%% Copyright (C) 1991, 1998 by Markus Rost.
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%% French text translated from the English by Jean-Pierre Serre.
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%% "A (mod 3) invariant for exceptional Jordan algebras"
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%% Comptes rendus de l'Academie des sciences - serie I - Mathematique
%% C. R. Acad. Sci. Paris S\'er. I Math. 313 (1991), no. 12, 823-827.
%% Note pres\'ent\'ee par Jean-Pierre Serre.
%% Note remise le 30 septembre 1991, accept\'ee le 1-er octobre 1991.
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\begin{document}

\title{A (mod 3) invariant for exceptional Jordan algebras}
\author{Markus Rost}
\address{NWF I - Mathematik,
  Universit\"at Regensburg,
  D-93040 Regensburg, Germany}
\email{markus.rost@mathematik.uni-regensburg.de}
\urladdr{http://www.physik.uni-regensburg.de/\~{}rom03516}
\renewcommand{\datename}{}
\date{The text appeared as: \frenchspacing C. R. Acad. Sci. Paris
S\'er. I Math. 313 (1991), no. 12, 823--827.}
\maketitle

\begin{Abstracts}{Abstract}
  We sketch the construction of an invariant in $H^3(F,\IZ/3)$ for
  simple exceptional Jordan algebras over a field~$F$ with $\car F
  \neq2$,~$3$.  This invariant vanishes if and only if the algebra has
  zero divisors.  In case $\mu_3\subset F^*$ the invariant can be
  lifted to Milnor's $K$-group $K_3^MF/3$.

  \resume{Un invariant (mod 3) pour les alg\`bres de Jordan
  exceptionelles}{R\'esum\'e} Nous d\'ecrivons la construction d'une
  invariante dans $H^3(F,\IZ/3)$ pour des alg\`ebres de Jordan simples
  exceptionelles sur un corps~$F$ de caracteristique $\neq 2$,~$3$.
  Cette invariante est\/~$0$ si et seulement si l'alg\`ebre a des
  diviseurs des zero.  Lorsque $F$ contient les racine cubiques de
  l'unit\'e, cet invariante provient d'un invariant \`a valeurs dans
  le groupe de Milnor $K_3^MF/3$.
\end{Abstracts}

\begin{VersFrancAbr}
  Soit $F$ un corps de caract\'eristique $\neq2$,~$3$.  Soit $J$ une
  $F$-alg\`ebre de Jordan centrale simple exceptionnelle, de
  dimension~$27$.  Nous d\'ecrivons la construction d'un
  \emph{invariant} (\`a isomorphisme pr\`es) $g(J)\in H^3(F,Z/3)$.
  Cet invariant est~$0$ si et seulement si $J$ a des diviseurs de
  z\'ero.  Le foncteur $J\mapsto g(J)$ d\'efinit une application
  \begin{equation*}
    H^1(F,G)\to H^3(F,\IZ/3),
  \end{equation*}
  o\'u $G$ est le groupe des automorphismes de l'alg\`ebre de Jordan
  exceptionnelle d\'ecompos\'ee (\emph{i.\kern.1eme.}\ un groupe de
  type~$F_4$ d\'eploy\'e).  L'\'existence d'une telle application
  avait \'et\'e conjectur\'ee par J.-P. Serre~\cite{3}.

  Cet invariant a des relations int\'eressantes avec le symbole
  galoisien
  \begin{equation*}
    h\colon K^M_3F/3\to H^3(F,\mu^{\otimes 3}_3).
  \end{equation*}
  Lorsque $F^*$ contient~$\mu_3$ on peut relever~$g$ \`a $K^M_3 F/3$,
  et en d\'eduire que l'application~$h$ ci-dessus est \emph{injective}
  sur le sous-ensemble des symboles de $K^M_3 F/3$.

  Dans ce qui suit, nous donnons le principe de la construction
  de~$g$\,; un expos\'e plus d\'etaill\'e est en pr\'eparation.

