%%% Date: Thu, 04 Oct 2001 16:18:31 +0200
%%% From: Thomas Unger <tpunger@pop3.ucd.ie>
%%% Subject: preprint submission
%%% To: Ulf Rehmann <rehmann@Mathematik.Uni-Bielefeld.DE>
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%%% Dear Ulf,
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%%% we would like to submit a preprint of ours for the preprint archive.
%%% Attached is the LaTeX file and also the corresponding postscript file to
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%%% 
%%% Best wishes,
%%% 
%%% David W. Lewis
%%% Thomas Unger
%%% ==========
%%% 
%%% Title: A local-global principle for algebras with involution and hermitian
%%% forms
%%% 
%%% Abstract:
%%% 
%%% Weakly hyperbolic involutions are introduced and a proof is given
%%% of the following local-global principle: a central simple algebra
%%% with involution of any kind is weakly hyperbolic if and only if
%%% its signature is zero for all orderings of the ground field.
%%% Also, the order of a weakly hyperbolic algebra with
%%% involution is a power of two, this being a direct consequence of a
%%% result of Scharlau. As a corollary an analogue of
%%% Pfister's local-global principle is obtained for the Witt group of
%%% hermitian forms over an algebra with involution.
%%% 
%%% 2000 Mathematics Subject Classification:  16K20, 11E39
%%% 
%%% Keywords and Phrases: Central simple algebras,
%%% involutions, hermitian forms, local-global principles
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%%% %
%%% %David W. Lewis, Thomas Unger
%%% %
%%% %A local-global principle for algebras
%%% %with involution and hermitian forms
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WITH INVOLUTION AND HERMITIAN FORMS}


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\title{A local-global principle for algebras with involution and hermitian forms}
\author{David W. Lewis}
\author{Thomas Unger}
\address{Department of Mathematics\\University College Dublin\\ Belfield\\ Dublin 4\\ IRELAND}
\email{david.lewis@ucd.ie}
\address{Department of Mathematics\\University College Dublin\\ Belfield\\ Dublin 4\\ IRELAND}
\email{thomas.unger@ucd.ie}
\date{\today}
%\thanks{Research supported by Enterprise Ireland.}
\keywords{Central simple algebras,
involutions, hermitian forms, local-global principles}
%\subjclass[2000]{16K20, 11E39}


\begin{document}

%\centerline{\emph{$**$ Preliminary Version $**$}}

\begin{abstract}
Weakly hyperbolic involutions are introduced and a proof is given
of the following local-global principle: a central simple algebra
with involution of any kind is weakly hyperbolic if and only if
its signature is zero for all orderings of the ground field.
Also, the order of a weakly hyperbolic algebra with
involution is a power of two, this being a direct consequence of a
result of Scharlau. As a corollary an analogue of
Pfister's local-global principle is obtained for the Witt group of
hermitian forms over an algebra with involution.
\end{abstract}



\maketitle

%2000 Mathematical Subject Classification: 16K20, 11E39


%=========================================================
\section{Introduction}
%=========================================================

Pfister's well-known local-global principle states that a
nonsingular quadratic form $q$ over a field $k$ (which we assume
to be of characteristic different from $2$)
is a torsion element in the
Witt ring $W(k)$ of $k$ if and only if the signature of $q$ is zero for
all orderings of $k$. Furthermore, every torsion element of $W(k)$
has $2$-power order.

If $W(A,\s)$ is the Witt group of hermitian
forms over some central simple $k$-algebra with involution of any kind
$(A,\s)$, then Scharlau showed that the torsion elements of
$W(A,\s)$ have $2$-power order.

In this paper, we complement
Scharlau's result by showing that $h\in W(A,\s)$ is a torsion
element if and only if $h$ has signature zero for all orderings of
the ground field, thus obtaining an analogue of Pfister's
local-global principle for hermitian forms. In fact, this will follow from
our main theorem which states that if $(A,\s)$ is a
central simple $k$-algebra with involution of any kind, then the
signature of $\s$ is zero for all orderings of $k$ if and only if
$(A,\s)$ is weakly hyperbolic. As a consequence of Scharlau's
theorem we also get that the order of $(A,\s)$ is a power of $2$
when $(A,\s)$ is weakly hyperbolic.

