over $F(\Psi)$ if and only if $\eta\in\Psi$.

We write $\Br(F)$ for the Brauer group $H^2(F,F_\sep^\times)$ of $F$. If $L/F$ is a field extension
and $\alpha \in \Br(F)$, we let $\alpha_L$ denote the image of $\alpha$ under the natural map $\Br(F)\to\Br(L)$. We say that $L$ is a \emph{splitting field} of $\alpha$ if $\alpha_L=0$.
The \emph{index} $\ind(\alpha)$ of $\alpha$ is the smallest degree of a splitting field of $\alpha$. The \emph{exponent}
$\exp(\alpha)$ is the order of $\alpha$ in $\Br(F)$. The integer $\exp(\alpha)$ divides $\ind(\alpha)$.

Let $A$ be a central simple $F$-algebra. The \emph{degree} of $A$ in the square root of $\dim(A)$.
We write $[A]$ for the class of $A$ in $\Br(F)$. The index of $[A]$ divides $\deg(A)$. If $\alpha\in\Br(F)$ and $n$ is a positive multiple of $\ind(\alpha)$, then there is a central simple $F$-algebra $A$ of degree $n$ with $[A]=\alpha$.

The cup-product
\[
\Ch(F)\tens F^\times=H^2(F,\Z)\tens H^0(F,F_{\sep}^\times)\to
H^2(F,F_{\sep}^\times)=\Br(F)
\]
takes $\chi\tens b$ to the class $\chi\cup(b)$ in $\Br(F)$ that is
split by $F(\chi)$. A class $\alpha\in\Br(F)$ is called \emph{$n$-cyclic} if
$\alpha=\chi\cup(b)$ for a character $\chi$ with $n\chi=0$. Such classes belong to $\Br_n(F)$. If $n$ is prime to $\ch(F)$, then $\Br_n(F)\simeq H^2(F,\mu_n)$, where $\mu_n$ is the $\Gamma_F$-module of all $n$-th roots of unity in $F_\sep$.


Let $n$ be prime to $\ch(F)$ and suppose that $F$ contains a
primitive $n$-th root of unity $\xi$. For any $a\in F^\times$, let
$\chi_a\in \Ch(F)$ be a unique character with values in
$\frac{1}{n}\Z/\Z\subset \Q/\Z$ such that
\[
\gamma(a^{1/n})=\xi^{(n\chi_a(\gamma))}a^{1/n}
\] 
for all $\gamma\in
\Gal(F_{\sep}/F)$. We write $(a,b)_n$ for $\chi_a\cup(b)$. The
symbol $(a,b)_n$ satisfies the following properties (see
\cite[Chap. XIV, Prop.4]{Serre}):

\smallskip

 $(a,b)_n +(a',b)_n = (aa',b)_n$,\\
\indent $(a,b)_n = -(b,a)_n$,\\
\indent $(a,-a)_n=0$.

\smallskip






\begin{comment}
$\mu_n$ for the Galois module of $n$th roots of
unity in $F_\sep$. There is a canonical isomorphism $H^{1}(F, \mu_{n})\simeq F^\times/(F^\times)^n$.
If $\mu_n\subset F$, we have the cup-product
\[
F^\times/(F^\times)^n\tens F^\times/(F^\times)^n\to H^{1}(F, \mu_{n})\tens H^{1}(F, \mu_{n})\to H^2(F,\mu_{n}^{\tens 2}).
\]
The choice of primitive $n$th root of unity identifies the target group with $H^2(F,\mu_{n})=\Br_{n}(F)$.
If $a,b\in F^\times$, we write $(a,b)_{n}$ in $\Br_{n}(F)$ for the image of $a(F^\times)^n\tens b(F^\times)^n$.

then $\cat{\Ch}_{n}(F)=H^{1}(F, \Z/n\Z)\simeq H^{1}(F, \gmu_{n})=F^\times/(F^\times)^n$ and we identify $\chi\in \cat{\Ch}_{n}(F)$ with $a\in F^\times/(F^\times)^n$ under this isomorphism. The cup-product
\[
H^{1}(F, \gmu_{n})\tens H^{1}(F, \gmu_{n})\to H^2(F,\gmu_{n}^{\tens 2})=H^2(F,\gmu_{n})=\Br_{n}(F)
\]
takes $a\tens b$ to the class $(a,b)_{n}$ of a cyclic algebra split by $F(\chi)$. The cyclic algebra $(a,b)_{n}$ is called \emph{$n$-cyclic}. Moreover, $\chi\cup(b)=(a,b)_{n}$ in $\Br_{n}(F)$ by \cite[Chap.XIV, Prop.5]{Serre}.
\end{comment}

For a finite subgroup $\Phi\subset \Ch(F)$ write
$\Br\(F(\Phi)/F\)_{\dec}$ for the \emph{subgroup of decomposable
elements} in $\Br\(F(\Phi)/F\)$ generated by the elements
$\chi\cup(a)$ for all $\chi\in\Phi$ and $a\in F^\times$. The
\emph{indecomposable relative Brauer group}
$\Br\(F(\Phi)/F\)_{\ind}$ is the factor group
$\Br\(F(\Phi)/F\)/\Br\(F(\Phi)/F\)_{\dec}$. Similarly, if $\Phi\subset \Ch_n(F)$ for some $n$, then $\Br_{n}(F(\Phi)/F\)_{\ind}$ is the \emph{indecomposable $n$-torsion relative Brauer group} defined as the factor group $\Br_{n}\(F(\Phi)/F\)/\Br\(F(\Phi)/F\)_{\dec}$.

Let $E$ be a complete field with respect to a discrete valuation $v$
and $K$ its residue field. Let $p$ be a prime integer different from $\ch(K)$. There is a
natural injective homomorphism $\Ch(K)\{p\}\to \Ch(E)\{p\}$ of the
$p$-primary components of the character groups that identifies
$\Ch(K)\{p\}$ with the character group of an unramified field
extension of $E$. For a character $\chi\in\Ch(K)\{p\}$, we write
$\widehat\chi$ for the corresponding character in $\Ch(E)\{p\}$.

By \cite[\S 7.9]{GMS}, there is an exact sequence
\[
0\to \Br(K)\{p\}\xra{i} \Br(E)\{p\}\xra{\partial_v} \Ch(K)\{p\}\to
0.
\]

If $\alpha\in\Br(K)\{p\}$, then we write $\widehat \alpha$ for the element
$i(\alpha)$ in $\Br(E)\{p\}$. For example, if $\alpha=\chi\cup(\bar u)$ for
some $\chi\in\Ch(K)\{p\}$ and a unit $u\in E$, then $\widehat
\alpha=\widehat\chi\cup(u)$. In the case $F$ contains a primitive $n$-th root of unity, where $n$ is a
power of $p$,
if $\alpha=(\bar a,\bar b)_n$ with $a$ and $b$ units in $E$, then $\widehat \alpha=(a,b)_n$.

If $\beta=\widehat \alpha+\(\widehat \chi\cup(x)\)$ for an element $\alpha\in
\Br(K)\{p\}$, $\chi\in\Ch(K)\{p\}$ and $x\in E^\times$ such that $v(x)$ is not divisible by $p$,
we have (cf. \cite[Prop. 2.4]{Tignol78})
\begin{equation}\label{indexx}
\ind(\beta)=\ind\(\alpha_{K(\chi)}\)\cdot\ord(\chi).
\end{equation}


\subsection{Representations of algebraic tori}\label{toritori}

Let $T$ be an algebraic torus over a field $F$, $L/F$ a finite
Galois splitting field for $T$ with Galois group $G$. The group $G$ is called the \emph{decomposition group} of $T$.
The
\emph{character group} $T^*:=\Hom_L(T_L,{\gm}_{,L})$ has the structure of a
$G$-module. The torus $T$ can be reconstructed from $T^*$ by
\[
T=\Spec \(L[T^*]^G\).
\]
A torus $P$ over $F$ split by $L$ is called \emph{quasi-split} if $P^*$ is a
\emph{permutation} $G$-module, i.e., if there exists a $G$-invariant
$\Z$-basis $X$ for $P^*$. The torus $P$ is canonically isomorphic
to the group of invertible elements of the \'etale $F$-algebra
$A=\Map_G(X,L)$. The torus $P$ acts linearly by multiplication on
the vector space $A$ over $F$ making $A$ a faithful $P$-space (a
linear representation of $P$) of dimension $\dim(P)$. It follows
that a homomorphism of algebraic tori $\nu:T\to P$ with $P$ a
quasi-split torus yields a linear representation of $T$ of
dimension $\dim(P)$ that is faithful if $\nu$ is injective.

Let $P$ be a split torus over $F$, and $P^*$ its character group. As
above, the choice of a $\Z$-basis $X$ for $P^*$ allows us to
identify $P$ with the group of invertible elements of a split
\'etale $F$-algebra $A$ and make $A$ a faithful $P$-space over
$F$. Let $\nu:T\to P$ be a homomorphism of split tori over $F$.
Suppose a finite group $G$ acts on $T$ and $P$ by tori
automorphisms so that $\nu$ is a $G$-equivariant homomorphism.
Then the map $\nu^*:P^*\to T^*$ is a $G$-module homomorphism.
Suppose that there is a $G$-invariant $\Z$-basis $X$ for $P^*$,
i.e., $P^*$ is permutation. Then $G$ acts on the algebra $A$ by
$F$-algebra automorphisms. The torus $T$ acts linearly on $A$ via
$\nu$. It follows that the semidirect product $T\rtimes G$ acts
linearly on $A$ making $A$ a $T\rtimes G$-space.

Let $L$ be a Galois $G$-algebra over $F$ (for example, $L/F$ is a Galois field extension with Galois group $G$). Then $\gamma:\Spec L\to\Spec F$ is a $G$-torsor. Twisting the split torus $T$ by the torsor $\gamma$, we get the torus
\[
T_\gamma=(T\times\Spec L)/G=\Spec \(L[T^*]^G\)
\]
that is split by $L$ and $T^*_\gamma$ is isomorphic to $T^*$ as $G$-modules.

By \cite[Prop. 28.11]{Book}, the fiber of $H^{1}(F, T\rtimes G)\to H^{1}(F, G)$ over the class of $\gamma$ is naturally bijective to the orbit set of the group $G_\gamma(F)$ in $H^{1}(F, T_{\gamma})$,
i.e.,
\begin{equation}\label{cop}
H^{1}(F, T\rtimes G)\simeq\coprod H^{1}(F, T_{\gamma})/G_\gamma(F),
\end{equation}
where the coproduct is taken over all $[\gamma]\in H^{1}(F, G)$.

