%Shree Maataaravindaarpanam

%Shree Maataaravindaarpanam

%Shree Paarthasaarathy Sharanam Mama
%Shree Paarthasaarathy Paahimaam
%Shree Paarthasaarathyaarpanam
%Shree Maathaaravinda Sharanam Mama
%Shree Maathaaravinda Paahimaam
%Shree Maathaaravindaarpanam Asthu

%According to requirements in Proceedings of the Indian- 
% -Academy of Sciences: Mathematical Sciences 


\documentclass[notitlepage,10pt]{article}



\textwidth125mm
\textheight185mm
\parindent8mm
\evensidemargin0pt
\oddsidemargin0pt
\frenchspacing

\pagestyle{myheadings}
\markboth{T. E. Venkata Balaji}{Degenerations of Rank 3 Quadratic Bundles and 
Rank 4 Azumaya Bundles over Schemes}

\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amscd}
\usepackage{latexsym}
\usepackage{fullpage}
\usepackage{epic}
\usepackage{eepic}
\usepackage{mathrsfs}
\usepackage[mathcal]{euscript}
\usepackage{eufrak}
\usepackage[all]{xy}

\begin{document}


%Shree Maataaravindaarpanam

\newtheorem{remark}{Remark}[section]
\newtheorem{remarks}[remark]{Remarks} 
\newtheorem{definition}[remark]{Definition} 
\newtheorem{lemma}[remark]{Lemma} 
\newtheorem{theorem}[remark]{THEOREM} 
\newtheorem{proposition}[remark]{PROPOSITION} 
\newtheorem{corollary}[remark]{Corollary} 
\newtheorem{recall}[remark]{Recall}
\newtheorem{notation}{Notation}[subsection]

%Shree Maataaravindaarpanam

\newcommand{\vq}{(  V,   q)}
\newcommand{\vprqpr}{(  V',   q')}
\newcommand{\vqpr}{(  V,   q')}

\newcommand{\covq}{  C_0(  V,  q)}
\newcommand{\covqpr}{  C_0(  V,  q')}
\newcommand{\covprqpr}{  C_0(  V',  q')}

\newcommand{\simvqvprqpr}{\hbox{\rm Sim}[\vq,\vprqpr]}
\newcommand{\isomvqvprqpr}{\hbox{\rm Iso}[\vq,\vprqpr]}

\newcommand{\simvqvqpr}{\hbox{\rm Sim}[\vq,\vqpr]}
\newcommand{\isomvqvqpr}{\hbox{\rm Iso}[\vq,\vqpr]}
\newcommand{\sisomvqvqpr}{\hbox{\rm S-Iso}[\vq,\vqpr]}

\newcommand{\simvqvq}{\hbox{\rm Sim}[\vq,\vq]}
\newcommand{\isomvqvq}{\hbox{\rm Iso}[\vq,\vq]}
\newcommand{\sisomvqvq}{\hbox{\rm S-Iso}[\vq,\vq]}

\newcommand{\isomcovqcovprqpr}{\hbox{\rm Iso}[\covq,\covprqpr]}

\newcommand{\isomcovqcovqpr}{\hbox{\rm Iso}[\covq,\covqpr]}
\newcommand{\isomcovqcovqprprime}{\hbox{\rm Iso}'[\covq,\covqpr]}
\newcommand{\sisomcovqcovqpr}{\hbox{\rm S-Iso}[\covq,\covqpr]}

\newcommand{\sovq}{{\hbox{\rm SO}}\vq} 
\newcommand{\ovq}{{\hbox{\rm O}\vq}} 
\newcommand{\govq}{{\hbox{\rm GO}\vq}}

\newcommand{\autvq}{{\hbox{\rm Aut}}\vq} 
\newcommand{\autcovq}{{\hbox{\rm Aut}}(\covq)}   
\newcommand{\sautcovq}{{\hbox{\rm S-Aut}}(\covq)}   
\newcommand{\autcovqprime}{{\hbox{\rm Aut}'}(\covq)}   

\newcommand{\isolambdatwovvpr}{{\hbox{\rm Iso}}[{\Lambda}^2(V),{\Lambda}^2(V')]}
\newcommand{\autlambdatwov}{\hbox{\rm Aut}(\Lambda^2(V))}
\newcommand{\autdetlambdatwov}{\hbox{\rm Aut}[\hbox{\rm det}(\Lambda^2(V))]}

\newcommand{\falgW}{\hbox{{\sf Alg}}_{{W}}}
\newcommand{\algW}{\hbox{\rm Alg}_{{W}}}
\newcommand{\faaWst}{\hbox{{\sf Id-Assoc}}_{{{W}}}}
\newcommand{\aaWst}{\hbox{{\rm Id-Assoc}}_{{{W}}}}
\newcommand{\faaWw}{{\hbox{\sf Id-}w\hbox{\sf -Assoc}_{{{W}}}}}
\newcommand{\aaWw}{{\hbox{\rm Id-}w\hbox{\rm -Assoc}_{{{W}}}}}
\newcommand{\stabw}{\hbox{\rm Stab}_w}
\newcommand{\gl}[1]{\hbox{\rm GL}_{#1}}
\newcommand{\fazuWw}{\hbox{\sf Id-}w\hbox{\sf -Azu}_{{W}}}
\newcommand{\azuWw}{\hbox{\rm Id-}w\hbox{\rm -Azu}_{{W}}}
\newcommand{\azuWiwi}{\hbox{\rm Id-}{w_i}\hbox{\rm -Azu}_{{W_i}}}
\newcommand{\azuWprwpr}{\hbox{\rm Id-}{w'}\hbox{\rm -Azu}_{{W'}}}
\newcommand{\fazuWst}{\hbox{\sf Azu}_{{W}}}
\newcommand{\azuWst}{\hbox{\rm Azu}_{{W}}}
\newcommand{\spazuWw}{\hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_{{W}}}
\newcommand{\spazuWiwi}{\hbox{\rm Id-}{w_i}\hbox{\rm -Sp-Azu}_{{W_i}}}
\newcommand{\spazuWprwpr}{\hbox{\rm Id-}{w'}\hbox{\rm -Sp-Azu}_{{W'}}}
\newcommand{\spazuWst}{\hbox{\rm Sp-Azu}_{{W}}}