Weakly hyperbolic involutions are a natural generalization of
torsion forms and are a special case of weakly isotropic
involutions. The latter were studied in \cite{LSU} as an
ingredient of a different local-global principle for algebras with
an involution of the first kind.

All involutions on central simple algebras considered in this paper are
arbitrary and all forms (quadratic, hermitian, \emph{etc}.) are
assumed to be nonsingular.
Standard references are \cite{L} and \cite{S} for the theory of
quadratic forms, \cite{BOI} for central simple algebras with an
involution and \cite{P} for real fields.

%=========================================================
\section{Preliminaries}
%=========================================================

Let $K$ be a field of characteristic different from two and let
$A$ be a central simple $K$-algebra, equipped with an involution
$\s$. We say that $\s$ is of the \emph{first kind} if $\s|_K=\1_K$
and of the \emph{second kind} otherwise. If $\s$ is of the second
kind, let $k$ denote its fixed field. Then $K$ is a quadratic
\'etale extension field of $k$, \emph{i.e.} either $K=k(\sqrt
\alpha)$ for some $\alpha \in k^\x$ or $K=k\x k$. In the latter
case, $A$ is no longer simple. Unless stated otherwise,
we will always assume that $K$ is a field, for reasons which will become
apparent later.
To cater for both kinds of involution, we take $k$ as the base field and
simply say that $(A,\s)$ is a central
simple $k$-algebra with involution, it being understood that the
center is $k$ or $K$, depending on the situation.

By Wedderburn's theorem, we can write
$$A\cong \End_D(V),$$
where $D$ is a central division algebra over $k$ (with center $K$) and $V$ is some
finite dimensional right $D$-vector space.  The \emph{degree}
of $A$ is $\deg A:=\sqrt{\dim_K A}$ and the
(\emph{Schur}) \emph{index} of $A$ is $\ind A:=\deg D$. We say that
$A$ is \emph{split} if $D=K$. Any field extension $L\supset k$ such
that $A\ox_k L$ is split, is called a \emph{splitting field} of
$A$. In particular, the separable closure $k_s$ of $k$
is a splitting field of $A$.

Let $n=\deg A$. If $\s$ is of the first kind, $\s$ is called
\emph{orthogonal} (resp. \emph{symplectic}) if the extended
involution $\s \ox \1_{k_s}$ on $A\ox_k k_s\cong M_n(k_s)$ is
adjoint to a symmetric (resp. skew-symmetric) bilinear form.
Involutions of the second kind are also called \emph{unitary
involutions}.

It is well-known that $\s$ is the adjoint involution
of some non-singular $\ve$-hermitian form ($\ve=\pm 1$) $h:V\x
V\to D$, with respect to some involution $\vt$ on $D$ which is of
the same kind as $\s$. (If $\s$ is of the second kind, then
$\ve=+1$ and $\vt(\alpha)=\s(\alpha) \forall \alpha \in K$.) So
$\s$ is of the form $\s_h$, where $\s_h$ is implicitly defined by
$$h\bigl(x, f(y)\bigr)=h\bigl(\s_h(f)(x),y\bigr)\quad \text{ for }
x,y\in V\text{ and } f\in \End_D(V).$$ The form $h$ is uniquely
determined up to multiplication with a $\vt$-invariant
factor in $K^\x$. (More generally, we can substitute a central
simple algebra  $E$ for $D$ and a finitely generated right
$E$-module $M$ for $V$, see \cite[4.A]{BOI}.)

