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\begin{document}

\title[Pfister forms]{Note on the cohomological invariant of Pfister forms}
 
\author{M.Tezuka and N.Yagita}
 


\address{ Department of mathematics, Faculty of Science, Ryukyu University, Okinawa, Japan\\
Department of Mathematics, Faculty of Education, 
Ibaraki University,
Mito, Ibaraki, Japan}
 
\email{ tez@sci.u-ryukyu.ac.jp,  yagita@mx.ibaraki.ac.jp, }
\keywords{Pfister forms, cohomological invariant, Dickson invariant}
\subjclass{Primary 11E72,  Secondary 12G05}

\begin{abstract}
The cohomological invariant ring of the $n$-Pfister forms is 
isomorphic to the invariant
ring 
in that of an elementary abelian $2$-group of rank $n$ under a
 $GL_n(\bZ/2)$-action.
\end{abstract}


\maketitle





\section{Introduction}
Let $G$ be an algebraic group over $k$ with $ch(k)\not =2$.
The cohomological invariant $Inv^*(G;\bZ/2)$ is (roughly speaking) the ring
of natural functors $H^1(F;G)\to H^*(F;\bZ/2)$ for the category of
finitely generated field $F$ over $k$.  (For details,  see the excellent book [Ga-Me-Se]). Moreover, we can define the cohomological invariant $Inv^*(Pfister_n;\bZ/2)$
of $n$-Pfister forms, while there does not exist the corresponding group $G$
for $n\ge 4$.
In Theorem 18.1 in [Ga-Me-Se], this invariant has been computed.  

In this note, we show that  this cohomological invariant is isomorphic to 
the invariant
ring in that of an elementary abelian $2$-group of rank $n$ under a
 $GL_n(\bZ/2)$-action, namely,
\[ Inv^*(Pfister_n;\bZ/2)\cong Inv^*((\bZ/2)^n;\bZ/2)^{GL_n(\bZ/2)}.\]


\section{motivic cohomology and cohomological invariant}
We recall the motivic cohomology $H^{*,*'}(X;\bZ/2)$
for a smooth scheme $X$ over $k$ with $ch(k)\not=2$.
By the Milnor conjecture (which is now solved by Voevosky), we know
$ H^{*,*'}(X;\bZ/2)\cong H_{et}^*(X;\bZ/2)$ for $*\le *'.$
Take $\tau \in H^{0,1}(Spec;\bZ/2)\cong \bZ/2$ as a nonzero element. 
It is known that
$ H^{*,*'}(X;\bZ/2)=0$ for $(*-*')>dim(X).$
Hence we have 
\[H^{*,*'}(Spec(k);\bZ/2)\cong H_{et}^*(Spec(k);\bZ/2)[\tau].\]  
Let us write $H^{*,*'}=H^{*,*'}(Spec(k);\bZ/2)$ and $H^*=H^{*}_{et}(Spec(k);\bZ/2)$
so that $H^{*,*'}\cong H^*[\tau].$

Let $BG$ be a classifying space of $G$ ([To], [Vo2]). 
Let $H^*(X;H_{\bZ/2}^{*'})$ be the sheaf cohomology 
 where $H_{\bZ/2}^n$ is the Zarisky sheaf induced from the
  presheaf $H_{et}^n(V;\bZ/p)$ for open subset $V$ of $X$. 
  Then Totaro proved that
  \[ Inv^*(BG;\bZ/2)\cong H^0(BG;H_{\bZ/2}^*)\]
  in a letter to Serre [Ga-Me-Se].
The Milnor conjecture implies the Beilinson and Lichtenbaum conjecture (see [Vo2,3]).
This fact  implies the following long exact sequences of 
motivic and sheaf cohomology theories
 (Lemma 3.1 in [Or-Vi-Vo], [Vo3])  
  \[  \to H^{m,n-1}(X;\bZ/2)\stackrel{\times \tau}{\to}
  H^{m,n}(X;\bZ/2)\qquad \qquad \qquad \]
  \[\qquad \qquad \to
H^{m-n}(X;H_{\bZ/2}^n)\to H^{m+1,n-1}(X;\bZ/2)\stackrel{\times \tau}{\to} .\]
Thus we have
\begin{thm} There is an additive isomorphism
\[ Inv^*(G;\bZ/2)\cong H^{*,*}(BG;\bZ/2)/(\tau)\oplus
Ker(\tau)|H^{*+1,*-1}(BG;\bZ/2).\]
\end{thm}

