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\author{\sc{N. Grenier-Boley}\\
                         \hspace{-.55cm}\sc{M.G. Mahmoudi}}
 \date{}
 \title{\textbf{Exact Sequences of Witt Groups}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{document}
\maketitle
 \begin{abstract}
An exact octagon of Witt groups of central
simple algebras with involution is constructed, extending an
exact sequence of Parimala, Sridharan and Suresh and
motivated by exact sequences obtained by Lewis.
From this, we derive relations between the cardinality 
of certain Witt groups.
An exact octagon of equivariant Witt groups is also
obtained, thus generalizing a similar octagon
constructed by Lewis for
quaternion algebras.
\footnotetext[1]{{\it 2000 Mathematics Subject Classification}. Primary 11E39; 
Secondary 11E81.}
\footnotetext[2]{{\it Key words and phrases.} Witt group, hermitian form,
central simple algebra with involution, exact sequence
of Witt groups,
isotropy, equivariant Witt group, Morita equivalence.}
\footnotetext[3]{The authors gratefully acknowledge support from 
the RT Network "K-theory, Linear Algebraic Groups and
Related Structures" (contract HPRN-CT-2002-00287)}
\end{abstract}
 \section{Introduction}\label{intro} 
\indent

 Base change is an important tool in the algebraic theory of quadratic
forms and of hermitian forms over division algebras. For a field
extension $L/K$ (of characteristic different from $2$), 
we can consider base change from $K$ to $L$; if moreover the extension has
finite
degree, then we also have the Scharlau transfer. The situation is
especially
well understood when $L/K$ is of odd degree or a quadratic extension: see
the book of Scharlau \cite{scharlau} for these basic notions and results.
 Another classical result, due to Kneser and Springer, concerns hermitian
 forms over quaternion algebras (see the appendix written  by
Springer in \cite{kneser} ).

The Witt group (and Witt ring for quadratic forms) gives a very useful
way to study quadratic and hermitian forms. The above results can be
expressed very efficiently in this framework.
 One of the basic result in the theory of 
 quadratic forms is a theorem of Pfister which determines
 the kernel of the restriction map $r_{L/K}^{*} : W(K) \rightarrow W(L)$ for 
 a quadratic extension $L/K$. More precisely, this kernel
 is the ideal generated by the form $<1, - \delta>$ 
where $L=K(\sqrt \delta)$. 
 One can express this result by the exactness of the
 sequence: 
\begin{equation}\label{exactpfister}W(K) \stackrel{t}{\longrightarrow} W(K) \stackrel{r_{L/K}^{*}}{\longrightarrow}W(L)
\end{equation}
 where $t$ is multiplication by the
 $2$-dimensional form $<1,-\delta>$. 
By a result of Elman-Lam, the Scharlau transfer map
$s_*:W(L)\longrightarrow W(K)$, can be used
to embed (\ref{exactpfister}) in the following 
exact triangle  (cf. \cite[Ch. 2, 5.10]{scharlau}):
\begin{equation}{\label{elman-lam}}\xymatrix@!{
W(K)\ar[rr]^{r^*}&&W(L)\ar[dl]^{s_*}\\
&W(K)\ar[ul]^{t}&
}
\end{equation}

 For a quadratic extension $L/K$ (resp. a quaternion division algebra 
 $(a,b)_{K}=D$) with nontrivial automorphism $-$ 
 (resp. with canonical involution $-$), one can consider the trace map 
 $W(L,-)\rightarrow W(K)$ (resp. $W(D,-) \rightarrow W(K)$). By a
 result of Jacobson, these maps are injective (cf. \cite[Ch. 10, 1.1, 1.2, 1.7]{scharlau}
 and \cite{jacobson}).

 In \cite[Appendix 2]{milnor}, Milnor and Husemoller construct the 
 following exact sequence of 
 $W(K)$-modules: 
\begin{equation}
0 \longrightarrow W(L,-) \longrightarrow W(K) \longrightarrow W(L)
\end{equation}
 where $-$ 
 is the nontrivial automorphism of the quadratic extension $L/K$.
 The results
concerning hermitian forms over quaternion division  algebras are given in the
papers of Lewis \cite{lewis0}, \cite{lewis1}. He found the following exact sequence:
\begin{equation}\label{suitequaternion} 
0\longrightarrow W(D,-)\longrightarrow W(L,-)\longrightarrow W^{-1}(D,-)\longrightarrow W(L)
\end{equation}
 where $L=K(\sqrt a)\subset D$
 is stable by $-$.
In fact, in this sequence, $D$ can be also split 
(cf. \cite[Ch. 10, 3.2]{scharlau}).

\begin{rem}
Note that Lewis uses $W(D,\wedge)$ instead of
$W^{-1}(D,-)$, where $\wedge$ is the orthogonal involution
of $D$ defined by $\widehat{i}=-i$ and $\widehat{j}=j$ 
where \{$1$, $i$, $j$, $ij$\} are usual generators of 
the quaternion algebra $D$. In fact $W(D,\wedge)\simeq
W^{-1}(D,-)$: this isomorphism is induced by the map which associates to
a hermitian form $h$ over $(D,\wedge)$, the skew-hermitian
form $ih$ over $(D,-)$.
\end{rem} 

 In \cite[Appendix 2]{bayer-parimala}, Parimala, Sridharan and Suresh obtain this 
 crucial exact sequence of Witt groups:
\begin{equation}\label{suitepss} 
W^{\varepsilon}(A,\sigma)\stackrel{\pi_1^{\varepsilon}}
{\longrightarrow} W^{\varepsilon} (\widetilde{A}, \sigma_{1})
\stackrel{\rho_1^\varepsilon}{\longrightarrow} 
 W^{-\varepsilon}(A, \sigma )\stackrel{\pi_2^{-\varepsilon}}
{\longrightarrow} W^{\varepsilon} (\widetilde{A}, \sigma_{2})
\end{equation}
 where $A$ is a central simple algebra with an involution $\sigma$ of 
 any kind and $\varepsilon=\pm1$ 
(for the notations, see the beginning of section \ref{definitions}). 
 The exact sequence 
(\ref{suitequaternion}) is a particular case of
 (\ref{suitepss}). In general the map 
$\pi_1^{\varepsilon}$ is not injective: in fact,
 \ref{injective} characterizes the cases where
$\pi_1^{\varepsilon}$ is injective.
 In \cite{bayer-parimala}, (\ref{suitepss}) is used to prove Serre's conjecture II for 
 classical groups.
 In \cite{lewis1}, Lewis has found a longer exact sequence:
\begin{equation}\label{suite-long}
\begin{array}{cc}0\rightarrow W(D,-)\rightarrow W(L,-)\rightarrow
W^{-1}(D,-)\rightarrow W(L) \rightarrow W^{-1}(D,-)\rightarrow \\ \\ 
\rightarrow
W(L,-)\rightarrow W(D,-)\rightarrow 0
\end{array}\end{equation}
 This sequence was one of our main motivations in order to embed the 
 exact sequence (\ref{suitepss}) in an even longer one. In fact we obtain an exact octagon:

 \begin{thm}\label{main1} 
 There is an exact sequence of Witt groups 
 (in fact of $W(K, \sigma|_{K})$-modules):
$$
\xymatrix@!{
&W^{\varepsilon}(A,\sigma)\ar[rr]^{\large\mbox{$\pi^{\varepsilon}_1$}} && W^{\varepsilon}(\widetilde{A},\sigma_1)\ar[dr]^{\large\mbox{$\rho_1^{\varepsilon}$}}\\
W^{-\varepsilon}(\widetilde{A},\sigma_2)\ar[ur]^{\large\mbox{$\rho_2^{-\varepsilon}$}}&&                              & & W^{-\varepsilon}(A,\sigma)\ar[d]^{\large\mbox{$\pi^{-\varepsilon}_2$}}\\
W^{\varepsilon}(A,\sigma)\ar[u]^{\large\mbox{$\pi^{\varepsilon}_2$}}&&                              & & W^{\varepsilon}(\widetilde{A},\sigma_2)\ar[dl]^{\large\mbox{$\rho_2^{\varepsilon}$}}\\
&W^{\varepsilon}(\widetilde{A},\sigma_1)\ar[ul]^{\large\mbox{$\rho_3^{\varepsilon}$}}                              && W^{-\varepsilon}(A,\sigma)\ar[ll]^{\large\mbox{$\pi_3^{-\varepsilon}$}}\\  
 }$$
 (for the definition of the maps, see section~\ref{definitions}). 
 \end{thm}

\begin{rem}
Note that the exact sequence (\ref{suite-long})
is a particular case of \ref{main1}.
\end{rem}

\begin{rem}
One can find a similar exact octagon of Witt groups of Clifford
algebras of quadratic forms with their canonical involution
 in \cite{lewis3}.
\end{rem} 

The proof of \ref{main1} can be found in section \ref{octagon}.
Because of the nature of Witt groups, the above octagon does not 
give, a priori, much information about the behavior of these maps
with respect to isotropy in the 
semigroups of isometry classes of nondegenerate
hermitian forms: in section \ref{calculs}, theorem \ref{pr2} 
describes this behavior in the case of division algebras.
We also deduce an alternative proof of \ref{main1}
in the case of division algebras.
 
 Section \ref{isotropy} contains the relation between the notion of
 isotropy over a central simple algebra $A$ and isotropy over the division
 algebra $D$ Brauer equivalent to $A$ via Morita equivalence as well as
 the Witt decomposition. 
In fact we point out that there exists a Witt decomposition
over a central simple algebra with involution. In particular,
this implies the existence of an anisotropic form in every
class of the Witt group.

In \cite{lewis2}, Lewis constructs an exact octagon of 
Witt groups of forms invariant under the action
of a finite group $G$ 
(Witt groups of equivariant forms) 
for quaternion division algebras.
in section \ref{octagong}, we show likewise that
the octagon of \ref{main1} is exact
if we replace Witt groups by equivariant Witt groups.
More precisely:
\begin{thm}\label{main2}
We suppose that $A$ satisfies the same hypotheses as at the beginning of
section \ref{definitions}. 
If we replace $W^{\pm \varepsilon}(A,\sigma)$ by $W^{\pm \varepsilon}(G,A,\sigma)$
and $W^{\pm \varepsilon}(\widetilde{A},\sigma_i)$ by 
$W^{\pm \varepsilon}(G,\widetilde{A},\sigma_i)$ (for $i=1, \ 2$) in theorem \ref{main1},
we obtain an exact octagon of $W(K, \sigma|_{K})$-modules.
\end{thm}
In section \ref{order}, by using the exact octagon, we obtain
 the following result: 

\begin{cor}{\label{order-first-kind}}
Let $A$ be a $K$-central simple algebra with an involution 
$\sigma$ of the first kind. Then we have
$|W^{\varepsilon}(A,\sigma)||W^{-\varepsilon}(A,\sigma)|=
|W(K)|$. In particular $W(K)$ is finite if and
only if $W^{\varepsilon}(A,\sigma)$ and 
$W^{-\varepsilon}(A,\sigma)$ are finite.
\end{cor}
\noindent 
This result was well known for quaternion algebras: see \cite{lewis1}.