%=========================================================
\subsection{Hyperbolic Involutions}
%=========================================================

A right ideal $I$ of $A$ is called \emph{isotropic} if $\s(I)I=0$.
The algebra with involution $(A,\s)$, or the involution $\s$
itself, is called \emph{isotropic} if $A$ contains an isotropic
right ideal and \emph{anisotropic} if $\s(x)x=0$ implies $x=0$,
$\forall x\in A$. We also say that $(A,\s)$ is \emph{weakly
isotropic} if there exist nonzero $x_1,\ldots, x_n\in A$ such that
$\s(x_1)x_1+\cdots + \s(x_n)x_n=0$ and \emph{strongly anisotropic}
otherwise.

The notion of a hyperbolic involution was first defined by
Bayer-Fluckiger \emph{et al.} \cite{Ba} (see also \cite[6.B]{BOI}):
an algebra with
involution $(A,\s)$, or the involution $\s$ itself, is called
\emph{hyperbolic} if either the center of $A$ is isomorphic to
$k\x k$, or $\s$ is the adjoint involution of some hyperbolic
$\ve$-hermitian form. In both cases $(A,\s)$ is hyperbolic if and
only if $A$ contains an idempotent $e$ such that $\s(e)=1-e$. This
is also equivalent with the existence of an isotropic right ideal
$I$ of dimension $\dim_k I=\frac{1}{2}\dim_k A$.

Note that hyperbolic involutions remain hyperbolic under arbitrary
scalar extensions and if $(A,\s)$ is hyperbolic and $(B,\tau)$ is
arbitary, then $(A\ox_k B,\s\ox \tau)$ is hyperbolic.

%=========================================================
\subsection{Signatures of Involutions}
%=========================================================


Suppose that $k$ is a real field. We fix some notation for the
rest of this paper. We denote the space of orderings of $k$ by
$X_k$. If $P\in X_k$, $k_P$ will denote the real closure of $k$
with respect to $P$. Furthermore, we set $A_P:=A\ox_k k_P$,
$\s_P=\s\ox \1_{k_P}$, \emph{etc}.

Let $P\in X_k$. The signature of
an involution was defined by Lewis and Tignol \cite{LT} for
involutions of the first kind and by Qu\'eguiner \cite{Q} for
involutions of the second kind as
$$\sig_P \s=
\begin{cases}
\sqrt{\sig_P T_\s} & \text{if $\s$ is of the first kind,}\\
\sqrt{\frac{1}{2}\sig_P T_\s} & \text{if $\s$ is of the second kind.}
\end{cases}$$
Here $T_\s$ denotes the \emph{trace quadratic form}
$T_\s(x):=\Trd_A(\s(x)x), \forall x\in A$, which takes values in $k$.

If $(A,\s)$ is split with orthogonal involution, $(A,\s)\cong
(\End_k(V), \s_q)$, $q$ being a  quadratic form over $k$, then Lewis
and Tignol showed that
$$\sig_P \s_q=|\sig_P q|.$$
Likewise, if $(A,\s)$ is split with unitary involution,
$(A,\s)\cong(\End_K(V), \s_h)$, $h$ being a $K/k$-hermitian form,
then Qu\'eguiner showed that
$$\sig_P \s_h=|\sig_P h|.$$
Clearly, the signature of $\s$ will be zero in the split-symplectic case.

%=========================================================
\subsection{The $n$-fold Orthogonal Sum}
%=========================================================

Let $f$ be a quadratic or $K/k$-hermitian
form, then the notion of $n$-fold orthogonal sum of $f$,
$$\perp^n f=n\x f=\<\underbrace{1,\ldots, 1}_n\> \ox f,$$
can be extended to the realm of algebras with an involution in a
straightforward way:

\begin{defi} Let $(A,\s)$ be a central simple $k$-algebra with
involution of any kind. The \emph{$n$-fold orthogonal sum}
$\bplus^n(A,\s)$ is defined by
$$\bplus^n(A,\s):=(M_n(K),*)\ox_k (A,\s),$$
with $*$  the conjugate transpose involution, defined by
$(a_{ij})^*=(\iota(a_{ij}))^t, \forall (a_{ij})\in M_n(K)$,
where $\iota$ is either the nontrivial automorphism of $K$
if $\s$ is of the second kind or the identity otherwise.
It is clearly again a
central simple $k$-algebra of degree $n\cdot \deg A$.
\end{defi}

