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\title{Essential $p$-Dimension of PGL$_n$}
\author
[A. Ruozzi]
{Anthony Ruozzi}

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

\email{aruozzi@math.ucla.edu}

\begin{document}
\maketitle
\section{Introduction}
$k$ a fixed base field of characteristic $\not = p$.  All fields mentioned in this paper are assumed to contain $k$.

Consider any functor $\mathcal{F}:\mathbf{Fields}/k \rightarrow \mathbf{Set}$.  We say that an element $a \in \mathit{F}(K)$ is defined over $k \subset K_0 \subset K$ if it is in the image of the map $\mathcal{F}(K_0) \rightarrow \mathcal{F}(K)$.  The essential dimension, $\mathrm{ed}_k(a)$, is the least transcendence degree$/k$ of a field of definition for $a$.  The essential dimension of $\mathit{F}$, $\mathrm{ed}_k(\mathcal{F}) = \mathrm{sup} \lbrace \mathrm{ed}_k (a) \rbrace$ where the supremum is taken over all $a \in K$ for all field extensions $K/k$.  The basic properties of this definition are outlined in [BF].

In this paper, we will be interested in the slightly simpler computation of essential $p$-dimension.  Here, we are allowed some extra flexibility: $\mathrm{ed}_k(a;p)$ is the minimum essential dimension of the image of $a$ in $\mathcal{F}(L)$ over all $L/K$ finite prime to $p$ extensions.  As above, we define $\mathrm{ed}_k(\mathcal{F};p)$ as the supremum of the essential $p$-dimensions over all elements and fields, $a \in K$.

A field $F/k$ is called $p$-closed if every finite extension of $F$ has degree prime to $p$.  For limit-preserving functors, the essential $p$-dimension of any element can be computed over a $p$-closure [LMMR Lemma 3.3], so we can always assume that our fields $F$ are $p$-closed.

A natural transformation of functors $\mathcal{F} \rightarrow \mathcal{G}$ will be called $p$-surjective if for any $K/k$, there is a finite extension $L/K$ of degree prime to $p$ such that $\mathcal{F}(L) \twoheadrightarrow \mathcal{G}(L)$.  More specifically, if $F/k$ is $p$-closed and $\mathcal{F}(F) \twoheadrightarrow \mathcal{G}(F)$, then the map is $p$-surjective and $\mathrm{ed}_k(\mathcal{F};p) \geq \mathrm{ed}_k(\mathcal{G};p)$ [LMMR Prop 3.4].

Of particular interest are the functors $\mathrm{H}^1(-,G)$ for an algebraic group $G/k$.  For ease of notation, the essential dimension of such functors will be denoted $\mathrm{ed}(G)$.  We will be studying the functor $\mathrm{H}^1(F,\mathrm{PGL}_n)$ which classifies central simple algebras$/F$ of degree $n$.  In what follows, this functor will be denoted by $\mathrm{Alg}_n(-)$.  Its essential dimension gives the least number of parameters needed to define a \textquotedblleft generic" central simple algebra of degree $n$.

Because we can apply prmiary decomposition to central simple algebras, the computation of $\mathrm{ed}(\mathrm{Alg}_n(-);p)$ reduces to a computation of $\mathrm{ed}(\mathrm{Alg}_{p^s}(-);p)$ where $p^s$ is the largest power of $p$ dividing $n$.  It is well-known that $\mathrm{ed}(\mathrm{Alg}_{p};p)=2$; cf. [R2].  Recently, Meyer and Reichstein gave an upper bound for $s \geq 2$ [MR2 Theorem 1.1]:
\[\mathrm{ed}(\mathrm{Alg}_{p^s};p) \leq 2p^{2s-2}-p^s+1\]
and conjectured that this bound is sharp.  The goal of this paper is to further strengthen this result.

\begin{thm}
$\mathrm{ed}(\mathrm{Alg}_{p^s}) \leq p^{2s-2} +1$ for $s \geq 2.$
\end{thm}

For $s=2$, Merkurjev showed that this bound is sharp [M], so we can consider $s \geq 3$.

