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



\author
[S. Baek] {Sanghoon Baek}


\address
{Department of Mathematics and Statistics, University of Ottawa, Canada}


\email {sbaek@uottawa.ca}



\begin{document}



\begin{abstract}
Let $1\leq m \leq n$ be integers with $m|n$ and $\cat{Alg}_{n,m}$ the class of central simple algebras of degree $n$ and exponent dividing $m$. In this paper, we find upper bounds for the essential ($2$)-dimension of $\cat{Alg}_{n,2}$. Moreover, we find a stronger upper bound for the essential $2$-dimension of $\cat{Alg}_{n,2}$ over a field $F$ of $\ch(F)\neq 2$. As a result, we show that $\ed_{2}(\cat{Alg}_{16,2})=24$ over a field $F$ of $\ch(F)\neq 2$.
\end{abstract}

\maketitle

\section{Introduction}

Let $\cT:\cat{Fields}/F\to\cat{Sets}$ be a functor from the category $\cat{Fields}/F$ of field extensions over $F$ to the category $\cat{Sets}$ of sets. For fields $E,E'\in \cat{Fields}/F$, a field homomorphism $f:E\to E'$ over $F$ and $\alpha\in\cT(E)$, we write $\alpha_{E'}$ for the image of $\alpha$ under the morphism $\cT(f):\cT(E)\to \cT(E')$.


Let $E\in\cat{Fields}/F$ and $K\subset E$ a subfield over $F$. An
element $\alpha\in \cT(E)$ is said to be \emph{defined over $K$} and
$K$ is called a \emph{field of definition of $\alpha$} if there
exists an element $\beta\in \cT(K)$ such that $\beta_{E}=\alpha$. The \emph{essential dimension
of $\alpha$} is $\ed(\alpha)=\min \{\td_{F}(K)\}$ over all fields of definition $K$ of $\alpha$. The
\emph{essential dimension of the functor $\cT$} is $\ed(\cT)=\sup \{\ed(\alpha)\}$, where the supremum is taken over all fields $E\in\cat{Fields}/F$ and all
$\alpha\in \cT(E)$. Hence, the essential dimension of an algebraic structure $\cT$ measures the complexity of the structure in terms of the smallest number of parameters required to define the structure over a field extension of $F$.




Let $p$ be a prime integer. The \emph{essential $p$-dimension of  $\alpha$} is $\ed_p(\alpha)=\min\{\ed(\alpha_{L})\}$, where $L$ ranges over all field extensions of $E$ of degree prime to $p$. In other words, $\ed_p(\alpha)=\min\{\td_{F}(K)\}$, where the minimum is taken over all field extensions $L/E$ of prime to $p$ and all subextensions $K/F$ of $L$ which are field of definition of $\alpha_{L}$. Hence, $\ed(\alpha)\geq\ed_p(\alpha)$ for all $p$. The \emph{essential $p$-dimension of $\cF$} is $\ed_p(\cT)=\sup \{\ed_p(\alpha)\}$, where the supremum ranges over all fields $E\in\cat{Fields}/F$ and all $\alpha\in \cT(E)$. 


Let $G$ be an algebraic group over $F$. The \emph{essential
dimension $\ed(G)$} (respectively, \emph{essential $p$-dimension $\ed_{p}(G)$}) of $G$ is defined to be $\ed(H^1(-,G))$ (respectively, $\ed_{p}(H^1(-,G))$), where $H^1(E,G)$ is the Galois cohomology set (equivalently, the set of isomorphism classes of $G$-torsors) over a field extension $E$ of $F$.


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 $m$. Then, for any field extension $E/F$, there is a natural bijection between $H^1(E,\gGL_{n}/\gmu_{m})$ and $\cat{Alg}_{n,m}(E)$ (see \cite[Example 1.1]{BM09}), thus $\ed(\cat{Alg}_{n,m})=\ed(\gGL_{n}/\gmu_{m})$ and $\ed_{p}(\cat{Alg}_{n,m})=\ed_{p}(\gGL_{n}/\gmu_{m})$.


In this paper, we compute upper bounds for the essential dimension of $\cat{Alg}_{n,2}$. By a theorem of Albert, a central simple algebra has exponent dividing $2$ if and only if it admits an involution of the first kind (see \cite[Theorem 3.1]{Book}). Thus, any algebra $A$ in $\cat{Alg}_{n,2}(K)$ for any field extension $K/F$ has involutions of the first kind. Moreover, such $A$ has involutions of both symplectic and orthogonal types (see \cite[Corollary 2.8(2)]{Book}). By the primary decomposition theorem and \cite[Section 6]{BM10}, we have $\ed(\cat{Alg}_{n,2})=\ed(\cat{Alg}_{2^{r},2})$ and $\ed_{2}(\cat{Alg}_{n,2})=\ed_{2}(\cat{Alg}_{2^{r},2})$, where $2^{r}$ is the largest power of $2$ dividing $n$. Hence, we may assume that $n$ is a power of $2$.