\newcommand{\spazuww}{\hbox{\rm Id-}w\hbox{\rm-Sp-Azu}_{W}}
\newcommand{\aaww}{{\hbox{\rm Id-}w\hbox{\rm -Assoc}_{W}}}
\newcommand{\azuww}{\hbox{\rm Id-}w\hbox{\rm -Azu}_{W}}
\newcommand{\aawst}{\hbox{\rm Id-Assoc}_{W}}
\newcommand{\azuwst}{\hbox{\rm Azu}_{W}}
\newcommand{\spazuwst}{\hbox{\rm Sp-Azu}_{W}}

\newcommand{\falgw}{\hbox{{\sf Alg}}_{W}}







\newcommand{\algw}{\hbox{Alg}_{W}}

\newcommand{\fazuww}{\hbox{{\sf Id}-w-\hbox{\sf Azu}}_{W}}



\newcommand{\faawst}{\hbox{{\sf Id-Assoc}}_{W}}



\newcommand{\fazuwst}{\hbox{{\sf Azu}}_{W}}



\newcommand{\fspazuwst}{\hbox{{\sf Sp-Azu}}_{W}}

\newcommand{\fspazuWst}{\hbox{{\sf Sp-Azu}}_{{W}}}




\newcommand{\spec}[1]{\hbox{Spec\,}\left({#1}\right)}




\newcommand{\sym}[2]{\hbox{\sf Sym}_{#1}\left[\left(
{{#2}_{#1}}^{\vee}\otimes_{#1}{{#2}_{#1}}^{\vee}\otimes_{#1}{{#2}_{#1}}\right)^{\vee}\right]}


\newcommand{\closure}[1]{\langle{#1}\rangle}

%Shree Maataaravindaarpanam


\title{\bf Degenerations of Rank 3 Quadratic Bundles\\
 and Rank 4 Azumaya Bundles over Schemes}


\author{
\begin{tabular}{c}
 {\sc T. E. Venkata Balaji}\\
 {\em  Mathematisches Institut, Georg-August-Universit\"at}\\
 {\em Bunsenstrasse 3-5, D-37073 G\"ottingen, Germany}\\
 {\tt  tevbal@uni-math.gwdg.de}\\
 \\
 \\ 
{\bf Dedicated to Professor Martin Kneser}\\
\\
\\  
\end{tabular}
}

\date{April 26, 2004}

\maketitle

\abstract{This paper uses scheme-theoretic methods to 
 address algebraic problems about rank 3 quadratic forms and rank 4 Azumaya 
 algebras.
 It extends well-known results on semiregular rank 3 quadratic bundles and rank 4 Azumaya algebra bundles to their degenerations 
by studying the scheme of specialisations of Azumaya algebra 
 bundle structures  introduced in Part A of
 \cite{tevb-paper1}. The Witt-invariant of a rank 3 quadratic bundle 
$\vq$, which by definition is 
 the isomorphism class (as algebra bundle) of its even-Clifford algebra
 $\covq$, is shown to determine the pair 
 $\vq$ upto tensoring by a discriminant line bundle.   
 The special, usual and the
 general orthogonal groups of $\vq$ are 
 computed and canonically 
 determined in terms of $\autcovq$, and it is shown that the 
 general orthogonal group is always a 
 semidirect product. Any element of $\autcovq$ can be lifted to a 
 self-similarity, and in fact to an element of the orthogonal group provided
 the determinant of the automorphism is a square. The special orthogonal 
 group and the group of determinant 1 automorphisms of $\covq$ 
 are naturally isomorphic.  
 If the base scheme $X$ is integral and 
 $q$ is semiregular at some point of $X$, then every automorphism 
 of $\covq$ has determinant 1 and is thus induced from a self-isometry; 
 the orthogonal group is also seen to be a semidirect product 
 in this case. If $X$ 
 is affine with coordinate ring a UFD, then every specialised algebra structure  on a rank 4 vector bundle over it arises as the even-Clifford algebra of 
 a global rank 3 quadratic bundle $\vq$, so that the set of rank 3 quadratic 
 bundles upto tensoring by a discriminant line bundle naturally corresponds to
 the set of isomorphism classes of specialised rank 4 algebra bundles.
 These results are seen as limiting versions of the natural bijection 
 ${\hbox{\rm\v H}}^1_{\hbox{\rm\tiny fppf}}(X, \hbox{\sf O}_3)/\hbox{\rm Disc}(X)\cong
 {\hbox{\rm\v H}}^1_{\hbox{\rm\tiny \'etale}}(X, \hbox{\sf PGL}_2).$
 The  multiplication table of every specialised algebra structure on any fixed
  free rank 4 vector bundle with fixed unit that is part of a global basis is 
 written down explicitly. For a connected proper scheme $X$ of finite type 
 over an algebraically closed field, the hypothesis of 
 self-duality on a unital associative 
  algebra bundle of square rank over $X$ forces the algebra to 
 be either globally Azumaya or to be nowhere-Azumaya. This implies that 
 the existence of a non-Azumaya specialisation which is Azumaya at some 
 point excludes the possibility of the existence of global Azumaya algebra 
 structures on that bundle. 
 The 
 use of the nice technical notion of semiregularity introduced by
 Kneser in \cite{Kneser} allows all of the above results to be valid  
 over an arbitrary base scheme $X$, some (or even all)
 of whose points may have residue fields of
  characteristic two.\\[2mm]
 {\bf Keywords:} semiregular, quadratic bundle, Azumaya bundle, Witt-invariant, Clifford algebra, discriminant bundle, orthogonal group, similarity, similitude.\\[2mm]
{\bf MSC Subject Classification Numbers:} 11Exx, 11E12, 11E20, 11E88, 
11R52, 14L15, 15A63, 16S, 16H05. 
}