\begin{remark} When $A$ is split, the involution $* \ox \s$
will be the adjoint of $n\x f$, for an appropriate form $f$.
\end{remark}

\begin{remark} If $(A,\s)$ is a central simple $k$-algebra with
involution of the first kind, the $n$-fold orthogonal sum
$\bplus^n(A,\s)$ was defined in a more intrinsic way in \cite{U}, so
as to conform with Dejaiffe's \cite{D} construction of an
orthogonal sum of two Morita equivalent algebras with involution of
the first kind.
\end{remark}

%=========================================================
\section{The Local-Global Principle for Algebras with Involution}
%=========================================================


\begin{defi} The algebra with involution $(A,\s)$ is called \emph{weakly
hyperbolic} if there exists an $n\in \N$ such that
$\bplus^n(A,\s)$ is hyperbolic. The \emph{order} of $(A,\s)$ is
the least integer $n$ such that $\bplus^n(A,\s)$ is hyperbolic.
\end{defi}

The main theorem of this paper reads:

\begin{theorem}\label{maintheorem}
Let $(A,\s)$ be a central simple $k$-algebra with
involution of any kind. Then $\sig_P \s=0, \forall P\in X_k$ if and
only if $(A,\s)$ is weakly hyperbolic. Furthermore, the order of
$(A,\s)$ is a power of two when $(A,\s)$ is weakly hyperbolic.
\end{theorem}

Note that when $k$ is not real, there is no notion of signature and
$(M_n(k),t)$ will be hyperbolic for some positive integer $n$, so that
$(A,\s)$ will be weakly hyperbolic. The main theorem will thus hold
trivially. Hence, we  assume from now on that $k$ is a real field.
In order to prove this theorem, we first need to establish a few
lemmas.

\begin{lemma}\label{splitcase}
The main theorem holds when $A$ is split; $A\cong
\End_k(V)$.
\end{lemma}

\begin{proof} When $A$ is split, $\s$ will be the adjoint of a
quadratic, a skew-symmetric bilinear, or a $K/k$-hermitian form $f$. If $f$ is
quadratic, we have proof by Pfister's local-global principle. If $f$
is skew-symmetric, the statement is trivial. If $f$ is
$K/k$-hermitian, $f$ is completely determined by its trace form
$q_f(x):=f(x,x)\in k, \forall x\in V$ (Jacobson's theorem, see \cite[Theorem~10.1.1]{S}), in which
case we have proof again by Pfister's local-global principle.
\end{proof}

\begin{lemma}\label{quatreal}
Let $(A,\s)$ be a central simple $k$-algebra with involution of
any kind and let $L$ be a real closed extension field of $k$ such that
$A_L\cong M_n\bigl((-1,-1)_L\bigr)$ for some positive integer $n$. Then
the main theorem holds for $(A_L,\s_L)$.
\end{lemma}

\begin{proof} If $\s_L$ is symplectic, then $\s_L$ is the conjugate transpose
involution $\bar{\phantom{x}}t$, where $\bar{\phantom{x}}$ is quaternion
conjugation. In this case, $(A_L,\s_L)$ is totally anisotropic and $\s_L$ is
positive definite, so the lemma holds.