By \cite[Remark 8.2 and Corollary 8.3]{BM10}, $\ed_{2}(\cat{Alg}_{4,2})=\ed(\cat{Alg}_{4,2})=4$ and $\ed_{2}(\cat{Alg}_{8,2})=\ed(\cat{Alg}_{8,2})=8$ over a field $F$ of $\ch(F)\neq 2$. In general, by \cite[Theorem]{BM10}, the following bounds were established over a field $F$ of $\ch(F)\neq 2$:
$$2^{r-1}(r-1)\leq \ed_{2}(\cat{Alg}_{2^{r},2}) \leq 2^{r-1}(2^{r-1}+1) \text{ for all } r\geq 2.$$


In the present paper, we find an upper bound $n(n-1)/2$ for the essential dimension of $\cat{Alg}_{n,2}$ in Corollary \ref{algnmupper}. Moreover, we find an upper bound $2^{2r-2}$ for the essential $2$-dimension of $\cat{Alg}_{2^{r},2}$ in Corollary \ref{normalizerofgl}. Both upper bounds are valid over an arbitrary field $F$. In particular, the bound $2^{2r-2}$ improves the bound $2^{2r-2}+2^{r-1}$ as above.

Using involutions of the first kind, we further improve the upper bound $2^{2r-2}$ as follows:

\smallskip

\noindent{\bf Theorem.} \emph{Let $F$ be a field of characteristic different from $2$. Then, for any integers $r\geq 3$,}
\[
\ed_{2}(\cat{Alg}_{2^{r},2})\ \leq\ 2^{r-1}(2^{r-3}+1).
\]

\smallskip


As a result, we find the essential $2$-dimension of $\cat{Alg}_{16,2}$:

\smallskip

\noindent{\bf Corollary.}
\emph{Let $F$ be a field of characteristic different from $2$. Then}
\[\ed_{2}(\cat{Alg}_{16,2})=24.\]

\smallskip

\begin{remark}
Recently, V.~Chernousov and A.~Merkurjev proved that \[\ed_{p}(\gSL_{p^r}/\gmu_{p^s})=\ed_{p}(\cat{Alg}_{p^r,p^s})+1\]
for any $0\neq s< r$ over a field of $\ch(F)\neq p$ (this result is communicated to the author by A.~Merkurjev). Therefore, the computation of essential $p$-dimension of split simple group of type $A_{p^r-1}$ is reduced to the computation of $\ed_{p}(\cat{Alg}_{p^r,p^s})$. In particular, we have $\ed_{2}(\gSL_{16}/\gmu_{2})=25$ over a field of $\ch(F)\neq 2$.
\end{remark}


\emph{Acknowledgements}: I am grateful to my advisor A.~Merkurjev for many useful discussions and support and to Z.~Reichstein for helpful comments. Section $2$ of this paper is based on the author's doctoral thesis at the University of California at Los Angeles.

\section{Upper bounds for the essential dimension of $\cat{Alg}_{n,2}$}
Let $G$ be a reductive algebraic group over $F$, let $T$ be a maximal torus of $G$ and let $N$ be the normalizer of $T$ in $G$. Then the canonical map
\[H^{1}(K,N)\to H^{1}(K,G)\]
is surjective for any field extension $K/F$ by Springer's Lemma (\cite[III.4 Lemma 6]{Serre97}). Therefore, we have
\begin{equation}\label{reductivenormalizer}
\ed(G)\leq \ed(N)
\end{equation}
by \cite[Proposition 1.3]{Merkurjev09}.

For any integer $n\geq 2$, consider a reductive group $\gGL_n/\gmu_{2}$ and the maximal torus $T_{n,2}:=\gm^{n}/\gmu_{2}$ in the group.



\begin{lemma}\label{simpletnm}
Let $F$ be an arbitrary base field and $S_{n}$ be the symmetric group on $n$ elements. Then for any $n\geq 3$, we have
\[
\ed(T_{n,2}\rtimes S_{n})\leq (n^{2}-n)/2.
\]
\end{lemma}
\begin{proof}
Note that the character group $(T_{n,2})^{*}$ is isomorphic to
\[\{(t_{1},\cdots,t_{n})\in \Z^{n}|\text{ }t_{1}+\cdots+t_{n}=0 \text{ in } \Z/2\Z\}.\]


Let $e_{i,j}=(0,\cdots, 1,\cdots, -1, 0)$ be an element of $(T_{n,2})^{*}$, where $1$ and $-1$ are placed in the $i$th and $j$th positions respectively for $1\leq i\neq j\leq n$ and $0$'s are placed in other positions. Similarly, let $f_{i,j}=(0,\cdots, 1,\cdots, 1, 0)$, where $1$'s are placed in the $i$th and $j$th positions for $1\leq i\neq j\leq n$ and $0$'s are placed in other positions and let $g_{k}=(0,\cdots, -2,\cdots, 0)$ as an element of $(T_{n,2})^{*}$, where $-2$ is placed in the $k$th position for $1\leq k\leq n$ and $0$'s are placed in other positions.


Let $X$ be a set consisting of $f_{i,j}$ and $g_{k}$ for all $1\leq i\neq j \leq n$ and all $1\leq k \leq n$. Then $X$ is a $S_n$-invariant subset of $(T_{n,2})^{*}$ and $|X|=|f_{i,j}|+|g_{k}|=(n^2-n)/2+n$.

It is clear that $e_{i,j}$ and $f_{i,j}$ generate $(T_{n,2})^{*}$ as an abelian group, as the indices $i$ and $j$ run over $1$ to $n$. Since $f_{i,j}+g_{j}=e_{i,j}$, $X$ generates $(T_{n,2})^{*}$ as an abelian group and hence we have a surjective $S_{n}$-equivariant homomorphism $\nu:\Z[X]\to (T_{n,2})^{*}$ taking $f_{i,j}$ and $g_{k}$ to themselves.