\tableofcontents

%Shree Maataaravindaarpanam

\section{\label{sec1}Overview of the Main Results}   
The smoothness of the schematic closure of Azumaya algebra
 structures on a fixed 
 vector bundle of rank 4 over any scheme was 
 obtained in Part A of \cite{tevb-paper1}. 
 Part B  of that work 
 had applied this result 
 to obtain desingularisations (with good specialisation properties) 
 of certain moduli spaces over fairly general 
 base schemes. The present work is concerned with applications 
 to the study of degenerations of rank 3 quadratic bundles over a scheme.

    
For a rank 3 vector bundle $ V$ over a scheme X with a 
 quadratic form $ q$ on $ V$ (taking
 values in ${\cal O}_X$), 
consider  
 the natural 
 functorial association of the pair 
$\vq$ 
to its 
 even Clifford algebra 
 bundle 
$\covq$ of rank 4 
over X.
  If $ q$ is a semiregular
  quadratic form, then $\covq$ 
 is an Azumaya algebra bundle over $X$ i.e., a twisted form for the  
 the $(2\times 2)$-matrix algebra over ${\mathcal O}_X$ for the \'etale 
 topology on $X.$
 (A generic quadratic form on a vector bundle of 
 odd rank is semiregular and 
 such forms are technically the ``good'' ones---they were introduced 
 by Kneser in \cite{Kneser} and are studied in detail by Knus in 
 his book  \cite{Knus}---coinciding
  with the usual regular quadratic forms when the residue 
 field of every point of $X$ has characteristic different from 2). 
 Therefore the association 
 is one that ``generically'' takes a rank 3 semiregular quadratic bundle to 
 a rank 4 Azumaya algebra bundle.
 This may also be described as the natural bijection 
 $${\hbox{\rm \v H}}^1_{\hbox{\rm\tiny fppf}}(X, \hbox{\sf O}_3)/\hbox{\rm Disc}(X)\cong
 {\hbox{\rm \v H}}^1_{\hbox{\rm\tiny \'etale}}(X, \hbox{\sf PGL}_2)$$
 where the left side classifies semiregular rank 3 quadratic bundles
 upto tensoring by a discriminant line bundle 
 on
 $X$, and the right side classifies Azumaya algebra bundles of rank 4 on $X.$
 If none of the residue fields of the points of $X$ 
 is of characteristic two, then the subscript `fppf' on the 
 left side may be replaced with `\'etale'. What follows is an attempt to 
 extend this bijection `to the limit' i.e., with degenerate (possibly 
 non-semiregular) quadratic bundles on the left and specialised (possibly 
 non-Azumaya) bundles on the right. We extend the left side by replacing it
 with 
 the set of quadratic bundles of rank 3 on $X$ upto tensoring by a discriminant
 line bundle and extend the right side by replacing it with
 algebra-isomorphism classes of  specialisations, of rank 4 
 Azumaya algebras, in 
 the sense of Part A, \cite{tevb-paper1}. We find that the resulting map 
 is always an injection; we are able to prove that it is a surjection in 
 certain cases, for example when $X$ is affine with coordinate ring a
 UFD, and hope that it will always be a bijection. 
 