  On consid\`ere d'abord des couples $(J$, $B)$, o\'u $B$ est une
  sous-alg\`ebre de Jordan de~$J$, de dimension~$9$, isomorphe \`a
  $M_3(F)^+$ sur une extension convenable de~$F$.  Par une analyse de
  la \flqq\,premi\`ere construction de Tits\,\frqq, on obtient un
  invariant de couples \`a isomorphisme pr\`es\,:
  \begin{equation*}
    g(J,B)\in H^3(F,\IZ/3).
  \end{equation*}
  Soit alors
  \begin{equation*}
    g(J)=g(J\otimes H, B_0)\in H^3(H,\IZ/3),
  \end{equation*}
  o\'u $B_0$ est une sous-alg\`ebre g\'en\'erique de dimension~$9$
  ayant le corps~$H$ pour corps de d\'efinition.  On montre que
  $\partial_v\bigl(g(J)\bigr)=0$ pour tous les homomorphismes
  bords~$\partial_v$ attach\'es aux valuations discr\`etes~$v$ de~$H$
  triviales sur~$F$.  Comme on peut supposer que $H$ est une extension
  transcendante pure de~$F$, cette propri\'et\'e entra\^ine que~$g(J)$
  appartient au sous-groupe $H^3(F,\IZ/3)$ de $H^3(H,\IZ/3)$.  Par
  sp\'ecialisation, on en d\'eduit que $g(J)=g(J,B)$ pour tout~$B$,
  d'o\'u l'invariant cherch\'e.
\end{VersFrancAbr}

\section{Introduction}

Let $F$ be a field of characteristic $\neq 2$,~$3$.  Let $J$ be an
exceptional Jordan algebra over~$F$ (we assume throughout $J$ to be
central simple and finite dimensional, so that $\dim J=27$).  We
sketch the construction of an invariant up to isomorphism:
\begin{equation*}
  g(J)\in H^3(F,\IZ/3).
\end{equation*}
Its basic property is that $g(J)=0$ if and only if $J$ has zero
divisors.  $g$ gives rise to a functorial map
\begin{equation*}
  H^1(F,G) \to H^3(F,\IZ/3)
\end{equation*}
where $G$ is the automorphism group of the split exceptional Jordan
algebra.  $G$ is of type~$F_4$ and is isomorphic to the automorphism
group of the cubic form in~$26$ variables
\begin{equation*}
  \widetilde N(x,y,z)=\det x+\det y+\det z -\mathop{\rm trace} xyz
\end{equation*}
where $x$, $y$,~$z$ are $3\times3$ matrices with $\mathop{\rm trace}
x=0$ (\cite{1}, Chapt.~VI, Theorem~7).  The invariant is also of
interest for questions related with the Galois symbol
\begin{equation*}
  h \colon K_3^MF/3\to H^3(F,\mu_3^{\otimes 3}).
\end{equation*}
In case $\mu_3\subset F^*$ the invariant can be lifted to $K^M_3 F/3$
and one can use this to show that $h$ is injective when restricted to
the subset of symbols in~$K^M_3 F/3$.

In this note we just describe briefly the principle of the
construction of~$g$; a more detailed paper is in preparation.

The existence of the invariant~$g$ has been suggested by J.-P.
Serre~\cite{3}.  I am indebted to J.-P. Serre for pointing out the
relation between exceptional Jordan algebras and $3$-symbols $\amod3$
and to T.~A. Springer for introducing me to the theory of Jordan
algebras.