We show that $S_{n}$ acts faithfully on $\Ker(\nu)$. Let $\sigma$ be a nontrivial element of $S_{n}$. Then there exists $1\leq i_{0} \leq n$ such that $\sigma(i_{0})\neq i_{0}$. Choose a $1\leq j_{0}\leq n$ which is different from $\sigma(i_{0})$ and $i_{0}$. Then $\sigma$ does not fix $2f_{i_{0},j_{0}}+g_{i_{0}}+g_{j_{0}}\in \Ker(\nu)$. By \cite[Lemma 3.3]{MR09}, $\ed(T_{n,2}\rtimes S_{n})\leq (n^2-n)/2+n-\rank((T_{n,2})^{*})=(n^2-n)/2$.

\end{proof}


By (\ref{reductivenormalizer}), we have an upper bound for $\ed(\cat{Alg}_{n,2})$ as follows:
\begin{corollary}\label{algnmupper}
Let $F$ be an arbitrary base field. Then for any $n\geq 3$,
\[
\ed(\cat{Alg}_{n,2})\leq (n^{2}-n)/2.
\]
\end{corollary}
\smallskip

Let $P_{n}$ be a Sylow $2$-subgroup of the symmetric group $S_{n}$ on $n$ elements. In the following Lemma, we compute an upper bound for the essential dimension of $T_{2^r,2}\rtimes P_{2^r}$.


\begin{lemma}\label{torsioncsa}
Let $F$ be an arbitrary base field. Then for any $r\geq 2$, we have
\[
\ed(T_{2^r,2}\rtimes P_{2^r})\leq 2^{2r-2},
\]
where $P_{2^r}$ is a Sylow $2$-subgroup of $S_{2^r}$.
\end{lemma}

\begin{proof}
Note that a Sylow $2$-subgroup $P_{2^r}$ of $S_{2^r}$ is isomorphic to $(P_{2^{r-1}})^{2}\rtimes \Z/2\Z$.

Consider the $e_{i,j}$, $f_{i,j}$ and $g_{k}$ as in the proof of Lemma \ref{simpletnm}. We divide the set of integers $\{1, 2,\cdots, 2^r\}$ into two subsets $\Lambda_{1}:=\{1, 2,\cdots, 2^{r-1}\}$ and $\Lambda_{2}:=\{2^{r-1}+1, 2^{r-1}+2,\cdots, 2^r\}$. Let $X$ be a set consisting of $f_{i,j}$ and $g_{k}$ for all $1\leq i\neq j \leq 2^r$ such that $i$ and $j$ are placed in different $\Lambda_{l}$'s and all $1\leq k \leq 2^{r}$, where $l$ is either $0$ or $1$. Then $X$ is a $P_{2^r}$-invariant subset of $(T_{2^r,2})^{*}$ and $|X|=2^{2r-2}+2^r$.

It is clear that $e_{i,j}$ and $f_{i,j}$ generate $(T_{2^r,2})^{*}$ as an abelian group, as the indices $i$ and $j$ run over $1$ to $2^r$. Note that $f_{i,j}=f_{i,k}+f_{j,k}+g_{k}$ for all $i$ and $j$ which are in the same $\Lambda_{l}$'s, where $l$ is either $1$ or $2$. As \[e_{i,j}=\begin{cases}
f_{i,j}+g_{j} & \text{if $i$ and $j$ are in different $\Lambda_{l}$'s},\\
f_{i,k}+f_{j,k}+g_{j}+g_{k} & \text{otherwise,}\\
\end{cases}
\]
$X$ generates $(T_{2^r,2})^{*}$ as an abelian group and hence we have a surjective $P_{2^r}$-equivariant homomorphism $\nu:\Z[X]\to (T_{2^r,2})^{*}$ taking $f_{i,j}$ and $g_{k}$ to themselves.


We show that $P_{2^r}$ acts faithfully on $\Ker(\nu)$. Note that the center of $P_{2^r}$, which is generated by $\sigma:=(1, 2)(3, 4)\cdots (2^{r}-1, 2^r)$ and it is enough to show that $\sigma$ acts faithfully on $\Ker(\nu)$. In fact, $\sigma$ does not fix the non-zero element $2f_{1,2^{r-1}+1}+g_{1}+g_{2^{r-1}+1}\in \Z[X]$. By \cite[Lemma 3.3]{MR09}, we have \[\ed(T_{2^r,2}\rtimes P_{2^r})\leq 2^{2r-2}+2^{r}-\rank((T_{2^r,2})^{*})=2^{2r-2}.\]
\end{proof}

As $(2,[T_{2^r,2}\rtimes S_{2^r}:T_{2^r,2}\rtimes P_{2^r}])=1$, we have $\ed_{2}(T_{2^r,2}\rtimes S_{2^r})=\ed_{2}(T_{2^r,2}\rtimes P_{2^r})$ by \cite[Lemma 4.1]{MR09}. Therefore, by Lemma \ref{torsioncsa}, we have the following Corollary:

\begin{corollary}\label{normalizerofgl}
Let $F$ be an arbitrary base field. Then for any $r\geq 2$,
\[
\ed_{2}(\cat{Alg}_{2^r,2})\leq 2^{2r-2}.
\]
\end{corollary}