A detailed analysis of the origins behind 
 the  smoothness result in Part A, \cite{tevb-paper1},  allows us to   
 extend many of the known results  for semiregular rank 3 quadratic bundles
 (and  rank 4 Azumaya bundles), as in Chap.V, \S 
  3 of the book of Knus \cite{Knus},  to degenerate 
 rank 3 quadratic bundles and 
 specialisations of rank 4 Azumaya algebra bundles.  
  For example, it turns out that 
 the Witt-invariant of $\vq$, which by definition is 
 the isomorphism class (as algebra bundle) of $\covq$, determines the pair 
 $\vq$ upto tensoring by a discriminant line bundle. For another example,  
 the orthogonal groups related to $\vq$ viz. the special, usual and the
 general orthogonal groups (contained in that order inside
 $\hbox{\rm Aut}( V)\times \Gamma(X, {\cal O}_X^*)=\hbox{\rm GL}( V)\times 
\Gamma(X, {\cal O}_X^*)$) may be 
 computed and canonically 
 determined in terms of $\autcovq.$ 

 Let us fix some notations and terminology before proceeding further.
In the following we switch between the language of (geometric) 
 vector bundles and the  
  (equivalent) language of locally-free 
 sheaves of finite constant positive rank; as a convention,
 sheaves corresponding to vector bundles $V, W, L\ldots$ are 
 respectively denoted $\mathcal{V, W, L}\ldots$ 
 Relative to 
 any fixed base scheme $B$,  consider the natural transformation 
 between two natural (covariant) functors 
 $$\hbox{\sf Witt}:\mathscr{Q}_3\longrightarrow\mathscr{A}_4,\hspace*{1cm}\mathscr{Q}_3,\mathscr{A}_4: 
(B-{\mathfrak{Schemes}})^{op}\longrightarrow \mathfrak{Sets}$$
 defined as follows.  If $X\longrightarrow B$ is a $B$-scheme, recall that 
 a pair $( L,  h)$ is said to be a discriminant line bundle on $X$ if 
 $ L$ is a line bundle and $ h: L\otimes_X  L
 \cong {\mathbb A}^1_X$ is an isomorphism of line bundles (i.e., 
 coming from an isomorphism of the 
 associated locally-free sheaves of ${\cal O}_X$-modules of rank 1). In other 
 words $ h$ is a non-singular 
 bilinear form on $ L$ with values in 
 the trivial line bundle (hence necessarily symmetric). Let 
 $\hbox{\sf Disc}(X)$ be the set of isometry classes of such discriminant 
 bundles. It is naturally an abelian group of exponent 2 under the 
  tensor product operation. It acts on the set of isometry classes of 
  quadratic bundles $\vq$ on $X$, since given $( L,  h)$
 there is the natural quadratic bundle $\vq\otimes( L,  h)$
 viz. the quadratic form $ q\otimes h$ on $ V\otimes 
  L$. Define $\mathscr{Q}_3(X)$
 to be the set of equivalence classes under 
 this action i.e., the set of rank 3 quadratic bundles on $X$ (isometric) 
 upto tensoring by a discriminant bundle.  
 Also define
  $\mathscr{A}_4(X)$ to be the set of isomorphism classes of associative 
 unital algebra structures on rank 4 vector bundles over $X$ (here
 unital means that there exists a global nowhere-vanishing section
 which serves as the unit for the algebra multiplication). Associating 
 to $\vq$ its Witt-invariant i.e., the isomorphism class of 
 its even Clifford algebra $\covq$  gives 
 rise to a functor in $X$:  
 $${\hbox{\sf Witt}}(X):\mathscr{Q}_3(X)\longrightarrow\mathscr{A}_4(X):[\vq]\mapsto[\covq]$$
since tensoring a quadratic bundle with a discriminant bundle (which includes
 replacing it with a similar---in particular with an isometric---quadratic bundle) does not affect the isomorphism 
 class of its even Clifford algebra (see Prop. \ref{isom-covq-covlqh}). Observe that by considering 
 only semiregular quadratic bundles we obtain a subfunctor 
 $\mathscr{Q}_3^{sr}\hookrightarrow\mathscr{Q}_3$ (since tensoring by 
 a discriminant bundle preserves semiregularity, see Prop.\ref{tensoring-with-sym-bil-module}). Also by considering 
 only Azumaya algebra bundle structures we obtain 
 a subfunctor $\mathscr{A}_4^{Azu}\hookrightarrow\mathscr{A}_4$ 
 so that  the transformation $\hbox{\sf Witt}$ takes 
 $\mathscr{Q}_3^{sr}$ into $\mathscr{A}^{Azu}_4.$ In fact, 
 $\hbox{\sf Witt}(X):\mathscr{Q}_3^{sr}(X)\longrightarrow
\mathscr{A}^{Azu}_4(X)$ 
 is bijective for each scheme $X$, and this
 may be deduced from the case of an affine $X$ which is proved in 
 \S 3, Chap.V, \cite{Knus} (see Theorem \ref{Witt-bijective-on-q3sr}).
 It follows from 
 the main result of Part A of \cite{tevb-paper1} that there exists  
 a subfunctor $\mathscr{A}^{Sp-Azu}_4\hookrightarrow\mathscr{A}_4$
 that contains $\mathscr{A}^{Azu}_4$ as well as 
  the image of $\hbox{\sf Witt}.$ This subfunctor 
 may be described as those algebra bundle structures that are scheme-theoretic 
 specialisations of Azumaya algebra structures, or also as those that are 
 locally (in the Zariski topology) the even Clifford algebras of 
 quadratic bundles. It turns out that these specialisations are 
 quaternion algebra bundles i.e., those that have a standard involution 
 (in the sense of Knus, para.1.3,  Chap.I, \cite{Knus}) but of 
 course even over an algebraically closed field there are quaternion 
 algebras that are not even-Clifford algebras of quadratic forms.

\begin{theorem}\label{injectivity}
For each $X$, the map $\hbox{\sf Witt}(X):\mathscr{Q}_3(X)\longrightarrow
\mathscr{A}^{Sp-Azu}_4(X)$ is injective. 
\end{theorem}