\section{Tits' first construction}

As general reference for Jordan algebras within this Note \emph{see}
\cite{1},~\cite{4}.  For an associative algebra~$A$ one denotes
by~$A^+$ the Jordan algebra with the same underlying space with Jordan
product
\begin{equation*}
  a\jdot b=\frac 12(ab+ba).
\end{equation*}
Let $F$ be a field ($\car F\neq 2$,~$3$), let $A$ be a central simple
algebra over~$F$ with $\dim_F A=9$ and let $\mu\in F^*$.  In Tits'
first construction one defines an exceptional central simple Jordan
algebra $(A,\mu)$ with underlying space $A_0\oplus A_1\oplus A_2$ by
an explicit multiplication table, where the $A_i$ are three copies
of~$A$ (\emph{see} \cite{1}, Ch.~IX, Sect.~12).  This multiplication
provides $A_0$ with the Jordan product of~$A^+$, so that $A^+$ is a
canonical Jordan subalgebra of~$(A,\mu)$.  The trace~$T_J$ and the
norm~$N_J$ on $J=(A,\mu)$ are given by
\begin{align*}
  T_J(a_0,a_1,a_2) &= T_A(a_0),\\
  N_J(a_0,a_1,a_2) &= N_A(a_0)+\mu
  N_A(a_1)+\mu^{-1}N_A(a_2)-T_A(a_0a_1a_2),
\end{align*}
where $T_A$ and~$N_A$ are the (reduced) trace and norm of~$A$
respectively (\cite{4}, 5.10).

The algebra~$(A,\mu)$ is an exceptional Jordan algebra over~$F$.  It
has zero divisors if and only if its normform has a nontrivial zero
and if and only if $\mu\in N_A(A^*)$ (\cite{1}, p.~227 and p.~416).

We consider pairs $(J,B)$ where $J$ is an exceptional Jordan algebra
over~$F$ and $B\subset J$ is a Jordan subalgebra which is isomorphic
to~$M_3(F)^+$ over the algebraic closure of~$F$.  Such subalgebras
always exist (\cite{1}, Ch.~IX, Lemma~2).  One knows that over a
separably closed field all such pairs are isomorphic (\cite{1},
Ch.~IX, in particular Theorem~3).  In the following we give an
explicit recipe how to identify any $(J,B)$ with some
$\bigl((A,\mu),A^+\bigr)$ at least after replacing the base field by a
quadratic extension.  Let
\begin{equation*}
  B^\perp =\{\,x\in J\mid T_J(b\jdot x)=0\text{ for $b\in B$}\,\}.
\end{equation*}
Then $B\jdot B^\perp= B^\perp$ and $J=B\oplus B^\perp$ (\cite{1},
p.~227 and p.~240).  We define $C\subset \text{End}_F(B^\perp)$ as the
subalgebra generated by all endomorphisms
\begin{equation*}
  B^\perp \to B^\perp,\quad x\mapsto b\jdot x
\end{equation*}
with $b\in B$.

\begin{proposition}
  \label{prop:1}
  The centre~$K$ of~$C$ is a separable quadratic extension of~$F$ and
  $C$ is an Azumaya algebra over~$K$ of rank~$9$.
\end{proposition}

We omit a proof here.  However note that it suffices to consider a
separably closed field~$F$ in which case $(J,B)$ is isomorphic to the
explicit model
\begin{equation*}
  \bigl(\bigl(M_3(F),1\bigr), M_3(F)^+\bigr).
\end{equation*}
A similar remark holds for proposition~\ref{prop:2} below.

Let $p_K\in K\otimes K$ be the canonical idempotent, that is $(1-p_K)
K\otimes K$ is the kernel of the multiplication map $K\otimes K\to K$.
Then $p_K\in K\otimes K\subset\text{End}_K(B^\perp \otimes K)$ and
there is a decomposition
\begin{equation*}
  J\otimes K=B\otimes K\oplus p_K(B^\perp\otimes K)\oplus
  (1-p_K)(B^\perp\otimes K).
\end{equation*}
This can be identified with the decomposition occuring in Tits' first
construction as follows.