\section{Algebras with involutions}

Let $A$ be a central simple algebra over $F$. For any $a\in A^{\times}$, we denote the inner automorphism of $A$ by $\Int(a)$: $\Int(a)(x)=axa^{-1}$ for all $x\in A$. For any subalgebra $B$ of $A$, we write $C_{A}(B)$ for the centralizer of $B$ in $A$. The following Lemma characterizes all involutions of the first kind on $A$.

\begin{lemma}\label{involdeg2}\cite[Proposition 2.7]{Book}
Let $F$ be a field of $\ch(F)\neq 2$, $A$ be a central simple algebra over $F$ and $\sigma$ be an involution of the first kind on $A$. Then every involution $\sigma'$ of the first kind on $A$ is of the form $\Int(a)\circ \sigma$ for some $a\in A^{\times}$ uniquely determined up to a factor in $F^{\times}$, such that $\sigma(a)=\pm a$. Moreover, $\sigma$ and $\sigma'$ are of the same type if and only if $\sigma(a)=a$.
\end{lemma}

We use the following Lemma for extension of involutions:

\begin{lemma}\label{extinvol}\cite[Theorem 4.14]{Book}
Let $F$ be a field of $\ch(F)\neq 2$, $A$ be a central simple algebra over $F$ with an involution $\sigma$ of the first kind, and $B$ be a simple subalgebra of $A$ with an involution $\tau$ such that $\tau |_{F}=\sigma |_{F}$. Then $A$ has involutions of both types whose restriction to $B$ is $\tau$, unless $\tau$ is of the first kind and $\deg(C_{A}(B))$ is odd.
\end{lemma}

From now we assume that the base field $F$ is $2$-closed (i.e., every finite extension of $F$ is separable of degree a power of $2$) and is of characteristic different from $2$.


\begin{proposition}\label{mainproposition}
Let $r\geq 3$ be an integer, $F$ a $2$-closed field such that $\ch(F)\neq 2$ and $D$ a division $F$-algebra of degree $2^{r}$ and exponent $2$. Then for any biquadratic field extension $K_{1}K_{2}/F$ in $D$ with quadratic field extensions $K_{1}/F$ and $K_{2}/F$ there exists a quadratic extension $K_{3}/F$ in $D$ such that $K_{1}K_{2}K_{3}/F$ is a triquadratic extension in $D$.
\end{proposition}
\begin{proof}
By \cite[Theorem 3.1(1)]{Book}, $D$ has an involution of the first kind $\sigma$. Let $\tau_{1}$ and $\tau_{2}$ be two distinct nontrivial automorphisms of the field $K_{1}K_{2}$. As $\sigma|_{F}=\tau_{i}|_{F}$ for any $i=1,2$, there are two distinct involutions $\sigma_{1}$ and $\sigma_{2}$ of the same type on $A$ such that $\sigma_{i}|_{K_{1}K_{2}}=\tau_{i}$ by Lemma \ref{extinvol}.

By Lemma \ref{involdeg2}, there exists $d\in D^{\times}$ such that $\sigma_{1}=\Int(d)\circ \sigma_2$ and $\sigma_{i}(d)=d$ for all $i=1,2$. In particular, $d^{2}$ commutes with $K_{1}$ and $K_{2}$ and $F(d^{2})\cap K_{1}K_{2}=F$. If $F(d^{2})\neq F$, then $F(d^{2})$ contains a quadratic extension $K_{3}$ over $F$ by \cite[Proposition 1.1]{RowSalt}. Hence we a triquadratic extension $K_{1}K_{2}K_{3}$ in $D$.

Suppose that $d^{2}\in F$. Then there exist quaternion subalgebras $Q_{1}:=(K_{1},d^{2})$ and $Q_{2}:=(K_{2},d^{2})$ of $D$. As $\ind(C_{D}(Q_{1}\tens Q_{2}))\geq 2$, $C_{D}(Q_{1}\tens Q_{2})$ contains a quadratic extension $K_{3}/F$ by \cite[Proposition 1.1]{RowSalt}. Therefore, we have $K_{1}K_{2}K_{3}=K_{1}\tens K_{2}\tens K_{3}\subset Q_{1}\tens Q_{2}\tens C_{D}(Q_{1}\tens Q_{2})=D$.
\end{proof}



\begin{corollary}\label{etalesubalgebra}
Let $r\geq 3$ be an integer and $F$ be a $2$-closed field such that $\ch(F)\neq 2$. Then for any division $F$-algebra $D$ of degree $2^{r}$ and exponent $2$ and an \'etale subalgebra $K_{1}K_{2}:=K_{1}\tens K_{2}$ of $D$ such that $\dim_{F}(K_{i})=2$ for $i=1,2$, there exists a maximal \'etale subalgebra $K_{1}K_{2}K:=K_{1}\tens K_{2}\tens K$ of $D$ with $\dim_{F}(K)=2^{r-2}$.
\end{corollary}
\begin{proof}
By Proposition \ref{mainproposition}, there exists a triquadratic field extension $K_{1}K_{2}K_{3}$ over $F$. Induction on $r$. If $r=3$, then $K=K_{3}$ satisfies the conclusion of Corollary. For $r\geq 3$, the centralizer $C_{D}(K_{3})$ is a division $K_{3}$-algebra of degree $2^{r-1}$. By the induction hypothesis with $K_{1}K_{3}/K_{3}$ and $K_{2}K_{3}/K_{3}$, $C_{D}(K_{3})$ contains a subfield $K/F$ with $[K:K_{3}]=2^{r-3}$. Hence $D$ contains a field extension $K_{1}K_{2}K$ over $F$ such that $\dim_{F}(K)=2^{r-3}\cdot 2$.
\end{proof}