\subsection{Generic torsors}\label{gentor}

Let $T$ be an algebraic torus split by a finite Galois field
extension $L/F$ with $G=\Gal(L/F)$. Let $P$ be a quasi-split
torus split by $L$ and containing $T$ as a subgroup. Set
$S=P/T$. Then the canonical homomorphism $\gamma:P\to S$ is a
$T$-torsor.

\begin{proposition}\label{gengen}
The $T$-torsor $\gamma$ is generic, i.e., for every field
extension $K/F$ with $K$ infinite, every $T$-torsor
$\gamma':E\to\Spec K$ and every nonempty open subset $W\subset S$,
there is a morphism $s:\Spec K\to S$ over $F$ with $\Im(s)\subset
W$ such that the $T$-torsors $\gamma'$ and $s^*(\gamma)=\gamma\times_S\Spec K$
over $K$ are isomorphic.
\end{proposition}

\begin{proof}
As $P$ is quasi-split, the last term in the exact sequence
\[
P(K)\xra{\gamma_K} S(K)\xra{\delta} H^1(K,T)\to H^1(K,P)
\]
is trivial. Then there is $s\in S(K)$ with $\delta(s)=[\gamma']$.
As $K$ is infinite, the $K$-points of $P$ are dense in $P$ and we
can modify $s$ by an element in the image of $\gamma_K$ so that
$s\in W(K)$, i.e., $\Im(s)\subset W$. Then the $T$-torsor
$\gamma'$ over $K$ with the class $\delta(s)$ satisfies the required property.
\end{proof}

\subsection{The algebraic tori $P^\Phi$, $S^\Phi$, $T^\Phi$,
${U}^\Phi$ and $V^\Phi$}\label{manytori}

Let $1\leq s\leq r$ be integers, $p$ a prime integer, $F$ a field with $\ch(F)\neq p$,
$\Phi$ a subgroup of $\Ch_{p}(F)$ of rank $r$ and
$L=F(\Phi)$. Let $G=\Gal(L/F)$. Choose a basis
$\chi_1,\chi_2,\dots,\chi_r$ for $\Phi$. Each $\chi_i$ can be viewed
as a character of $G$, i.e., as a homomorphism $\chi_i:G\to\Q/\Z$.
Let $\sigma_1,\sigma_2,\dots,\sigma_r$ be the dual basis for $G$,
i.e.,
\[
\chi_i(\sigma_j)= \left\{
  \begin{array}{ll}
    (1/p)+\Z, & \hbox{if $i=j$;} \\
    0, & \hbox{otherwise.}
  \end{array}
\right.
\]Let $R$ be the group ring $\Z[G]$. Consider the surjective $G$-modules homomorphism $\bar\varepsilon:R\to \Z/p^{s}\Z$, defined by $\bar\varepsilon(x)=\varepsilon(x)+p^s\Z$, where
$\varepsilon: R\to\Z$ is the \emph{augmentation homomorphism} given by $\varepsilon(\rho)=1$ for all
$\rho\in G$. Set $J:=\Ker(\bar\varepsilon)$, thus, we have an exact sequence
\begin{equation*}
0\to J\to R\xra{\bar{\varepsilon}}\ \Z/p^{s}\Z \to 0.
\end{equation*}
Moreover, the $G$-module $J$ is generated by $I$ and $p^{s}$, where $I:=\Ker(\varepsilon)$ is the \emph{augmentation ideal} in $R$.


Consider the
$G$-module homomorphism $h:R^{r+1}\to R$ taking the $i$-th canonical basis
element $e_i$ to $\sigma_i-1$ for $1\leq i \leq r$ and $e_{r+1}$ to $p^{s}$. The image of $h$ coincides with $J$.




Set $N:=\Ker(h)$ and write $w_i=1+\sigma_i+\sigma_i^2+\dots +\sigma_i^{p-1}\in R$ for $1\leq i \leq r$. Consider the following elements in $N$:
\[
e_{ij}=(\sigma_i-1)e_j-(\sigma_j-1)e_i,\quad
f_i=w_ie_i,\quad \text{and} \quad g_i=-p^{s}e_i+(\sigma_{i}-1)e_{r+1}
\]
for all $1\leq i, j \leq r$.

\begin{lemma}\label{gene}
The $G$-module $N$ is generated by $e_{ij}$, $f_i$ and $g_i$.
\end{lemma}
\begin{proof}
Consider the surjective morphism $k:R^{r}\to I$ taking $e_i$ to $\sigma_i-1$ and set $N':=\Ker(k)$.
Then we have the following commutative diagram
\begin{equation*}
\xymatrix{
N'  \ar@{_{(}->} [d]\ar@{_{(}->}[r] & R^{r} \ar@{_{(}->}[d]\ar@{->}^{k}[r] & I \ar@{_{(}->>} [d]  \\
N\ar@{->>}[d]\ar@{_{(}->}[r]  &  R^{r+1} \ar@{->>}[d]\ar@{->>}^h[r] & J \ar@{->>}^{\varepsilon'}[d]\\
I \ar@{_{(}->}[r] & R \ar@{->>}[r]^{\varepsilon} & \Z  \\
}
\end{equation*}
where $R^{r+1}\to R$ is the projection morphism to the last coordinate and $\varepsilon':J\to \Z$ is given by $\varepsilon'(j)=\varepsilon(j)/p^s$.

By the exactness of the first column of the diagram, $N$ is generated by $N'$ and the liftings $g_i$ of $\sigma_i-1$ in $N$. The module $N'$ is generated by $e_{ij}$ and $f_i$ by \cite[Lemma 3.5]{Merkurjev09b}. This completes the proof.
\end{proof}


Let $\varepsilon_i:R^{r+1}\to\Z$ be the $i$-th projection followed by
the augmentation map $\varepsilon$. It follows from Lemma \ref{gene}
that $\varepsilon_i(N)=p\Z$ for every $i=1,\dots, r$. Moreover, the
$G$-homomorphism
\[
q:N\to\Z^r,\quad x\mapsto
\(\varepsilon_1(x)/p,\dots,\varepsilon_r(x)/p\)
\]
is surjective. Set $M:=\Ker(q)$ and $Q:=R^{r+1}/M$.


Let $P^\Phi$, $S^\Phi$, $T^\Phi$, ${U}^\Phi$ and $V^\Phi$ be the
algebraic tori over $F$ with the character $G$-modules $R^{r+1}$, $Q$,
$M$, $J$ and $N$,  respectively. The diagram of homomorphisms of
$G$-modules with the exact columns and rows
\begin{equation}\label{diagmod}
\xymatrix{
M  \ar@{_{(}->} [d]\ar@{=}[r] & M \ar@{_{(}->}[d] &  \\
N\ar@{->>}_q[d]\ar@{_{(}->}[r]  &  R^{r+1} \ar@{->>}[d]\ar@{->>}^h[r] & J \ar@{=}[d]\\
\Z^r \ar@{_{(}->}[r] & Q \ar@{->>}[r] & J  \\
}
\end{equation}
yields the following diagram of homomorphisms of the tori
\begin{equation}\label{diag}
\xymatrix{
T^\Phi  \ar@{=}[r] & T^\Phi &  \\
{V}^\Phi\ar@{->>}[u]  &  P^\Phi \ar@{->>}[u]\ar@{->>}_{\gamma}[l] & U^\Phi \ar@{_{(}->}[l] \ar@{=}[d]\\
\gm^r \ar@{_{(}->}[u] & S^\Phi \ar@{_{(}->}[u]\ar@{->>}[l] & U^\Phi  \ar@{_{(}->}[l]\\
}
\end{equation}



\medskip

Let $K/F$ be a field extension and set $KL:=K\tens_F L$. The commutative diagram
\begin{equation*}
\begin{CD}
0  @>>> I  @>>> R @>>> \Z @>>> 0       \\
@.    @VVV  @|  @VVV @. \\
0  @>>> J  @>>> R @>>> \Z/p^{s}\Z @>>> 0         \\
\end{CD}
\end{equation*}
induces the commutative diagram of homomorphisms of algebraic groups
\begin{equation}\label{escho}
\begin{CD}
1  @>>> \mu_{p^s}  @>>> R_{L/F}(\gml) @>>> U^\Phi @>>> 1       \\
@.    @VVV  @|  @VVV @. \\
1  @>>> \gm  @>>> R_{L/F}(\gml) @>>> U'^\Phi @>>> 1         \\
\end{CD}
\end{equation}
and then the commutative diagram
\begin{equation}\label{eschoescho}
\begin{CD}
0  @>>> H^1(K,U^\Phi) @>>>  H^2(K,\mu_{p^s}) @>>>  H^{2}(KL,\gm)        \\
@.   @VVV   @VVV    @|      \\
0  @>>> H^1(K,U'^\Phi) @>>>  H^2(K,\gm) @>>>  H^2(KL,\gm).        \\
\end{CD}
\end{equation}
Hence
\begin{equation}\label{hu}
H^1(K,U^\Phi)\simeq \Br_{p^s}(KL/K)\quad\text{and}\quad H^1(K,{U'}^\Phi)\simeq \Br(KL/K).
\end{equation}

\begin{lemma}\label{compute2}
The map $H^1(K,U^\Phi)\to H^1(K,S^\Phi)$ induces an isomorphism
$H^1(K,S^\Phi)\simeq \Br_{p^s}(KL/K)_{\ind}$.
\end{lemma}

\begin{proof}
Consider the following commutative diagram
\begin{equation*}
\begin{CD}
1  @>>> U^\Phi  @>>> S^\Phi @>>> \gm^{r} @>>> 1       \\
@.    @VVV  @VVV  @| @. \\
1  @>>> U'^\Phi  @>>> S'^\Phi @>>> \gm^{r} @>>> 1,         \\
\end{CD}
\end{equation*}
where the bottom row is induced by the bottom row of the diagram $(4)$ in \cite{Merkurjev09b}.
This yields a commutative diagram
\[
\begin{CD}
(K^\times)^r  @>>>  H^1(K,U^\Phi)  @>>>  H^{1}(K,S^\Phi)  @>>>  0     \\
@|    @VVV  @VVV    \\
(K^\times)^r  @>\lambda>>  H^1(K,U'^\Phi)  @>>>  H^{1}(K,S'^\Phi)  @>>>  0     \\
\end{CD}
\]
with the exact rows.
The homomorphism $\lambda$ takes
$(x_1,\dots,x_r)$ to $\sum_{i=1}^r \((\chi_i)_K\cup(x_i)\)$ by \cite[Lemma 3.6]{Merkurjev09b}, whence the result.
\end{proof}




\section{Essential dimension of algebraic tori}

Let $S$ be an algebraic group over $F$. The \emph{essential
dimension $\ed(S)$} (respectively, \emph{essential $p$-dimension $\ed_p(S)$}) of $S$ is defined to be the essential ($p$-)dimension of the functor taking a field extension $K/F$ to the set of isomorphism classes $S$-$\tors
(K)$ of $S$-torsors over $K$. Note that the functor $S$-$\tors$ is isomorphic to the functor taking $K$ to the set $H^1(K,S)$.