\section{Essential Dimension and Tori}
Throughout this section, let $F/k$ be an arbitrary field extension.  Fix a finite group $G$ and a finite $G$-set, $X$, of $n$ elements.  Consider the augmentation exact sequence of $G$-modules:
\[0 \rightarrow I \rightarrow \mathbb{Z}[X] \rightarrow \mathbb{Z} \rightarrow 0.\]
Construct any resolution of $I$
\[0 \rightarrow M \rightarrow P \rightarrow I \rightarrow 0\]
where $P$ is a permutation module.  Fixing bases and using the usual anti-equivalence of categories, this sequence corresponds to an exact sequence of algebraic tori split over $F$:
\[1 \rightarrow T \rightarrow U \rightarrow S \rightarrow 1\]
where $U=\mathrm{GL}_1(E)$ is the diagonal subgroup in $\mathrm{GL}_F(E)$ for the split \'{e}tale algebra $E/F$ corresponding to $P$.  Thus we have a faithful representation $T \hookrightarrow \mathrm{GL}_F(E)$.

Since $T$ acts on $E$ via this representation, $G$ acts on $E$ by algebra automorphisms, and the above representation is $G$-equivariant, this extends to a representation $T \rtimes G \rightarrow \mathrm{GL}_F(E)$.  We will construct an upper bound for the essential dimension using the following important result:

\begin{thm} If $G$ acts faithfully on $M$ in the resolution of $I$ constructed above, then $\mathrm{ed}(T \rtimes G) \leq \mathrm{rank}(P) - \mathrm{rank}(I) = \mathrm{rank}(M).$
\end{thm}
\begin{proof}
This result is a combination of [R Theorem 3.4] and [MR1 Lemma 3.3].
\end{proof}

Since we a interested in essential $p$-dimension, the following result will also be useful:
\begin{prop}
Let $H=\mathrm{Syl}_p(G)$.  Then $\mathrm{ed}(T \rtimes G;p)=\mathrm{ed}(T \rtimes H;p)$.
\end{prop}
\begin{proof}
This is just a special case of [MR1 Lemma 4.1].
\end{proof}

\section{Division Algebras}
In what follows we are interested in computing the essential $p$-dimension, so we can assume that $F \supset k$ is $p$-closed.  Let $D/F$ be a central division algebra of degree $n=p^s$ with $s \geq 2$.  The following results will be useful:

\begin{prop}
$D$ contains degree $p$ cyclic extension of $F$.
\end{prop}
\begin{proof}
See [RS Prop 1.1].
\end{proof}

\begin{thm}
Let $L_1 \subset D$ be a degree $p$ cyclic extension$/F$.  Then there is another degree $p$ cyclic exentsion $L_2/F$ contained in $D$ such that $L_1L_2 \subset D$ is a bicyclic extension.
\end{thm}
\begin{proof}
Fix a generator $<\sigma> = \mathrm{Gal}(L_1/F)$.  The Skolem-Noether theorem gives an element $y \in D^{\times}$ such that $yxy^{-1}=\sigma(x)$ for all $x \in L_1$.  There are two possibilities:

$\mathbf{y^p \in F}$:
In this case, $y^p = a$ defines a cyclic algebra$/F$ $B = (L_1,a)$.  By the double centralizer theorem $C_DB$ is division algebra$/F$ of degree $\geq p$.  Thus, $C_DB \subset D$ has a cyclic subfield $L_2$.  Since $L_1 \otimes L_2 \subset B \otimes C_DB \simeq D$ is a subfield, $L_1 \otimes L_2 \simeq L_1L_2$ is bicyclic as desired.

$\mathbf{y^p \not \in F}$:
Let $K = F(y^p)$.  By the proposition, $K$ contains a cyclic subfield $L_2$.  Any element of $L_2$ commutes with $F(y)$.  Then $x \in L_1 \cap L_2$ commuting with $y$ implies that $\sigma$ acts trivially, so $L_1$ and $L_2$ are disjoint and give $L_1L_2$ bicyclic.  
\end{proof}

\begin{cor}
For $A$ any central simple algebra$/F$ of degree $p^s \geq p^2$ and $L_1 \subset A$ an \'{e}tale sub-algebra of degree $p$, there is a maximal \'{e}tale sub-algebra $K \subset A$ that can be written as $K=L_1 \otimes_F L_2$  for $L_2/F$ an \'{e}tale algebra of dimension $p^{s-1}$.
\end{cor}

\begin{proof}
First, consider the case where $A$ is a division algebra.  By the theorem, $A$ contains two distinct degree $p$ cyclic extensions $L_1$ and $L$ over $F$.  Proceed by induction on $s$.

If $s=2$ , then taking $L_2=L$ we are done.  Otherwise, assume we have the result for any division algebra of degree $p^{s-1}$ and any degree $p$ subfield $L'$.  The centralizer $C_AL$ is a division algebra over $L_1$ of degree $p^{s-1}$.  By definition it contains the degree $p$ subfield $L_1L/L$, so by induction hypothesis,  $C_AL$ has subfield $L_2/L$ disjoint from $L_1L$ of degree $p^{s-2}$.