If $\s_L$ is orthogonal, then $\s_L$ is the conjugate transpose
involution $\hat{\phantom{x}}t$, where $\hat{\phantom{x}}$ is the orthogonal
involution on $(-1,-1)_L$, determined by $\hat{1}=1$, $\hat{i}=-i$,
$\hat{j}=j$, $\hat{k}=k$. Now $\sig \s_L=0$ and $\bplus^2(A_L,\s_L)$ is
hyperbolic, since $\bplus^2\bigl((-1,-1)_L,\hat{\phantom{x}}\bigr)$ is
hyperbolic (the element $e:= \frac{1}{2} \bigl(\begin{smallmatrix}
1 & j^{-1}\\
j & 1
\end{smallmatrix}\bigr)$ satisfies $e^2=e$ and $\hat{e}^t=1-e$.)

If $\s_L$ is unitary, then the center of $A_L$ is $L(\sqrt{-1})$, which is
algebraically closed, since $L$ is real. Hence $A_L\cong
M_{2n}\bigl(L(\sqrt{-1})\bigr)$ and $\s_L$ is the conjugate transpose
involution $*$, defined earlier.
Now $(A_L,\s_L)$ is totally anisotropic and $\s_L$ is positive
definite, so that we are done.
\end{proof}



\begin{lemma}\label{lemma5.4}
Suppose $\s$ is an anisotropic involution of any kind on a central
simple $k$-algebra $A$ and let $L=k(\sqrt \gd)$ be a quadratic field
extension of $k$. If the involution $\s_L=\s\ox \1_L$ on $A\ox_k
L$ is hyperbolic, then there exists $r\in A$ such that $r^2=\gd$ and
$\s(r)=-r$.
\end{lemma}

This is in fact \cite[Lemma~3.2]{Ba} which holds for involutions
of any kind, not just for those of the first kind (as it is stated
in \cite{Ba}).
Another result from \cite{Ba} which holds for arbitrary
involutions, is their Theorem~3.3:

\begin{theorem}\label{thm3.3hyperb}
Let $(A,\s)$ be a  central simple $k$-algebra with involution of
any kind and let $L=k(\sqrt \gd)$ be a quadratic field extension of
$k$. If there exists $r\in A$ such that $r^2=\gd$ and $\s(r)=-r$,
then $(A\ox_k L, \s_L)$ is hyperbolic. The converse holds except
in the case where $A$ is split, $\s$ is orthogonal and its
associated quadratic form has odd Witt index.
\end{theorem}


\begin{proof}
If $\s$ is an involution of the first kind, we refer to
\cite[Theorem~3.3]{Ba} for a proof, which can be made to work for
involutions of the second kind quite easily, as we will proceed to
do now. (The crucial ingredients, Witt cancellation and Witt
decomposition, also work in this situation.) So, suppose that $\s$
is unitary and note that the exceptional case does not occur now.
Recall that $K=k(\sqrt \alpha)$ for some $\alpha \in k^\x$.

Given $r$, let $t=\gd^{-1}r\ox \sqrt{\gd} \in A\ox_k L$. Note that
$t^2=1$ and $\s_L(t)=-t$. Therefore, the element $e=\frac{1}{2}
(1+t)$ satisfies $e^2=e$ and $\s_L(e)=1-e$, so that $(A\ox_k L,
\s_L)$ is hyperbolic.

Conversely, let $V$ be an irreducible left $A$-module and
$D=\End_A(V)$, so that $A=\End_D(V)$. Choose an involution of the
second kind $\bar{\phantom{x}}$ on $D$ and let $h$ be a hermitian
form on $V$ with respect to which $\s$ is the adjoint involution.

Consider a Witt decomposition
$$(V,h) \cong (V_1,h_1)\perp (V_0,h_0),$$
where $(V_1,h_1)$ is hyperbolic and $(V_0,h_0)$ is anisotropic.
Let $\s_i$ denote the adjoint involution with
respect to $h_i$ on $\End_D(V_i)$ for $i=0,1$.
By Witt cancellation, $(V_0,h_0)$ becomes
hyperbolic over $L$, hence  Lemma~\ref{lemma5.4} yields an element
$r_0\in \End_D(V_0)$ such that $r_0^2=\gd$ and $\s_0(r_0)=-r_0$.