As an application, we first consider the case $G=\bZ/2$.  The $mod(2)$ motivic cohomology is computed
in [Vo1,2].
\[H^{*,*'}(B\bZ/2;\bZ/2)\cong  H^{*,*'}[y]\otimes \Delta(x) \]
with $\beta(x)=y$, hence $deg(y)=(2,1)$ and $deg(x)=(1,1)$.  Here 
Voevodsky shows ([Vo1,2])  
\[x^2=\tau y+\rho x\quad with \ \rho=(-1)\in H^1=k^*/(k^*)^2.\]


Next consider their product $G=(\bZ/2)^n$.  
The cohomology $H^{*,*'}(B\bZ/2;\bZ/2)$ has the Kunneth formula
(also by Voevodsky).
Hence the motivic cohomology
is given 
\[  H^{*,*'}(BG;\bZ/2)\cong H^{*,*'}[y_1,...,y_n]\otimes \Delta(x_1,..,x_n)\]
where $\beta(x_i)=y_i$ and $x_i^2=\tau y_i+\rho x_i$.
Hence from Theorem 2.1,  we get (as stated in [Ga-Me-Se])
\begin{lemma} Let $G$ be an elementary abelian $2$-group of $rank(G)=n$. Then
$ Inv^*(G;\bZ/2)\cong H^{*}\otimes\Delta (x_1,...,x_n)$ with $x_i^2=\rho x_i$. 
\end{lemma}









\section{Dickson invariants}


Recall that the mod $2$ (topological) cohomology
\[ H^*(B(\bZ/2)^n;\bZ/2)\cong \bZ/2[x_1,...,x_n]\quad |x_i|=1.\]
It is well known that the invariant ring under the $GL_n(\bZ/2)$-action  is 
 the Dickson algebra
\[H^*(B(\bZ/2)^n;\bZ/2)^{GL_n(\bZ/2)}\cong \bZ/2[d_{n,0},...,d_{n,n-1}]\]
where each generator $d_{n,i}$ is given by
\[ w_t(x)=\Pi_{\ep_i=0\ or\ 1}(t+\ep_1x_1+...\ep_nx_n)\]
\[     =t^{2^n}+d_{n,n-1}t^{2^{n-1}}+d_{n,n-2}t^{2^{n-2}}+...+d_{n,0}t.\]

{\bf Examples.} Let us write by $w_i$ the elementary symmetric function for $x_j$
in $H^*(B(\bZ/2)^n;\bZ/2)$.  Then
\[ \begin{cases} d_{2,1}=x_1^2+x_1x_2+x_2^2=w_1^2+w_2\\
               d_{2,0}=x_1^2x_2+x_1x_2^2=w_1w_2
               \end{cases}. \]
% $d_{3,2}=w_1^4+w_2^2+w_3w_1$.  
%We want to know invariants in the motivic cohomology $H^{*,*'}(B(\bZ/2)^n;\bZ/2)$.
%For example, the following elements are invariant for $n=2$  
%\[ d_1'=\tau y_1+x_1x_2+\tau y_2, \quad d_2'=y_1x_2+y_2x_1=\tau^{-1}w_1w_2.\]
%(Note that $y_1+x_1x_2+y_2$ is not a homogeneous element in $H^{*,*'}(B(\bZ/2)^2;\bZ/2).$)