\section{Essential $2$-dimension of $\cat{Alg}_{n,2}$}\label{ed2sec}

Let $n\geq 2$ be an integer, $G$ be a subgroup of $S_{n}$ and $X$ be a $G$-set of $n$ elements ($G$ acts on $X$ by permutation). For any divisor $m$ of $n$, we consider the surjective $G$-modules homomorphism $\bar\varepsilon:\Z[X]\to \Z/m\Z$, defined by $\bar\varepsilon(x)=\varepsilon(x)+m\Z$, where $\varepsilon: \Z[X]\to\Z$ is the \emph{augmentation homomorphism} given by $\varepsilon(x)=1$ for all
$x\in X$. Set $J=\Ker(\bar\varepsilon)$. Then we have an exact sequence
\begin{equation}\label{JXsequence}
0\to J\to \Z[X]\xra{\bar{\varepsilon}}\ \Z/m\Z \to 0.
\end{equation}
We shall need the following lemma (see also the proof of \cite[Theorem
8.1]{BM10}):

\begin{lemma}\label{lemmabrau}
Let $F$ be a field of $\ch(F)\mathrel{\not |}n$ and $T=\Spec F[J]$
be the split torus with the character group $J$. Then \[H^{1}(F,
T\rtimes G)=\coprod_{\Gal(E/F)=G}\Br_{m}(E/F),\] where the disjoint
union is taken over all isomorphism classes of Galois $G$-algebras
$E/F$.
\end{lemma}
\begin{proof}
Let $T_{\gamma}$ (respectively, $G_{\gamma}$) be the twist of $T$
(respectively, $G$) by the $1$-cocycle $\gamma\in Z^{1}(F, G)$.
Then by \cite[Proposition 28.11]{Book}, there is a natural bijection
between the fiber of $H^{1}(F, T\rtimes G)\to H^{1}(F,G)$ over
$[\gamma]$ and 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 $[\gamma]\in H^{1}(F, G)$.

Let $E$ be the corresponding Galois $G$-algebra over $F$ to
$\gamma$. From (\ref{JXsequence}), we have the
corresponding exact sequence of algebraic groups
\[1\to \gmu_{m}\to \gm^{n} \to T\to 1\] and then the exact sequence
\begin{equation}\label{ccoopp}
1\to \gmu_{m}\to R_{E/F}(\gme)\to T_{\gamma}\to 1,
\end{equation}
each term of which is twisted by $\gamma$.
The exact sequence (\ref{ccoopp}) induces an exact sequence of Galois cohomology
\begin{equation}\label{cohomolnothing}
1\to H^{1}(F, T_{\gamma})\to H^{2}(F,\gmu_{m})=\Br_{m}(F)\to
H^{2}(E,\gme)=\Br(E)
\end{equation} by Eckmann-Faddeev-Shapiro's Lemma and Hilbert's $90$.
The $G$-action on $R_{E/F}(\gme)$ restricts to the trivial action on
the subgroup $\gmu_{m}$. Let $\sigma\in G_\gamma(F)$ acts on
$T_{\gamma}=R_{E/F}(\gme)/\gmu_{m}$. The action of $\sigma$ and
(\ref{cohomolnothing}) induce the following diagram
\begin{equation*}
\xymatrix{
H^1(F,T_{\gamma}) \ar@{->}^{\sigma^{\ast}}[d]\ar@{^{(}->}[r] & H^{2}(F,\gmu_{m}) \ar@{=} [d]  \\
H^1(F,T_{\gamma})\ar@{^{(}->}[r]  & H^{2}(F,\gmu_{m}).\\
}
\end{equation*}
Therefore, $G_\gamma(F)$ acts trivially on $H^1(F,T_{\gamma})$, hence the result follows by (\ref{cop}).
\end{proof}



Let $r\geq 3$ be an integer. Let $G_{r}=S_{2}\times S_{2}\times S_{2^{r-2}}$ be a subgroup of the symmetric group $S_{2^{r}}$ on $2^{r}$ elements and let $H_{r}=S_{2}\times S_{2}\times P_{2^{r-2}}$ be a Sylow $2$-subgroup of $G_{r}$, where $P_{2^{r-2}}$ is a Sylow $2$-subgroup of $S_{2^{r-2}}$. Let $X_{r}$ be a $G_{r}$-set of $2^{r}$ elements ($G_{r}$ acts on $X_{r}$ by permutations). The action of $H_{r}$ may be described as follows: we subdivide the integers $1,2,\cdots, 2^{r}$ into four blocks $B_{1},B_{2},B_{3},B_{4}$ such that each block consists of $2^{r-2}$ consecutive integers. The $P_{2^{r-2}}$ permutes the elements of $B_{i}$ for all $1\leq i \leq 4$, $S_{2}$ interchanges $B_{2i-1}$ and $B_{2i}$ for all $i=1,2$, and another $S_{2}$ interchanges $B_{1}\cup B_{2}$ and $B_{3}\cup B_{4}$.