By \cite[Theorem~2.2]{Ba}, $\End_D(V_1)$ contains a
$\s_1$-invariant subalgebra $M$ such that $(M,\s_1|_M)\cong
(M_2(K), \Theta)$, where $\Theta$ is defined by
$$\Theta \left(\begin{array}{cc}
  a & b \\
  c & d
\end{array}\right)=
\left(\begin{array}{cc}
  \bar d & \bar b \\
  \bar c & \bar a
\end{array}\right), \quad \forall a,b,c,d \in K.$$
Let $r_1\in M \subset \End_D(V_1)$ denote the image of $\left(\begin{array}{cc}
   0 & \gd\sqrt \alpha \\
  \frac{1}{\sqrt\alpha} & 0
\end{array}\right)$, then $\s_1(r_1)=-r_1$ and $r_1^2=\gd$.

Finally, regarding $V=V_1\oplus V_0$, define $r\in A=\End_D(V)$ by
$$r(x,y)=\bigl(r_1(x),r_0(y)\bigr).$$
It is straightforward to check that $\s(r)=-r$ and $r^2=\gd$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{maintheorem}]
Suppose first that $(B,\tau):=\bplus^n(A,\s)$ is hyperbolic for
some $n$. Since hyperbolic involutions remain hyperbolic under
arbitrary scalar extensions, $(B_P,\tau_P)$ is hyperbolic for every
ordering $P$ of $k$. Since $(B_P,\tau_P)$ is either split or a
matrix algebra over $(-1,-1)_{k_P}$ and since signatures do
not change under scalar extension, we obtain $\sig_P \tau=0$,
$\forall P\in X_k$ from Lemmas~\ref{splitcase} and \ref{quatreal}.
This implies $\sig_P \s=0$, $\forall P\in X_k$.

Conversely, suppose for the sake of contradiction that $(A,\s)$ has zero
signature at each ordering of the field $k$, but that $(A,\s)$ is
not weakly hyperbolic. By Zorn's Lemma, there exists a maximal
algebraic extension field $L$ of $k$ for which $(A_L,\s_L)$ is not
weakly hyperbolic, see Remark~\ref{zorn}.

Observe that $L$ must be real, because if not, then $(M_n(L),t)$
will be hyperbolic for some positive integer $n$, so that
$\bplus^n(A_L,\s_L)$
is hyperbolic. Also, $L$ cannot be a
splitting field for $A$, because the local-global principle works
in the split case (Lemma~\ref{splitcase}).

Next we will show that $L$ is euclidean by showing that $\pm 1$ are the only
two square classes in $L$.
Suppose for the sake of contradiction that
there exists $\alpha \in L$ such that $\pm \alpha $ is not a
square in $L.$ Then $L(\sqrt{\alpha})$ and $L(\sqrt{-\alpha})$
are each quadratic extension fields of $L$. By maximality of $L$ we have
that $(A,\sigma )$ must become weakly hyperbolic on extension to each of
these two fields. Thus there exists an integer $n$ for which the
$n$-fold orthogonal sum of
$(A,\sigma )$ becomes hyperbolic over both $L(\sqrt{\alpha})$ and
$L(\sqrt{-\alpha})$. Let us write $(B,\tau ):=\bplus^n(A_{L},\sigma _{L}).$

Applying Theorem~\ref{thm3.3hyperb} to the two quadratic
extensions above, yields two elements $r,s\in B$ such that
$r^2=\alpha$, $\tau(r)=-r$, $s^2=-\alpha$ and $\tau(s)=-s$.
Now we will show that the
sum of two copies of $(B,\tau )$ is hyperbolic (which is a contradiction
and hence $L$ must be euclidean). To show this, note that the sum of two
copies of $(B,\tau )$ is $(M_{2}(B),t\ox\tau)$ and let $e=\frac{1}{2}\left(
\begin{array}{cc}
1 & rs^{-1} \\
sr^{-1} & 1
\end{array}
\right) $. Then it is easy to check that $e^{2}=e$ and
$t\ox\tau(e)=1-e$. Therefore, the involution $t\ox\tau$ is hyperbolic.
Thus we have proved that $L$ is euclidean.