We thank Eva Bayer-Fluckiger, Detlev Hoffmann and
Emmanuel Lequeu for useful conversations about this work
and for their comments on earlier versions of this paper.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Notation and Definition of the Maps}\label{definitions}
 Let $K$ be a field of characteristic different from $2$. All the modules in 
 this paper are supposed to be right modules which are finitely generated 
 and all the $\varepsilon$-hermitian forms are supposed to be nondegenerate. 
 Let $A$ be a central simple algebra over $K$ with an involution $\sigma$ 
 (of any kind). 
For $\varepsilon = \pm 1$, let $S^{\varepsilon}(A,\sigma)$  
 denote
the semigroup of isometry classes of $\varepsilon$-hermitian forms over 
$(A,\sigma)$ and let $W^{\varepsilon}(A,\sigma)$
be the Witt 
 group of $(A,\sigma)$ (i.e., the quotient of the
 Grothendieck group corresponding to $S^{\varepsilon}(A,\sigma)$
 by the subgroup generated by metabolic forms, 
 an $\varepsilon$-hermitian form $(V,h)$ being metabolic
 if there exists an $A$-submodule $W$ of $V$ such that
 $W=W^{\bot}$ for $h$).
 \begin{rem}
 As $A$ is simple, there is of course no difference between
 the notions of metabolic and hyperbolic
 hermitian forms. We use any of these two notions
 subsequently (except in section \ref{octagong}).
 \end{rem}
 First, we define the different maps involved in 
 \ref{main1}.As in \cite{bayer-parimala}, we suppose that
  there exist $\lambda, \ \mu \in A^{*}$ 
 such that $\sigma (\lambda) = - \lambda$, $\sigma (\mu) = -\mu$, $\mu \lambda = - 
 \lambda \mu$ and such that $L= K(\lambda)$ is a quadratic extension of $K$.
 We write $\widetilde{A}$ for the commutant of $L$ in $A$: this is a central 
 simple algebra over $L$ . One can easily verify that 
 $\mu \widetilde{A} = \widetilde{A} \mu$, $\mu^{2} \in \widetilde{A}$, 
 $\sigma (\widetilde{A}) = \widetilde{A}$ and 
 $A = \widetilde{A} \oplus \mu \widetilde{A}$.
 We define two involutions on $\widetilde{A}$ in the following way: let 
 $\sigma_{1} = \sigma |_{\widetilde{A}}$ and let 
 $\sigma_{2} = \intt(\mu^{-1}) \circ \sigma_{1}$ 
 (where $\intt(\mu^{-1}) (x) = \mu^{-1} x \mu$).
  \\\\
 $\diamond$ \underline{\textbf{Definition of $\pi^{\varepsilon}_{1}$ and 
 $\pi^{\varepsilon}_{2}$}}\\
 We have two $L$-linear projections: 
 $\pi_{1} : A \rightarrow \widetilde{A} : a_{1} + \mu a_{2} \mapsto a_{1}$ 
 and $\pi_{2} : A \rightarrow \widetilde{A} : a_{1} + \mu a_{2} \mapsto a_{2}$.
 If $h : V \times V \rightarrow A$ is an $\varepsilon$-hermitian space over 
 $(A, \sigma)$, we define (for $i=1,2$) $\pi^{\varepsilon}_{i}(h) : V \times V \rightarrow \widetilde{A}$ by 
 $\pi^{\varepsilon}_{i}(h)(x,y) = \pi_{i}(h(x,y))$. One readily verifies that
 $\pi^{\varepsilon}_{1}(h)$ is an $\varepsilon$-hermitian space over $(\widetilde{A}, \sigma_{1})$ and 
 that $\pi^{\varepsilon}_{2}(h)$ is a $-\varepsilon$-hermitian space over $(\widetilde{A}, \sigma_{2})$.
 In order to see that $\pi^{\varepsilon}_{1}$ and $\pi^{\varepsilon}_{2}$ 
 induce homomorphisms of Witt groups we have to prove that these maps respect 
 regularity, isometry classes, orthogonality and hyperbolicity. 
All of these properties come from the fact that:
$$h(x,y)=0\quad\forall y \iff \pi^{\varepsilon}_{1}(h)(x,y)=0\quad\forall y\iff \pi^{\varepsilon}_{2}(h)(x,y)=0\quad\forall y $$ 
 Hence, $\pi^{\varepsilon}_{1}$ and $\pi^{\varepsilon}_{2}$ induce 
 homomorphisms of semigroups of isometry classes of nondegenerate 
 hermitian forms and 
 homomorphisms of Witt groups (again denoted by $\pi^{\varepsilon}_{1}$ and 
 $\pi^{\varepsilon}_{2}$) 
 \[
 \begin{array}{ll}
  \pi^{\varepsilon}_{1} : S^{\varepsilon}(A, \sigma) \rightarrow S^{\varepsilon}(\widetilde{A},\sigma_{1});&  
  \pi^{\varepsilon}_{1} : W^{\varepsilon}(A, \sigma) \rightarrow W^{\varepsilon}(\widetilde{A},\sigma_{1})\\
  \pi^{\varepsilon}_{2} : S^{\varepsilon}(A, \sigma) \rightarrow S^{-\varepsilon}(\widetilde{A}, \sigma_{2});&
 \pi^{\varepsilon}_{2} : W^{\varepsilon}(A, \sigma) \rightarrow W^{-\varepsilon}(\widetilde{A}, \sigma_{2})
 \end{array}
 \]
 $\diamond$ \underline{\textbf{Definition of $\rho_1^{\varepsilon}$}}\\
 Let $(V,f)$ be an $\varepsilon$-hermitian space over 
 $(\widetilde{A}, \sigma_{1})$. We associate to it 
 $(V \otimes_{\widetilde{A}} A, \rho_1^{\varepsilon}(f))$ where 
 $\rho_1^{\varepsilon}(f) (x \otimes \alpha, y \otimes \beta) = 
 \sigma(\alpha) \lambda f(x,y) \beta$ for $x, \ y \in V$ and $\alpha, \ \beta \in A$.
 We can easily verify that $\rho_1^{\varepsilon}$ is well defined and that 
 $(V \otimes_{\widetilde{A}} A, \rho_1^{\varepsilon}(f))$ is a 
 $-\varepsilon$-hermitian space over $(A,\sigma)$.
 Moreover $\rho_1^{\varepsilon}$ induces homomorphisms: 
\[
\begin{array}{ll}
 \rho_1^{\varepsilon} : S^{\varepsilon}( \widetilde{A}, \sigma_{1}) \rightarrow S^{-\varepsilon} (A, \sigma); &
 \rho_1^{\varepsilon} : W^{\varepsilon}( \widetilde{A}, \sigma_{1}) \rightarrow W^{-\varepsilon} (A, \sigma)
\end{array}
\]
 $\diamond$ \underline{\textbf{Definition of $\rho_2^{\varepsilon}$}}\\
 Let $(V,f)$ be an $\varepsilon$-hermitian space over 
 $(\widetilde{A},\sigma_{2})$. We associate to it
  $(V \otimes_{\widetilde{A}} A, \rho_2^{\varepsilon}(f))$ 
  where $\rho_2^{\varepsilon}(f)( x \otimes \alpha, y \otimes \beta ) = \sigma (\alpha) \lambda \mu f(x,y) \beta$ for $x, \ y \in V$ and $\alpha, \ \beta \in A$.
 One can verify that 
 $\rho_2^{\varepsilon}$ is well defined, that $(V \otimes_{\widetilde{A}} A,
  \rho_2^{\varepsilon}(f))$ is a $-\varepsilon$-hermitian space over 
  $(A, \sigma)$ and that $\rho_2^{\varepsilon}$ induces homomorphism: 
  \[\begin{array}{ll}
    \rho_2^{\varepsilon} : 
  S^{\varepsilon}(\widetilde{A}, \sigma_{2}) \rightarrow S^{-\varepsilon}(A, \sigma);&
  \rho_2^{\varepsilon} : 
  W^{\varepsilon}(\widetilde{A}, \sigma_{2}) \rightarrow W^{-\varepsilon}(A, \sigma)
  \end{array}\]
 $\diamond$ \underline{\textbf{Definition of $\pi_3^{\varepsilon}$}}\\
 We define $\pi_3^{\varepsilon}$ to be $\lambda\pi^{\varepsilon}_{1}$, so we obtain homomorphisms:
\[\begin{array}{ll}
  \pi_3^{\varepsilon} :  S^{\varepsilon}(A, \sigma) \rightarrow S^{-\varepsilon}(\widetilde{A},\sigma_{1});&
  \pi_3^{\varepsilon} :  W^{\varepsilon}(A, \sigma) \rightarrow W^{-\varepsilon}(\widetilde{A},\sigma_{1})
 \end{array}\] 
 $\diamond$ \underline{\textbf{Definition of $\rho_3^{\varepsilon}$}}\\
 We define: $\rho_3^{\varepsilon} (f) = \rho_1^{-\varepsilon} (\lambda^{-1}f)$,  i.e.,
 $\rho_3^{\varepsilon}(f)(x\otimes\alpha,y\otimes\beta)=\sigma(\alpha)f(x,y)\beta$ 
 for $x,y\in V$ and $\alpha,\beta\in A$.
By a straightforward verification we obtain homomorphisms:
\[\begin{array}{ll}
\rho_3^{\varepsilon} : S^{\varepsilon}(\widetilde{A}, \sigma_{1}) \rightarrow S^{\varepsilon} (A, \sigma);&
   \rho_3^{\varepsilon} : W^{\varepsilon}(\widetilde{A}, \sigma_{1}) \rightarrow W^{\varepsilon} (A, \sigma)
\end{array}\]
\begin{rem}
Note that, in these definitions, $\varepsilon$ is arbitrary so for example 
$\pi_1^{-\varepsilon}$ will be a homomorphism of Witt groups from
 $W^{-\varepsilon} (A, \sigma)$ to  
$W^{-\varepsilon}(\widetilde{A}, \sigma_{1})$ and so on for
the other maps.
\end{rem}
A summary of these definitions can be found in table~\ref{table-de-def}.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{The Behavior of the Maps for Division Algebras}\label{calculs}
In this section $(D,\tau)$ is a division algebra with an involution $\tau$ of any kind. 
We assume that $(D,\widetilde{D},\tau,\tau_1,\tau_2)$ satisfies the same hypotheses as the ones mentioned in section 
\ref{definitions} for $(A,\widetilde{A},\sigma,\sigma_1,\sigma_2)$.

 \begin{prop}\label{pr1}
 {\rm(i)} If $h=<\delta>$ is a one dimensional $\varepsilon$-hermitian form over $(D,\tau)$ (with $\delta=d_{1} + \mu d_{2}$, $d_{1}, \ d_{2} \in \widetilde{D}$) then the matrix of $\pi^{\varepsilon}_{1}(h)$ over $(\widetilde{D},\tau_{1})$ with respect to the basis $\{ 1,\mu \}$ is $$\left(
                                                                                                                                                                 \begin{array}{cc}
                                                                                                                                                   d_{1} & \mu d_{2} \mu \\
                                                                                                                                           -\mu^{2}d_{2} & -\mu d_{1} \mu 
                                                                                                                                          \end{array} \right)$$
 {\rm(ii)} If $P$ is the matrix of  an $\varepsilon$-hermitian form $f$ over $(\widetilde{D},\tau_{1})$ with respect to a basis B, then $\lambda P$ is the matrix of $\rho_1^{\varepsilon}(f)$ over $(D, \tau)$ with respect to the basis $B\otimes 1$.\\
 {\rm(iii)} If $h=<\delta>$ is a one dimensional $-\varepsilon$-hermitian form over 
 $(D,\tau)$ (with $\delta=d_{1} + \mu d_{2}$, $d_{1}, \ d_{2} \in \widetilde{D}$) 
 then the matrix of $\pi^{-\varepsilon}_{2}(h)$ over $(\widetilde{D},\tau_{2})$ with respect to the basis $\{1,\mu\}$ is $$\left(
                                                                                                                                                                 \begin{array}{cc}
                                                                                                                                                   d_{2} & \mu^{-1} d_{1} \mu \\
                                                                                                                                           -d_{1} & -\mu d_{2} \mu 
                                                                                                                                          \end{array} \right)$$
 {\rm(iv)} If $P$ is the matrix of an $\varepsilon$-hermitian form $f$ over $(\widetilde{D},\tau_{2})$ with respect to a basis B, then $\lambda \mu P$ is the matrix of $\rho_2^{\varepsilon}(f)$ over $(D, \tau)$ with respect to the basis $B \otimes1$.\\
 {\rm(v)}  For a $-\varepsilon$-hermitian form $h$ over $(D,\tau)$, the matrix of $\pi_3^{-\varepsilon} (h)$ with respect to a basis B is $\lambda$ times the matrix of $\pi^{\varepsilon}_{1}(h)$ with respect to the basis
 $B \cup\mu B$.\\
 {\rm(vi)} If $P$ is the matrix of an $\varepsilon$-hermitian form 
 $f$ over $(\widetilde{D},\tau_{1})$ with respect to a basis B, then $P$ is the matrix of $\rho_3^{\varepsilon}(f)$ over $(D, \tau)$ with respect to the basis $B \otimes 1$.\\
 \end{prop}
 \begin{proof}
 {\rm(i)} We have :\\
 $h(1,1) = d_{1} + \mu d_{2}$, $h(1,\mu) = d_{1} \mu + \mu d_{2} \mu$, $h(\mu,1) = - \mu d_{1} - \mu^{2} d_{2}$ and $h(\mu,\mu) = -\mu d_{1} \mu -\mu^{2} d_{2} \mu$.                           
 As $d_{1}, \ d_{2} \in \widetilde{D}$, we can find the given matrix by applying $\pi^{\varepsilon}_{1}$ to these relations.\\
 {\rm(ii)} If $(V,f)$ is an $\varepsilon$-hermitian space over $(\widetilde{D}, \tau_{1})$ and if $\lbrace e_{1},\ldots,e_{n} \rbrace$ is a basis of $V$ over $\widetilde{D}$ then $\lbrace e_{1} \otimes 1,\ldots,e_{n}\otimes 1 \rbrace$ is a basis of $V \otimes_{\widetilde{D}} D$ over $D$.
 With the definition of $\rho_1^{\varepsilon}$, we have: $\rho_1^{\varepsilon}(f)(e_{i} \otimes 1, e_{j} \otimes 1) = \lambda f(e_{i},e_{j}) \ \forall \ i, j =1,\ldots,n$.\\
 {\rm(iii)} By applying $\pi^{-\varepsilon}_{2}$ to the four relations found in {\rm(i)}, we deduce the given matrix for $\pi^{-\varepsilon}_{2}(h)$.\\
 {\rm(iv)} If $(V,f)$ is an $\varepsilon$-hermitian space over $(\widetilde{D}, \tau_{2})$ and if $\lbrace e_{1},\ldots,e_{n} \rbrace$ is a basis of $V$ over $\widetilde{D}$ then $\lbrace e_{1} \otimes 1,\ldots,e_{n}\otimes 1 \rbrace$ is a basis of $V \otimes_{\widetilde{D}} D$ over $D$.
 With the definition of $\rho_2^{\varepsilon}$, we have: $\rho_2^{\varepsilon}(f)(e_{i} \otimes 1, e_{j} \otimes 1) =  \lambda \mu f(e_{i},e_{j}) \ \forall \ i, j =1,\ldots,n$.\\
 {\rm(v)} This is obvious by the definition of $\pi_3^{-\varepsilon}$.\\
 {\rm(vi)} If $(V,f)$ is an $\varepsilon$-hermitian space over $(\widetilde{D}, \tau_{1})$ and if $\lbrace e_{1},\ldots,e_{n} \rbrace$ is a basis of $V$ over $\widetilde{D}$ then $\lbrace e_{1} \otimes 1,\ldots,e_{n}\otimes 1 \rbrace$ is a basis of $V \otimes_{\widetilde{D}} D$ over $D$.
 By the definition of $\rho_3^{\varepsilon}$, we have: $\rho_3^{\varepsilon}(f)(e_{i} \otimes 1, e_{j} \otimes 1) = f(e_{i},e_{j}) \ \forall \ i, j =1,\ldots,n$.
\end{proof}\\
 The summary of the previous proposition can be found in 
table~\ref{tabl1}.