For two quadratic bundles $\vq$ and $\vprqpr$
 denote by $\simvqvprqpr$ the set of similarities (also called 
 similitudes) from $\vq$ to
 $\vprqpr$ (with multipliers 
 being global sections of ${\mathcal O}^{*}_X$) and by 
 $\isomvqvprqpr$ the subset of isometries (i.e., similarities with 
 trivial multipliers). When $ V= V'$, the 
 subset of isometries with trivial determinant is denoted 
 $\sisomvqvqpr.$ On taking $ q= q'$
  these sets 
 naturally become subgroups of $\hbox{\rm Aut}(V)\times\Gamma(X, {\cal O}_X^*)
=\hbox{\rm GL}(V)\times\Gamma(X, {\cal O}_X^*)$
 and we get  
$$\simvqvq=\govq\supset\isomvqvq
=\ovq\supset\sisomvqvq=\sovq.$$  

Of course, $\ovq$ and $\sovq$ may as usual be considered as subgroups of 
 $\hbox{\rm GL}(V)\equiv\hbox{\rm GL}(V)\times\{1\}.$
 Since in general   
 a quadratic bundle $\vq$ on a non-affine scheme $X$
may not be induced from a global bilinear 
 form, one is unable to identify the $(\mathbb Z/2.\mathbb Z)$--
 graded vector bundle 
 underlying its Clifford algebra bundle with that underlying the  
 exterior algebra bundle of $V$ (which is the same as the Clifford algebra 
 bundle of the zero quadratic form). 

\begin{proposition}\label{transfer-to-lambda2}
Every isomorphism of algebra-bundles $\phi:\covq\cong\covprqpr$
is naturally associated to an 
 isomorphism of bundles $\phi_{\Lambda^2}:\Lambda^2(V)
\cong\Lambda^2(V')$ which induces a map 
$$\zeta_{\Lambda^2}:\isomcovqcovprqpr\longrightarrow\isolambdatwovvpr: \phi\mapsto\phi_{\Lambda^2}$$ 
where  $\isomcovqcovprqpr$ is the set of algebra bundle isomorphisms.  
When $ V= V'$, we may thus denote  
  the subset of those $\phi$ for which
 $\hbox{\rm det}(\phi_{\Lambda^2})\in\autdetlambdatwov\cong
\Gamma(X, {\cal O}_X^*)$ is 
 a square (respectively $=1$) by $\isomcovqcovqprprime$ (respectively
 by    the smaller subset $\sisomcovqcovqpr$). 
 Taking $ q= q'$ in these sets and replacing `` {\rm Iso}'' by ``{\rm Aut}''
 in their notations respectively defines the  
 groups $\autcovq\supset\autcovqprime\supset\sautcovq.$  
\end{proposition}



\begin{theorem}\label{lifting-of-isomorphisms}
 For rank 3 quadratic bundles $\vq$ and $\vqpr$ with the same 
 underlying bundle $ V$ on a scheme $X$, one has the following 
 commuting diagram of natural maps of sets with the downward arrows 
 being the canonical inclusions, the horizontal arrows being surjective 
 and the top horizontal arrow being bijective:  
$$\begin{CD}
\sisomvqvqpr @>{\cong}>> \sisomcovqcovqpr \\
 @V{\hbox{\rm inj}}VV @VV{\hbox{\rm inj}}V \\
\isomvqvqpr @>{\hbox{\rm onto}}>> \isomcovqcovqprprime \\
  @V{\hbox{\rm inj}}VV @VV{\hbox{\rm inj}}V \\
\simvqvqpr @>{\hbox{\rm onto}}>> \isomcovqcovqpr 
\end{CD}$$
With respect to the surjections of the 
 horizontal arrows in the diagram above, we further have the following (where 
 $l$  is the function that associates to any similitude its 
 multiplier, $\hbox{\rm det}(g,l):=\hbox{\rm det}(g)$ for a similitude with 
 multiplier $l$ and $\zeta_{\Lambda^2}$ is the map of
 Prop.\ref{transfer-to-lambda2} above):
\begin{description}
\item[(a)] there is a family of sections
 $s_{2k+1}: \isomcovqcovqpr\longrightarrow\simvqvqpr$ indexed by the integers 
 such that $l\circ s_{2k+1}
=\hbox{\rm det}^{2k+1}\circ\zeta_{\Lambda^2}$ and such 
 that $(\hbox{\rm det}^2
\circ s_{2k+1})\times(l^{-3}\circ s_{2k+1})=\hbox{\rm det}
\circ\zeta_{\Lambda^2}$;
\item[(b)] there is also a 
 section $s':\isomcovqcovqprprime\longrightarrow \isomvqvqpr$ 
 such that $\hbox{\rm det}^2\circ s'=\hbox{\rm det}\circ\zeta_{\Lambda^2};$
\item[(c)]  there is a family of sections 
 $s_{2k+1}^{+}:\isomcovqcovqpr\longrightarrow\simvqvqpr$ indexed by 
 the integers which is 
 multiplicative when followed by the natural inclusions
 into $\hbox{\rm GL}(V)\times \Gamma(X, {\cal O}_X^*)$, i.e., 
 if $\phi_i\in\hbox{\rm Iso}[C_0(V,q_i), C_0(V,q_{i+1})]$ 
 then $s_{2k+1}^{+}(\phi_2\circ\phi_1)=s_{2k+1}^{+}(\phi_2)\circ s_{2k+1}^{+}(\phi_1)\in\hbox{\rm GL}(V)\times\Gamma(X, {\cal O}_X^*).$
 Further, $l\circ s_{2k+1}^{+}
=\hbox{\rm det}^{2k+1}\circ\zeta_{\Lambda^2}$ and  $(\hbox{\rm det}^2
\circ s_{2k+1}^{+})\times(l^{-3}\circ s_{2k+1}^{+})=\hbox{\rm det}
\circ\zeta_{\Lambda^2}.$ 
\item[(d)] The maps $s_{2k+1}$ and 
 $s'$  above  may not be 
 multiplicative  but are
 mutliplicative upto $\mu_2(\Gamma(X, {\mathcal O}_X))$
 i.e., these followed  by the quotient map, on taking the quotient of 
 $\hbox{\rm GL}(V)\times\Gamma(X, {\cal O}_X^*)$ by 
 $\mu_2(\Gamma(X, {\mathcal O}_X))\times\{1\}$,  
 become multiplicative. 
\end{description}
\end{theorem}