\begin{proposition}
  \label{prop:2}
  There exist $Z\in p_K(B^\perp\otimes K)$ such that
  $\lambda=N_{J\otimes K}(Z)\in K$ is a unit.  For any such~$Z$ one
  has
  \begin{enumerate}
    \renewcommand{\labelenumi}{(\theenumi)}
    \renewcommand{\theenumi}{\roman{enumi}}
    \item
    \label{item:2.i}
    $p_K(B^\perp\otimes K)=C\jdot Z $ and $N_{J\otimes
    K}\bigl(p_K(B^\perp\otimes K)\bigr)= N_C(C^*)\jdot \lambda$.
    \item
    There is a unique isomorphism
    \begin{equation*}
      \psi\colon J\otimes K\to (C,\lambda)
    \end{equation*}
    of Jordan algebras over~$K$ with $\psi(B\otimes K)=C^+$ such that
    $\psi(Z)=(0,1,0)$ and $\psi(b)=\varphi(b)$ for $b\in B$, where
    \begin{equation*}
      \varphi\colon B\to C\subset \text{End}_F(B^\perp)
    \end{equation*}
    is given by $\varphi(b)(x)=\bigl(T_J(b)-2b\bigr)\jdot x$.\qed
  \end{enumerate}
\end{proposition}

\section{The invariant $g(J)$}

As usual let
\begin{equation*}
  (c)\in H^1(F,\mu_3)=F^*/(F^*)^3 \quad\text{and}\quad [A]\in
  H^2(F,\mu_3)\subset Br(F)
\end{equation*}
be the classes of $c\in F^*$ and a $9$-dimensional algebra~$A$
respectively.  Note that $\mu_3\otimes \mu_3=\IZ/3$ as Galois module.

\begin{theorem}
  There is an invariant up to isomorphism
  \begin{equation*}
    g(J)\in H^3(F,\IZ/3)
  \end{equation*}
  which is natural with respect to base change and such that
  \begin{equation*}
    g\bigl((A,\mu)\bigr)=[A]\cup (\mu).
  \end{equation*}
\end{theorem}

By a result of A.~S. Merkuriev and A.~A. Suslin~\cite{2} one has
$[A]\cup (\mu)=0$ if and only if $\mu\in N_A(A^*)$.  Moreover a cubic
form has a nontrivial zero if this holds over a quadratic extension.
From this and the remarks above it follows that $g(J)=0$ if and only
if $J$ has zero divisors.

A proof of the theorem can be briefly sketched as follows.  One first
defines a relative invariant for the pairs $(J,B)$ by the formula
\begin{equation*}
  g(J,B)=-\mathop{\rm cor}\nolimits_{K/F}
  \bigl([C]\cup(\lambda)\bigr)\in H^3(F,\IZ/3)
\end{equation*}
where $C$, $K$,~$\lambda $ are as in proposition~\ref{prop:1} and
proposition~\ref{prop:2}.  This invariant is welldefined by
proposition~\ref{prop:2}~\eqref{item:2.i}.  One can show that $[C]\cup
(\lambda)$ is invariant under $\text{Aut}(K/F)$ and therefore
\begin{equation*}
  [C]\cup (\lambda)=g(J,B)_K.
\end{equation*}
For a pair obtained from Tits' first construction it turns out that
\begin{equation*}
  g\bigl((A,\mu),A^+\bigr)=[A]\cup (\mu).
\end{equation*}
The invariant $g(J,B)$ does not depend on the choice of~$B$.  To prove
this one uses the following two lemmas.