We want to know the Dickson invariant ring in the cohomological invariant.
Let us consider 
\[ U=Inv^*_{k=\bR}((\bZ/2)^n;\bZ/2)\cong H^{*,*'}(B(\bZ/2)^n|_{\bR};\bZ/2)/(\tau)\]
\[ \cong\bZ/2[\rho]\otimes \Delta(x_1,...,x_n),\quad x_i^2=\rho x_i.\]
For example,  in $U$,  we see $ d_{2,0}=\rho x_1x_2+x_1 \rho x_2=0,$  and
\[d_{2,1}=\rho x_1+x_1x_2+\rho x_2=\rho w_1+w_2.\]

\begin{lemma}
In $U$, we have $d_{n,i}=0$ for $i<n-1$ and
\[ d_{n,n-1}=\sum _{i\ge 1}^nw_i\rho^{2^{n-1}-i}=
(\rho+x_1)...(\rho+x_n)\rho^{2^{n-1}-n}+\rho^{2^{n-1}}.\]
\end{lemma}
\begin{proof} Decompose that
\[ w_t(x)=\Pi(t+\ep_1x_1+...\ep_{n-1}x_{n-1})\times \Pi(t+x_n+\ep_1x_1+...\ep_{n-1}x_{n-1}).\]
By induction on $n$, we assume this element is
\[ (t^{2^{n-1}}+d_{n-1,n-2}t^{2^{n-2}})((t+x_n)^{2^{n-1}}+d_{n-1,n-2}(t+x_n)^{2^{n-2}}).\]
Letting $d_{n-1,n-2}=d$, $t^{2^{n-2}}=T$ and $x^{2^{n-2}}=X$, we see that the above formula is
\[ =(T^2+dT)(T^2+X^2+dT+dX)\]
\[ =T^4+(d^2+dX+X^2)T^2+(d^2X+dX^2)T.\]
Here note $X^2=\rho^*X=\rho^{*'}x_n$, $d^2=\rho^*d$ (since $(\rho+x)^2=\rho(\rho+x)$).  So $(d^2X+dX^2)=0$.
Let $a=a_{n-1}=(\rho+x_1)...(\rho+x_{n-1})$.  Then  we have
\[  d^2+dX+X^2=\rho^*d+\rho^{*-1}dx_n+\rho^{*'}x_n\]
\[ =\rho^{\sharp}(a+\rho^{*''})+\rho^{\sharp-1}(a+\rho^{*''})x_n+\rho^{*'}x_n\]
\[=\rho^{\sharp-1}a(\rho+x_n)+\rho^{\sharp''}  =\rho^{\sharp-1}a_{n}+\rho^{\sharp''}\]
as desired.
\end{proof}

\begin{cor} Let us write
\[e_{n}=\rho^{-2^{n-1}+n}d_{n,1}=\sum_{i\ge 1}^{n}w_i\rho^{n-i}=
(\rho+x_1)...(\rho+x_{n})+\rho^n.\]
Then we have 
\[Inv^*((\bZ/2)^n;\bZ/2)^{GL_n(\bZ/2)}\cong H^*\{1,e_n\}.\]
\end{cor}
\begin{proof}
By $Ideal(\rho)$, we consider the associated graded algebra
\[gr(H^*\otimes \Delta(x_1,...,x_n))\cong gr(H^*)\otimes \Lambda(x_1,...,x_n)\quad (x_i^2=0).\]
Note $e_n=w_n$ in the above graded algebra.
We can see
\[\Lambda(x_1,...,x_n)^{GL_n(\bZ/2)}\cong \bZ/2\{1,w_n\}\]
as following arguments. 

Since $x_i^2=0$, the only possibility of invariant is $w_s$.
Suppose $s<n$.  Write
\[w_s=x_1(\sum_{1\not =i_k}x_{i_1}...x_{i_{s-1}})+
(\sum_{1\not =i_k}x_{i_1}...x_{i_{s}}).\]
Consider the action $x_{12}:x_1\mapsto x_1+x_2$ but
$x_{12}:x_i\mapsto x_i$  for $i>1$.  Then
\[(x_{12}-1)w_s=x_2(\sum_{1\not =i_k}x_{i_1}...x_{i_{s-1}})
=x_2x_3...x_{s+1}+...\not =0 \quad in\  \Lambda(x_1,...,x_n).\]

All elements in $H^*$ and $e_n$ are really invariants in 
$H^*\otimes \Delta(x_1,...,x_n)$.  Thus we have the corollary.
\end{proof}

Let $n=2$ and $ G=SO_3$ or $n=3$ and $G=G_2$ the exceptional group.
Then $G$ has only  one conjugacy class $A_n$ of maximal elementary abelian
$2$ groups of rank $n$.  The Weyl group $W_G(A_n)\cong GL_n(\bZ/2)$.
Hence we have the restriction map
\[ Inv^*(G;\bZ/2)\to Inv^*(A_n;\bZ/2)^{W_G(A_n)}\cong H^*\{1,e_n\}.\]
The result in [Ga-Me-Se] shows this map is an isomorphism.