\begin{theorem}\label{lifting-of-automorphisms}
For a rank 3 quadratic bundle $\vq$ on a scheme $X$, 
 one has the following natural 
 commutative diagram of groups with exact rows, where the downward arrows 
 are the  canonical inclusions and where 
 $l$ 
 is the function that associates to any orthogonal similitude its 
 multiplier:
\begin{displaymath}
\xymatrix{ 
  &  & \sovq \ar[d]_{\hbox{\rm inj}} \ar[r]^{\cong\hspace*{1cm}}  &
 \sautcovq \ar[d]^{\hbox{\rm inj}} & \\
 1\ar[r] & \mu_2(\Gamma(X, {\mathcal O}_X)) \ar[r]  \ar[d]_{\hbox{\rm inj}} & 
 \ovq \ar[r] \ar[d]_{\hbox{\rm inj}} & \autcovqprime \ar[d]^{\hbox{\rm inj}} 
 \ar[r]  & 1\\
 1\ar[r] & \Gamma(X, {\mathcal O}_X^{*}) \ar[r] & 
 \govq \ar[r] \ar[d]_{\hbox{\rm det}^2\times l^{-3}} & \autcovq \ar[d]^{\hbox{\rm det}} 
  \ar[r] 
  & 1\\
 & & \Gamma(X, {\mathcal O}_X^{*}) \ar@{=}[r]  & \Gamma(X, {\mathcal O}_X^{*}) & 
}
\end{displaymath}
Further, we have
\begin{description}
\item[(a)] There are splitting homomorphisms 
 $s_{2k+1}^{+}: \autcovq\longrightarrow\govq$ such that 
$l\circ s_{2k+1}^{+}=\hbox{\rm det}^{2k+1}$ and $(\hbox{\rm det}^2
\circ s_{2k+1}^{+})\times(l^{-3}\circ s_{2k+1}^{+})=\hbox{\rm det}.$ The 
 restriction of $s_{2k+1}^{+}$ to $\autcovqprime$ does not necessarily take 
 values in $\ovq$, but the further restriction to $\sautcovq$ does 
 take values in $\sovq.$ In particular, $\govq$ is a semidirect product.
 The maps $s_{2k+1}$ and 
 $s'$ of Theorem \ref{lifting-of-isomorphisms} above (under the 
 current hypotheses) may not be 
 homomorphisms  but are homomorphisms upto $\mu_2(\Gamma(X, {\mathcal O}_X)).$ 
\item[(b)] Suppose $X$ is integral and $q\otimes\kappa(x)$ is semiregular
 at some point $x$ of 
 $X$ with residue field $\kappa(x).$
 Then any automorphism of $\covq$ has determinant 1 so that  
   $\autcovq=\autcovqprime=\sautcovq$ and in particular, $\ovq$ is the 
 semidirect product of $\mu_2(\Gamma(X, {\cal O}_X))$ and 
 $\sovq.$
\end{description}
\end{theorem}

We next turn to the question whether the map
$\hbox{\sf Witt}(X):\mathscr{Q}_3(X)\longrightarrow
\mathscr{A}^{Sp-Azu}_4(X)$ is 
 always surjective. This is the same as 
 asking if, given a specialisation $ A$ of
 rank 4 Azumaya bundles on $X$, 
 there exists a rank 3 quadratic bundle $\vq$ such that $\covq\cong A$
 (as  algebra bundles)---of course
 Theorem \ref{injectivity} will guarantee that 
 $\vq$ is unique upto tensoring by a discriminant bundle. It is this 
 question of surjectivity that seems to involve the geometry of $X.$
 