Let $S$ be an algebraic torus over $F$ split by $L$ with
$G=\Gal(L/F)$. We assume that $G$ is a group of order $p^r$, where $p$ is a prime integer and $r\geq 2$. Let $X$ be
the $G$-module of characters of $S$.  Define the group $\overline X:=X/(pX+IX)$, where $I$ is the
augmentation ideal in $R=\Z[G]$. For any subgroup $H\subset G$,
consider the composition $X^H\hookrightarrow X\to \overline X$. For
every $k$, let $V_k$ denote the subgroup generated by images of the homomorphisms
$X^H\to\overline X$
over all subgroups $H$ with $[G:H]\leq
p^k$. We have the sequence of subgroups
\[
0=V_{-1}\subset V_0\subset\dots\subset V_r=\overline X.
\]
A \emph{p-presentation} of $X$ is a $G$-homomorphism $P\to X$ with $P$ a permutation $G$-module and finite cokernel of order prime to $p$. A $p$-presentation with the smallest $\rank(P)$ is called \emph{minimal}.
The essential $p$-dimension of algebraic tori was determined in
\cite[Th. 1.4]{LMMR} in terms of a minimal $p$-presentation $P\to X$:
\begin{equation}\label{reich}
\ed_p(S)=\rank(P)-\dim(S).
\end{equation}



We have the following explicit formula for the essential ($p$-)dimension of $S$ (cf. \cite[Th. 4.3]{Merkurjev09b}):

\begin{theorem}\label{edtori}
Let $S$ be a torus over a field $F$ and $p$ a prime integer different from $\ch(F)$. If the decomposition group $G$ of $S$ is a $p$-group, then
\[
\ed(S)=\ed_p(S)=\sum_{k=0}^{r}(\rank V_k-\rank V_{k-1})p^k -\dim(S).
\]
\end{theorem}
\begin{proof}
The second equality was proven in \cite[Th. 4.3]{Merkurjev09b}. Let $\nu:P\to X$ be a minimal $p$-presentation.
By definition, the index $[X:\Im(\nu)]$ is prime to $p$. Let
$T$ and $U$ be algebraic groups of multiplicative type split by $L$ with the
character $G$-modules $\Im(\nu)$ and $X/\Im(\nu)$,
respectively, hence we have an exact sequence
\[
1\to U\to S\to T\to 1.
\]
Let $K/F$ be a field extension. 
By assumption, the group $U(KL)=\Hom(X/\Im(\nu), KL^\times)$ has
order prime to $p$. We have an exact sequence
\begin{equation*}
H^{1}\(G,U(KL)\)\to H^{1}\(G,S(KL)\)\to H^{1}\(G,T(KL)\)\to H^{2}\(G,U(KL)\).
\end{equation*}
As the order of $U(KL)$ is prime to $p$ and $G$ is a $p$-group, the groups
$H^{i}\(G,U(KL)\)$ are trivial for $i\geq 1$, hence the
homomorphism $S\to T$ induces an isomorphism of functors
$S$-$\tors\iso$ $T$-$\tors$. It follows that $\ed(S)=\ed(T)$. The
surjection $P\to \Im(\nu)$ yields a generically free
representation of $T$ by
\cite[Lemma 3.3]{MR09}. Hence, by \cite[Prop. 4.11]{BerhuyFavi03} and
(\ref{reich}), we have
\[
\ed_{p}(S)\leq \ed(S)=\ed(T)\leq \rank(P)-\dim(T)=\rank(P)-\dim(S)=\ed_{p}(S),
\]
therefore, $\ed(S)=\ed_p(S)$.
\end{proof}



Let $F$ be a field, $\Phi$ a subgroup of $\Ch_{p}(F)$ of rank $r\geq 2$, $L=F(\Phi)$ and $G=\Gal(L/F)$. In this section we compute the essential ($p$-)dimension of the algebraic tori $U^\Phi$ and $S^\Phi$ defined by (\ref{diag}). For any subgroup $H$ of $G$, we write $n_{H}:=\sum_{\tau\in H} \tau$ in $R=\Z[G]$. An element $x\in R$ is \emph{decomposable} if $x=yz$ with $y,z\in R$, and $\varepsilon(y),\varepsilon(z)\in p\Z$.

\begin{lemma}\label{deco}
Let $H\subset G$ be a nontrivial subgroup and $x\in R$ such that $\varepsilon(n_{H}x)\in p^2\Z$. Then $n_Hx$ is decomposable.
\end{lemma}
\begin{proof}
If $|H|=p$, then $\varepsilon(x)\in p\Z$ and hence $n_Hx$ is decomposable. Otherwise $H=H'\times H''$ for nontrivial subgroups $H'$ and $H''$. As $n_{H}=n_{H'}\cdot n_{H''}$, the element $n_{H}$ and therefore, $n_{H}x$ is decomposable.
\end{proof}



\begin{lemma}\label{decomposable}
If $x\in R$ is decomposable, then $x\equiv \varepsilon(x)$ modulo $pI+I^2$.
\end{lemma}

\begin{proof}
Let $y=\varepsilon(y)+u$ and $z=\varepsilon(z)+v$ for some $u,v\in I$. Then we have $yz-\varepsilon(yz)=\(\varepsilon(y)v+\varepsilon(z)u\)+uv\in pI+I^2$.
\end{proof}

\begin{lemma}\label{twocas}
The group $V_k$ is generated by
\begin{enumerate}
  \item the elements $\overline{n_{H}x}$ such that $|H|\geq p^{r-k}$ and $\varepsilon(n_{H}x)\in p^{s}\Z$ if $r-k< s$,
  \item the elements $\overline{n}_{H}$ such that $|H|\geq p^{r-k}$ if $r-k\geq s$.
\end{enumerate}
\end{lemma}

\begin{proof}
The statement follows from the equality $J^H=R^H\cap J=n_HR\cap J$.
\end{proof}



\begin{lemma}\label{krs}
If $k<r-s$, then $V_k=0$.
\end{lemma}
\begin{proof}
By Lemma \ref{twocas}(2), $V_k$ is generated by $\overline{n}_{H}$ with $|H|\geq p^{r-k}$. As $n_{H}$ is decomposable and $|H|>p^{s}$, in view of Lemma \ref{decomposable}, we have $\overline{n}_{H}=\overline{\varepsilon(n_H)}=\overline{|H|}=0$ as $|H|\in pJ$.
\end{proof}

\begin{lemma}\label{rsk}
If $s\geq 2$ and $r-s\leq k \leq r-1$, then $\dim(V_{k})=1$.
\end{lemma}

\begin{proof}
By Lemma \ref{twocas}, $V_k$ is generated by $\overline{n_{H}x}$ with $H$ nontrivial and $\varepsilon(n_{H}x)\in p^s\Z$. As $s\geq 2$, the element $n_{H}x$ is decomposable by Lemma \ref{deco}. In view of Lemma \ref{decomposable}, $\overline{n_{H}x}=\overline{\varepsilon(n_{H}x)}$, hence $V_{k}$ is generated by $\overline{p^s}$.
\end{proof}

\begin{lemma}\label{s1podd}
If $s=1$ and $p$ is odd, then $\dim(V_{r-1})=1$.
\end{lemma}

\begin{proof}
We claim that $V_{r-1}$ is generated by $\overline{p}$.
By Lemma \ref{twocas}(2), $V_{r-1}$ is generated by $\overline{n}_{H}$ with $|H|\geq p$. If $|H|\geq p^2$, then by Lemma \ref{deco}, $\overline{n}_{H}$ is decomposable and in view of Lemma \ref{decomposable}, $\overline{n}_{H}=\overline{\varepsilon({n}_{H})}=0$.


Suppose $|H|=p$ and let $\sigma \in H$ be a generator. We have $n_{H}-p=(\sigma-1)m$, where $m=\sum_{i=0}^{p-2}(p-1-i)\sigma^i$, so $\varepsilon(m)=p(p-1)/2$. As $p$ is odd, $\varepsilon(m)\in p\Z$. Hence, $m\in pR+I$, therefore, $n_{H}-p\in pI+I^2$ and $\overline{n}_{H}=\overline{p}$ in $\overline{J}$.
\end{proof}

\begin{lemma}\label{s1peven}
If $s=1$ and $p=2$, then $V_{r-1}=\overline{J}$.
\end{lemma}

\begin{proof}
By Lemma \ref{twocas}(2), $V_{r-1}$ is generated by $\overline{n}_{H}$ with $|H|\geq 2$. Take nontrivial elements $\sigma\neq\tau$ in $G$. Then $\overline{2}=\overline{(1+\sigma\tau)}-\sigma\overline{(1+\tau)}+\overline{(1+\sigma)}\in V_{r-1}$. Also, for any $\sigma \in G$, $\overline{\sigma-1}=\overline{1+\sigma}-\overline{2}\in V_{r-1}$. The group $\overline{J}$ is generated by $\overline{2}$ and $\overline{\sigma-1}$ over all $\sigma\in G$.
\end{proof}

\begin{proposition}\label{edU}
We have
\[
\ed(U^\Phi)=\ed_{p}(U^\Phi)= \begin{cases}
(r-1)2^{r-1} & \text{if $p=2$ and $s=1$},\\
(r-1)p^r+p^{r-s}& \text{otherwise}.
\end{cases}
\]
\end{proposition}

\begin{proof}

Note that $V_{r}=\overline{J}$, $\rank(\overline J)=\rank(V_{r})=r+1$ and $\dim(U^\Phi)=p^r$.

\smallskip

{\it Case} $1$: $p$ is odd or $p=2$ and $s\geq 2$. By Lemmas \ref{krs}, \ref{rsk} and \ref{s1podd}, we have
\[
\rank V_{k}= \begin{cases}
r+1 & \text{if $k=r$},\\
1 & \text{if $r-s\leq k<r$},\\
0 & \text{if $0\leq k < r-s$}.
\end{cases}
\]
Since the decomposition group $G$ of $U^\Phi$ is a $p$-group, by Theorem \ref{edtori},
\[
\ed(U^\Phi)=\ed_{p}(U^\Phi)=rp^{r}+p^{r-s}-\dim(U^{\Phi})=rp^{r}+p^{r-s}-p^r=(r-1)p^{r}+p^{r-s}.
\]

\smallskip

{\it Case} $2$: $p=2$ and $s=1$. By Lemmas \ref{krs} and \ref{s1peven}, we have
\[
\rank V_{k}= \begin{cases}
r+1 & \text{if $k=r-1$ or $k=r$},\\
0 & \text{if $0\leq k\leq r-2$}.
\end{cases}
\]
Again by Theorem \ref{edtori},
\[
\ed(U^\Phi)=\ed_{2}(U^\Phi)=(r+1)2^{r-1}-\dim(U^{\Phi})=(r-1)2^{r-1}.\qedhere
\]
\end{proof}

\begin{remark}\label{surmin}
One can construct a surjective minimal $p$-presentation $\nu:P'\to J$ as follows.

\smallskip

{\it Case} $1$: $p$ is odd or $p=2$ and $s\geq 2$. Let $H$ be a subgroup of $G$ of order $p^s$ and $P':=R^r\oplus \Z[G/H]$. We define $\nu$ by
\[
\nu(x_1,\dots, x_n, y)=\sum_{i=1}^{r}(\sigma_i-1)x_i+n_Hy.
\]
The image of $\nu$ contains $I$ and $n_H$. As $n_H\equiv p^s$ modulo $I$, we have $p^s\in\Im(\nu)$, hence $\nu$ is surjective. Note that $e_{ij}=(\sigma_i-1)e_j-(\sigma_j-1)e_i\in\Ker(\nu)$. As $\sigma_i e_{ij}\neq e_{ij}$ for every $j\neq i$, the group $G$ acts faithfully on $\Ker(\nu)$.