 \begin{lem}\label{comm}
 If $\widetilde{D}$ is commutative then $D$ is a quaternion division algebra, say $D=(a,b)_K$, $\widetilde{D}=L=K(\lambda)$ with
 $\lambda^2=a \in K$ and $\mu^2=b \in K$.
 \end{lem}
 \begin{proof}
 $\widetilde{D}$ is the commutant of $L$ in $D$, so it is a central simple algebra of center $L$. 
 As $\widetilde{D}$ is commutative, we have $\widetilde{D}=L$ and the lemma readily follows.
 \end{proof}

 \begin{lem}\label{noncomm}
 If $E$ is a noncommutative central simple algebra and $\star$ an involution over $E$,
 then for $\varepsilon= \pm1$, the set of nonzero $\varepsilon$-hermitian elements 
 with respect to $\star$ is nonempty.
 \end{lem}
 \begin{proof}
 This is a consequence of \cite[Ch. 8, 7.5]{scharlau}.
 \end{proof}\\ 
 The following theorem determines completely the behavior of the maps
 $\pi_1$, $\rho_1$, $\pi_2$, $\rho_2$,
  $\pi_3$ and $\rho_3$ with respect to isotropy.

 \begin{thm}\label{pr2}
 {\rm(i)} Let $h\in S^\varepsilon(D,\tau)$. If $\widetilde{D}$ is commutative and $\varepsilon=1$,
then $\pi_1^{\varepsilon}(h)\in S^{\varepsilon}(\widetilde{D},\tau_1)$ is isotropic if and only if $h$ is isotropic.
 Otherwise, $\pi^{\varepsilon}_1(h)$ is isotropic if and only if
 there exists $c \in \widetilde{D}^*$ with $\tau_2 (c)= -\varepsilon c$
 such that $h$  contains a subform isometric to $<\mu c>$.\\
 {\rm(ii)} Suppose that $f\in S^{\varepsilon}(\widetilde{D},\tau_1)$.
If
$\widetilde{D}$ is commutative, $\varepsilon=1$ and $\dim(f)=2$, then
$\rho_1^{\varepsilon}(f)\in S^{-\varepsilon}(D,\tau)$ is isotropic
if and only if $f$ is isotropic or $f$ is isometric to the 
two dimensional anisotropic form $<c,-bc>$ where $c\in K^*$
and $b=\mu^2$ (see \ref{comm}).
 Otherwise $\rho_1^{\varepsilon}(f)$ is
 isotropic if and only if there exist $d_1,d_2\in\widetilde{D}$ such that
 $\tau_1(d_1)=\varepsilon d_1$, $\tau_2(d_2)=-\varepsilon d_2$ and $f = f_{1} \bot f_{2}$
 for some nondegenerate \
$$f_{1} \simeq \left(\begin{array}{cc}
                     d_{1} & \mu d_{2} \mu \\
                    -\mu^{2}d_{2} & -\mu d_{1} \mu 
                    \end{array} \right).$$\\
 {\rm(iii)} For $h\in  S^{-\varepsilon}(D,\tau)$,  
 $\pi^{-\varepsilon}_{2}(h)\in S^{\varepsilon}(\widetilde{D},\tau_2)$ 
 is isotropic if and only if there exists $c \in \widetilde{D}^*$ with 
$\tau_1(c)=-\varepsilon c$
 such that $h$ contains a subform isometric to $<c>$.\\
 {\rm(iv)} For $f\in S^{\varepsilon}(\widetilde{D},\tau_2)$, 
  $\rho_2^{\varepsilon}(f)\in S^{-\varepsilon}(D,\tau)$ is isotropic 
if and only if there exist $d_1,d_2 \in \widetilde{D}$ 
  such that $\tau_1(d_1)=-\varepsilon d_1$, $\tau_2(d_2)=\varepsilon d_2$ 
and 
  $f =f_{1} \bot f_2$ for some nondegenerate 
  $$f_{1} \simeq \left(\begin{array}{cc}
            d_{2} & \mu^{-1} d_{1} \mu \\
            -d_{1} & -\mu d_{2} \mu 
          \end{array} \right).$$
 {\rm(v)} Let $h \in S^{-\varepsilon}(D,\tau)$.
If $\widetilde{D}$ is commutative and $\varepsilon=-1$, then  
$\pi_3^{-\varepsilon}(h)\in S^{\varepsilon}(\widetilde{D},\tau_1)$ 
is isotropic if and only if $h$ is isotropic.
Otherwise, $\pi_3^{-\varepsilon}(h)$ is isotropic if and only if
 there exists $c \in \widetilde{D}^*$ with $\tau_2(c)=\varepsilon c$
 such that $h$ contains a subform isometric to $<\mu c>$.\\
 {\rm(vi)} 
Suppose that $f\in S^{\varepsilon}(\widetilde{D},\tau_1)$.
If
$\widetilde{D}$ is commutative, $\varepsilon=-1$ and $\dim(f)=2$, then
$\rho_3^{\varepsilon}(f)\in S^{\varepsilon}(D,\tau)$ is isotropic
if and only if $f$ is isotropic or $f$
 is isometric to the 
two dimensional anisotropic form
$<\lambda c,-\lambda bc>$ 
where $c\in K^*$
and $b=\mu^2$ (see \ref{comm}).
Otherwise
 $\rho_3^{\varepsilon}(f)$ is isotropic
if and only if there exist $d_1,d_2 \in \widetilde{D}$ such that
 $\tau_1(d_1)=-\varepsilon d_1$, $\tau_2(d_2)=\varepsilon d_2$ and 
$f =f_{1} \bot f_2$ 
for some nondegenerate $$f_{1} \simeq \left(\begin{array}{cc}
\lambda d_{1} & \lambda \mu d_{2} \mu \\
 -\lambda \mu^{2} d_{2} & -\lambda \mu d_{1} \mu 
      \end{array} \right).$$
\end{thm}
 \begin{proof}Let $\H_{\varepsilon}$
 denote an $\varepsilon$-hyperbolic plane.\\
 \noindent
 {\rm(i)} First, suppose that $\widetilde{D}$ is 
commutative and $\varepsilon=1$. 
 By \ref{comm}, we know that $D$ is a quaternion division algebra. 
 The equivalence comes from the fact that, in this case,
 the trace form of $h$ is isotropic if and only if $h$ is isotropic.

 Next, suppose that the previous case is excluded.
 If $h$ is anisotropic and $\pi^{\varepsilon}_{1}(h)$ is isotropic then 
we can find $x \in V-\lbrace 0 \rbrace$ such that
 $\pi_1^{\varepsilon}(h(x,x))=0$, i.e., $h(x,x)= \mu c$ with 
$c \in \widetilde{D}^*$. As $\tau(h(x,x))=\varepsilon h(x,x)$, we conclude
 that $\tau_2(c)=-\varepsilon c$. It readily follows that $h$ contains 
$<\mu c>$.
Now consider the case where $h$ is isotropic. We have 
$\H_{\varepsilon}\simeq<\mu c,-\mu c>$ for all $c\in \widetilde{D}^*$ with 
$\tau_2(c)=-\varepsilon c$, provided such a $c$ exists.
If $\varepsilon=-1$ then we can take $c=1$.
If $\varepsilon=1$ then we only have to show that
there exists $c \in \widetilde{D}$ such 
that $\tau_2(c)=-c$.
 By assumption $\widetilde{D}$ is noncommutative, 
 and one can apply \ref{noncomm} 
to conclude.
 Conversely, suppose that $h$ contains a subform isometric to $<\mu c>$ as 
in the assertion.
By applying \ref{pr1} {\rm(i)} to the form $<\mu c>$, we easily deduce that
$\pi_1^{\varepsilon}(h)$ is isotropic.\\ 
 \noindent 
 {\rm(ii)} 
First suppose that $\widetilde{D}$ is commutative, 
$\varepsilon=1$ and $\dim(f)=2$.
We are in the situation of \ref{comm}.
 If $f$ is
isotropic or $f\simeq<c,-bc>$, it is obvious that
$\rho_1^{\varepsilon}(f)$ is isotropic. Conversely
suppose that
for a two dimensional form $f=<c,d>$, \ 
 $\rho_1^{\varepsilon}(f)=<\lambda c,\lambda d>$ is isotropic.
 So there exists
$q\in D^*$ such that 
\begin{equation}\label{z1z2}
\tau(q)\lambda cq+\lambda d=0.
\end{equation}
Write $q=z_1+\mu z_2$
with $z_1,\ z_2\in\widetilde{D}$.
By replacing $q$  with $z_1+ \mu z_2$
in (\ref{z1z2}), by using the fact that
$\lbrace 1,\mu \rbrace$ is an $L$-basis of $D$ and
by remarking that $\tau_2=\id_L$, we obtain the following system:
 $$\left \lbrace \begin{array}{l}
                  \tau_1(z_1)cz_1+\tau_1(z_2)bcz_2+d=0\\
                   z_1z_2=0
   \end{array} \right.$$
If $z_2=0$, then  $\tau_1(z_1)cz_1+d=0$; this means
that $f$ is isotropic. If $z_1=0$ then 
$d=-\tau_1(z_2)bcz_2$, so
$f\simeq<c,-\tau_1(z_2)bcz_2>\simeq<c,-bc>$.

Next, suppose that the previous case is excluded.
 If $f$ is anisotropic and $\rho_1^{\varepsilon}(f)$ is isotropic,
 let $z=x_1\otimes 1+y_1\otimes \mu$ be a nonzero isotropic  
 vector for $\rho_1^\varepsilon(f)$. By the definition of 
$\rho_1^{\varepsilon}$,
 we obtain 
$(\lambda f(x_1,x_1)-\mu\lambda f(y_1,y_1)\mu)+(\lambda f(x_1,y_1)\mu-\mu
 \lambda f(y_1,x_1))=0$
and so we obtain the following system:
  \begin{equation}\label{syst1}
  \left \lbrace \begin{array}{l}
                            f(x_{1},x_{1}) + \mu f(y_{1},y_{1}) \mu = 0\\
                            \mu f(y_{1},x_{1}) + f(x_{1},y_{1}) \mu = 0
                            \end{array} \right.
  \end{equation}                          
 As $f$ is anisotropic, thanks to this system, we can suppose that both $x_1$ and $y_1$ are nonzero.
Moreover, $x_1$ and $y_1$ are linearly independent over 
$\widetilde{D}$. In fact, if  $x_1$ and $y_1$ are linearly dependent then
  $x_1=y_1d$ with $d \in \widetilde{D}^*$ and by replacing
  $x_1$ with $y_1 d$ in (\ref{syst1}), we obtain the following system:
  \begin{equation}\label{syst2}
  \left \lbrace \begin{array}{l}
                            \tau(d)f(y_{1},y_{1})d + \mu f(y_{1},y_{1}) \mu = 0\\
                            \mu f(y_{1},y_{1})d + \tau(d)f(y_{1},y_{1}) \mu = 0
                            \end{array} \right.
  \end{equation} 
From the second equation of (\ref{syst2}), we obtain that
$\tau(d) f(y_1,y_1)=-\mu f(y_1,y_1)d \mu^{-1}$. By replacing
$\tau(d) f(y_1,y_1)$ by $-\mu f(y_1,y_1)d \mu^{-1}$ in the first equation of (\ref{syst2}), we obtain
$\mu f(y_1,y_1)(-d \mu^{-1}d+ \mu)=0$. As the second factor
is non zero for all $d \in \widetilde{D}$, $f(y_1,y_1)=0$ which
is a contradiction with the anisotropy of $f$.
 Now, $y_1$ and $x_1$ span a two dimensional subspace $W$ over
$\widetilde{D}$ and if we denote 
 $d_1=f(y_1,y_1)$, $d_2=\mu^{-1}f(y_1,x_1)\mu^{-1}$, the matrix $M$
 of $f\vert_{ W}$ with respect to the basis $\lbrace y_1,x_1 \rbrace$ is 
exactly
 the one given in the proposition. As $f_1=f|_W$ is nondegenerate (since $f$ is anisotropic), we
 can write $f=f_1 \bot f_2$ so $f$ contains the given form.
 Now consider the case where $f$ is isotropic.
 If $\widetilde{D}$ is noncommutative, we take $d_1=0$ and
we can find 
$d_2 \in \widetilde{D}$ such that $\tau_2(d_2)=-\varepsilon d_2$
 and it is  obvious that $$\H_\varepsilon \simeq \left(\begin{array}{cc}  0 & \mu d_2 \mu\\
                                                          -\mu^2 d_2 & 0
                                                          \end{array} \right),$$ so
$f$ contains the given form.
 If $\widetilde{D}$ is commutative and $\varepsilon=-1$,
 we take $d_1=0$ and $d_2=1$ and
 $\H_{-1}$ is isometric to the matrix given 
in the proposition.
 If $\widetilde{D}$ is commutative, $\varepsilon=1$ 
and $\dim(f) \geq 3$ then $f \simeq \H_1 \bot f_1$
 with $\dim(f_1) \geq 1$.
 If $f_1 \simeq <a,...>$ then 
$f \simeq < \mu a\mu,-\mu a\mu,a,...>$ so, for $d_1=a$, $d_2=0$, $f$
contains the given form.
 Conversely, with the same notations as {\rm(ii)},
  \ref{pr1} {\rm(ii)} and a straightforward calculation show that 
 $(\mu,1)$ is an isotropic vector for $\rho_1^{\varepsilon}(f_1)$.
 \\
\noindent
 {\rm(iii)} If $h$ is anisotropic and $\pi^{-\varepsilon}_{2}(h)$ 
is isotropic
 then we can find $x \in V- \lbrace 0 \rbrace$
 such that $\pi^{-\varepsilon}_{2}(h(x,x)) = 0$, that is
 $h(x,x)=c \in \widetilde{D}^*$. We conclude as in {\rm(i)}.
If $h$ is isotropic and $\varepsilon=1$ then 
$h \simeq <\lambda, -\lambda> \bot h_1$ and we can take $c=\lambda$.
 If $h$ is isotropic and $\varepsilon=-1$, $h\simeq\H_1\bot h_1$
 and all we have to do is to find $c \in \widetilde{D}^*$ such that
 $\tau_1(c)=c$; 
  we can take $c=1$.