We set $J_{r}=\Ker(\Z[X_{r}]\xra{\bar{\varepsilon}} \Z/2\Z)$, where $\bar{\varepsilon}$ is the map with $m=2$ as in (\ref{JXsequence}). Applying Lemma \ref{lemmabrau} with $n=2^{r}$, $m=2$, $G=G_{r}$, $X=X_{r}$, $J=J_{r}$, and $T=T_{r}:=\Spec(F[J_{r}])$, we have a morphism \[\theta:H^{1}(-,T_{r}\rtimes G_{r}) \to \cat{Alg}_{2^{r},2}\] defined by $\theta(N)([A])=B$ for a field extension $N$ over $F$, where $[A]\in \Br_{2}(L/N)$ for some field extension $L/N$ with $\Gal(L/N)=G_{r}$ and $B$ is the central simple $N$-algebra of degree $2^{r}$ such that $[A]=[B]$ in $\Br_{2}(L/N)$.

We also have a morphism
\begin{equation}\label{thetaa}
\Theta:H^{1}(-,T_r\rtimes G_r) \coprod \big(\coprod_{1\leq i \leq r-1}\cat{Alg}_{2^{i},2}\big) \to \cat{Alg}_{2^{r},2}
\end{equation}
defined by \[[A]\mapsto \theta(N)([A]),\quad A_{i}\mapsto M_{2^{r-i}}(A_{i})\] over a field extension $N$ over $F$, where $A_{i}\in \cat{Alg}_{2^{i},2}(N)$ for $1\leq i \leq r-1$.

\begin{lemma}\label{surjectlemm}
If the base field $F$ is $2$-closed and is of characteristic different $2$, then $\Theta$ is surjective.
\end{lemma}
\begin{proof}
We show that $\Theta(N)$ is surjective for a field extension $N/F$. By the definition of $\Theta$, we only need to check the surjectivity for a division $N$-algebra $D$ of degree $2^r$ and exponent $2$. By \cite[Theorem 1.2]{RowSalt}, there exists an \'etale subalgebra $K_{1}K_{2}$ in $D$ such that $\dim_{N}(K_{i})=2$ for $i=1,2$. By Corollary \ref{etalesubalgebra}, there exists a maximal \'etale subalgebra $K_{1}K_{2}K$ in $D$ such that $\dim_{N}(K)=2^{r-2}$. Hence $\theta$ is surjective, so is $\Theta$.
\end{proof}

\begin{example}(see \cite[Remark 3.10]{BM10})\label{firstexamp}
Let $r=3$. Then $G_3=H_3=S_{2}\times S_{2}\times S_{2}:=\langle\tau_{1}\rangle \times \langle\tau_{2}\rangle \times \langle\tau_{3}\rangle$. As the action of $H_3$ on $X_3$ is simply transitive, $X_3\simeq H_3$ as $H_3$-sets, hence $J_3$ is generated by $2$ and $\tau_{i}-1$ for $i=1,2,3$. Set $\Lambda_3:=\Z[H_3/\langle\tau_{1}\rangle]\oplus\Z[H_3/\langle\tau_{2}\rangle]\oplus\Z[H_3/\langle\tau_{3}\rangle]\oplus\Z[H_3/\langle\tau_{1}\tau_{2}\rangle]$. Define a map $\rho:\Lambda_3\to J_3$ by \[\rho(\overline{x_{1}},\overline{x_{2}},\overline{x_{3}},\overline{x_{4}})=\sum_{i=1}^{3}(\tau_{i}+1)x_{i}+(\tau_{1}\tau_{2}+1)x_{4}.\]
As $2=(\tau_1\tau_2+1)-\tau_1(\tau_2+1)+(\tau_1+1)$, $\rho$ is surjective. It is easy to check that $H_3$ acts on $\Ker(\rho)$ faithfully. Therefore, by \cite[Lemma 3.3]{MR09} and \cite[Corollary 4.2]{Merkurjev09}, $\ed_{2}(\cat{Alg}_{8,2})\leq 4+4+4+4-2^{3}=8$.

\end{example}



For an $x\in X_r$, let $H_{r,x}$ be the stabilizer of $x$ in $H_{r}=S_{2}\times S_{2}\times P_{2^{r-2}}:=\langle \tau_{1}\rangle \times \langle \tau_{2}\rangle \times P_{2^{r-2}}$. We set $P_{2^{r-2}}=(P_{2^{r-3}})^{2}\rtimes \langle \tau_{r}\rangle$.

\begin{lemma}\label{usss}
For any $r\geq 3$ and any $x\in X_r$, we have
\begin{enumerate}
  \item $H_{r,x}=H_{r-1,x}\times P_{2^{r-3}}$.
  \item $H_{r}=\langle \tau_{1},\tau_{2}, \tau_{r}, H_{r,x}\rangle$.
  \item $J_r=\langle 2x, \tau_{1}x-x, \tau_{2}x-x, \tau_{r}x-x \rangle$.
  \item $\tau_{r}H_{r,x}\tau_{r}\cap H_{r,x}=H_{r-1,x}\times H_{r-1,x}$.
\end{enumerate}
\end{lemma}
\begin{proof}
$(1)$ The stabilizer of $x$ in $H_{r}$ is the stabilizer of $x$ under the action of $P_{2^{r-2}}$ on the block $B_{i}$ containing $x$ for some $i$. As $P_{2^{r-2}}=(P_{2^{r-3}})^{2}\rtimes \langle \tau_{r}\rangle$, the stabilizer of $x$ in $P_{2^{r-2}}$ is $H_{r-1,x}\times P_{2^{r-3}}$.