\smallskip

{\it Case} $2$: $p=2$ and $s=1$. Let $H_i$ be the subgroup of $G$ generated by $\sigma_i$ and $H=\langle \sigma_1\sigma_2\rangle$. Set $P'=\coprod_{i=1}^{r}\Z[G/H_i]\oplus\Z[G/H]$.  We define $\nu$ by
\[
\nu(x_1,\dots, x_n, y)=\sum_{i=1}^{r}(\sigma_i+1)x_i+(\sigma_1\sigma_2+1)y.
\]
The image of $\nu$ contains $\sigma_i+1$ and $2=(\sigma_1\sigma_2+1)-\sigma_1(\sigma_2+1)+(\sigma_1+1)$, hence $\nu$ is surjective. Note that $h_{ij}:=(\sigma_i+1)e_j-(\sigma_j+1)e_i\in\Ker(\nu)$. As $\sigma_k h_{ij}\neq h_{ij}$ for distinct $i, j$ and $k$, the group $G$ acts faithfully on $\Ker(\nu)$ if $r\geq 3$. In fact, $G$ acts trivially on $\Ker(\nu)$ if $r=2$.
\end{remark}


\begin{corollary}\label{edS}
We have
\[
\ed(S^\Phi)=\ed_{p}(S^\Phi)= \begin{cases}
(r-1)2^{r-1}-r & \text{if $p=2$ and $s=1$},\\
(r-1)p^r+p^{r-s}-r & \text{otherwise}.
\end{cases}
\]
\end{corollary}
\begin{proof}
By (\ref{reich}) and Proposition \ref{edU}, there is a minimal $p$-presentation $\nu: P\to J$ such that
\begin{equation}\label{forfo}
\rank(P)=\begin{cases}
(r+1)2^{r-1} & \text{if $p=2$ and $s=1$},\\
rp^{r}+p^{r-s} & \text{otherwise}.
\end{cases}
\end{equation}

The exact sequence
\[
0\to \Z^{r}\to Q\to J\to 0
\]
in the bottom row of (\ref{diagmod}) yields an exact sequence
\[
\Hom_{G}(P,Q)\to \Hom_{G}(P,J)\to \Ext^{1}_{G}(P,\Z^{r}).
\]
As $P$ and $\Z^{r}$ are permutation $G$-modules, $\Ext^{1}_{G}(P,\Z^{r})=0$, hence the homomorphism $\nu$ factors through a morphism $\nu':P\to Q$. 

Recall that we write $\overline X=X/(pX+IX)$ for a $G$-module $X$.
As $\overline{\Z^{r}}\simeq (\Z/p\Z)^{r}\to \overline{Q}$ is zero map, the natural homomorphism $\overline Q\to \overline J$ is an isomorphism, hence $\nu'$ is a minimal $p$-presentation of $Q$. Note that $G$ is the decomposition group of $S^\Phi$ and $\dim(S^{\Phi})=p^r+r$. By Theorem \ref{edtori}, $\ed(S^\Phi)=\ed_{p}(S^{\Phi})=\rank(P)-\dim(S^{\Phi})$, hence the result follows by (\ref{forfo}).
\end{proof}




\section{Degeneration}

In this section we relate the essential $p$-dimensions of $\cat{Alg_{p^r,p^s}}$ and of the torus $S^\Phi$ by means of the iterated  degeneration (Proposition \ref{f2f3}). The latter is a method of comparison of the essential $p$-dimension of an object (a central simple algebra in our case) over a complete discrete valued field and of its specialization over the residue field.




\subsection{A simple degeneration}
Let $F$ be a field, $p$ a prime integer different from $\ch(F)$ and
$\Phi\subset \Ch_{p}(F)$ a finite subgroup. For integers $k\geq
0$, $s\geq 1$ and a field extension $K/F$, let
\begin{equation}\label{definitionB}
\cS_{k,s}^\Phi(K)=\{\alpha\in \Br(K)\{p\}\quad | \quad\ind
(\alpha_{K(\Phi)})\leq p^k , \exp(\alpha)\leq p^{s} \}.
\end{equation}
We say that two elements $\alpha$ and $\alpha'$ in $\cS_{k,s}^\Phi(K)$ are \emph{equivalent}
if $\alpha-\alpha'\in\Br\(K(\Phi)/K\)_{\dec}$. Write $\widetilde\cS_{k,s}^\Phi(K)$ for the
set of equivalence classes in $\cS_{k,s}^\Phi(K)$. To simplify notation, we shall write $\alpha$ for the equivalence class of an element $\alpha\in\cS_{k,s}^\Phi(K)$ in $\widetilde\cS_{k,s}^\Phi(K)$. We view $\cS_{k,s}^\Phi$ and $\widetilde\cS_{k,s}^\Phi$ as functors from $\cat{Fields}/F$ to $\cat{Sets}$.

In particular, if $k=0$, then $\cS_{0,s}^\Phi(K)$ and $\widetilde\cS_{0,s}^\Phi(K)$ are bijective to $\Br_{p^s}(K(\Phi)/K)$ and $\Br_{p^s}(K(\Phi)/K)_{\ind}$, respectively. Hence, by (\ref{hu}) and Lemma \ref{compute2}, \begin{equation}\label{BUS}
\cS_{0,s}^\Phi\simeq U^\Phi\text{-}\tors \quad\text{ and }\quad \widetilde\cS_{0,s}^\Phi\simeq S^\Phi\text{-}\tors.
\end{equation}
Moreover, if $\Phi=0$, then \begin{equation}\label{AlgB}
\cS_{k,s}^\Phi= \widetilde\cS_{k,s}^\Phi \simeq \cat{Alg_{p^k,p^s}}.
\end{equation}

Let $\Phi'\subset\Phi$ be a subgroup of index $p$ and
$\eta\in\Phi\setminus \Phi'$, hence $\Phi=\langle\Phi',\eta\rangle$.
Let $E/F$ be a field extension such that $\eta_E\notin\Phi'_E$ in
$\Ch(E)$. Choose an element $a\in\cS_{k,s}^\Phi(E)$, i.e.,
$\alpha\in\Br(E)\{p\}$ such that $\ind(\alpha_{E(\Phi)})\leq p^k$ and $\exp(\alpha)\leq p^s$.

Let $E'$ be a field extension of $F$ that is complete with respect
to a discrete valuation $v'$ over $F$ with residue field $E$ and set
\begin{equation}\label{bbb}
\alpha':=\widehat \alpha+\(\widehat\eta_E\cup (x)\)\in \Br(E'),
\end{equation}
for some $x\in {E'}^\times$ such that $v'(x)$ is prime to
$p$. As $\eta_{E(\Phi')}\neq 0$, it follows from (\ref{indexx}) that
\[
\ind(\alpha'_{E'(\Phi')})=p\cdot\ind(\alpha_{E(\Phi)})\leq p^{k+1}\quad\text{and}\quad\exp(\alpha')=\lcm\(\exp(\alpha),p\)\leq p^s,
\]
hence $\alpha'\in \cS_{k+1,s}^{\Phi'}(E')$.

In the case the condition $\exp(\alpha)\leq p^{s}$ in (\ref{definitionB}) is dropped, the following proposition was proved in \cite[Prop. 5.2]{Merkurjev09b}:

\begin{proposition}\label{f2f3}
Suppose that for any finite field extension $N/E$ of degree prime to
$p$ and any character $\rho\in\Ch(N)$ of order $p^2$ such that
$p\rho\in \Phi_N\setminus\Phi'_N$, we have $\ind
\(\alpha_{N(\Phi',\rho)}\)\geq p^{k}$. Then
\[
\ed_p^{\widetilde\cS_{k+1,s}^{\Phi'}}(\alpha')\geq \ed_p^{\widetilde\cS_{k,s}^{\Phi}}(\alpha)+1.
\]
\end{proposition}

\begin{proof}
The proof of \cite[Prop. 5.2]{Merkurjev09b} still works with the following modification.


Let $M/E'$ be a finite field extension of degree prime to $p$,
$M_0\subset M$ a subfield over $F$ and $\alpha'_0\in
\cS_{k+1,s}^{\Phi'}(M_0)$ such that $(\alpha'_0)_M=\alpha'_M$ in
$\widetilde\cS_{k+1,s}^{\Phi'}$ and $\td_F(M_0)=\ed_p^{\widetilde\cS_{k+1,s}^{\Phi'}}(\alpha')$. We extend the discrete valuation $v'$ on $E'$ to a
(unique) discrete valuation $v$ on $M$ and let $N$ be its residue field. Let $n_0$ be the residue field of the restriction of $v$ on $M_0$.
It was shown in the proof of \cite[Prop. 5.2]{Merkurjev09b} that there exist $\alpha_0\in\Br(N_0)\{p\}$ with $\ind(\alpha_{0})_{N_{0}(\Phi)}\leq p^{k}$, a prime element $\pi_0$ in $M_0$, and $\eta_0\in\Ch_p(N_0)$ such that
\begin{equation}\label{tempp}
(\alpha'_0)_{\widehat M_0}=\widehat \alpha_0+\(\widehat\eta_0\cup(\pi_0)\) \text{ in } \Br(\widehat M_0)
\end{equation}
and
\begin{equation}\label{temppp}
\alpha_N-(\alpha_0)_N\in\Br\(N(\Phi)/N\)_{\dec}.
\end{equation}
By (\ref{tempp}), we have
\[
\exp(\alpha_0)=\exp(\widehat \alpha_0)\leq\lcm\(\exp(\alpha'_0)_{\widehat M_0},p\)\leq\lcm\(\exp(\alpha'_0),p\)\leq p^s,
\]
hence $\alpha_{0}\in \cS_{k,s}^{\Phi}(N_{0})$. Therefore, the class of $\alpha_N$ in $\widetilde\cS_{k,s}^{\Phi}(N)$ is defined over $N_{0}$ by (\ref{temppp}). It follows that
\[
\ed_p^{\widetilde\cS_{k+1,s}^{\Phi'}}(\alpha')=\td_F(M_0)\geq\td_F(N_0)+1\geq \ed_p^{\widetilde\cS_{k,s}^{\Phi}}(\alpha)+1.\qedhere
\]
\end{proof}



\subsection{A technical lemma}\label{degen}

In this subsection we prove Lemma \ref{cond} that will allow us to apply Proposition \ref{f2f3}.

Until the end of this subsection we assume that the base field $F$ contains a
primitive $p^2$-th root of unity.

Let $\chi_1,\chi_2,\dots,\chi_r$ with $r\geq 2$ be linearly independent characters
in $\Ch_{p}(F)$ and $\Phi=\langle\chi_1,\chi_2,\dots,\chi_r\rangle$.
Let $E/F$ be a field extension such that $\rank(\Phi_E)=r$ and let
$\alpha\in\Br(E)\{p\}$ be an element that is split by $E(\Phi)$ and $\exp(\alpha)\leq p^s$.