In fact, $L$ is real closed, since $L$ cannot have any proper odd
degree extensions by \cite[Corollary~6.16]{BOI}.
But then $Q:=(-1,-1)_L$ is the only division algebra over $L$.
Hence,  $A_L$ is an endomorphism algebra over $Q$ (since $L$ is
not a splitting field of $A$). Since the local-global principle
holds for such algebras (Lemma~\ref{quatreal}), we have a contradiction and
conclude that $(A,\s)$ has to be weakly hyperbolic.

The second statement is a direct consequence of Scharlau's
result \cite{S2} that the torsion in the Witt group of
central simple algebras with involution is $2$-primary.
\end{proof}




\begin{remark}
Whenever there exists an invertible element $y\in B$ with
$\tau (y)=-y^{-1}$ one can get that the sum of two copies of $(B,\tau )$ is
hyperbolic by defining $e=\frac{1}{2}\left(
\begin{array}{cc}
1 & y^{-1} \\
y & 1
\end{array}
\right)$. When $B$ is split and $\tau $ is the adjoint of some quadratic
form $\phi $, the condition $\tau (y)=-y^{-1}$ reduces to the fact that $\phi
$ is isometric to $-\phi $, which of course implies that the sum of two
copies of $\phi $ is hyperbolic.
\end{remark}

\begin{remark}\label{zorn}
The reasoning behind the use of Zorn's Lemma is analogous to what
happens in the proof of the ``(iv)$\implies$(v)'' direction of
\cite[Theorem~2.7.1]{S}: Let $\bar k$ be an algebraic closure of
$k$ and let $\mathcal{M}$ be the set of all intermediate fields
$M$ such that $A_M:=A\ox_k M$ is not  weakly hyperbolic.
The set $\mathcal{M}$ is ordered by
inclusion. For a totally ordered chain of subfields $M_i$ in
$\mathcal{M}$, we let $A_i:=A\ox_k M_i$ and $M:=\bigcup M_i$.

Assume for the sake of contradiction that $(A_M,\s_M)$ is
weakly hyperbolic. This means that there exists a square
matrix $X$ with entries in $A_M$ such that $X^2=X$ and $\s_M\ox *
(X)=I-X$. These are matrix equations with finitely many
coefficients from $A_M$.
So all coefficients must lie in $A_i$ for some $i$. Thus $M_i$ would not
belong to $\mathcal{M}$, a contradiction.
Therefore $M\in\mathcal{M}$ and we have
found an upper bound. By Zorn's Lemma, $\mathcal{M}$ has a maximal
element.
\end{remark}

%=========================================================
\section{The Local-Global Principle for Hermitian Forms}
%=========================================================

Let $k$ be a real closed field and let $K$ be either
$k(\sqrt{-1})$ or $(-1,-1)_k$. Let $W(K,\bar{\phantom{x}})$ denote
the Witt group of hermitian forms over $K$ with respect to the
canonical involution on $K$ (\emph{i.e.} the nontrivial
automorphism of $k(\sqrt{-1})$ or quaternion conjugation). Given
$h\in W(K,\bar{\phantom{x}})$, let $q_h$ denote the trace form of
$h$, defined by $q_h(x):=h(x,x)$ for all $x\in K$. In both cases
$q_h$ takes values in the ground field $k$. Let $\<\a_1,\ldots,
\a_n\>$ be a diagonalization of $h$. Then $\a_i\in k^\x$ $(1\leq
i\leq n)$ in both cases and
$$q_h=
  \begin{cases}
    2\x \<\a_1,\ldots, \a_n\> & \text{if $K=k(\sqrt{-1})$}, \\
    4\x \<\a_1,\ldots, \a_n\> & \text{if $K=(-1,-1)_k$}.
  \end{cases}$$
As is well-known (see \emph{e.g.} \cite[10.1.6]{S}), the signature
of $\<\a_1,\ldots, \a_n\>$ is an invariant of the hermitian form
$h$ in both cases, aptly called the \emph{signature} of $h$. To
summarize,
$$\sig h=
  \begin{cases}
    \frac{1}{2}\sig q_h & \text{if $K=k(\sqrt{-1})$}, \\
    \frac{1}{4}\sig q_h & \text{if $K=(-1,-1)_k$}.
  \end{cases}$$