\smallskip

$(2)$ Induction on $r$. The case $r=3$ comes from Example \ref{firstexamp}. By induction hypothesis we have $P_{2^{r-3}}=\langle \tau_{r-1}, H_{r-1,x}\rangle$. As $\tau_{r-1}$ is generated by $\tau_{r}$ and $P_{2^{r-3}}$, the result follows immediately.

\smallskip

$(3)$ As $H_{r}$ acts on $X_r$ transitively, the result follows from Lemma \ref{usss} $(2)$ and the sequence $(\ref{JXsequence})$.

\smallskip

$(4)$ As $\tau_{r}H_{r,x}\tau_{r}=H_{r,\tau_{r}(x)}$, the result follows from $(1)$.
\end{proof}


\begin{theorem}\label{mainthm}
For any $r\geq 3$, there exists a $2^{r-1}(2^{r-3}+3)$-dimensional generically free representation for $T_r\rtimes H_r$. Hence $\ed_{2}(\cat{Alg}_{2^{r},2})\leq 2^{r-1}(2^{r-3}+1)$.
\end{theorem}
\begin{proof}
For $r\geq 3$ and $x\in X_r$, we set \begin{align*}
\Lambda_{r} &:= \Z[H_{r}/\langle\tau_{1}\rangle\times H_{r,x}]\oplus \Z[H_{r}/\langle\tau_{2}\rangle\times H_{r,x}]\oplus\Z[H_{r}/\langle\tau_{1}\tau_{2}\rangle\times H_{r,x}] \\
& \oplus\Z[H_{r}/(\tau_{r}H_{r,x}\tau_{r}\cap H_{r,x})\rtimes \langle\tau_{r}\rangle].
\end{align*}
Define a map $\rho:\Lambda_{r}\to J_r$ by taking a generator of the first component (respectively, the second component) of $\Lambda_{r}$ to $\tau_{1}x+x$ (respectively, $\tau_{2}x+x$), a generator of the third component of $\Lambda_{r}$ to $\tau_{1}\tau_{2}x+x$, and a generator of the last component of $\Lambda_{r}$ to $\tau_{r}x+x$. By construction, this map is well defined. As $2x=(\tau_1\tau_2x+x)-\tau_1(\tau_2x+x)+(\tau_1x+x)$, $\rho$ is surjective by Lemma \ref{usss} $(3)$.





As $H_{r}$ acts faithfully on $\Ker(\rho)$ by Lemma \ref{faithful}, there exists a generically free representation for $T_r\rtimes H_r$ by \cite[Lemma 3.3]{MR09}. Therefore, by \cite[Corollary 4.2]{Merkurjev09}, we have \begin{align*}
\ed_{2}(T_r\rtimes H_r)&\leq \rank(\Z[H_{r}/\langle\tau_{1}\rangle\times H_{r,x}])+\rank(\Z[H_{r}/\langle\tau_{2}\rangle\times H_{r,x}])\\
& +\rank(\Z[H_{r}/\langle\tau_{1}\tau_{2}\rangle\times H_{r,x}])+\rank(\Z[H_{r}/(\tau_{r}H_{r,x}\tau_{r}\cap H_{r,x})\rtimes \langle\tau_{r}\rangle])\\
&-\rank(J_r)\\
& =2^{r-1}+2^{r-1}+2^{r-1}+2^{r+(r-1)-2-1}-2^{r} \text{ (by Lemma \ref{usss}(1),(4)) }\\
& =2^{r-1}+2^{2r-4}.
\end{align*}




By \cite[Lemma 4.1]{MR09}, $\ed_{2}(T_r\rtimes G_r)=\ed_{2}(T_r\rtimes H_r)$. As the morphism $\Theta$ in (\ref{thetaa}) is surjective by Lemma \ref{surjectlemm}, we get \[\ed_{2}(\cat{Alg}_{2^{r},2})\leq \max\{\ed_{2}(T_r\rtimes G_r), \ed_{2}(\cat{Alg}_{2,2}), \cdots, \ed_{2}(\cat{Alg}_{2^{r-1},2})\}.\] By induction on $r$, we finally have $\ed_{2}(\cat{Alg}_{2^{r},2})\leq \ed_{2}(T_r\rtimes G_r)\leq 2^{r-1}+2^{2r-4}$.



\end{proof}


\begin{lemma}\label{faithful}
Let $\rho:\Lambda_{r}\to J_r$ be the morphism in proof of Theorem \ref{mainthm}. The action of $H_{r}$ on $\Ker(\rho)$ is faithful.
\end{lemma}