Let $E_0=E$, $E_1,\dots, E_r$ be field extensions of $F$ such that
for any $k=1,2,\dots, r$, the field $E_k$ is complete with respect
to a discrete valuation $v_k$ over $F$ and $E_{k-1}$ is its residue
field. For any $k=1,2,\dots, r$, choose elements $x_k\in E_k^\times$
such that $v_k(x_k)$ is prime to $p$ and define the elements
$\alpha_k\in\Br(E_k)\{p\}$ inductively by $\alpha_0:=\alpha$ and
\[
\alpha_k:=\widehat{\alpha_{k-1}}+\(\widehat{(\chi_k)}_{E_{k-1}}\cup(x_k)\).
\]

Let $\Phi_k$ be the subgroup of $\Phi$ generated by
$\chi_{k+1},\dots,\chi_r$. Thus, $\Phi_0=\Phi$, $\Phi_r=0$ and
$\rank(\Phi_k)=r-k$. Note that the character
$(\chi_k)_{E_{k-1}(\Phi_k)}$ is not trivial. It follows from (\ref{indexx}) that
\[
\ind (\alpha_k)_{E_k(\Phi_k)}=p\cdot\ind (\alpha_{k-1})_{E_{k-1}(\Phi_{k-1})}
\]
for any $k=1,\dots, r$. As $\ind \alpha_{E(\Phi)}=1$, we have $\ind
(\alpha_k)_{E_k(\Phi_k)}=p^k$ for all $k=0,1,\dots,r$. Moreover, as $\exp(\alpha)\leq p^s$, we have $\exp(\alpha_{k})=\lcm(\exp(\alpha_{k-1}),p)\leq p^s$. Therefore,
$\alpha_k\in\cS_{k,s}^{\Phi_k}(E_k)$.

The followings lemma assures that under a certain restriction on the
element $\alpha$, the conditions of Proposition \ref{f2f3} are satisfied
for the fields $E_k$, the groups of characters $\Phi_k$ and the
elements $\alpha_k$. This lemma is similar to \cite[Lemma 5.4]{Merkurjev09b}.

\begin{lemma}\label{cond}
Suppose that for any subgroup $\Psi\subset \Phi$ with $[\Phi:\Psi]=p^2$ and any field extension $L/E(\Psi)$ of degree prime to $p$, the element $\alpha_{L}$ is not $p^2$-cyclic . Then for every $k=0,1,\dots, r-1$, and any finite field extension $N/E_k$ of degree prime to $p$ and any character $\rho\in\Ch(N)$ of order $p^2$ such
that $p\rho\in\(\Phi_k)_N\setminus (\Phi_{k+1}\)_N$, we have
\begin{equation}\label{iiii}
\ind (\alpha_k)_{N(\Phi_{k+1},\rho)}\geq p^{k}.
\end{equation}
\end{lemma}

\begin{proof}
Let $k,N$ and $\rho$ satisfy the conditions of the lemma.
We construct a new sequence of
fields $\tilde E_0, \tilde E_1,\dots, \tilde E_r$ such that each
$\tilde E_i$ is a finite extension of $E_i$ of degree prime to $p$
as follows. We set $\tilde E_{k}=N$. The fields $\tilde E_j$ with $j<k$ are
constructed by descending induction on $j$. If we have constructed
$\tilde E_j$ as a finite extension of $E_j$ of degree prime to $p$,
then we extend the valuation $v_j$ to $\tilde E_j$ and let $\tilde
E_{j-1}$ to be its residue field. The fields $\tilde E_m$ with $m>k$ are
constructed by ascending induction on $m$. If we have constructed
$\tilde E_m$ as a finite extension of $E_m$ of degree prime to $p$, then let $\tilde E_{m+1}$ be an
extension of $E_{m+1}$ of degree $[\tilde E_{m}:E_{m}]$ with residue
field $\tilde E_{m}$. Replacing $E_i$ by $\tilde E_i$
and $\alpha_i$ by $(\alpha_i)_{\tilde E_i}$, we may assume that $N=E_{k}$.

We proceed by induction on $r$. The case $r=1$ is obvious.

\smallskip

$(r-1)\Rightarrow r$: First suppose that $k< r-1$. Consider the fields $F'=F(\chi_{r})$, $E'=E(\chi_{r})$, $E'_i=E_i(\chi_{r})$, the sequence of characters $\chi'_i=(\chi_i)_{F'}$, and the sequence of elements
$\alpha'_i:=(\alpha_i)_{E_i'}\in\Br(E'_i)$ for $i=0,1,\dots, r-1$.
Let $\Phi'=\langle \chi'_1,\chi'_2,\dots,\chi'_{r-1}\rangle\subset \Ch(F')$, let $\Phi'_i$ be the subgroup of $\Phi'$ generated by
$\chi'_{i+1},\dots,\chi'_{r-1}$ and $\rho'=\rho_{E'_k}$.

We check the conditions of the lemma for the new datum. Let $\Psi'$ be a subgroup of $\Phi'$ of index $p^2$. Then
the pre-image $\Psi$ of $\Psi'$ under the map $\Ch(F)\to\Ch(F')$ is a subgroup of $\Phi$ of index $p^2$ and $E'(\Psi')=E(\Psi)$. Let $L'/E'(\Psi')$ be a field extension of degree prime to $p$. By assumption, the element $\alpha'_{L'}=\alpha_{L'}$ is not $p^2$-cyclic. We also have $p\rho'=p\rho_{E'_k}\in\(\Phi_k)_{E'_k}=\(\Phi'_k)_{E'_k}$. Suppose that $p\rho'\in\(\Phi'_{k+1})_{E'_k}$, i.e., $p\rho_{E'_k}=p\rho'=\eta_{E'_k}$ for some $\eta\in\(\Phi_{k+1})_{E_k}$. It follows that $p\rho-\eta\in\Ker\(\Ch(E_k)\to\Ch(E'_k)\)=\langle(\chi_{r})_{E_k}\rangle$ and therefore, $p\rho\in (\Phi_{k+1}\)_{E_k}$, a contradiction, hence $p\rho'\in\(\Phi'_k)_{E'_k}\setminus (\Phi'_{k+1}\)_{E'_k}$.

By the induction hypothesis, the inequality (\ref{iiii}) holds for $\alpha'_k$, i.e,
\[
\ind (\alpha'_k)_{E'_k(\Phi'_{k+1},\rho')}\geq p^{k}.
\]
As
\[
(\alpha'_k)_{E'_{k}(\Phi'_{k+1},\rho')}=(\alpha_k)_{E_{k}(\Phi_{k+1},\rho)},
\]
the inequality (\ref{iiii}) holds for $\alpha_k$. Therefore, it remains to show the inequality (\ref{iiii}) in the case $k=r-1$. Note that is this case
$p\rho$ is a nonzero multiple of $(\chi_r)_{E_{r-1}}$ and $\Phi_{k+1}=\Phi_r=0$.


\smallskip

{\it Case} $1$: The character $\rho$ is unramified with respect to
$v_{r-1}$, i.e., $\rho=\widehat\mu$ for a character
$\mu\in\Ch(E_{r-2})$ of order $p^2$. Note that
$p\mu$ is a nonzero multiple of $(\chi_r)_{E_{r-2}}$.



By (\ref{indexx}),
\begin{equation}\label{swsw}
\ind(\alpha_{r-2})_{E_{r-2}(\chi_{r-1},\mu)}=\ind(\alpha_{r-1})_{E_{r-1}(\rho)}/p.
\end{equation}




Consider the fields $F'=F(\chi_{r-1})$, $E'=E(\chi_{r-1})$,
$E'_i=E_i(\chi_{r-1})$, the new sequence of characters
$\chi'_1=(\chi_1)_{F'},\dots,\chi'_{r-2}=(\chi_{r-2})_{F'},\chi'_{r-1}=(\chi_r)_{F'}$, the group of characters $\Phi'=\langle\chi'_1,\chi'_2,\dots,\chi'_{r-1}\rangle$ and the elements $\alpha'_i\in\Br(E'_i)$
for $i=0,1,\dots, r-1$ defined by $\alpha'_i=(\alpha_i)_{E_i'}$ for $i\leq
r-2$ and $\alpha'_{r-1}=\widehat \alpha_{r-2}+\(\widehat\chi_r\cup(x_{r-1})\)$
over $E'_{r-1}$, and the character $\mu$. The new datum satisfy the conditions of the lemma. By the induction hypothesis, the inequality (\ref{iiii}) holds for $\alpha'_{r-2}$, i.e,
\[
\ind (\alpha'_{r-2})_{E'_{r-2}(\mu)}\geq p^{r-2}.
\] As
\[
(\alpha'_{r-2})_{E_{r-2}'(\mu)}=(\alpha_{r-2})_{E_{r-2}(\chi_{r-1},\mu)},
\]
the inequality (\ref{iiii}) holds for $\alpha_{r-1}$ in view of the equality (\ref{swsw}).




\smallskip

{\it Case} $2$: The character $\rho$ is ramified.
Assume that
inequality (\ref{iiii}) does not hold for $\alpha_{r-1}$, i.e., we have
\[
\ind (\alpha_{r-1})_{E_{r-1}(\rho)}\leq p^{r-2}.
\]
By \cite[Lemma 2.3(2)]{Merkurjev09b}, there exists a unit $u\in E_{r-1}$ such
that $E_{r-2}(\chi_r)=E_{r-2}(\bar u^{1/p})$ and
\[
\ind\(\alpha_{r-2}-(\chi_{r-1}\cup(\bar
u^{1/p}))\)_{E_{r-2}(\chi_r)}=\ind (\alpha_{r-1})_{E_{r-1}(\rho)}\leq
p^{r-2}.
\]
By descending induction on $j=0,1,\dots, r-2$ we show that there
exist an element $u_j$ in $E_{j}^\times$ and a subgroup $\Psi_j\subset\Phi$
of rank $r-j-2$ such that $\langle\chi_1,\dots,\chi_j,\chi_{r-1},\chi_{r}\rangle\cap \Psi_j=0$,\
$E_j(\chi_r)=E_j(u_j^{1/p})$ and
\begin{equation}\label{laastineq}
\ind\(\alpha_j-(\chi_{r-1}\cup(u_j^{1/p}))\)_{E_j(\Theta_j)}\leq
p^j,
\end{equation}
where $\Theta_j:=\langle\Psi_{j},\chi_r\rangle$.
We set $\Psi_{r-2}=0$ and $u_{r-2}=\bar u$.