  The converse is an easy consequence of \ref{pr1} {\rm(iii)}.\\ 
 \noindent
 {\rm(iv)} If $f$ is anisotropic and $\rho_2^\varepsilon(f)$ is isotropic,
 let $z= x_{1} \otimes 1 + y_{1} \otimes \mu $ be
  an isotropic vector for $\rho_2^{\varepsilon}(f)$. With a 
straightforward computation we find the same system as in the proof of~{\rm(ii)}.
Proceeding in the same way,
  we can suppose that $x_1$ and $y_1$ are nonzero and span 
  a two dimensional subspace $W$ over $\widetilde{D}$.
  If $d_{1} = -f(x_{1},y_{1})$ and $d_{2}= f(y_{1},y_{1})$ then 
the matrix of $f|_{ W}$ with respect to the basis $\{y_1,x_1\}$ is 
exactly the one
  given in the proposition.
  Now consider the case where $f$ is isotropic.
   If  $\varepsilon=1$, we take $d_2=0$ and $d_1=\lambda$.
   If $\varepsilon=-1$, we take $d_2=0$ and $d_1=1$.

Conversely, $(\mu,1)$ is an isotropic vector for $\rho_2^\varepsilon(f_1)$.\\
  \noindent {\rm(v)} If $\widetilde{D}$ is commutative and $\varepsilon=-1$ then
  the equivalence between $\pi_3^{-\varepsilon}(h)$ being isotropic
  and $h$ being isotropic readily comes from {\rm(i)}.

  Next, we suppose that the previous case is excluded.
   If $h$ is anisotropic and $\pi_3^{-\varepsilon}(h)$
 is isotropic we can conclude as in {\rm(i)} and {\rm(iii)}.
 If $h$ is isotropic then $h\simeq<\mu c,-\mu c>\bot h_1$ for all $c\in\widetilde{D}^*$
  such that $\tau_2(c) =\varepsilon c$ and we only have to find such a $c$.
  If $\widetilde{D}$ is noncommutative
this is clear. If $\widetilde{D}$ is commutative then $\tau_2=\id_L$.
As $\varepsilon=1$ we can take $c=1$.
Conversely, we only have to apply \ref{pr1} {\rm(v)}.\\
  \noindent 
{\rm(vi)} 
First, suppose that
 $\widetilde{D}$ is commutative, $\varepsilon=-1$ and
$\dim(f)=2$ then we are in the situation of
\ref{comm}. The proof goes as in {\rm(ii)}. We leave
it to the reader.
  
Next suppose that the previous case is 
excluded.
If $f$ is anisotropic and 
$\rho_3^\varepsilon (f)$ is isotropic, 
let $z= x_{1} \otimes 1 + y_{1} \otimes \mu $ be an isotropic vector 
  for $\rho_3^{\varepsilon}(f)$. We have the following system:
$$\left \lbrace
                                                                                                                                                                                 \begin{array}{l}
                                                                                                                                                                                   f(x_{1},x_{1}) - \mu f(y_{1},y_{1}) \mu       = 0\\
                                                                                                                                                                       f(x_{1},y_{1}) \mu - \mu f(y_{1},x_{1})  = 0
                                                                                                 \end{array} \right.$$
 Now, let $d_{1} = \lambda^{-1} f(y_{1},y_{1})$ and $d_{2} = \mu^{-1}
 \lambda^{-1} f(y_{1},x_{1}) \mu^{-1}$. 
Let $W$ be the two dimensional $\widetilde{D}$-subspace generated by $x_1$ and $y_1$
(the proof of the fact that $x_1$ and $y_1$ are linearly independent
over $\widetilde{D}$ is similar to {\rm(ii)}).
The matrix of $f|_W$ with respect to the basis $\{y_1,x_1\}$ is
exactly the one given in the proposition (the form $f|_W$
is nondegenerate because $f$ is anisotropic).
\indent Now consider the case where $h$ is isotropic.
 If $\widetilde{D}$ is noncommutative, we take $d_1=0$ 
and $d_2 \in \widetilde{D}$ such that
 $\tau_2(d_2)=\varepsilon d_2$.
 If $\widetilde{D}$ is commutative and $\varepsilon=1$, 
$f \simeq\H_1 \bot f_1$ and we take $d_2=1$ and $d_1=0$.
 If $\widetilde{D}$ is commutative, $\varepsilon=-1$ and 
$\dim(f) \geq 3$, we conclude as in {\rm(ii)}.
Conversely, $(\mu,1)$ is an isotropic vector for $\rho_3^\varepsilon(f_1)$.
\end{proof}\\
In particular, if $f$ and $h$ are anisotropic, we obtain table \ref{tabl2}.
\noindent From \ref{pr1} and \ref{pr2}, we obtain the following result: 

\begin{cor}\label{princ}
The octagon of \ref{main1} is exact when $A$ is a division algebra.
\end{cor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Isotropy of Hermitian Forms over Central Simple~Algebras}\label{isotropy}
The goal of this section is to prove that there exists a Witt decomposition
for nondegenerate $\varepsilon$-hermitian forms over a central simple algebra
with involution $(A,\sigma)$, i.e.,
for any form $h$, 
 $h\simeq h_1 \bot h_2$ with $h_1$ 
hyperbolic and
$h_2$ anisotropic (a notion that we have to define) unique up to isometry.

 The notion of isotropy that we are going to use is the following:

\begin{defi}\label{iso}
A nondegenerate $\varepsilon$-hermitian space $(V,h)$ over $(A,\sigma)$
is said to be isotropic if there exists $x \in V-\{0\}$ such that
$h(x,x)=0$.
\end{defi}

\begin{rem}
In the case where $A$ is a division algebra, this notion is exactly
the one usually given in the literature.  
\end{rem}
Now, we recall a crucial notion: Morita-equivalence. The following
definition is based upon \cite{dejaiffe}. 

\begin{defi}\label{Morita}
 Let $(A,\sigma)$ and $(B,\tau)$ be two central simple algebras
over $K$ such that $\sigma$ and $\tau$ are two $K/k$-involutions.
 Let $\delta=1$ if $\sigma$ and $\tau$ are of the second kind or of the first kind 
 and of the same type and $\delta=-1$ if $\sigma$ and $\tau$ are of the first kind
  and of different type.\\
 A $\delta$-Morita equivalence $((A, \sigma ), (B, \tau), M, N, f, g, \nu)$ between the 
algebra with involutions $(A, \sigma)$ and $(B, \tau)$ is a tuple consisting 
of :\\
 $\bullet$ an $(A,B)$-bimodule  $M$ (i.e., a left $A$-module and a right $B$-module with compatible structures);\\
 $\bullet$ a $(B,A)$-bimodule $N$;\\
 $\bullet$ two nonzero bimodule homomorphisms 
 $f : M \otimes_{B} N \rightarrow A$ and $g : N \otimes_{A} M \rightarrow B$ 
which are associative, i.e., 
 $f( m \otimes n). m^{\prime} = m. g(n \otimes m^{\prime})$ et $g(n \otimes m).
  n^{\prime} = n. f(m \otimes n^{\prime})$ for all $m, \ m^{\prime} \in M$, $n, \ n^{\prime} \in N$;\\
 $\bullet$ a linear bijective map $\nu : M \rightarrow N$ which
  verifies $\nu(amb) = \tau (b) \nu(m) \sigma (a)$ for all $a\in A$, $m\in M$, $b \in B$.
 \end{defi}

 \begin{rem}
Note that we do not suppose that $\sigma$ and $\tau$ are of the same type as in \cite{dejaiffe}:that
is why we call this notion $\delta$-Morita equivalence.
\end{rem}


\begin{rem}\label{bimo}
 In fact one can prove that $f$ (resp. $g$) is a  bimodule isomorphism
 between $M\otimes_BN$ and $A$ (resp. $N\otimes_AM$ and $B$), see \cite[\S 1.1]{dejaiffe}.
\end{rem}
\noindent Now, we suppose that $B=D$ denotes the division algebra Brauer equivalent to $A$.
By Albert's theorem, we know that there exists an involution $\tau$ over $D$ such that
$\tau$ is of the same kind as $\sigma$. By \cite[\S 1.4]{dejaiffe}, one can find $M, \ N, \ f, \ g$ and $\nu$
as in \ref{Morita} and $\delta \in \{\pm1\}$ such that $((A,\sigma),(D,\tau),M,N,f,g,\nu)$
 is a $\delta$-Morita equivalence. We can define semigroup homomorphisms:
 $$F:S^\varepsilon(A,\sigma) \rightarrow S^{\delta \varepsilon}(D,\tau)
 ;\quad(V,h) \mapsto (V\otimes_AM, b_0 h)$$
 $$G:S^{\delta \varepsilon}(D,\tau)\rightarrow S^\varepsilon(A,\sigma)
 ;\quad(W,\phi)\mapsto (W \otimes_DN, b^{'}_0 \phi)$$
\noindent where: 
\[\begin{array}{llll}
(b_{0}h) (v \otimes m, v^{\prime}\otimes m^{\prime} ) &=&
g (\nu (m) \otimes h(v,v^{\prime}) m^{\prime})& \forall v, \ v^{\prime} \in V
 , m, \ m^{\prime} \in M\\
 (b_{0}^{\prime}\phi)(w \otimes n, w^{\prime}\otimes n^{\prime})
  &=& f ( \nu^{-1}(n) \otimes \phi(w,w^{\prime}) n^{\prime})
 &\forall w, \ w^{\prime} \in W, n, \ n^{\prime} \in N.
 \end{array}\]
  In fact, we can prove that $F$
 is a semigroup isomorphism and $G$ is its inverse and that they induce
 isomorphisms of Witt groups.
 The details of proofs
 can be found in \cite[{\rm{I.9}}, 3.5]{knus}. 
\begin{rem}\label{morita-theory}
From now on, when we will use an argument involving
 Morita theory, we will implicitly refer to 
\cite[{\rm{I.9}}]{knus}. 
\end{rem} 
 We have:
\begin{lem}\label{part}
{\rm(i)} $M$ is a simple left $A$-module and $N$ is a simple right $A$-module.
\noindent{\rm(ii)} The maps $F$ and $G$ respect the rank of hermitian spaces
(recall that the rank of a hermitian space $(V,h)$ over $(A,\sigma)$, where
$V$ is a right (resp. left) $A$-module, is defined to be the positive integer
$n$ such that $V\simeq T^n$ where $T$ is a simple right (resp. left) $A$-module).
\end{lem}
 \begin{proof}
 {\rm(i)} We prove it for $N$, the proof for $M$ being similar. As
 $A$ is a simple algebra, $N$ is a semisimple right $A$-module and we can write
 $N \simeq T^n$ where $n\in\N-\{0\}$ and $T$ is a simple right $A$-module. But, we know from \ref{bimo}
 that $N\otimes_AM\simeq D$ as $(D,D)$-bimodules so:
 $$D \stackrel{g^{-1}}{\simeq} N\otimes_AM \simeq (T\otimes_A M)^n$$
 as right $D$-vector spaces. A dimension argument shows that $n=1$ and $N$ is simple as 
 a right $A$-module.\\
 {\rm(ii)} We prove it for $F$, the proof for $G$ is similar.
 If $(V,h)$ is a right $A$-module of rank $n$ then we have $V \simeq T^n$ where
 $T$ is a simple right $A$-module; by {\rm(i)}, we can take $T=N$.
 As $D$ is a division algebra, $D$ is a simple right $D$-module
 and we have:$$V\otimes_AM \simeq (N\otimes_AM)^n \stackrel{g}{\simeq} D^n.$$
 So we deduce that the rank of $V\otimes_AM$ is $n$.
 \end{proof}\\
 Now, we can prove (for fixed $A$, $\sigma$ and $D$ as before):