Since $L_2 \cap L_1L = L$ and $L$ is disjoint from $L_1$, $L_2 \cap L_1 = F$.  It follows that $L_2/F$ is a degree $p^{s-1}$ extension disjoint from $L_1$ and $L_1 \otimes L_2 \subset A$.

Suppose $A$ is not a division algebra.  If $A$ is split, the result is immediate.  Otherwise, choose a division algebra $D \sim A$.  $\mathrm{deg}(D) = p^t \leq p^s = \mathrm{deg}(A)$.  If $t = 1$, then any maximal subfield $K \subset D$ gives the desired \'{e}tale algebra $K \otimes F^{\times p^{s-1}} \subset A$, so suppose $t \geq 2$.  By the above argument, $D$ has a subfield $K=L_1L_2 \simeq L_1 \otimes L_2$ where $[L_1:F]=p$ and $[L_2:F]=p^{t-1}$.  Set $L=K^{\times p^{s-t}}$ an \'{e}tale sub-algebra of $B$.  Since $L$ has dimension $p^s$, it is maximal.  But $L \simeq L_1 \otimes L_2^{\times p^{s-t}}$, so $L_1$ and $L_2^{\times p^{s-t}}$ are \'{e}tale algebras of the desired degree.
\end{proof}

Consider the functor $\mathrm{H}^1(F, S_n)$ which classifies $n$-dimensional \'{e}tale algebras up to isomorphism [KMRT 29.9].  Let $G = S_p \times  S_{p^{s-1}}$.  We have the usual isomorphism
\[\mathrm{H}^1(F,S_p) \times \mathrm{H}^1(F,S_{p^{s-1}}) \rightarrow \mathrm{H}^1(F,G). \]
Converting this to the laguange of algebras, $\mathrm{H}^1(F,G)$ can be identified as pairs $(L_1, L_2)$ of \'{e}tale algebras of dimensions $p$ and $p^{s-1}$, respectively.  Under the natural inclusion
\[\mathrm{H}^1(F,G) \rightarrow \mathrm{H}^1(F,S_{p^s}),\]
the image of such a pair is the \'{e}tale algebra $L_1 \otimes L_2$.  Let $K=L_1 \otimes L_2 \subset A$ be given for a central simple algebra $A$ as in the corollary.  Using these identifications, any such $K \subset A$ can be viewed as an element of $\mathrm{H}^1(F,G)$.

Recall the notation of section 2.  Consider the split torus $T$ for $G$ as above and $X$ a $G$-set of $p^s$ elements.  For its cohomology group, we have a disjoint union of fibers
\[\mathrm{H}^1(F,T \rtimes G) = \coprod_{\gamma \in \mathrm{H}^1(F,G)}\mathrm{H}^1(F,T_{\gamma})/G_{\gamma}^{\Gamma}\]
where $T_{\gamma}$ denotes $T$ with action twisted by the cocycle $\gamma$ and $\Gamma=\mathrm{Gal}(F^{\mathrm{sep}}/F)$ [KMRT 28.C].  $T_{\gamma}$ is also a torus with character module $I$.  In particular, if $X$ corresponds to a $p^s$-dimensional \'{e}tale algebra $N$ representated by $\gamma$ then the usual anti-equivalence of categories gives an exact sequence
\[1 \rightarrow \mathbb{G}_m \rightarrow \mathrm{R_{N/F}(\mathbb{G}_{m,N})} \rightarrow T_{\gamma} \rightarrow 1.\]
Passing to cohomology and applying Hilbert theorem 90 gives
\[1 \rightarrow \mathrm{H}^1(F,T_{\gamma}) \rightarrow \mathrm{H}^2(F,\mathbb{G}_m) \rightarrow \mathrm{H}^2(F,R_{N/F}(\mathbb{G}_m))\]
showing that $\mathrm{H}^1(F,T_{\gamma}) \simeq \mathrm{Br}(N/F)$.  Now, $G_{\gamma}^{\Gamma}$ acts on $\mathbf{G}_m$ trivially, so given $g \in G_{\gamma}^{\Gamma}$, we get a diagram:

\[\begin{CD}
1 @>>> \mathrm{H}^1(F,T_{\gamma}) @>>> \mathrm{H}^2(F,\mathbb{G}_m)\\
@. @VVgV @|\\
1 @>>> \mathrm{H}^1(F,T_{\gamma}) @>>> \mathrm{H}^2(F,\mathbb{G}_m)
\end{CD}\]
and the commutativity implies that the action of $g$ must be trivial for all $g \in G_{\gamma}^{\Gamma}$.  
Combining these observations with the above, 
\[\mathrm{H}^1(F,T \rtimes G) = \coprod_{N}\mathrm{Br}(N/F),\]
and thus we can define a map $\phi_G(F):\mathrm{H}^1(F,T \rtimes G) \rightarrow \mathrm{Alg}_{p^s}(F)$ which sends $[B] \in \mathrm{Br}(N/F)$ to the unique (up to isomorphism) $C \sim B$ with degree $C = p^s$.  Since any $A \in \mathrm{Alg}_{p^s}$ is split over $K= L_1 \otimes L_2$, it is in the image of this map.  We have proven:  

\begin{prop}
$\phi_{G}$ is $p$-surjective.
\end{prop}

Let $G_s = \mathrm{Syl}_p(S_p \times S_{p^{s-1}})=\mathrm{Syl}_p(S_p) \times \mathrm{Syl}_p(S_{p^{s-1}}) := \Sigma_1 \times \Sigma_{s-1}$.  Using Proposition 2.2 and the property of $p$-surjective maps stated in the introduction, 
\[\mathrm{ed}_k(T \rtimes G_s;p)= \mathrm{ed}_k(T \rtimes (S_p \times S_{p^{s-1}});p) \geq \mathrm{ed}_k(\mathrm{Alg}_{p^s}(-);p).\] 
We can therefore reduce to the computation of the essential dimension of $T \rtimes G_s$. The calculations for this case will be done in the next section.


\section{Construction}
We will produce an upper bound for the essential dimension of $T \rtimes G_s$.  By section 2, this requires finding a faithful $G_s$-module, $M$, in a resolution of $I$ of smallest rank.  In what follows, we denote $G_s=\Sigma_{s-1} \times \mathbb{Z}/p = \Sigma_{s-1} \times <\sigma>$, where as above $\Sigma_s = \mathrm{Syl}_p(S_{p^s})$.

First observe that $G_s \subset S_{p^s}$ acts by permutation on a set $X_s$ of $p^s$ elements.  This action can be descrbibed as an action on $p$ blocks of $p^{s-1}$ elements where $\sigma$ cyclically permutes the blocks and $\Sigma_{s-1}$ acts as usual on a block of $p^{s-1}$ elements.    In particular, the action is transitive, so if $H_s = \mathrm{Stab}(x)$ for some $x \in X_s$, then $X_s \simeq G_s/H_s$ as $G_s$-sets.
$\\$
$\\$
\noindent Begin with the case $s=2$.
$\\$
$G_2 = \Sigma_1 \times \Sigma_1 = <\tau_1> \times <\sigma>$.  No non-trivial element of $G_2$ fixes $x \in X_2$, so $H_2 =1$.  Identifying $X_2 \simeq G_2$, $I$ is generated by $\sigma - 1$ and $\tau_1 - 1$ as a $G_2$-module.  Since $G_2$ is abelian, the map
\[\mathbb{Z}[G_2] \oplus \mathbb{Z}[G_2] \rightarrow I \]
defined by sending a generator of the first term to $\sigma - 1$ and a generator of the second to $\tau_1 - 1$ is a well-defined $G_2$-module homomorphism.  It is surjective by definition.  The kernel of this map, $M$, has rank:
\[\mathrm{rank}(M)= 2 \; \mathrm{rank}(\mathbb{Z}[G_2])-\mathrm{rank}(I)= 2p^2 - (p^2 -1) = p^2 +1.\]
\noindent For the general case,
$\\$
$\Sigma_{s-1}=(\Sigma_{s-2})^p \rtimes \mathbb{Z}/p = (\Sigma_{s-2})^p \rtimes <\tau_{s-1}>$. We will show inductively that $H_s = H_{s-1} \times (\Sigma_{s-2})^{p-1}$ and $G_s=<\sigma,\tau_{s-1},H_s>$.  