$j\Rightarrow (j-1)$: The field $E_j(u_j^{1/p})=E_j(\chi_r)$ is
unramified over $E_j$, hence $v_j(u_j)$ is divisible by $p$.
Modifying $u_j$ by a $p^2$-th power, we may assume that $u_j=vx_j^{mp}$ for a unit $v\in E_j$, $x_j\in E_j^\times$ and an integer $m$.
Then
\[
\(\alpha_j-(\chi_{r-1}\cup(u_j^{1/p}))\)_{E_j(\Theta_j)}=\widehat \beta
+\(\widehat\eta\cup(x_j)\)_{E_j(\Theta_j)},
\]
where $\eta=\chi_j-m\chi_{r-1}$ and
$\beta=\(\alpha_{j-1}-(\chi_{r-1}\cup(u_{j-1}^{1/p}))\)_{E_{j-1}(\Theta_j)}$, where $u_{j-1}=\bar v$. As $\eta$ is not contained
in $\Theta_j$, the character $\eta_{E_{j-1}(\Theta_j)}$ is not
trivial. Set $\Psi_{j-1}=\langle\Psi_j,\eta\rangle$. It follows
from (\ref{indexx}) and the induction hypothesis that
\[
\ind(\beta_{E_{j-1}(\Theta_{j-1})})=\ind \(\alpha_j-(\chi_{r-1}\cup(u_j^{1/p}))\)_{E_j(\Theta_j)}/p\leq p^{j-1}.
\]


Applying the inequality (\ref{laastineq}) in the case $j=0$, we have
\[
\alpha_{E(\Theta_0)}=\(\chi_{r-1}\cup(w^{1/p})\)_{E(\Theta_0)}
\]
for an element $w\in E^\times$ such that $E(w^{1/p})=E(\chi_r)$. Hence
\[
\alpha_{E(\Psi_{0})(w^{1/p^{2}})}=(\alpha_{E(\Theta_0)})_{E(\Theta_0)(w^{1/p^{2}})}=0 \text{ in }\Br(E(\Psi_{0})(w^{1/p^{2}})).
\]
Since $\alpha_{E(\Psi_{0})}$ is split by a cyclic extension $E(\Psi_{0})(w^{1/p^{2}})/E(\Psi_{0})$ of degree $p^{2}$, $\alpha_{E(\Psi_{0})}$ is $p^2$-cyclic. As $[\Phi:\Psi_{0}]=p^{2}$, this contradicts the assumption. Hence, the inequality (\ref{iiii}) holds for $\alpha_{r-1}$.
\end{proof}

\section{Non-cyclicity of the generic element}\label{cyclicgeneric}

The aim of this section is the technical Lemma \ref{corelemma} that
will allow us to apply later Lemma \ref{cond} and Proposition
\ref{f2f3}.

In this section we assume that the base field $F$ contains a
primitive $p^3$-th root of unity. The choice of a primitive $p^2$-th root of unity $\xi$ allows us to define the symbol $(a,b)_{p^2}$ as in Section \ref{chandbr}. As $-1$ is a $p^2$-th power in $F^\times$, we have $(a,-1)_{p^2}=0$, hence
$(a,a)_{p^2}=0$ for all $a\in F^\times$. We shall write $(a,b)_{p}$ for $p(a,b)_{p^2}=(a^p,b)_{p^2}$.



\begin{lemma}\label{simplecyclic}
Let $E$ be a field extension of $F$ that is complete with respect
to a discrete valuation $v$ with residue field $K$ and $\alpha\in \Br(K)$. Set $\beta=\widehat \alpha+(a,x)_{p}$ for a unit $a\in {E}$ and $x\in E^{\times}$ such that $\bar a\notin K^{\times p}$ and $v(x)$ is prime to
$p$. If $\beta$ is $p^2$-cyclic, then $\alpha=(\bar a,z)_{p^2}$ in $\Br(K)$ for some $z\in K^\times$.
\end{lemma}

\begin{proof}
Suppose that $\beta=(u\pi^{i},w\pi^{j})_{p^2}$ and write $x=t\pi^{k}$ for a prime element $\pi$, integers $i,j,k=v(x)$ and units $u,w,t$ in $E$. Then we have
\[
\widehat{\alpha}+(a^{p},w\pi^{k})_{p^2}=\beta=(u\pi^{i},v\pi^{j})_{p^2}=(u,v)_{p^2}+(u^j/v^{i},\pi)_{p^2}.
\]
Applying the residue map $\partial_{v}$, we get $\bar a^{pk}=\bar u^j/\bar v^i$ in $K^{\times}/K^{\times p^{2}}$ and
\[
\alpha=(\bar u,\bar v)_{p^2}-(\bar a,\bar w^{p})_{p^2}.
\]

Suppose that $i/j$ is a $p$-integer (the other case is similar). As $k$ is not divisible by $p$ and $\bar a$ is not a $p$-th power in $K^\times$, $j$ is not divisible by $p^2$. It follows that
$\bar  u\in \langle \bar a,\bar  v \rangle$ in $K^{\times}/K^{\times p^{2}}$ and then $\bar  u\in\bar a^r \bar v^sK^{\times p^{2}}$  for some $r$ and $s$. Hence
$\alpha=(\bar a,\bar v^r/\bar w^{p})_{p^2}$.
\end{proof}

\begin{corollary}\label{simplecorop}
Let $x$,$y$ be independent variables over $F$ and $a,b\in F^{\times}$. If $(a,b)_{p}\neq 0$ in $\Br(F)$, then for any field extension $M/F(x,y)$ of degree prime to $p$, the element $(a,x)_{p}+(b,y)_{p}$ in $\Br(M)$ is not $p^2$-cyclic.
\end{corollary}

\begin{proof}
Let $M/F(x,y)$ be a field extension of degree prime to $p$ and $\beta=(a,x)_{p}+(b,y)_{p}$ over $M$.
As the degree of $M/F(x,y)$ is prime to $p$, by \cite[Lemma 6.1]{Merkurjev09}, there exists a field extension $E$ of the fields $F(\!(y)\!)(\!(x)\!)$ and $M$ over $F$ such that the degree of $E/F(\!(y)\!)(\!(x)\!)$ is finite and prime to $p$. The discrete valuation $v_x$ on the complete field $F(\!(y)\!)(\!(x)\!)$ extends uniquely to a discrete valuation $v$ of $E$. The ramification index of $E/F(\!(y)\!)(\!(x)\!)$ is prime to $p$, hence $v(x)$ is prime to $p$.
The residue field $K$ of $v$ is an extension of $F(\!(y)\!)$ of degree prime to $p$.

Let $v'$ be the valuation on $K$ extending the discrete valuation $v_y$ on $F(\!(y)\!)$. The ramification index $e'$ of $K/F(\!(y)\!)$ is prime to $p$. The residue field $N$ of $v'$ is a finite extension of $F$ of degree prime to $p$.


Let $\alpha=(b,y)_{p}$ over $K$, so $\beta_E=\widehat \alpha+(a,x)_{p}$. Suppose that $\beta$ is $p^2$-cyclic over $M$. Then $\beta_{E}$ is also $p^2$-cyclic. By Lemma \ref{simplecyclic}, applied to $\beta_{E}$ over $E$, we have $\alpha=(a,z)_{p^2}$ for some $z\in K^{\times}$, hence $(b^{p},y)_{p^2}=(a,z)_{p^{2}}$. Taking the cup product with $(a)_{p^2}\in K^\times/K^{\times p^2}$, we get
\[
(a)_{p^2}\cup (b^{p},y)_{p^2}=(a)_{p^2}\cup (a,z)_{p^{2}}=(a,a)_{p^2}\cup (z)_{p^2}=0.
\]
Applying the residue map $\partial_{v'}$, we find that $e'(a,b)_p=e'(a,b^p)_{p^{2}}=0$ over $N$, hence $(a,b)_{p}=0$ in $\Br(N)$. Taking the corestriction map $\Br(N)\to \Br(F)$, we see that  $(a,b)_{p}=0$ in $\Br(F)$, a contradiction.
\end{proof}

\begin{lemma}\label{betachosen}
For any integer $r\geq 2$, there exist a field extension $F'/F$ and a subgroup $\Phi\subset\Ch_p(F')$ of rank $r$ such that for any subgroup $\Psi\subset\Phi$ of index $p^2$, there is an element $\beta\in\Br_p\(F'(\Phi)/F'\)$ with the property that any field extension $M/F'(\Psi)$ of degree prime to $p$, the element $\beta_M$ is not $p^2$-cyclic.
\end{lemma}

\begin{proof}
Let $a_{1},a_{2},\ldots,a_{r},x,y$ be independent variables over $F$ and set $F':=F(a_{1},a_{2},\ldots,a_{r},x,y)$.
For every $i=1,\dots, r$, let $\chi_i\in\Ch_p(F')$ be a character such that $F'(\chi_{i})=F'(a_{i}^{1/p})$
and set $\Phi:=\langle \chi_{1},\chi_{2},\ldots,\chi_{r}\rangle$. Let $\Psi$ be a subgroup of $\Phi$ of index $p^2$. Choose a basis $\eta_{1},\eta_{2},\ldots,\eta_r$ for $\Phi$ such that $\Psi=\langle \eta_{1},\eta_{2},\ldots,\eta_{r-2}\rangle$ and the elements $b_{1},b_{2},\ldots,b_{r}$ in $F'$ such that $F(\eta_{i})=F(b_{i}^{1/p})$ for all $i=1,\dots, r$ and $F(b_{1},b_{2},\ldots,b_{r})=F(a_{1},a_{2},\ldots,a_{r})$.
Clearly, $b_{1},b_{2},\ldots,b_{r}$ are algebraically independent over $F$ and $F'(\Psi)=L(x,y)$, where $L:= F(b_{1}^{1/p},\dots, b_{r-2}^{1/p},b_{r-1},b_{r})$ with the generators algebraically independent over $F$.


Let $\beta=(b_{r-1},x)_{p}+(b_{r},y)_{p}$ in $\Br_p\(F'(\Phi)/F'\)$ and $M/F'(\Psi)$ a field extension of degree prime to $p$. As $\partial_{v}\((b_{r-1},b_{r})_p\)=\bar b_{r-1}$, where $v$ is the discrete valuation on $L$ associated with $b_r$, is nontrivial, we have $(b_{r-1},b_{r})_{p}\neq 0$ in $\Br(L)$. The result follows from Corollary \ref{simplecorop}.
\end{proof}

Let $F'/F$ be the field extension and $\Phi\subset\Ch_p(F')$ the subgroup of rank $r$ as in Lemma \ref{betachosen}. Consider the algebraic tori $P^\Phi$, $S^\Phi$, $T^\Phi$, ${U}^\Phi$ and $V^\Phi$ over $F'$ defined in Section \ref{manytori}. The morphism $\gamma:P^\Phi\to V^\Phi$ in the diagram (\ref{diag}) is a ${U}^\Phi$-torsor. Denote by $\delta$ the image of the class of $\gamma$ under the composition
\[
H_{\acute{e}t}^{1}(V^\Phi,U^\Phi)\to H_{\acute{e}t}^{1}(V^\Phi,U'^\Phi)\to H_{\acute{e}t}^{2}(V^\Phi,\gm),
\]
induced by the diagram (\ref{escho}). We write $\delta_{gen}$ for the image of $\delta$ under the homomorphism
\[
H_{\acute{e}t}^{2}(V^\Phi,\gm)\to H^{2}\(F(V^\Phi),\gm\)=\Br\(F'(V^\Phi)\)
\]
induced by the generic point morphism $\Spec\(F'(V^\Phi)\)\to V^\Phi$. It follows from (\ref{eschoescho}) that $\delta_{gen}\in \Br_{p^s}\(F'(V^\Phi)\)$.


\begin{lemma}\label{corelemma}
Let $K=F'(V^\Phi)$ and $\Psi\subset \Phi$ a subgroup with $[\Phi:\Psi]=p^2$. Then for any field extension $M/K(\Psi)$ of degree prime to $p$, the element $(\delta_{gen})_{M}$ is not $p^{2}$-cyclic.
\end{lemma}

\begin{proof}
Suppose that there exist a subgroup $\Psi\subset \Phi$ with $[\Phi:\Psi]=p^2$ and a field $M/K(\Psi)$ of degree prime to $p$ such that $(\delta_{gen})_{M}=\chi \cup (a)$ for some $\chi\in H^2(M,\Z)=\Ch(M)$ with $p^2\chi=0$ and $a\in H^0(M,\gm)=M^\times$. Choose an integral scheme $X$ over $F'$ such that $F'(X)=M$ together with a dominant $F'$-morphism
\[
f:X\to V^{\Phi}(\Psi):=(V^{\Phi})_{F'(\Psi)}
\]
of degree prime to $p$ that induces the embedding of the function field $K(\Psi)$ into $M$. Let $h:X\to V^{\Phi}$ be the composition of $f$ with the natural morphism $g:V^{\Phi}(\Psi)\to V^{\Phi}$. Replacing $X$ by a nonempty open set, we may assume that $h^{\ast}(\delta)=\chi_{0}\cup (a_{0})$ for some $\chi_{0}\in H^2_{\acute{e}t}(X,\Z)$ with $p^2\chi_0=0$ and $a_{0}\in H^0_{\acute{e}t}(X,\gm)$.