 \begin{prop}
 $(V,h)$ is isotropic over $(A,\sigma)$ if and only if for every $\delta$-Morita
 equivalence $((A,\sigma),(D,\tau),M,N,f,g,\nu)$, $F(V,h )$
 is isotropic over $(D,\tau)$.
 \end{prop}
 \begin{proof}
 Let $x \in V-\{0\}$ be such that $h(x,x)$ = 0 and $((A,\sigma),(D,\tau),M,N,f,g,\nu)$
 be a $\delta$-Morita equivalence. We can easily see that there
 exists $m\in M$ such that $x \otimes m\neq 0$; in fact if $x\otimes m=0$ for 
 all $m\in M$ then we have $V_1\otimes_AM=0$ where $V_1=xA\neq 0$. 
 Now $N$ is a simple right $A$-module by \ref{part},
 so we have $V_1=\oplus_{i=1}^dN$, $d \geqslant 1$. Therefore we conclude that
 $0=V_1\otimes_AM\simeq\oplus_{i=1}^d(N\otimes_AM)\stackrel{g}{\simeq}D^d$ which is a 
 contradiction. Now $x\otimes m$ is clearly an isotropic vector for $b_{0}h$  
 so $F(V,h)$ is isotropic.
 If $F(V,h)$ is isotropic and if $((A,\sigma),(D,\tau),M,N,f,g,\nu)$ 
 is a $\delta$-Morita equivalence, let $y\neq 0$ be 
 an isotropic vector for $b_{0}h$. By the same argument as before one can find
 $n\in N$ such that $y\otimes n\neq 0$. Using the definition of $F^{-1}(=G)$ we  
 see that $y \otimes n$ is an isotropic vector for $b_{0}^{\prime}b_{0}h$ 
 for all $n \in N$. 
 But $(V\otimes_{A} M \otimes_{D} N, b_{0}^{\prime}b_{0}h)$ is isometric
  to $(V,h)$ so we can conclude the existence of an $x \in V-\{0\}$ such that 
  $h(x,x) =0$.
\end{proof}\\
Using this proposition and the fact that the Witt decomposition exists over $(D,\tau)$,
we conclude the existence of a Witt decomposition over $(A,\sigma)$.
Namely, if $(V,h)$ is an $\varepsilon$-hermitian space over $(A,\sigma)$,
then $F(V,h) \simeq \phi_1 \bot \phi_2$ where $\phi_1$ is hyperbolic and $\phi_2$ anisotropic
 over $(D,\tau)$.
We have : $$(V,h) \simeq (G \circ F)(V,h) \simeq G(\phi_1) \bot G(\phi_2)$$
By the previous proposition, $G(\phi_1)$ is hyperbolic and we can show that
$G(\phi_2)$ is anisotropic. By the same type of argument, we can show that this
decomposition is unique up to isometry because it is the case over $(D,\tau)$.
So:

\begin{cor}\label{wittdeco}
{\rm(i)} There exists a Witt decomposition over $(A,\sigma)$.\\
{\rm(ii)} For all $\lbrack h \rbrack \in W^{\varepsilon}(A,\sigma)$, there exists an
anisotropic form $h_0$ over $(A,\sigma)$ such that $\lbrack h\rbrack=\lbrack h_0\rbrack$
 in $W^\varepsilon(A,\sigma)$.
\end{cor}

\begin{rem}
This result should be well known,
but we could not find an explicit statement of it in the literature.
\end{rem} 
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
 \section{Exactness of the Octagon for Central Simple Algebras}\label{octagon}
 Let $(A, \sigma)$ be a central simple algebra over $K$ ($\car K \not= 2$) 
 with involution. We make the same hypotheses as in \ref{definitions}: 
 namely we suppose that there exist 
 $\lambda, \ \mu \in A^{*}$ such that 
 $\sigma (\lambda) = - \lambda$, 
 $\sigma (\mu) = -\mu$, $\mu \lambda = - \lambda \mu$ and that $L= K(\lambda)$ 
 is a quadratic extension of $K$.
 We write $\widetilde{A}$ for the commutant of $L$ in $A$. We have 
 $\mu \widetilde{A} = \widetilde{A} \mu$, $\mu^{2} \in \widetilde{A}$, 
 $\sigma (\widetilde{A}) = \widetilde{A}$ and 
 $A = \widetilde{A} \oplus \mu \widetilde{A}$.
 We define two involutions on $\widetilde{A}$ in this way: 
 let $\sigma_{1} = \sigma |_{\widetilde{A}}$ and let 
 $\sigma_{2} = \intt(\mu^{-1}) \circ \sigma_{1}$.

\noindent \\
\textbf{Proof of theorem \ref{main1}:}
 thanks to \cite[Appendix 2]{bayer-parimala}, we already know that
 $\im\pi^{\varepsilon}_1=\ker\rho_1^{\varepsilon}$ and $\im\rho_1^{\varepsilon}=\ker\pi^{-\varepsilon}_2$.
 First, we prove that this sequence is a complex from $\pi_2^{-\varepsilon}$ up 
to $\rho_3^{\varepsilon}$.

$\rho_2^{\varepsilon}\pi_2^{-\varepsilon}=0:$
  let $(V,h)$ be a ($-\varepsilon$)-hermitian space over $(A, \sigma)$, so
  $\pi_2^{-\varepsilon}(h)$ is an $\varepsilon$-hermitian form over 
  $(\widetilde{A},\sigma_2)$ and $\rho_2^{\varepsilon}\pi_2^{-\varepsilon}(h)$
  will be a ($-\varepsilon$)-hermitian form over $(A,\sigma)$. It is enough
  to find a self orthogonal right $A$-submodule of $V\otimes_{\widetilde{A}}A$
  with respect to $\rho_2^{\varepsilon}\pi_2^{-\varepsilon}(h)$.   
 Let 
 \begin{equation}\label{space1}
 W = \lbrace x.\mu \otimes 1 + x \otimes \mu | \ x \in V \rbrace.
 \end{equation}
 Now, $W$ is readily seen to be a right $A$-submodule of 
  $V_{\widetilde{A}} \otimes_{\widetilde{A}} A$ and an easy calculation 
  shows that this space is a totally isotropic subspace of $\rho_2^{\varepsilon} \pi^{-\varepsilon}_{2} (h)$. 
  By dimension count over $K$, we have $W = W^{\bot}$ (with respect to 
  $\rho_2^{\varepsilon}\pi_2^{-\varepsilon}(h)$) and so 
  $\rho_2^{\varepsilon} \pi^{-\varepsilon}_{2} (h)$ is hyperbolic.

$\pi_3^{-\varepsilon} \rho_2^{\varepsilon}=0:$
  let $(V,h)$ be an $\varepsilon$-hermitian space over 
 $(\widetilde{A}, \sigma_{2})$. Let 
\begin{equation}\label{space2}
 W'= \lbrace x \otimes 1 | \ x \in V \rbrace\subset V\otimes_{\widetilde{A}}A. 
\end{equation} 
 Then $W'$ is a $\widetilde{A}$-submodule of $V_{A} \otimes_{\widetilde{A}} A$ 
 and it is a totally isotropic subspace for $\pi_3^{-\varepsilon} \rho_2^{\varepsilon} (h)$. 
 By a dimension argument one has $W'={W'}^{\bot}$ and so
 $\pi_3^{-\varepsilon} \rho_2^{\varepsilon} (h)$ is hyperbolic.

$\rho_3^{\varepsilon} \pi_3^{-\varepsilon}=0:$
 it is  obvious from the definition that $\rho_3^{\varepsilon} \pi_3^{-\varepsilon} =0$.\\
Next we prove that the sequence is exact from $\pi_2^{-\varepsilon}$
 up to $\pi_1^\varepsilon$.

$\ker(\rho_2^{\varepsilon}) \subset  \im(\pi_2^{-\varepsilon}):$
let $(W,f)$ be an $\varepsilon$-hermitian form over $(\widetilde{A},\sigma_2)$
such that $\rho_2^{\varepsilon}(f)$ is hyperbolic.
We may assume that $f$ is anisotropic thanks to \ref{wittdeco}. 
There exists an $A$-submodule $W_1$ of $W\otimes_{\widetilde{A}}A$ such that
$W_1^\bot=W_1$ (with respect to $\rho_2^{\varepsilon}(f)$).
Let $W\otimes \mu= \lbrace w\otimes \mu |w \in W \rbrace$.
Let $w_1\in W_1\cap(W\otimes\mu)$ ($w_1=w\otimes\mu$ with $w_1\in W_1$
and $w\in W$). As $\rho_2^{\varepsilon}(f)(w_1,w_1)=0$, $f(w,w)=0$ and
so $w_1=w\otimes\mu=0$ since $f$ is anisotropic.
Moreover $\dim_KW_1=\frac{1}{2}\dim_KW\otimes_{\widetilde{A}}A=
\dim_KW\otimes\mu$ so $W\otimes_{\widetilde{A}}A=W_1\oplus(W\otimes\mu)$
as $\widetilde{A}$-modules.
This implies that for all $w\in W$, there exists $w'\in W$ such that
$w\otimes 1+w'\otimes\mu\in W_1$. Since $A$ is a free $\widetilde{A}$-module, 
$w'$ is uniquely determined by $w$ and we write 
$J(w):=w'$. By definition of $J$, we have
$J^2(w)=w\mu^{-2}$ and $J(wa)=J(w)\mu a\mu^{-1}$ for all
$w\in W$ and $a\in \widetilde{A}$.
As $W_1=W_1^\bot$, 
$\rho_2^{\varepsilon}(f)(x\otimes1+J(x)\otimes\mu,
y\otimes 1+J(y)\otimes\mu)=0$ for all $x,\ y\in W$ and we 
obtain the following system:
\begin{equation}\label{Jsystem} 
\left \lbrace \begin{array}{l}
   f(x,y)+\mu f(J(x),J(y))\mu=0\\
   f(x,J(y))\mu+\mu f(J(x),y)=0
   \end{array} \right.
\end{equation} 
By means of $J$, we define an
$A$-module structure over $W$ by 
$$w.\mu=J(w)\mu^2$$
for all $w\in W$. We denote by $W_J$
the $A$-module $W$ equipped with this
new action. Let $h$ be the map defined by 
\begin{equation}\label{ante1}
h(x,y)=\mu f(x,J(y))\mu+\mu f(x,y)
\end{equation} 
for all $x,\ y\in W_J$. By the definition of $J$
and (\ref{Jsystem}), we conclude that 
$(W_J,h)$ is a $(-\varepsilon)$-hermitian space over 
$(A,\sigma)$. Let us show that $h$ is sesquilinear on the left with
respect to $\sigma$: $h$ is clearly biadditive, so it suffices
to show this fact for $\mu$ and for elements of $\widetilde{A}$.
We have:
\[
\begin{array}{ll}
h(x.\mu,y) &=h(J(x)\mu^2,y)\\
           &=\mu f(J(x)\mu^2, J(y))\mu+\mu f(J(x)\mu^2,y)\\
           &=\mu^3 f(J(x),J(y))\mu+\mu^3f(J(x),y)\\
           &=-\mu^2 f(x,y)-\mu^2f(x,J(y))\mu \quad (\mathrm{using} \ (\ref{Jsystem}))\\
           &=\sigma(\mu)h(x,y) 
\end{array}\]
Sesquilinearity on the left for elements of $\widetilde{A}$ and
linearity on the right are done
in the same way. Let us prove that $h$ is $-\varepsilon$-hermitian
with respect to $\sigma$:
\[\begin{array}{ll}
h(y,x) &=\mu f(y,J(x)) \mu +\mu f(y,x)\\
       &=\varepsilon  \sigma(f(J(x),y))\mu^2+ \varepsilon\sigma(f(x,y))\mu\\
       &=-\varepsilon\sigma(\mu^{-1}f(x,J(y))\mu)\mu^2-\varepsilon\sigma
(\mu f(x,y)) \quad (\mathrm{using} \ (\ref{Jsystem}))\\
       &=-\varepsilon \sigma(h(x,y))
\end{array}\]
If $h$ is degenerate, by (\ref{ante1}), there exists a non zero $x \in W$
such that $f(x,y)=0$ for all $y \in W$. In particular, this implies
that $f$ is isotropic which is a contradiction.
Now, $(W_J,h)$ is 
the antecedent of $(W,f)$ by $\pi_2^{-\varepsilon}$,
i.e. 
$((W_J)_{\widetilde{A}}, \pi_2^{-\varepsilon}(h))$
is isometric to $(W,f)$ (the isometry
is given by the identity map).
We conclude that 
$\ker(\rho_2^{\varepsilon})\subset\im(\pi_2^{-\varepsilon})$.