The case $s=2$ was verified above.  Suppose the formula holds for $s-1 \geq 2$. $\Sigma_{s}$ acts on a set $X_{s}$ of $p^{s}$ elements, and $H_s=\mathrm{Stab}(x)$.  As above, this action can be thought of as an action on $p$ blocks of $p^{s-1}$ elements.  On any of these blocks, $\Sigma_{s-1} \subset G_s$ acts as usual by considering it as a collection of $p$ blocks of $p^{s-2}$ elements.  Therefore, we see that to be the stabilizer of $x$ is to stabilize the block containing $x$ and allow the others to be permuted freely.  That is, $H_s=H_{s-1} \times (\Sigma_{s-2})^{p-1}$.

Now, $\tau_{s-2} \in <\tau_{s-1},(\Sigma_{s-2})^{p-1}>$ and since $<\tau_{s-2},H_{s-1}>=\Sigma_{s-2}$ by assumption, we can conclude that $G_s = <\sigma,\tau_{s-1},H_s>$.  By an easy argument, $I$ is then generated by $\sigma x - x$ and $\tau_{s-1} x - x$; cf. [MR2 proof of Theorem 4.1].  Setting $H'_s=\tau_{s-1} H \tau_{s-1}^{-1} \cap H \simeq H_{s-1} \times H_{s-1} \times (\Sigma_{s-2})^{p-2}$, we can define a map as above
\[\mathbb{Z}[G_s/H_s] \oplus \mathbb{Z}[G_s/H'_s] \rightarrow I\]
by sending a generator of the first to $\sigma x - x$ and a generator of the second to $\tau_{s-1} x - x$.  This is well-defined since $H'_s$ is exactly the subset of $G_s$ that fixes $\tau_{s-1} x - x$.  We then have constructed a surjective $G_s$-module map with $\mathrm{rank}(M)=$
\[\mathrm{rank}(\mathbb{Z}[G_s/H_s]) + \mathrm{rank}(\mathbb{Z}[G_s/H'_s]) -\mathrm{rank}(I)= p^s + p^{s+(s-2)} - (p^s -1) = p^{2s-2}+1.\]

\section{Conclusions}
We now complete the proof of the theorem stated in the introduction. Recall that we are assuming that the base field $k$ has characteristic $\not = p$.

\begin{thm} 
$\mathrm{ed}(\mathrm{Alg}_{p^s};p) \leq p^{2s-2} +1$ for $s \geq 2$.
\end{thm}
\begin{proof}
For $F$ $p$-closed, the construction in the previous section produced a $G_s$-module $M$ of rank $p^{2s-2}+1$.  By section 2, it remains to show that the $G$-action on $M$ is faithful (cf. [MR Lemma 3.2] for a more general argument).  Faithfulness can be checked over $\mathbb{Q}$, so we have the split exact sequences:

\[0 \rightarrow I \otimes \mathbb{Q} \rightarrow \mathbb{Q}[G_s/H_s] \rightarrow \mathbb{Q} \rightarrow 0\]
\[0 \rightarrow N \rightarrow \mathbb{Q}[G_s/H'_s] \rightarrow \mathbb{Q}[G_s/H_s] \rightarrow 0\]
\[0 \rightarrow M \otimes \mathbb{Q} \rightarrow \mathbb{Q}[G_s/H_s] \oplus \mathbb{Q}[G_s/H'_s] \rightarrow I \otimes \mathbb{Q} \rightarrow 0\]

Combining these together,
\begin{eqnarray}
 (M \otimes \mathbb{Q}) \oplus (I \otimes \mathbb{Q})  
   &\simeq& \mathbb{Q}[G_s/H_s] \oplus \mathbb{Q}[G_s/H'_s]      \nonumber \\
   &\simeq& \mathbb{Q}[G_s/H_s] \oplus N \oplus \mathbb{Q}[G_s/H_s] \nonumber \\
   &\simeq& \mathbb{Q}[G_s/H_s] \oplus N \oplus \mathbb{Q} \oplus (I \otimes \mathbb{Q}). \nonumber 
\end{eqnarray}
Therefore, $\mathbb{Q}[G_s/H_s]$ is a direct summand of $M \otimes \mathbb{Q}$, so it suffices to check that the $G_s$ action on $\mathbb{Q}[G_s/H_s]$ is faithful.  However, if the coset $gH_s$ is fixed by every element in $G_s$, then $g \in \bigcap_{g_s \in G_s} g_sH_sg_s^{-1}$.  A quick induction argument shows that for all $s \geq 2$ this group is trivial.

Since then the action is faithful, we have the bound:
\[ p^{2s-2} +1 \geq \mathrm{ed}(T \rtimes G_s; p) \geq \mathrm{ed}(\mathrm{Alg}_{p^s}; p). \qedhere \]
\end{proof}
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\end{document}