\begin{proof}
Note that $J_r\tens \Q=\Q[X_r]$ by the exact sequence (\ref{JXsequence}). Hence, by the exact sequence \[\Ker(\rho)\to \Lambda_{r}\xra{\rho} J_r,\] we have
\begin{align}\label{eeeE}
\Q[X_r]\oplus (\Ker(\rho))_{\Q} &=\Q[H_{r}/\langle\tau_{1}\rangle\times H_{r,x}]\oplus \Q[H_{r}/\langle\tau_{2}\rangle\times H_{r,x}]\oplus \Q[H_{r}/\langle\tau_{1}\tau_{2}\rangle\times H_{r,x}] \nonumber \\
& \oplus \Q[H_{r}/(\tau_{r}H_{r,x}\tau_{r}\cap H_{r,x})\rtimes \langle\tau_{r}\rangle].
\end{align}
By the actions of $\tau_{1}$ and $\tau_{2}$, the natural map \[i:\Z[X_r]\to \Z[X_r/\langle \tau_{1}\rangle]\oplus \Z[X_r/\langle \tau_{2}\rangle] \oplus \Z[X_r/\langle \tau_{1}\tau_{2}\rangle]\] is injective, hence we get the exact sequence \[0\to \Z[X_r]\xra{i} \Z[X_r/\langle \tau_{1}\rangle]\oplus \Z[X_r/\langle \tau_{2}\rangle] \oplus \Z[X_r/\langle \tau_{1}\tau_{2}\rangle]\to \Coker(i)\to 0\] and
\begin{equation}\label{faithfulexact}
\Q[X_r]\oplus (\Coker(i))_{\Q}=\Q[X_r/\langle \tau_{1}\rangle]\oplus \Q[X_r/\langle \tau_{2}\rangle] \oplus \Q[X_r/\langle \tau_{1}\tau_{2}\rangle].
\end{equation}
By (\ref{eeeE}) and (\ref{faithfulexact}), we have \[(\Ker(\rho))_{\Q}=(\Coker(i))_{\Q}\oplus \Q[H_{r}/(\tau_{r}H_{r,x}\tau_{r}\cap H_{r,x})\rtimes \langle\tau_{r}\rangle],\] thus it is enough to show that $H_{r}=\langle\tau_{1}\rangle \times \langle \tau_{2}\rangle \times [(P_{2^{r-3}})^{2}\rtimes \langle \tau_{r}\rangle]$ acts faithfully on $Y_r:=\Q[H_{r}/(\tau_{r}H_{r,x}\tau_{r}\cap H_{r,x})\rtimes \langle\tau_{r}\rangle]$:
\smallskip

Case $1$: $h=\tau_{i}h'$ or $\tau_{1}\tau_{2}h''\in H_{r}$ for $i=1,2$ and $h',h''\in (P_{2^{r-3}})^{2}\rtimes \langle \tau_{r}\rangle$.

If $h=\tau_{i}h'\in H_{r}$ for some $h'\in (P_{2^{r-3}})^{2}\rtimes \langle \tau_{r}\rangle$ and $i=1,2$, then $h\overline{\tau_{i}}=\overline{h'}\neq \overline{\tau_{i}}$ in $Y_r$. Similarly, if $h=\tau_{1}\tau_{2}h''\in H_{r}$ for some $h''\in (P_{2^{r-3}})^{2}\rtimes \langle \tau_{r}\rangle$, then $h\overline{\tau_{1}\tau_{2}}=\overline{h''}\neq \overline{\tau_{1}\tau_{2}}$ in $Y_r$.

\smallskip

Case $2$: $h\in (P_{2^{r-3}})^{2}\rtimes \langle \tau_{r}\rangle\backslash (\tau_{r}H_{r,x}\tau_{r}\cap H_{r,x})\rtimes \langle\tau_{r}\rangle$.

If $h\in (P_{2^{r-3}})^{2}\rtimes \langle \tau_{r}\rangle\backslash (\tau_{r}H_{r,x}\tau_{r}\cap H_{r,x})\rtimes \langle\tau_{r}\rangle$, then $h\overline{1}=\overline{h}\neq \overline{1}$ in $Y_r$.

\smallskip

Case $3$: $h\in (\tau_{r}H_{r,x}\tau_{r}\cap H_{r,x})\rtimes \langle\tau_{r}\rangle\backslash (\tau_{r}H_{r,x}\tau_{r}\cap H_{r,x})$.

Let $h=h_{1}h_{2}\tau_{r}\in (H_{r-1,x}\times H_{r-1,x})\rtimes \langle\tau_{r}\rangle$. We may assume that $h_{1}\neq 1$. Choose a transposition $\delta\in P_{2^{r-3}}$ which does not fix $x$. Then $h\overline{\delta}=\overline{h_{1}h_{2}\tau_{r}\delta}=\overline{h_{1}h_{2}\delta'\tau_{r}}=\overline{\delta'h_{1}h_{2}\tau_{r}}=\overline{\delta'}\neq \overline{\delta}$, where $\delta'$ is the transposition such that $\tau_{r}\delta=\delta'\tau_{r}$.

\smallskip

Case $4$: $h\in \tau_{r}H_{r,x}\tau_{r}\cap H_{r,x}$.

Let $h=h_{1}h_{2}\in H_{r-1,x}\times H_{r-1,x}=\tau_{r}H_{r,x}\tau_{r}\cap H_{r,x}$. We may assume that $h_{1}\neq 1$. Let $\tau_{r-1}$ be the permutation which acts on the same block with $h_{1}$. Then $h\overline{\tau_{r-1}}=\overline{h_{1}\tau_{r-1}}\neq \overline{\tau_{r-1}}$ in $Y_r$.

\smallskip
This completes the proof of faithfulness.

\end{proof}


\begin{corollary}
Let $F$ be a field of characteristic different $2$. Then \[\ed_{2}(\cat{Alg}_{16,2})=24.\]
\end{corollary}
\begin{proof}
The lower bound $24\leq \ed_{2}(\cat{Alg}_{16,2})$ follows from \cite[Theorem 6.1]{BM10}. Therefore, the statement follows from Theorem \ref{mainthm}.
\end{proof}

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\end{document}