$\ker(\pi_3^{-\varepsilon})\subset\im(\rho_2^\varepsilon)$:
let $(V,h)$ be a $(-\varepsilon)$-hermitian space over
$(A,\sigma)$ such that $\pi_3^{-\varepsilon}(h)$ is hyperbolic. We can assume that $h$ 
is anisotropic thanks to \ref{wittdeco}. There exists an $\widetilde{A}$-submodule $W$ of
 $V_{\widetilde{A}}$ such that $W^{\bot}=W$ (with respect to $\pi_3^{-\varepsilon}(h)$).
 This implies that $h(x,y) \in \mu\widetilde{A}$, for all $x, \ y \in W$.
We define a map 
$f: W \times W \rightarrow \widetilde{A}$ where:
\begin{equation}\label{ante2}
f(x,y)= \mu^{-1}\lambda^{-1}h(x,y)
\end{equation} for $x, \ y \in W$.
 Since $h$ is anisotropic,
$f$ is nondegenerate, we can easily
see that $(W,f)$ is an $\varepsilon$-hermitian form over
$(\widetilde{A},\sigma_2)$ and $(W\otimes_{\widetilde{A}}A,\rho_2^\varepsilon(f))$
is isometric to $(V,h)$ via $\Phi(w\otimes a)=wa$ for all $w \in W$, $a\in A$. 
 We conclude
 that $\ker\pi_3^{-\varepsilon}\subset\im \rho_2^{\varepsilon}$.\\
The exactness of $$W^{-\varepsilon}(A, \sigma)\stackrel{\pi_3^{-\varepsilon}}{\rightarrow}W^{\varepsilon}(\widetilde{A},\sigma_1)\stackrel{\rho_3^{\varepsilon}}{\rightarrow}W^{\varepsilon}(A,\sigma)$$
  follows from theexactness of
  $W^{\varepsilon}(A, \sigma)\stackrel{\pi_1^{\varepsilon}}{\rightarrow}W^{\varepsilon}(\widetilde{A},\sigma_1)\stackrel{\rho_1^{\varepsilon}}{\rightarrow}W^{-\varepsilon}(A,\sigma)$;
 in fact we have:
\begin{center} 
$\ker(\rho_3^\varepsilon)=\lambda\ker(\rho_1^{-\varepsilon})
=\lambda\im(\pi_1^{-\varepsilon})=\im(\pi_3^{-\varepsilon}).$
\end{center}
 The exactness of 
 $$W^{\varepsilon}(\widetilde{A}, \sigma_1)\stackrel{\rho_3^{\varepsilon}}{\rightarrow}W^{\varepsilon}(A,\sigma)\stackrel{\pi_2^{\varepsilon}}{\rightarrow}W^{-\varepsilon}(\widetilde{A},\sigma_2)$$ 
 follows from the exactness 
 of $$W^{\varepsilon}(\widetilde{A}, \sigma_1)\stackrel{\rho_1^{\varepsilon}}{\rightarrow}W^{-\varepsilon}(A,\sigma)\stackrel{\pi_2^{-\varepsilon}}{\rightarrow}W^{\varepsilon}(\widetilde{A},\sigma_2).$$
The exactness at $W^{-\varepsilon}(\widetilde{A}, \sigma_2)$
follows from the exactness at $W^{\varepsilon}(\widetilde{A}, \sigma_2)$.
\\ Finally, we have 
$\ker\pi^{\varepsilon}_1=\lambda^{-1}\ker\pi_3^{\varepsilon}=\lambda^{-1}\im\rho_2^{-\varepsilon}=\im\rho_2^{-\varepsilon}$
which shows the exactness of $$W^{-\varepsilon}(\widetilde{A},\sigma_2)\stackrel{\rho_2^{-\varepsilon}}{\rightarrow}W^{\varepsilon}(A,\sigma)\stackrel{\pi_1^{\varepsilon}}{\rightarrow}W^{\varepsilon}(\widetilde{A},\sigma_1)$$ 
thus completing the proof.
\hfill$\Box$\bigskip


\begin{rem}
The proof of \ref{main1} is based on the
one given in \cite{lewis2} for quaternion algebras. Nevertheless, one can
also give a proof of it by an induction argument
as in \cite[Appendix 2]{bayer-parimala}.
\end{rem}

We need the following trivial but useful lemma:

\begin{lem}\label{morita-split}
Suppose that $B$ is a central simple algebra over $K$ with an involution
$\tau$. Then \\
{\rm(i)} $W(B,\tau)=0$ if and only if
$B$ is split and $\tau$ is of the first kind and of symplectic type.\\
{\rm(ii)} $W^{-1}(B,\tau)=0$ if and only if
$B$ is split and $\tau$ is of the first kind and of orthogonal type.
\end{lem}
\begin{proof} For (i), if $B$ is split and $\tau$ is of the 
first kind and of symplectic type it is clear that 
$W(B,\tau)=0$. Conversely let suppose that
 $W(B,\tau)=0$. First we show that $B$ is split. In fact
 let $B_0$ be the Brauer-equivalent division algebra of $B$.
 By Albert's theorem there exists an involution $\tau_0$ on $B_0$
 of the same kind than $\tau$. By Morita theory,
 $W(B,\tau)\simeq W^{\varepsilon}(B_0,\tau_0)$,
 where $\varepsilon=1$ if $\tau$ and $\tau_0$ are
both of the second kind or of the first kind and of
 the same type, otherwise $\varepsilon=-1$. So we
 conclude that  $W^{\varepsilon}(B_0,\tau_0)=0$. If 
 $B$ is not split then the same holds for $B_0$. But in this case,
 \ref{noncomm} implies that the set of the nondegenerate
 $\varepsilon$-hermitian forms of dimension $1$ is nonempty,
 in particular $W^{\varepsilon}(B_0,\tau_0)\neq 0$ which
 is a contradiction. So $B$ is split.  Now we prove that
 $\tau$ is of the first kind. In fact if $\tau$ is of the 
 second kind then by Morita theory
 we have $W(B,\tau)\simeq W(K,\tau|_K)$. So we conclude
 that $W(K,\tau|_K)=0$ which is impossible. Thus $\tau$
 is of the first kind. Finally $\tau$ is of symplectic type, otherwise
 by Morita theory we obtain
 $0=W(B,\tau)\simeq W(K)$ which is impossible. The proof
 of (ii) is similar.
\end{proof}

\begin{cor}\label{injective}
The map $\pi^{\varepsilon}_1$ is injective 
if and only if $L$ is a splitting field of $A$, $\sigma$ is of the first kind and either $\varepsilon=1$ and $\sigma$ is
symplectic, or $\varepsilon=-1$ and $\sigma$ is orthogonal.  
\end{cor}
\begin{proof}
First, one can easily see that $\sigma$ and $\sigma_2$ are of the same
kind and of different type when they are of the
first kind.
If $L$ is a splitting field of 
$A$,  then by \cite[Ch. 8, 5.4]{scharlau}, $\widetilde{A}$
is split. 
If $\varepsilon=1$ and $\sigma$ is of the first kind and
of symplectic type or $\varepsilon=-1$ and $\sigma$ is of
the first kind and of orthogonal type then, by Morita theory,
we conclude that $W^{-\varepsilon}(\widetilde{A},\sigma_2)=0$.
 By the exactness of the octagon
at $W^{\varepsilon}(A,\sigma)$,  $\pi^{\varepsilon}_1$ becomes injective.

Conversely if $\pi_1^{\varepsilon}$ is injective then again by the exactness of the octagon 
at $W^{\varepsilon}(A,\sigma)$,
$\im\rho_2^{-\varepsilon}=0$. We show that 
$W^{-\varepsilon}(\widetilde{A},\sigma_2)=0$.
 If $W^{-\varepsilon}(\widetilde{A},\sigma_2)\neq 0$,
 and as $\delta$-Morita equivalence  preserves the rank by \ref{part}, we can find a
 nondegenerate $-\varepsilon$-hermitian form of rank one
 over $(\widetilde{A},\sigma_2)$ by considering the division algebra 
Brauer-equivalent to $\widetilde{A}$.
 As $\rho_2^{-\varepsilon}$ preserves the rank we conclude that there exists
a nondegenerate $\varepsilon$-hermitian form of rank one in the image of 
$\rho_2^{-\varepsilon}$, which is
a contradiction, since the image of $\rho_2^{-\varepsilon}$ in 
$W^{\varepsilon}(A,\sigma)$
is zero.
So $W^{-\varepsilon}(\widetilde{A},\sigma_2)=0$. Now 
the result follows from \ref{morita-split}.
\end{proof}\

\begin{rem}
If $A$ is a
split quaternion algebra and $\sigma$ is symplectic, then  the 
exact octagon of \ref{main1} becomes:
$$0\rightarrow W(L,-)\rightarrow W(K)\rightarrow W(L)\rightarrow
W(K)\rightarrow W(L,-)\rightarrow 0$$ where 
 $L/K$ is a quadratic extension with a nontrivial
 automorphism $-=\sigma|_L$. If $A$ is a quaternion
division algebra and $\sigma$ is symplectic
then \ref{main1} becomes:
\begin{equation*}
\begin{array}{cc}
0\rightarrow W(A,\sigma)\rightarrow W(L,\sigma|_L)\rightarrow
W^{-1}(A,\sigma)\rightarrow W(L)\rightarrow W^{-1}(A,\sigma)\rightarrow\\ \\
\rightarrow
W(L,\sigma|_L)\rightarrow W(A,\sigma)\rightarrow 0
\end{array}
\end{equation*}
where $L$ is a maximal subfield of $A$
which is stable by the involution. These sequences can
be found in \cite{lewis1}.
   \end{rem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Exact octagon of Witt groups of equivariant forms}\label{octagong}   
Let $K$ be a field of characteristic different from $2$ and
$G$ be a finite group. Let $A$ be
a central simple algebra over $K$ with an involution $\sigma$.
We denote by $A[G]$ the group algebra of $G$ over $A$.
Let $\varepsilon=\pm 1$.
\begin{defi}
We say that an $\varepsilon$-hermitian space $(M,h)$ over $(A,\sigma)$ is a $G$-space if:\\
$\bullet$ $G$ acts on $M$ on the right and for all $g \in G$, the map
$M\rightarrow M: m\mapsto m.g$ is $A$-linear on the right,\\
$\bullet$ we have $h(m.g,n.g)=h(m,n)$ for all $m, \ n \in M$
and for all $g \in G$.\\
In this case, we will say that $h$ is a $G$-form. It is obvious
that $M$ is a right $A[G]$-module.
\end{defi}
\noindent
We say that two $\varepsilon$-hermitian $G$-spaces $(M,h)$ and $(M',h')$
are isometric if there
exists an isomorphism $\Phi:M \rightarrow M'$ of right $A[G]$-modules
such that 
$$h'(\Phi(m),\Phi(n))=h(m,n),$$ for all $m, \ n \in M$.
If $M$ is a right $A[G]$-module, $M^*=\Hom_A(M,A)$ has a natural structure of 
right $A[G]$-module: $(f.g)(m)=f(m.g^{-1})$ if $f \in M^*$, $g\in G$
and $m \in M$.
A $G$-space $(M,h)$ over $(A,\sigma)$
is said to be nondegenerate if 
$M\rightarrow M^*:\ x\mapsto h(x,-)$ is an
isomorphism of $A$-modules.
One can define
the hyperbolic $\varepsilon$-hermitian $G$-space associated
 to such an $M$ by $(M \oplus M^*, \h_M)$, where 
$$\h_M(m \oplus f, m'\oplus f')= f(m') + \varepsilon \sigma(f'(m)),$$
for all $m, \ m' \in M$ and $f, \ f' \in M^*$.
\begin{rem}\label{metabolic}
If $\car K \nmid |G|$ then, by Maschke's theorem, the group algebra
$A[G]$ is semisimple. Thanks to that, we can show that
an $\varepsilon$-hermitian $G$-space $(M,h)$ is hyperbolic if and only if
it is metabolic (i.e. if there exists a right $A[G]$-submodule $N$ of $M$
such that $N=N^{\bot}$ for $h$): the proof can be adapted from
\cite[Corollary 1.4]{bayer-lenstra}.
This fact will not be used here.
\end{rem}
\noindent
Now, one can construct a group (as for $\varepsilon$-hermitian forms)
called the Witt group of $\varepsilon$-hermitian $G$-forms
(also called the Witt group of equivariant forms)
which will be denoted by $W^{\varepsilon}(G,A,\sigma)$
(i.e., the quotient of the
 Grothendieck group corresponding to isometry classes
of nondegenerate $\varepsilon$-hermitian $G$-forms
by the subgroup generated by metabolic forms).
An element of this group is denoted by $[(M,h)]$ where $(M,h)$ is
an $\varepsilon$-hermitian $G$-space over $(A,\sigma)$.

We can easily see that the maps involved in the octagon \ref{main1}
induce group homomorphisms between the corresponding Witt groups of hermitian $G$-forms: if $W$ is an $\widetilde{A}[G]$-module then $W\otimes_{\widetilde{A}}A$ is an $A[G]$-module, where $G$ acts on $W\otimes_{\widetilde{A}}A$ by
$(w\otimes a).g=w.g\otimes a$ for $w \in W, \ a \in A, \ g \in G$.  

The notion of anisotropy for $G$-forms that we will use
is the following (as in
 \cite[p. 29]{cibils}):
\begin{defi}\label{anis}
An $\varepsilon$-hermitian $G$-space $(M,h)$ over $(A,\sigma)$ is said to be
anisotropic if for all $A[G]$-submodules $N$ of $M$, we have
$N\cap N^{\bot}=0$ (for $h$).
\end{defi}
\begin{rem}
Note that this notion of anisotropy coincides with the usual notion
of anisotropy (see section \ref{isotropy}) when $(M,h)$ is an
$\varepsilon$-hermitian space over a simple algebra with involution.
But in the case of $\varepsilon$-hermitian $G$-forms, this notion of
anisotropy is weaker than the usual one. For example, 
let $q :\mathbb{C}\times \mathbb{C} \rightarrow \mathbb{C}$ be the quadratic
form defined by $q(x,y)=x^2+y^2$ and $G=\lbrace 1, \theta \rbrace$ where
$\theta$ is the reflection in the hyperplane orthogonal to $(1,0)$.
Then, $q$ is a quadratic $G$-form which is isotropic as a
quadratic form but anisotropic as a $G$-form in the sense of \ref{anis}.
\end{rem}
\noindent
Now, one can prove a proposition analogous to \cite[proposition 2]{cibils}:
\begin{prop}\label{repr}
If $[(M,h)] \in W^{\varepsilon}(G,A,\sigma)$, $[(M,h)] \neq 0$ then we can
find an anisotropic $\varepsilon$-hermitian $G$-space $(M',h')$ over $(A,\sigma)$ such that
$[(M,h)]=[(M',h')]$.
\end{prop}
\begin{proof}
The proof goes as in \cite[proposition 2]{cibils}.
\end{proof}\\
Now, we prove theorem \ref{main2}:\\
\textbf{Theorem}
\textit{We suppose that $A$ satisfies the same hypotheses as at the beginning of
section \ref{definitions}. 
If we replace $W^{\pm \varepsilon}(A,\sigma)$ by $W^{\pm \varepsilon}(G,A,\sigma)$
and $W^{\pm \varepsilon}(\widetilde{A},\sigma_i)$ by 
$W^{\pm \varepsilon}(G,\widetilde{A},\sigma_i)$ (for $i=1, \ 2$) in theorem \ref{main1},
we obtain an exact octagon of $W(K, \sigma|_{K})$-modules.}\\
\begin{proof}
We only have to adapt the proof of \ref{main1}. Let us keep the
same notations.
The fact that the considered octagon is a complex readily
follows from the proof of \ref{main1} as we can easily
verify that there exist right $G$-modules structures over $W$ and $W'$
(see (\ref{space1}) and (\ref{space2}) for the definition of these spaces).