\section{Pfister forms}

The most important quadratic forms are Pfister forms.
Given $a=(a_1,...,a_n)\in (k^*/(k^*)^2)^{\times n}$,
the $n$-th Pfister form $P_a$ is the quadratic form defined as
\[P_a=\la\la a_1,...,a_n\ra\ra =\la 1,-a_1\ra\otimes  ...\otimes \la 1,-a_n\ra \]
\[=\oplus_{1\le i_1<...<i_s\le n} \la (-a_{i_1})...(-a_{i_s}) \ra.\]

Given a quadratic form $q_a=\la a_1,...,a_n\ra$, the total Stiefel-Whitney class
is given by 
\[w_t(q_a)=\Pi (t+a_i).\]
Hence we know
\[ w_t(P_a)=\Pi_{\ep_i=0\ or\ 1}(t+\ep_1(\rho+x_1)+...+\ep_n(\rho+x_n))\]
identifying $x_i=(a_i)$.
Hence the following proposition follows the preceding lemma. 
(Substitute $x_i+\rho$ for $x_i$ in the right hand side of the euation in
Lemma 3.1.)
\begin{prop}
Let $x_i=(a_i)\in k^*/(k^*)^2$ and $w_n=x_1...x_n$.  Then
\[w_t(P_a)=t^{2^n}+(w_n+\rho^n)\rho^{2^{n-1}-n}t^{2^{n-1}}.\]
\end{prop}

Next consider the map
from $(k^*/(k^*)^2)^{\times n}$ to the set of $n$-th Pfister forms $Pfist_n$
defined by
\[p:a=(a_1,...,a_n)\mapsto P_{-a}=\la\la -a_1,...,-a_n\ra\ra=\la 1,a_1\ra \otimes...\otimes
\la 1,a_n\ra.\]
 This map induces the map
of cohomological invariants
\[ p^*:Inv^*(Pfister_n;\bZ/2)\to Inv^*((k^*/(k^*)^2)^{\times n};\bZ/2).\]
Here the last invariant is isomorphic to
\[Inv^*((\bZ/2)^n;\bZ/2)\cong H^*\otimes \Delta(x_1,...,x_n).\]

On $ H^*\otimes \Delta(x_1,...,x_n)$, we can define the usual $GL_n(\bZ/2)$-action.
This action is also written as follows.
Consider the Bruhat decomposition
\[GL_n(\bZ/2)=\coprod_{w\in S_n} BwB\]
 where $B$ is the Borel group generated by
upper triangular matrices, and $S_n$ is the $n$-th symmetric group generated by
transition matrices.  The group $B$ is generated by $x_{ij}=1+{e_{ij}}$ ; the elementary matrix
with $(i,j)$ entries $1$ with the relations 
\[ x_{ij}^2=1,\quad [x_{ij},x_{kl}]=\begin{cases} x_{il}\quad if\ j=k\\
  0\quad otherwise.\end{cases}\] 
  
  Define $w(x_i)=x_{w(i)}$ for $w\in S_n$ and
 \[ x_{ij}(x_i)=x_i+x_j,\quad x_{ij}(x_k)=x_k\ \ for\ i\not =k.\]
 Then the $GL_n(\bZ/2)$-action is decided
 on   $Inv^*((\bZ/2)^n;\bZ/2)$.