\begin{theorem}\label{structure-of-specialisation}
Let $X$ be a scheme and $ A$ a specialisation of rank 4 Azumaya algebra 
 bundles on $X.$ Let $\cal A$ denote the locally-free ${\cal O}_X$-algebra  
 corresponding to $ A.$ 
 Let ${\cal O}_X. 1_{\cal A}\hookrightarrow {\cal A}$ 
 denote the image of the canonical morphism 
${\cal O}_X\longrightarrow {\cal A}$ defined by the nowhere-vanishing 
 global section of $\cal A$ corresponding to the unit
 for algebra multiplication.
\begin{description}
\item[(a)]
 There exists a rank 3 vector bundle 
 $V$ on $X$ (with corresponding ${\cal O}_X$-module ${\cal V}$) such 
 that the following hold: 
\begin{description}
\item[(1)] $\hbox{\rm det}({\cal A})\otimes \Lambda^2({\cal V})\cong 
{\cal A}/{\cal O}_X. 1_{\cal A},$ from which the following 
 may be deduced: 
\item[(2)]  $\hbox{\rm det}(\Lambda^2({\cal V}))\cong{(\hbox{\rm det}({\cal A}))}^{\otimes-2};$
\item[(3)]  ${\cal V}\cong{({\cal A}/{\cal O}_X. 1_{\cal A})}^\vee\otimes
\hbox{\rm det}({\cal V})\otimes\hbox{\rm det}({\cal A});$
\item[(4)] $\hbox{\rm det}({\cal A}^\vee)\cong {(\hbox{\rm det}({\cal A}))}^{\otimes-3}\otimes {(\hbox{\rm det}({\cal V}))}^{\otimes-2}$ which 
implies that $\hbox{\rm det}({\cal A})\otimes\hbox{\rm det}({\cal A}^\vee)\in
2.\hbox{\rm Pic}(X).$ 
\end{description}
\item[(b)] If there exists a quadratic bundle $\vq$ on $X$ such that 
 $A\cong\covq$ then ${\cal A}/{\cal O}_X.1_{\cal A}\cong\Lambda^2(V)$  
 which implies that $\hbox{\rm det}({\cal A}^\vee)\in
2.\hbox{\rm Pic}(X)$, and therefore using (4) of (a) above, that
 $\hbox{\rm det}({\cal A})\in
2.\hbox{\rm Pic}(X).$ 
\end{description}   
\end{theorem}

A bilinear form $b$ (with values in ${\cal O}_X$) 
 on a vector bundle $V$ on $X$ 
induces a quadratic form $q_b$ given on sections by 
 $x\mapsto b(x,x).$
 Further, $b$ also defines a $(\mathbb Z/2.{\mathbb Z})$-graded 
 linear isomorphism $\psi_b:C(V,q_b)\cong \Lambda(V)$ which is 
 unique with respect to certain properties
 (see (2d), Theorem \ref{Bourbaki}). When $X$ is affine, every (global) 
quadratic  form on a vector bundle is induced from a (global) bilinear form. 
Therefore the following theorem is optimal for affine $X.$  

\begin{theorem}\label{surjectivity-limited}
With $X$ and $A$ as in Theorem \ref{structure-of-specialisation} above, 
 there exists a rank 3 quadratic bundle $\vq$ (on $X$) induced from a
 rank 3 bilinear-form bundle $( V,  b)$ such that 
 $\covq\cong A$ (isomorphism as algebra bundles) iff  the 
 following two conditions hold:
\begin{description}
\item[(1)] the determinant bundle of $ A$ is an element of 
$2.\hbox{\rm Pic}(X)$;
\item[(2)] ${\cal O}_X.1_A$ is an ${\cal O}_X$-direct summand of $A.$
\end{description}
The hypothesis {\rm\bf(2)} is satisfied
  in the following cases:
\begin{description}
\item[(i)] $X$ is an affine scheme;
\item[(ii)] the residue field of every point
 of $X$ has characteristic $\neq 2.$ 
\end{description}
The hypothesis {\rm\bf(1)} is satisfied when 
 $A$ is itself an Azumaya algebra bundle of rank 4 over $X.$
 It is also satisfied  whenever $\hbox{\rm det}({\cal A})\cong {\cal O}_X$
 (for example, when $A$ is  (globally) free) and 
  whenever $\hbox{\rm Pic}(X)=2.\hbox{\rm Pic}(X)$---a condition which 
 is trivially satisfied when $\hbox{\rm Pic}(X)=0$, for example  
  when $X=\hbox{\rm Spec}(R)$ and 
   $R$ is a UFD or a local ring or a field. It follows therefore 
  that if $X=\hbox{\rm Spec}(R)$ is an affine scheme 
 such that $\hbox{\rm Pic}(X)=2.\hbox{\rm Pic}(X)$, 
 then the map $\hbox{\sf Witt}(X): 
 {\mathscr Q}_3(X)\longrightarrow {\mathscr A}_4^{Sp-Azu}(X)$ is surjective, 
 and hence a bijection. 
  \end{theorem}


Before proceeding further, we
 need to recall a few concepts from Part A of \cite{tevb-paper1}. 
For a rank $n^2$ vector bundle $W$ 
 on a scheme $X$ and $w\in\Gamma(X,W)$ 
 a nowhere-vanishing global section, 
 recall that if  
  $\hbox{\rm Id-}w\hbox{\rm -Azu}_W$ is 
  the open $X$--subscheme of Azumaya 
 algebra structures on $ W$ with identity $ w$ then 
 its schematic image (or the scheme of specialisations or the limiting 
 scheme) in the bigger $X$--scheme 
$\hbox{\rm Id-}w\hbox{\rm -Assoc}_W$ of associative 
 $ w$-unital algebra structures on $ W$  is the $X$--scheme  
 $\hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_W.$ 
By definition, the set of distinct specialised $w$-unital 
 algebra structures on 
 $W$ corresponds precisely to 
the set of global sections of this last scheme over 
 $X.$