By \cite[Lemma 6.2]{Merkurjev09}, there is a nonempty open set $W'\subset V^\Phi(\Psi)$ such that for every $x'\in W'$ there exists a point $x\in X$ with $f(x)=x'$ and the degree $[F'(x):F'(x')]$ prime to $p$. Let $Z=V^\Phi(\Psi)\setminus W'$. As $g$ is finite, $g(Z)\neq V^\Phi$, hence the open set $W:=V^\Phi\setminus g(Z)$ is not empty. We have
$g^{-1}(W)\subset W'$.

Consider the element $\beta\in\Br_p\(F'(\Phi)/F'\)$ constructed in Lemma \ref{betachosen}. Let $\gamma'\in H^{1}(F',U^{\Phi})$ be the corresponding class of $U^{\Phi}$-torsors over $F'$ under the isomorphism $H^{1}(F',U^{\Phi})\simeq \Br_{p^{s}}\(F'(\Phi)/F'\)$ by (\ref{hu}). As $\gamma$ is a generic $U^\Phi$-torsor, there exists an $F'$-morphism $v:\Spec F'\to V^{\Phi}$ such that $v^{\ast}(\gamma)=\gamma'$ and $\Im(v)\subset W$ (see Section \ref{gentor}). From the commutativity of the diagram
\[
\begin{CD}
H^1_{\acute{e}t}(V^\Phi,U^\Phi)  @>v^*>>  H^1(F',U^\Phi)        \\
@VVV    @VVV      \\
H^2_{\acute{e}t}(V^\Phi,\gm)  @>v^*>>  H^2(F',\gm)        \\
\end{CD}
\]
we find that $v^{\ast}(\delta)=\beta$.


Let $v':\Spec F'(\Psi)\to V^{\Phi}(\Psi)$ be the morphism $v_{F'(\Psi)}$.
Note that $\Im(v')\subset g^{-1}(W)\subset W'$.

By the definition of $W'$, there is a point $x\in X$ such that the degree of the field extension $F'(x)$ over the residue field of (the only) point in $\Im(v')$ is prime to $p$. By \cite[Lemma 6.1]{Merkurjev09}, there exist a field extension $M/F'(\Psi)$ of degree prime to $p$ and a morphism $w:\Spec(M)\to X$ such that the diagram
\[
\begin{CD}
\Spec(M)  @>>>  \Spec\(F'(\Psi)\)   @>>>  \Spec(F')     \\
@V{w}VV    @V{v'}VV  @VvVV    \\
X  @>f>>  V^{\Phi}(\Psi)  @>g>>  V^{\Phi}      \\
\end{CD}
\]
is commutative. It follows that
\[
\beta_M=v^*(\delta)_M=w^*h^*(\delta)=w^*\(\chi_{0}\cup (a_{0})\)=w^*(\chi_{0})\cup w^*(a_{0}),
\]
i.e., $\beta_M$ is $p^2$-cyclic. This contradicts Lemma \ref{betachosen}.
\end{proof}




\section{A lower bound for $\ed_p(\cat{Alg}_{p^{r},p^{s}})$}\label{mainpf}

Let $n\geq 1$ be an integer, $m$ a divisor of $n$ and $p$ a prime integer. Let $p^r$ (respectively, $p^s$) be the largest power of $p$ dividing $n$ (respectively, $m$). If $A\in \cat{Alg}_{n,m}(K)$ for some field extension $K/F$, then there is a finite field extension $E/K$ of degree prime to $p$ such that $\ind(A_E)$ is a $p$-power. Hence $\ind(A_E)$ divides $p^r$ and $\exp(A_E)$ divides $p^s$ as it divides $m$ and $\ind(A_E)$, i.e., $A_E\in \cat{Alg}_{p^{r},p^{s}}(E)$. It follows that the embedding functor $\cat{Alg}_{p^{r},p^{s}}\to \cat{Alg}_{n,m}$ is $p$-surjective and hence $\ed_p(\cat{Alg}_{n,m})\leq\ed_p(\cat{Alg}_{p^{r},p^{s}})$ by \cite[Sec. 1.3]{Merkurjev09}. Conversely, if $A\in \cat{Alg}_{n,m}(K)$, then the $p$-primary component $A_p$ of $A$ satisfies $A_p\in \cat{Alg}_{p^{r},p^{s}}(K)$, hence the morphism of functors $\cat{Alg}_{n,m}\to\cat{Alg}_{p^{r},p^{s}}$, taking $A$ to $A_p$ is surjective and therefore, $\ed_p(\cat{Alg}_{n,m})\geq\ed_p(\cat{Alg}_{p^{r},p^{s}})$. We proved that
\[
\ed_p(\cat{Alg}_{n,m})=\ed_p(\cat{Alg}_{p^{r},p^{s}}).
\]

\begin{theorem}\label{main}
Let $F$ be a field and $p$ a prime integer different from $\ch(F)$.
Then, for any integers $r$ and $s$ with $1\leq s\leq r$,
\[
\ed_{p}(\cat{Alg}_{p^{r},p^{s}})\geq \begin{cases}
(r-1)2^{r-1} & \text{if $p=2$ and $s=1$},\\
(r-1)p^r+p^{r-s} & \text{otherwise}.
\end{cases}
\]
\end{theorem}

\begin{proof}
By \cite[Prop.1.5]{Merkurjev09}, we can replace the base field by any field extension. Hence we may assume that $F$ contains a primitive $p^3$-th root of unity. Moreover, we can replace $F$ by the field $F'$ in Lemma \ref{betachosen}. Let $V^\Phi$ be the algebraic torus constructed in Section \ref{manytori}. Set $E=F(V^\Phi)$ and let $\alpha:=\delta_{gen}\in \Br_{p^{s}}(E(\Phi)/E)$ be the element defined in Section \ref{cyclicgeneric}. Let $E_k$ be the fields and $\alpha_k\in\cS_{k,s}^{\Phi_k}(E_k)$ the elements constructed in Section \ref{degen}, so that $E_0=E$ and $\alpha_0=\alpha$. By Lemma \ref{corelemma}, $\alpha_{M}$ is not $p^2$-cyclic for any subgroup $\Psi\subset \Phi$ with $[\Phi:\Psi]=p^2$ and any field extension $M/E(\Psi)$ of degree prime to $p$, hence $\alpha$ satisfies the condition of Lemma \ref{cond}. It follows that we can apply Proposition \ref{f2f3}. By the iterated application of this proposition, we have
\begin{multline}\label{mul}
\ed_p^{\cat{Alg_{p^r,p^s}}}(\alpha_r)=\ed_p^{\widetilde\cS_{r,s}^{\Phi_{r}}}(\alpha_r)\geq
\ed_p^{\widetilde\cS_{r-1,s}^{\Phi_{r-1}}}(\alpha_{r-1})+1\geq\dots\\\geq
\ed_p^{\widetilde\cS_{1,s}^{\Phi_{1}}}(\alpha_1)+(r-1)\geq
\ed_p^{\widetilde\cS_{0,s}^{\Phi_{0}}}(\alpha_0)+r=\ed_p^{\widetilde\cS_{0,s}^{\Phi}}(\alpha)+r.
\end{multline}

Consider the commutative diagram with exact rows:
\[
\begin{CD}
1  @>>>  U^\Phi  @>>>  P^\Phi   @>\gamma>>  V^\Phi   @>>>  1    \\
@.  @VVV    @VVV  @|     \\
1  @>>>  S^\Phi  @>>>  P^\Phi\times \gm^r   @>\gamma'>>  V^\Phi  @>>>  1,  \\
\end{CD}
\]
where $P^\Phi\to P^\Phi\times \gm^r$ takes $x$ to $(x,1)$ and $S^\Phi\hookrightarrow P^\Phi\times \gm^r$
is the product of $S^\Phi\hookrightarrow P^\Phi$ and $S^\Phi\twoheadrightarrow \gm^r$.

The element $\alpha$ considered in $\cS_{0,s}^\Phi(E)$ corresponds to
the generic fiber of the $U^\Phi$-torsor $\gamma$ under the
bijection $\cS_{0,s}^\Phi(E)\simeq U^\Phi\text{-}\tors(E)$ in (\ref{BUS}). Hence,
by the diagram, the class of $\alpha$ in $\widetilde\cS_{0,s}^\Phi(E)$
corresponds to the generic fiber $\gamma'_{gen}$ of the
$S^\Phi$-torsor $\gamma'$ under the bijection
$\widetilde\cS_{0,s}^\Phi(E)\simeq S^\Phi\text{-}\tors(E)$. As
$P^\Phi\times \gm^r$ is a quasi-split torus, $\gamma'$ is a
generic $S^{\Phi}$-torsor by Proposition \ref{gengen}, hence
\begin{equation}\label{2222}
\ed_p^{\widetilde\cS_{0,s}^{\Phi}}(\alpha)=\ed_p^{S^\Phi\text{-}\tors}(\gamma'_{gen})=\ed_p(S^\Phi)
\end{equation}
by \cite[Th. 2.9]{Merkurjev09}.
The essential $p$-dimension of $S^\Phi$ was calculated in
Corollary \ref{edS}.
From (\ref{mul}),(\ref{2222}) and this corollary, we have
\[
\ed_p\(\cat{Alg_{p^r,p^s}}\)\geq \ed_p^{\cat{Alg_{p^r,p^s}}}(\alpha_r)\geq\ed_p(S^\Phi)+r=
 \begin{cases}
(r-1)2^{r-1} & \text{if $p=2$ and $s=1$},\\
(r-1)p^r+p^{r-s} & \text{otherwise}.
\end{cases}
\]
This concludes the proof.
\end{proof}



\section{An upper bound for $\ed_p(\cat{Alg}_{p^{r},p^{s}})$}\label{upup}

\begin{lemma}\label{upperpbd}
Let $F$ be a field and $p$ a prime. Then, for any integers $r$ and $s$ with $1\leq s\leq r$,
\[\ed_p(\cat{Alg}_{p^{r},p^{s}})\leq \ed_p(\cat{Alg}_{p^{r}})+p^{r-s}-1.\]
\end{lemma}

\begin{proof}
Let $A\in \cat{Alg}_{p^{r},p^{s}}(K)\subset \cat{Alg}_{p^{r}}(K)$ for a field extension $K/F$. There exist a field extension $K'/K$ of degree prime to $p$, a subfield $K_0\subset K'$ over $F$ and $B\in \cat{Alg}_{p^{r}}(K_{0})$ such that
$\td_F(K_0)\leq \ed_p(\cat{Alg}_{p^{r}})$ and $A\tens_{K} {K'}\simeq B\tens_{K_{0}} K'$.