  
   
 
\begin{thm}
The above map $p^*$ induces the isomorphism
\[Inv^*(Pfister_n;\bZ/2)\cong H^*\otimes \Delta(x_1,...,x_n)^{GL_n(\bZ/2)}\cong
H^*\{1,e_n\}.\]
\end{thm}
\begin{proof}
On $(k^*/(k^*)^2)^{\times n}$, we can define the $GL_n(\bZ/2)$-action  by
  \[ x_{ij}(a_1,...a_n)=(a_1,...,a_{i-1},a_ia_j,a_{i+1},...,a_{n}),\]
  \[w(a_1,...,a_n)=(a_{w(1)},...,a_{w(n)}).\]
  This induces the action on $Inv^*((\bZ/2)^n;\bZ/2)$ by $w(x_i)=x_{w(i)}$ and 
   \[ x_{ij}(x_i)=x_i+x_j,\quad x_{ij}(x_k)=x_k\ \ for\ i\not =k.\]
   
 Define a $GL_n(\bZ/2)$ action on $Pfister_n$ by $x_{ij}p(a)=p(x_{ij}(a))$.
Then this action is invariant,  indeed,
   \[ x_{12}\la\la a_1,a_2\ra\ra=px_{12}(-a_1,-a_2)=p(a_1a_2,-a_2)\]
   \[=\la\la -a_1a_2,a_2\ra\ra =\la 1, a_1a_2,-a_2,-a_1a_2^2\ra\]
   \[ =\la 1,a_1a_2,-a_2,-a_1\ra=\la\la a_1,a_2\ra\ra.\]
  Hence     we have the map
   \[ q^*: Inv^*(Pfister_n;\bZ/2)\to H^*\otimes \Delta(x_1,...,x_n)^{GL_n(\bZ/2)}.\]
   Since $P_{-a}\mapsto e_n=w_n(P_{-a})+\rho^n$ is a cohomology invariant and hence the above map is epic.
   
   For each finitely generated field $F$ over $k$, the map
   $p: (K^*/(K^*)^2)^n\to Pfister_n|_{K}$ is of course an epimorphism.  Hence 
the induced map  
  \[p^*(x): (K^*/(K^*)^2)^n\to Pfister_n\stackrel{x}{\to} H^n(K;\bZ/2)\]
   is always injective.
    \end{proof}
    If we consider the map
    \[q:a=(a_1,...,a_n)\mapsto P_{a}=\la\la a_1,...,a_n\ra\ra,\]
   then the map $q^*$ also induces the isomorphism
    \[Inv^*(Pfister_n;\bZ/2)\cong H^*\otimes \Delta(x_1,...,x_n)^{GL_n(\bZ/2)_{II}}\cong
H^*\{1,w_n\}\]
where $GL(\bZ/2)_{II}$ is the unusual action defined by
$x_{ij}(x_i)=\rho+x_i+x_j$.
    



 
 
  \begin{thebibliography}{Ra-Wi-Ya}
  
  
  
 


\bibitem[Ga-Me-Se]{Ga-Me-Se} 
S.Garibaldi, A.Merkurjev and J.P.Serre.
\newblock  Cohomological invariants in Galois cohomology.
\newblock \emph{University lect. series vol(28) AMS}  (2003).



 \bibitem[Or-Vi-Vo]{Or-Vi-Vo} 
D.Orlov,A.Vishik and V.Voevodsky.
\newblock An exact sequence for Milnor's K-theory with applications to quadric forms.
\newblock \emph{Ann. of Math.} \textbf{165} (2007) 1-13.





\bibitem[To]{To} 
B.~Totaro.  
\newblock The Chow ring of classifying spaces. 
\newblock \emph{Proc.of Symposia in Pure Math. "Algebraic K-theory" 
(1997:University of Washington,Seattle)} \textbf{67} (1999), 248-281.



\bibitem[Vo1]{Vo1} 
V.~Voevodsky. 
\newblock Motivic cohomology with $\bZ/2$ coefficient.
\newblock \emph{Publ. Math. IHES}
\textbf{98} (2003), 59-104.

\bibitem[Vo2]{Vo2} 
V.Voevodsky. 
\newblock Reduced power operations in motivic cohomology.
\newblock \emph{Publ.Math. IHES}
\textbf{98} (2003),1-57.


\bibitem[Vo3]{Vo3} 
V.Voevodsky. 
\newblock On motivic cohomology with $\bZ/l$-coefficients.
\newblock \emph{www.math.uiuc.edu/K-theory/0631}    (2003).







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