To show that this sequence
is exact, we use proposition \ref{repr} to exhibit
an anisotropic representative (in the sense of definition \ref{anis})
of each form in the considered kernel.

Let us show that $\ker(\rho_2^{\varepsilon}) \subset \im(\pi_2^{-\varepsilon})$.
We only point at the changes to the original proof.
So let $(W,f)$ be an anisotropic $\varepsilon$-hermitian $G$-space
(in the sense of \ref{anis}) lying
in the kernel of $\rho_2^{\varepsilon}$. 
Thanks to \ref{metabolic}, we know that there exists an $A[G]$-submodule
$W_1$ of $W\otimes_{\widetilde{A}}A$ such that $W_1 = W_1^{\bot}$.
 To show that $W_1 \cap (W \otimes \mu)=0$,
we consider the following subspace: $V= \lbrace w \in W | w\otimes \mu \in W_1\rbrace$.
It is an $\widetilde{A}$-submodule of $W$ and if $w\in W$, $g\in G$ we have:
$w.g\otimes \mu=(w\otimes \mu).g \in W_1$ as $W_1$ is a right $G$-module. So, $V$
is an $\widetilde{A}[G]$-submodule of $W$. Now, if $v, \ v'\in V$ then
$\rho_2^{\varepsilon}(f)(v\otimes \mu, v'\otimes \mu)=0$ and we have
$f(v,v')=0$. So $V \subset V^{\bot}$ and as $(W,f)$ is anisotropic, we
deduce that $V=0$ and that $W_1 \cap (W \otimes \mu)=0$.
Now, $W\otimes_{\widetilde{A}}A=W_1 \oplus(W\otimes \mu)$
as $\widetilde{A}[G]$-modules and
this implies that for all $w\in W$, there exists $w'\in W$ such that
$w\otimes 1+w'\otimes\mu\in W_1$.
One can define the map $J:W \rightarrow W$ as in the proof
of \ref{main1} by $J(w)=w'$(the map $J$ is well-defined because
$f$ is anisotropic).
Thanks to the previous uniqueness,
we have $J(w.g)=J(w).g$ for all $w \in W$, $g\in G$.
Thanks to that, we easily show that $W_J$ is an $A[G]$-module. 
We define $h$ as in (\ref{ante1}).
If $h$ is degenerate then there exists $x \neq 0$ in $W_J$ such that
$h(x,y)=0$ for all $y \in W_J$. We deduce that $f(x,y)=0$ for all $y \in W$
and this shows that $x \in W \cap W^{\bot}$ which is a contradiction to
the anisotropy of $(W,f)$. We can conclude as in \ref{main1}.

Now let us show that $\ker(\pi_3^{-\varepsilon})\subset\im(\rho_2^\varepsilon)$.
Let $(V,h)$ be an anisotropic $\varepsilon$-hermitian $G$-space
lying in the kernel of $\pi_3^{-\varepsilon}$.
Then there exists an $\widetilde{A}[G]$-submodule $W$ of $V_{\widetilde{A}}$ such that
$W=W^{\bot}$.
 As in the proof of \ref{main1},
we define a map $f$ as in (\ref{ante2}) and all we have
to do is to show that the $\varepsilon$-hermitian form
$f$ is nondegenerate. If $f$ is degenerate, let $U$ be the
right $A[G]$-module generated by $W$. Then $U$ is
an $A[G]$-submodule of $V$. Now there exists $x \in W$ such that
$h(x,y)=0$ for all $y \in W$ and this implies that $h(x,y)=0$ for all
$y \in U$ and we have $x \in U \cap U^{\bot}$ which is a contradiction
to the anisotropy of $(V,h)$ thus completing the proof.
\end{proof}
\begin{rem}
If the group $G$ is trivial, then \ref{main1} is a special
case of \ref{main2}.
\end{rem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Order of Witt group}\label{order}
Let $L/K$ be a finite extension. One can ask for the relation 
between the orders of $W(K)$ and $W(L)$. If $L/K$ is an
extension of odd order then by the weak version of Springer's
theorem there is a canonical injection
$W(K)\hookrightarrow W(L)$ so the finiteness of $W(L)$ implies
that of $W(K)$. If $L/K$ is an extension of even degree, then
this property fails. However, for a quadratic extension
$L/K$ the finiteness of $W(K)$ implies that of $W(L)$; 
what is easy to see is that
by the exact triangle of Elman-Lam (\ref{elman-lam})
one has: 
 $|W(L)|\leqslant|W(K)|^2$. In \cite{lewis1} the
defect of this inequality is calculated;
in fact, in
 \cite{lewis1} 
the following relation has been proved: 
\begin{equation}{\label{order-lewis-commutatif}}
|W(L)||W(L,-)|^2=|W(K)|^2
\end{equation}
 where $-$ is the nontrivial 
automorphism of $L/K$.
We have the same situation for a quaternion algebra $Q$ with its 
symplectic involution $-$. In this case the finiteness
of $W(K)$ implies that of $W(Q,-)$ by the exact sequence of
Jacobson. 
In fact by \cite{lewis1}  
we have
\begin{equation}{\label{order-lewis-noncommutatif}}
|W^{\varepsilon}(Q,-)||W^{-\varepsilon}(Q,-)|=|W(K)|
\end{equation}
More generally, as stated in \ref{order-first-kind}, we have:\\
\textbf{Corollary}
\textit{Let $A$ be a $K$-central simple algebra with an involution 
$\sigma$ of the first kind. Then we have
$|W^{\varepsilon}(A,\sigma)||W^{-\varepsilon}(A,\sigma)|=
|W(K)|$. In particular $W(K)$ is finite if and
only if $W^{\varepsilon}(A,\sigma)$ and 
$W^{-\varepsilon}(A,\sigma)$ are finite.}\\
\begin{proof}
 By Merkurjev's theorem $A$ is similar to a multi-quaternion algebra,
say $A\thicksim Q_1\otimes\ldots\otimes Q_n$. By Morita theory
we have $W^{\varepsilon}(A,\sigma)\simeq 
W^{\varepsilon'}(Q_1\otimes\ldots\otimes
Q_n,\sigma_1\otimes\ldots\otimes
\sigma_n)$ where
$\varepsilon'=\varepsilon$ or $\varepsilon'=-\varepsilon$ and
 $\sigma_i$ is the canonical involution
of $Q_i$ for $1\leqslant i\leqslant n$. So in order
to prove the statement, we can suppose that:
$$
(A,\sigma)=
(Q_1\otimes\ldots\otimes Q_n,\sigma_1\otimes\ldots\otimes\sigma_n).$$
We proceed by induction on $n$.
If $n=0$, i.e., $A$ is split, the statement becomes
$|W^{\varepsilon}(K)||W^{-\varepsilon}(K)|=
|W(K)|$ which is true because 
$\{W^{\varepsilon}(K)$, $W^{-\varepsilon}(K)\}
=\{W(K),0\}$.
If $n=1$ the statement is a consequence
of (\ref{order-lewis-noncommutatif}). 
Suppose that $n>1$. Suppose that 
$Q_n=(a,b)_K$ where $(a,b)_K$ is the quaternion algebra
generated by $i$ and $j$ with
$i^2=a$, $j^2=b$ and $ij=-j i$ where
$a,\ b\in K$. Take the split quaternion algebra
$Q_n'=(a,1)_K$ generated by ${i}'$ and ${j}'$ with
${i'}^2=a$, ${j'}^2=1$ and  ${i'}j'=-{j'}{i'}$.
If we compare the exact octagon of \ref{main1} for
$(A,\sigma)=(Q_1\otimes\ldots\otimes Q_{n-1}\otimes Q_n,
\sigma_1\otimes\ldots\otimes\sigma_{n-1}\otimes\sigma_n)$ with
$\lambda=1\otimes\ldots\otimes1\otimes i$
and $\mu=1\otimes\ldots\otimes1\otimes j$
and also for 
$(A',\sigma')=(Q_1\otimes\ldots\otimes Q_{n-1}\otimes Q_n',
\sigma_1\otimes\ldots\otimes\sigma_{n-1}\otimes{\sigma'}_n)$
(${\sigma'}_n$ is the canonical involution of $Q_n'$)
with $\lambda'=1\otimes\ldots\otimes1\otimes i'$
and  $\mu'=1\otimes\ldots\otimes1\otimes j'$ we
deduce that 
$|W^{\varepsilon}(A,\sigma)||W^{-\varepsilon}(A,\sigma)|
=|W^{\varepsilon}(A',\sigma')||W^{-\varepsilon}(A',\sigma')|$.
By Morita theory we have 
$W^{\varepsilon}(A',\sigma')\simeq W^{-\varepsilon}(A'',\sigma'')$
and 
$W^{-\varepsilon}(A',\sigma')\simeq W^{\varepsilon}(A'',\sigma'')$
where 
 $$(A'',\sigma'')=
(Q_1\otimes\ldots\otimes
Q_{n-1},\sigma_1\otimes\ldots\otimes
\sigma_{n-1})$$
 So by induction hypothesis
we obtain
$|W^{\varepsilon}(A,\sigma)||W^{-\varepsilon}(A,\sigma)|=|W(K)|$.
\end{proof}

\begin{rem}
If $A$ is a quaternion algebra over $K$, and
$\sigma$ is the canonical involution of $A$, then
for $K=\mathbb{Q}_p$, both groups  $W^{\pm1}(A,\sigma)$ are finite.
For $K=\mathbb{R}$, the group  $W^{-1}(A,\sigma)$ is finite and
 $W^{1}(A,\sigma)$ is infinite.
\end{rem}

\begin{cor}{\label{ordre-quaternion}}
Let $A$ be a quaternion algebra over $K$ with an involution 
involution $\sigma$ of the second kind.
Then  
$|W(A,\sigma)|=
|W(K,\sigma|_K)|$.
\end{cor}
\begin{proof}
By a theorem of 
Albert \cite[Ch. 8, 11.2]{scharlau}
$(A,\sigma)=(A_0\otimes_kK,\sigma_0\otimes\sigma|_K)$
where $k$ is the fixed field of $\sigma$ in $K$, $A_0$ is
quaternion algebra over $k$ and $\sigma_0$ is its canonical
involution. Suppose that $A_0=(a,b)_k$ where $(a,b)_k$ is 
the quaternion algebra generated by $i$ and $j$ with
$i^2=a$, $j^2=b$ and $ij=-ji$ where $a,\ b\in k$.
Take the split quaternion algebra 
$A_0'=(a,1)_k$ generated by $i'$ and $j'$ with
${i'}^2=a$, ${j'}^2=1$ and $i'j'=-j'i'$.
Let $\sigma_0'$ be the canonical involution of 
$A_0'$. If we compare the exact octagon of
\ref{main1} for $(A_0\otimes_kK, \sigma_0\otimes\sigma|_K)$ with
$\lambda=i\otimes 1$ and $\mu=j\otimes 1$ and for
$(A',\sigma'):=(A_0'\otimes_kK, \sigma_0'\otimes\sigma|_K)$ with
$\lambda'=i'\otimes 1$ and $\mu'=j'\otimes 1$, we deduce that
$|W^{\varepsilon}(A,\sigma)||W^{-\varepsilon}(A,\sigma)|
=|W^{\varepsilon}(A',\sigma')||W^{-\varepsilon}(A',\sigma')|$.
As $W^{\varepsilon}(A,\sigma)\simeq W^{-\varepsilon}(A,\sigma)\simeq
W(A,\sigma)$
and
$W^{\varepsilon}(A',\sigma')\simeq W^{-\varepsilon}(A',\sigma')\simeq
W(A',\sigma')$ we deduce that
$|W(A,\sigma)|=|W(A',\sigma')|$. By Morita theory
$W(A',\sigma')\simeq W(K,\sigma|_K)$ because $A'$ is split.
This implies
the result.
\end{proof}

\begin{cor}\label{multi-Q}
Let $A=Q_1\otimes_K\ldots\otimes_K Q_n$ be a 
a multi-quaternion algebra over $K$ with
the involution
$\sigma=\sigma_1\otimes\ldots\otimes\sigma_n$
where
$\sigma_i$ is an involution of
$Q_i$ of the second kind for
$i=1,\ldots,n$. Then
$|W(A,\sigma)|=|W(K,\sigma|_K)|$. 
\end{cor}
\begin{proof}
The argument is similar to \ref{order-first-kind}:
we use an induction on $n$ and the case
$n=1$ has already been proved in \ref{ordre-quaternion}.
\end{proof}