By \cite[Lemma 5.6]{Saltman}, $\ind(B^{\tens p^{s}})$ divides $p^{r-s}$. Choose a central simple algebra $C$ of degree $p^{r-s}$ over $K_0$ in the Brauer class of $B^{\tens p^{s}}$ in $\Br(K_{0})$ and consider the \emph{Severi-Brauer variety} $X:=\SB(C)$ of $C$. Since $\exp(A)$ divides $p^{s}$, the algebra $C$ is split over $K'$, hence $X(K')\neq \emptyset$. This implies that there exists $x\in X$ such that $K_{0}(x)\subset K'$ and $X\(K_{0}(x)\)\neq \emptyset$. Therefore, $C_{K_{0}(x)}$ is split, hence $\exp(B_{K_{0}(x)})$ divides $p^s$, i.e., $B_{K_{0}(x)}\in \cat{Alg}_{p^{r},p^{s}}\(K_{0}(x)\)$. Since $\dim(X)=p^{r-s}-1$, we have
\begin{multline*}
\ed_{p}^{\cat{Alg}_{p^{r},p^{s}}}(A)\leq \td_F(K_{0}(x))=\td_F(K_{0})+\td_{K_{0}}(K_{0}(x))\leq\\
\ed_p(\cat{Alg}_{p^{r}})+\dim(x)\leq\ed_p(\cat{Alg}_{p^{r}})+(p^{r-s}-1).\qedhere
\end{multline*}
\end{proof}

By \cite[Th.1.1]{MR09a},
\[
\ed_p(\cat{Alg}_{p^{r}})\leq 2p^{2r-2}-p^{r}+1,
\]
if $r\geq 2$, therefore, by Lemma \ref{upperpbd}, we have the
following upper bound for $\ed_p(\cat{Alg}_{p^{r},p^{s}})$:
\begin{theorem}\label{cuupperpbd}
Let $F$ be a field and $p$ a prime integer. Then, for any integers $r\geq 2$ and
$s$ with $1\leq s\leq r$,
\[\ed_p(\cat{Alg}_{p^{r},p^{s}})\leq 2p^{2r-2}-p^{r}+p^{r-s}.\]
\end{theorem}


\section{Essential dimension of $\cat{Alg_{L/F}}$, $\cat{Alg_{G}}$ and $\cat{ALG_{G}}$.}\label{verylast}

Let $G$ be an elementary abelian group of order $p^r$ and $K/F$ a field extension. Consider the subset $\cat{Alg_{G}}(K)$ of $\cat{Alg_{p^{r},p^{s}}}(K)$ consisting of all classes that have a splitting Galois $K$-algebra $E$ with $\Gal(E/K)\simeq G$.

Let $L/F$ be a Galois field extension with $\Gal(L/F)\simeq G$. Consider the subset $\cat{Alg_{L/F}}(K)$ of $\cat{Alg_{G}}(K)$ consisting of all classes split by the field extension $KL/K$. We have the subfunctors of $\cat{Alg_{p^{r},p^{s}}}$:
\[
\cat{Alg_{L/F}}\subset \cat{Alg_{G}}\subset \cat{Alg_{p^{r},p^{s}}}.
\]
We write $\cat{ALG_{G}}(K)$ for the set of pairs $(A,E)$, where $A\in \cat{Alg_{G}}(K)$ and $E$ is a Galois $G$-algebra splitting $A$. We have an obvious surjective morphism of functors $\cat{ALG_{G}}\to \cat{Alg_{G}}$.


\begin{theorem}\label{crossed}
Let $F$ be a field, $p$ a prime integer different from $\ch(F)$, $G$ an
elementary abelian group of order $p^r$ with $r\geq 2$, and $L/F$ a
Galois field extension with $\Gal(L/F)\simeq G$.
Let an integer $s$ satisfy $1\leq s\leq r$. Suppose that $r\geq 3$ if $p=2$ and $s=1$. Let $\cF$ be one of
the three functors: $\cat{Alg_{L/F}}$, $\cat{Alg_{G}}$ or $\cat{ALG_{G}}$. Then
\[
\ed_{p}(\cF)=\ed(\cF)= \begin{cases}
(r-1)2^{r-1} & \text{if $p=2$ and $s=1$},\\
(r-1)p^r+p^{r-s}& \text{otherwise}.
\end{cases}
\]
\end{theorem}

\begin{proof}
Let $\Phi$ be a subgroup of $\Ch_{p}(F)$ of rank $r$ such that
$L=F(\Phi)$. By (\ref{hu}), we have $\cat{Alg_{L/F}}\simeq U^\Phi\text{-}\tors$. It follows from Proposition \ref{edU} that  \[
\ed_{p}(\cat{Alg_{L/F}})=\ed(\cat{Alg_{L/F}})=d_{p,r,s}:= \begin{cases}
(r-1)2^{r-1} & \text{if $p=2$ and $s=1$},\\
(r-1)p^r+p^{r-s}& \text{otherwise}.
\end{cases}
\]
Let $\alpha_r\in \Br(E_{r})$ be as in the proof of Theorem \ref{main}. By construction, $\alpha_{r}$ is split by $E_{r}(\Phi)$, hence $\alpha_r\in \cat{Alg_{G}}(E_{r})$. Note that $\ed_{p}^{\cS}(\beta)\leq \ed_{p}^{\cH}(\beta)$ for any subfunctor $\cH$ of a functor $\cS$ and any $\beta\in \cH(K)$. Hence, by the proof of Theorem \ref{main}, we have
\[
\ed_{p}(\cat{Alg_{G}})\geq \ed_{p}^{\cat{Alg_{G}}}(\alpha_r)\geq \ed_{p}^{\cat{Alg_{p^r,p^s}}}(\alpha_r)\geq d_{p,r,s}.
\]

Let $J$ be the $G$-module defined in the Section \ref{manytori} and $T:=\Spec F[J]$ the split torus with the character group $J$. Consider the minimal surjective $p$-presentation $\nu:P'\to J$ as in Remark \ref{surmin}. As explained in Section \ref{toritori}, a choice of a $G$-invariant basis of $P$ yields a linear $T\rtimes G$-space $V$ with $\dim(V)=\rank(P')$. By Remark \ref{surmin}, $G$ acts faithfully on $\Ker(\nu)$.
It follows from \cite[Lemma 3.3]{MR09} that the
action of $T\rtimes G$ on $V$ is generically free in this case, hence, by \cite[Prop. 4.11]{BerhuyFavi03},
\begin{align*}
\ed(T\rtimes G) &\leq \dim(V)-\dim(T\rtimes G) \\
&= \rank(P')-\rank(J) \\
&= \rank\(\Ker(\nu)\)\\
&=  d_{p,r,s}.
\end{align*}

Let $\gamma\in H^{1}(F,G)$ and let $L$ be the corresponding Galois $G$-algebra over $F$. Since $G$ is an abelian group, we have $G=G_{\gamma}$.
The $G$-action on $R_{L/F}(\gml)$ restricts to the trivial action on the subgroup $\mu_{p^{s}}$. As $T_{\gamma}=R_{L/F}(\gml)/\mu_{p^{s}}$, the connecting map
\[
H^{1}(F, T_{\gamma})\to H^{2}(F,\mu_{p^s})=\Br_{p^s}(F)
\]
is injective, hence the group $G_{\gamma}(F)=G$ acts trivially on $H^{1}(F, T_{\gamma})$.
By (\ref{cop}),
\[
H^{1}(F, T\rtimes G)=\coprod_{\Gal(E/F)=G}\Br_{p^s}(E/F),
\]
where the disjoint union is taken over all isomorphism classes of Galois $G$-algebras $E/F$.
Hence we have a surjective morphism of functors $T\rtimes G\text{-}\tors \to \cat{ALG_{G}}$. As $\cat{ALG_{G}}$ surjects on $\cat{Alg_{G}}$, we have
\[
\ed_{p}(\cat{Alg_{G}})\leq\(\ed_{p}(\cat{ALG_{G}})\ \ \text{or}\ \ed(\cat{Alg_{G}})\)\leq \ed(\cat{ALG_{G}})\leq\ed(T\rtimes G)\leq d_{p,r,s}.\qedhere
\]
\end{proof}

\begin{remark}\label{lastst}
Suppose that $p=r=2$ and $s=1$ and $F$ is a field of characteristic different from $2$. By \cite[Th.1]{RST} or \cite[Sec.2.4]{BM09}, there exists a nontrivial cohomological invariant of degree $4$ for $\cat{Alg_{G}}$ over $F(i)$, where $i$ is a primitive $4$-th root of unity. Hence, $\ed_{2}(\cat{Alg_{G}})\geq \ed_{2}(\cat{Alg_{G}})_{F(i)}\geq 4$ by \cite[Lemma 6.9]{Reichstein00a}. Moreover, by the structure theorem on central simple algebras split by a biquadratic field extension \cite[Cor.2.8]{Tignol81}, every $(A,E)\in \cat{ALG_{G}}(K)$ is of the form $E=K(a^{1/2},b^{1/2})$ and $[A]=(a,x)_2+(b,y)_2$ for some $a,b,x,y\in K^\times$. Hence $\ed(\cat{ALG_{G}})\leq 4$. As $\cat{ALG_{G}}$ surjects on $\cat{Alg_{G}}$, we have \[
4\leq \ed_{2}(\cat{Alg_{G}}) \leq \(\ed_{2}(\cat{ALG_{G}})\ \ \text{or}\ \ed(\cat{Alg_{G}})\)\leq \ed(\cat{ALG_{G}})\leq 4, \]
hence the essential (2)-dimension of $\cat{Alg_{G}}$ and $\cat{ALG_{G}}$ is equal to $4$.
\end{remark}

\begin{corollary}\label{deg8}
Let $F$ be a field of characteristic $\neq 2$. Then
\[\ed_{2}(\cat{Alg}_{8,2})=\ed(\cat{Alg}_{8,2})=8.\]
\end{corollary}
\begin{proof}
As any central simple algebra of degree $8$ and exponent $2$ has a triquadratic splitting field by \cite{Rowcsa}, we have $\cat{Alg}_{8,2}=\cat{Alg_{G}}$ for the elementary abelian group $G$ of order $8$, hence the statement follows from Theorem \ref{crossed}.
Note that the inequality $\ed_{2}(\cat{Alg}_{8,2})\geq 8$ is also proven in
Theorem \ref{main} and the opposite inequality $\ed(\cat{Alg}_{8,2})\leq 8$ was
shown in \cite[Th.2.12]{BM09}.
\end{proof}


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