\begin{lemma}
  \label{lem:1}
  For $A$, $\mu$ as above let $\Sigma$ be the norm variety over~$F$
  with coordinates $x\in A$ and equation $N_A(x)=\mu$.  Then the
  restriction map
  \begin{equation*}
    r\colon H^2(F,\mu_3)\to H^2(F(\Sigma),\mu_3)
  \end{equation*}
  is injective.
\end{lemma}

\begin{proof}[Proof \textup{(\emph{see} also \cite{5})}]
  We may assume
  \begin{equation*}
    \mu_3\subset F^*\quad\text{and}\quad [A]=(\lambda)\cup (\rho)
  \end{equation*}
  in $H^2(F,\mu_3)\simeq H^2(F,\IZ/3)$ for some $\lambda$, $\rho\in
  F^*$.  Let $\nu=\lambda^k\rho^\ell$ with $(k,\ell)\not\in 3\IZ\times
  3\IZ$ and let $\Omega$ be the Brauer-Severi variety for the algebra
  with invariant $(\nu)\cup (\mu)$.  Then $\Sigma$ has a
  $F(\Omega)$-rational point.  Hence the kernel of~$r$ is contained in
  the kernel of
  \begin{equation*}
    H^2(F,\mu_3)\to H^2\bigl(F(\Omega),\mu_3\bigr)
  \end{equation*}
  which is generated by $(\nu)\cup (\mu)$.  However the intersection
  of these groups over all~$\nu$ is trivial.
\end{proof}

\begin{lemma}
  \label{lem:2}
  Let $H/F$ be an extension of fields and let $v\colon H^*\to\IZ$ be a
  discrete valuation trivial on~$F$.  Then for $J$ over~$F$ and a pair
  $(J\otimes H,B)$ over~$H$ the element $g(J\otimes H,B)$ vanishes
  under the boundary map
  \begin{equation*}
    \partial_v\colon H^3(H,\IZ/3)\to H^2\bigl(\kappa(v),\mu_3\bigr).
  \end{equation*}
\end{lemma}

\begin{proof}
  We may assume that $J=(A,\mu)$ after replacing $F$ by some quadratic
  extension, see proposition~\ref{prop:2}.  By lemma~\ref{lem:1} we
  may replace $F$ by~$F(\Sigma)$.  But then $J$ has zero divisors and
  $g(J\otimes H,B)=0$.
\end{proof}

Now let $X$,~$Y$ be two independent generic elements of~$J$; their
field of definition~$H$ is rational in a doubled set of coordinate
functions on~$J$.  Let $B_0\subset J\otimes H$ be the Jordan
subalgebra generated by $X$ and~$Y$.  Then $(J\otimes H,B_0)$ is a
pair over~$H$ as above (\cite{1}, Ch.~IX, Lemma~2).  Since $H$ is
rational over~$F$ it follows from lemma~\ref{lem:2} that $g(J\otimes
H,B_0)$ is in the image of the injective restriction map
\begin{equation*}
  H^3(F,\IZ/3)\to H^3(H,\IZ/3).
\end{equation*}
Now take for $g(J)$ the preimage of $g(J\otimes H,B_0)$.  By
construction $g(J)$ is an invariant of~$J$ up to isomorphism.  Using
an appropriate specialization one finds $g(J)=g(J,B)$ for any choice
of~$B$.

\begin{thebibliography}{1}

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  Publications XXXIX, Amer. Math. Soc., Providence, R.~I., 1968.

  \bibitem{2} {\sc A. S. Merkurjev} and {\sc A. A. Suslin},
  $K$-Cohomology of Severi-Brauer varieties and the norm residue
  homomorphism, {\it Izv. Akad. Nauk S.S.S.R.}, 46, 1982, pp.
  1011--1046 (={\it Math. U.S.S.R. Izvestiya}, 21, 1983, pp. 307-340).

  \bibitem{3} {\sc J.-P. Serre}, {\it R\'esum\'e des cours de
  l'ann\'ee} 1990--1991, Coll\`ege de France.

  \bibitem{4} {\sc T. A. Springer}, {\it Jordan algebras and algebraic
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  Springer-Verlag, 1973.

  \bibitem{5} {\sc A. A. Suslin}, $K$-Theory and $K$-cohomology of
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  Math.~4, Amer. Math. Soc., Providence, R.~I., 1991, pp.~53--74
%% Preprint 90-026, Universit\"at Bielefeld, 1990.

\end{thebibliography}

\end{document}