Now let $k$ be a real field and let
$(A,\s)$ be a central simple $k$-algebra (with involution of any
kind). Let $W(A,\s)$ denote the Witt group of
hermitian forms over $A$ with respect to $\s$. The previous
observations allow us to define the signature of $h$ with respect
to the ordering  $P$, $\sig_P h$, by going over to the real
closure $k_P$ of $k$ with respect to $P$ and using Morita theory.
We will not elaborate
this here, but refer instead to \cite{BP} for a recent exposition.
It turns out that $\sig_P h$ can be defined in a meaningful way in
the following situations:
\begin{itemize}
\item $\s$ unitary;
\item $\s$ orthogonal and $A_P$ split;
\item $\s$ symplectic and $A_P\cong$ matrices over $(-1,-1)_{k_P}$.
\end{itemize}
In the other cases, $\sig_P h$ is defined to be equal to zero.
(This includes the situation where $K_P=k_P\x k_P$, which,
together with the fact that algebras with involution with center
$k\x k$ are by definition hyperbolic, prompted us to exclude this
possibility from our considerations.)

The main point for us is the observation that
$$\sig_P h=0 \qquad \iff \qquad \sig_P \s_h=0,$$
where $\s_h$ is the adjoint involution of $h$. This can be deduced quite
easily from the work of Lewis--Tignol and Qu\'eguiner mentioned earlier.

Given $h\in W(A,\s)$, it is now a simple matter of going over to
the adjoint involution $\s_h$ on $\End_A(V)$, in order to obtain
the following consequence of our main theorem:

\begin{theorem}
Let $h\in W(A,\s)$. Then $h$ is a torsion element if and only if
$\sig_P h=0$ for all $P\in X_k$. Every torsion element of
$W(A,\s)$ has $2$-power order.
\end{theorem}

The first statement of the theorem can be translated into a familiar looking
exact sequence:
$$0\too W_t(A,\s)\too W(A,\s){\overset{(\sig_P)}{\too}}
\prod_{P\in X_k}W(A_P,\s_P),$$
where $W_t(A,\s)$ denotes the torsion subgroup of $W(A,\s)$ and $W(A_P,\s_P)$
is either $0$, $\Z$ or $\Z/2\Z$, depending on what happens at
the ordering $P$.


We note again that the second statement of the theorem is nothing new.
It is due to Scharlau \cite{S2}, a fact which we already
acknowledged in the proof of our main theorem. To be more precise, Scharlau
proved that if $A$ is a finite-dimensional \emph{semisimple} $k$-algebra with
involution $\s$ such that $\s|_k=\1_k$, then $W_t(A,\s)$ is $2$-primary
\cite[Theorem~5.1(i)]{S2}. If
we denote the Grothendieck group of hermitian
forms over $(A,\s)$ by $\widehat W(A,\s)$ in this situation,
then the following exact sequence
can be deduced from \cite[Theorem~5.2]{S2}:
$$0\too \widehat W_t(A,\s)\too \widehat W(A,\s){\overset{r^*}{\too}}
\prod_{P\in X_k}\widehat W(A_P,\s_P),$$
where $r^*$ is the canonical extension homomorphism.





%=========================================================
\section*{Acknowledgements}
%=========================================================

Part of this research is funded by the
TMR research network (ERB~FMRX~CT-97-0107) on
 ``$K$-theory and algebraic groups''.


%=========================================================
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\end{document}