The previous corollary gives the motivation to ask
the following question for which we do not know the answer.

\begin{question}
Let $A$ be a $K$-central simple algebra with an involution
$\sigma$ of the second kind. Is it true that
$|W(A,\sigma)|=
|W(K,\sigma|_K)|$?
\end{question}

\begin{rem}
Using \ref{main2} and  applying the same type
of arguments, we can show that, if $A$ is
a central simple algebra with an involution $\sigma$
of the first kind, then
$|W^{\varepsilon}(G,A,\sigma)||W^{-\varepsilon}(G,A,\sigma)|=
|W(G,K)||W^{-1}(G,K)|$. In \ref{multi-Q} one can replace
the Witt groups by equivariant Witt groups.
\end{rem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\section{APPENDIX}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%table de definitions%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{tabl}\label{table-de-def}
\footnotesize
\begin{center}
\begin{tabular}{|l|l|}   \hline
Map & Definition\\ \hline
$
\begin{array}{cc} 
S^{\varepsilon}(A,\sigma) \stackrel{\pi^{\varepsilon}_{1}}{\rightarrow} 
S^{\varepsilon} (\widetilde{A}, \sigma_{1})
&\hspace{.23cm}  (V_A,h)\mapsto (V_{\widetilde{A}},\pi_1^{\varepsilon}(h))
\end{array}
$ &
$\pi_1^{\varepsilon}(h)(x,y)=\pi_{1}(h(x,y))$\\
\hline
$\begin{array}{ll}
S^{\varepsilon}(A, \sigma ) \stackrel{\pi^{\varepsilon}_{2}}{\rightarrow} 
S^{-\varepsilon} (\widetilde{A}, \sigma_{2})
& (V_A,h)\mapsto (V_{\widetilde{A}},\pi_2^{\varepsilon}(h))
\end{array}$
 &
$\pi_2^{\varepsilon}(h)(x,y)=\pi_{2}(h(x,y))$
\\
\hline
$\begin{array}{ll}
S^{\varepsilon}(A, \sigma ) \stackrel{\pi_3^{\varepsilon}}{\rightarrow} 
S^{-\varepsilon}(\widetilde{A}, \sigma_{1})&
 (V_A,h)\mapsto (V_{\widetilde{A}},\pi_3^{\varepsilon}(h))
\end{array}$
&
 $\pi_3^{\varepsilon}(h)(x,y)=\lambda\pi^{\varepsilon}_{1}(h(x,y))$
\\
\hline
$\begin{array}{ll}
S^{\varepsilon} (\widetilde{A}, \sigma_{1}) \stackrel{\rho_1^{\varepsilon}}
{\rightarrow} S^{-\varepsilon}(A, \sigma )&
(W,f)\mapsto (W\otimes_{\widetilde{A}}A,\rho_1^{\varepsilon}(f))
\end{array}$&
 $\rho_1^{\varepsilon}(f) (x \otimes \alpha, y \otimes \beta) = 
 \sigma(\alpha) \lambda f(x,y) \beta$
\\
\hline
$\begin{array}{ll}
S^{\varepsilon} (\widetilde{A}, \sigma_{2}) \stackrel{\rho_2^{\varepsilon}}
{\rightarrow} S^{-\varepsilon}(A, \sigma )&
(W,f)\mapsto (W\otimes_{\widetilde{A}}A,\rho_2^{\varepsilon}(f))
\end{array}$&
 $\rho_2^{\varepsilon}(f)( x \otimes \alpha, y \otimes \beta ) = 
\sigma (\alpha) \lambda \mu f(x,y) \beta$
\\
\hline
$\begin{array}{ll}
S^{\varepsilon}(\widetilde{A}, \sigma_{1}) \stackrel{\rho_3^{\varepsilon}}
{\rightarrow} S^{\varepsilon}(A, \sigma)&\hspace{.23cm}
(W,f)\mapsto (W\otimes_{\widetilde{A}}A,\rho_3^{\varepsilon}(f))
\end{array}$&
 $\rho_3^{\varepsilon}(f)(x\otimes\alpha,y\otimes\beta)=\sigma(\alpha)f(x,y)\beta$ 
\\
\hline
\end{tabular}
\end{center}
\end{tabl}

\begin{tabl}{\label{tabl1}}
\scriptsize
\begin{center}
\begin{tabular}{|l|l|l|l|}   \hline
Map & Form & Conditions & Image \\
\hline
 $S^{\varepsilon}(D,\tau) \stackrel{\pi^{\varepsilon}_{1}}{\rightarrow} S^{\varepsilon} (\widetilde{D}, \tau_{1})$ &\begin{tabular}{l}$<\delta_1,\ldots,\delta_n>$; $\delta_i=d_{2i-1}+\mu d_{2i},$\\ $d_{2i-1},d_{2i}\in \widetilde{D}$\end{tabular}&\begin{tabular}{l}$\tau_1(d_{2i-1})=\varepsilon d_{2i-1}$\\$\tau_2(d_{2i})=-\varepsilon d_{2i}$\end{tabular}&  $\bigoplus_{i=1}^n\left(\begin{array}{cc} d_{2i-1} & \mu d_{2i} \mu \\ -\mu^{2}d_{2i} & -\mu d_{2i-1} \mu  \end{array} \right)$
 \\ \hline
$S^{\varepsilon} (\widetilde{D}, \tau_{1}) \stackrel{\rho_1^{\varepsilon}}{\rightarrow} S^{-\varepsilon}(D, \tau )$ &\begin{tabular}{l}$<\gamma_1,\ldots,\gamma_n>$;\\$\gamma_i\in\widetilde{D}$ \end{tabular}&\begin{tabular}{l}$\tau_1(\gamma_i) =\varepsilon\gamma_i$\end{tabular}&  $<\lambda \gamma_1,\ldots,\lambda \gamma_n>$ \\   \hline
$S^{-\varepsilon}(D, \tau ) \stackrel{\pi^{-\varepsilon}_{2}}{\rightarrow} S^{\varepsilon} (\widetilde{D}, \tau_{2})$ &\begin{tabular}{l}$<\delta_1,\ldots,\delta_n>$; $\delta_i=d_{2i-1}+\mu d_{2i},$\\ $d_{2i-1},d_{2i}\in \widetilde{D}$\end{tabular}&\begin{tabular}{l}$\tau_1(d_{2i-1})=-\varepsilon d_{2i-1}$\\$\tau_2(d_{2i})=\varepsilon d_{2i}$\end{tabular}& $\bigoplus_{i=1}^n\left(
                                                                                                                                                                 \begin{array}{cc}
                                                                                                                                                   d_{2i} & \mu^{-1} d_{2i-1} \mu \\
                                                                                                                                           -d_{2i-1} & -\mu d_{2i} \mu 
                                                                                                                                          \end{array} \right)$ \\ \hline
$S^{\varepsilon} (\widetilde{D}, \tau_{2}) \stackrel{\rho_2^{\varepsilon}}{\rightarrow} S^{-\varepsilon}(D, \tau )$ &\begin{tabular}{l}$<\gamma_1,\ldots,\gamma_n>$;\\$\gamma_i\in\widetilde{D}$ \end{tabular}&\begin{tabular}{l}$\tau_2(\gamma_i)=\varepsilon \gamma_i$\end{tabular}&    $<\lambda\mu \gamma_1,\ldots,\lambda\mu \gamma_n>$ \\   \hline
$S^{-\varepsilon}(D, \tau ) \stackrel{\pi_3^{-\varepsilon}}{\rightarrow} S^{\varepsilon}(\widetilde{D}, \tau_{1})$ &\begin{tabular}{l}$<\delta_1,\ldots,\delta_n>$; $\delta_i=d_{2i-1}+\mu d_{2i},$\\ $d_{2i-1},d_{2i}\in \widetilde{D}$\end{tabular}&\begin{tabular}{l}$\tau_1(d_{2i-1})=-\varepsilon d_{2i-1}$\\$\tau_2(d_{2i})=\varepsilon d_{2i}$\end{tabular}& $\bigoplus_{i=1}^n\left(\begin{array}{cc} \lambda d_{2i-1} & \lambda\mu d_{2i} \mu \\ -\lambda\mu^{2}d_{2i} &-\lambda\mu d_{2i-1} \mu  \end{array} \right)$
\\ \hline
$S^{\varepsilon}(\widetilde{D}, \tau_{1}) \stackrel{\rho_3^{\varepsilon}}{\rightarrow} S^{\varepsilon}(D, \tau)$&\begin{tabular}{l}$<\gamma_1,\ldots,\gamma_n>$;\\$\gamma_i\in\widetilde{D}$ \end{tabular}&\begin{tabular}{l}$\tau_1(\gamma_i)=\varepsilon\gamma_i$\end{tabular}& $<\gamma_1,\ldots,\gamma_n>$\\ \hline
\end{tabular}
\end{center}
\normalsize
\end{tabl}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%    Table   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{tabl}\label{tabl2}
\footnotesize
\begin{center}\footnote{provided that $f$ and $h$ are anisotropic.} 
\begin{tabular}{|l|l|l|}  \hline
  $\pi_1^{\varepsilon}(h)$ is hyperbolic $\Rightarrow h\simeq<\mu \gamma_1,\ldots,\mu \gamma_n>$&$\begin{array}{l}\tau_2(\gamma_i)=-\varepsilon \gamma_i\end{array}$&$S^{\varepsilon}(D,\tau)\stackrel{\pi_1^{\varepsilon}}{\rightarrow}S^{\varepsilon}(\widetilde{D},\tau_1)$\\ \hline
 $\rho_1^{\varepsilon}(f)$ is hyperbolic $\Rightarrow f\simeq\bigoplus_{i=1}^n\left(\begin{array}{cc} d_{2i-1} & \mu d_{2i} \mu \\ -\mu^{2}d_{2i} & -\mu d_{2i-1} \mu  \end{array} \right)$&$\begin{array}{l} \tau_1(d_{2i-1})=\varepsilon d_{2i-1}\\ \tau_2(d_{2i})=-\varepsilon d_{2i}\end{array}$  &$S^{\varepsilon} (\widetilde{D}, \tau_{1})\stackrel{\rho_1^{\varepsilon}}{\rightarrow} S^{-\varepsilon}(D, \tau )$
 \\ \hline
 $\pi_2^{-\varepsilon}(h)$ is hyperbolic $\Rightarrow h\simeq<\gamma_1,\ldots,\gamma_n>$&$\begin{array}{l}\tau_1(\gamma_i)=-\varepsilon \gamma_i\end{array}$&$S^{-\varepsilon}(D, \tau ) \stackrel{\pi^{-\varepsilon}_{2}}{\rightarrow} S^{\varepsilon} (\widetilde{D}, \tau_{2})$ \\   \hline
 $\rho_2^{\varepsilon}(f)$ is hyperbolic $\Rightarrow f\simeq\bigoplus_{i=1}^n\left(
                                                                                                                                                                 \begin{array}{cc}
                                                                                                                                                   d_{2i} & \mu^{-1} d_{2i-1} \mu \\
                                                                                                                                           -d_{2i-1} & -\mu d_{2i} \mu 
                                                                                                                                          \end{array} \right)$&$\begin{array}{l} \tau_1(d_{2i-1})=-\varepsilon d_{2i-1}\\ \tau_2(d_{2i})=\varepsilon d_{2i}\end{array}$ &$S^{\varepsilon} (\widetilde{D}, \tau_{2}) \stackrel{\rho_2^{\varepsilon}}{\rightarrow} S^{-\varepsilon}(D, \tau)$  \\ \hline
 $\pi_3^{-\varepsilon}(h)$ is hyperbolic $\Rightarrow h\simeq<\mu \gamma_1,\ldots,\mu \gamma_n>$& $\begin{array}{l}\tau_2(\gamma_i)=\varepsilon \gamma_i\end{array}$ &$S^{-\varepsilon}(D, \tau ) \stackrel{\pi_3^{-\varepsilon}}{\rightarrow} S^{\varepsilon}(\widetilde{D}, \tau_{1}) $ \\   \hline
 $\rho_3^{\varepsilon}(f)$ is hyperbolic $\Rightarrow f\simeq\bigoplus_{i=1}^n\left(\begin{array}{cc} \lambda d_{2i-1} & \lambda\mu d_{2i} \mu \\ -\lambda\mu^{2}d_{2i} &-\lambda\mu d_{2i-1} \mu  \end{array} \right)$&$\begin{array}{l} \tau_1(d_{2i-1})=-\varepsilon d_{2i-1}\\ \tau_2(d_{2i})=\varepsilon d_{2i}\end{array}$&$S^{\varepsilon}(\widetilde{D}, \tau_{1}) \stackrel{\rho_3^{\varepsilon}}{\rightarrow} S^{\varepsilon}(D, \tau)$
\\ \hline
\end{tabular}
\end{center}
\normalsize
\end{tabl}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
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\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \normalsize
 \noindent Laboratoire de Math\'ematiques\\
Universit\'e de Franche-Comt\'e\\
25030 Besan\c{c}on CEDEX\\
FRANCE\\
\quad       {\tt grenier@math.univ-fcomte.fr\\
mahmoudi@math.univ-fcomte.fr}\\
 \end{document}
 