\begin{theorem}\label{relations-with-self-duality}
Let $X$ be a connected proper scheme of finite type 
  over an algebraically closed field and let 
  $W$ be a  vector bundle on $X$ of rank $n^2$ for some $n\geq 2.$  
\begin{description}
\item[(a)] If $W$ is self-dual ($\Leftrightarrow W\cong W^\vee$)
  and $A$ is an associative unital algebra 
   structure on $W$ such that 
  ${\cal A}\otimes_X {\cal O}_{X,x}$
  is Azumaya even for a single point $x\in X$, then $A$ 
 is Azumaya at every point of $X.$ This 
 may also be stated as follows: if a section to $\aaWw$ over $X$ topologically 
 meets $\azuWw$, then it
factors as a morphism through the open subscheme 
 $\azuWw$, where 
 $w:=1_A$ and $A$ corresponds to the given section.
\item[(b)] Let the rank of $W$ be 4. 
 If there exists an associative unital algebra structure on $W$ which
 is not globally Azumaya but is Azumaya atleast at one point of $X$,  
 then there does 
 not exist any global Azumaya algebra structure on $W.$ Thus, 
 if there is a section over $X$ of $\aaWw$ that topologically meets both the 
 open subscheme $\azuWw$ and its complement (with 
 $w=1_A$ where $A$ corresponds to the given section),
 then the $X$-schemes $\hbox{\rm Id-}{w'}\hbox{\rm -Assoc}_W$ (with 
 $w'$ global nowhere-vanishing) cannot have sections that land 
 topologically inside $\hbox{\rm Id-}{w'}\hbox{\rm -Azu}_W$ and 
 hence in particular the $X$-schemes 
  $\hbox{\rm Id-}{w'}\hbox{\rm -Azu}_W$ have 
  no sections over $X.$ 
\end{description}
\end{theorem}

 Recall that an integral separated Noetherian
 scheme is said to be locally-factorial if each of its 
 local rings is a unique factorisation domain (=UFD=factorial ring). 


\begin{theorem}\label{surjectivity-affine-limited}
Let $X$ be a scheme and  $W$ a rank 4 vector bundle on $X$ with  
 a global nowhere-vanishing section $w.$ Let $D_X$ denote  
 the closed subset $\spazuWw\backslash\azuWw.$
\begin{description}
\item[(a)] $X$ is irreducible iff $\spazuWw$ is irreducible  iff 
 $\azuWw$ is irreducible iff $D_X$ is irreducible. The set of 
 irreducible components of $X$ is locally finite---for example this 
 happens when  $X$ is locally noetherian---iff the same is 
 true of the corresponding set for $\spazuWw$ or for $\azuWw.$ If 
 $X$ is noetherian and finite-dimensional then the same are true for 
 $\spazuWw$ and  
 $\azuWw.$
\item[(b)] If $X'\longrightarrow X$ is a morphism of schemes,  
 and if $(W',w')$ denotes the pullback of $(W,w)$, then we have  a 
 canonical isomorphism 
$$\azuWprwpr\cong\azuWw\times_{\spazuWw} \spazuWprwpr$$
In particular, the topological image of $D_{X'}$ is $D_{X}.$
Moreover, when $X'\longrightarrow X$ is a homeomorphism onto its 
 topological image---which is for example the case when it is a closed 
 or an open immersion, then $D_X\cap\spazuWprwpr$ can be identified with 
 $D_{X'}.$ 
\item[(c)] 
If $X$ is a scheme which is finite dimensional and 
 whose set of irreducible components is locally finite,
 then the closed subset $D_X$ is a divisor i.e., it has codimension 
 1 in $\spazuWw.$
\item[(d)]  $X$ is affine iff 
 $\hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_W$ is affine iff 
$\hbox{\rm Id-}w\hbox{\rm -Azu}_W$
 is affine. If $X$ is regular in codimension 1 (respectively locally-factorial)
 then so are $\spazuWw$ and $\azuWw.$ 
\item[(e)] 
 Assume that $X$ is 
 locally-factorial and $W$ is self-dual (i.e., $W\cong W^\vee$).  
 Then the (Weil) divisor $n.(D_X)$ is principal for some positive integer 
 $n$,   
 so that the natural homomorphism given by restriction of line bundles 
 $\hbox{\rm Pic}(\hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_W)\longrightarrow 
 \hbox{\rm Pic}(\hbox{\rm Id-}w\hbox{\rm -Azu}_W)$ is an isomorphism iff 
 $n=1.$ The integer 
 n is divisible by 2 iff  $\hbox{\rm det}(W\otimes_X\spazuWw)
\in 2.\hbox{\rm Pic}(\spazuWw)$ so that in this case 
 if there exists a specialised algebra structure on $W$ 
 with unit $w$, then $\hbox{\rm det}(W)\in 2.\hbox{\rm Pic}(X)$; if 
 in addition, ${\cal O}_X.w$ is an ${\cal O}_X$-direct summand of 
 $W$, then $W$ is of the form $\Lambda^{even}(V)$ for a rank 3 
 vector bundle $V$ on $X$, and  for every $X$-scheme $T$ and 
 for every specialisation $A_T$ on $W_T:=(W\otimes_X T)$ with unit 
 $w_T:=(w\otimes_X T)$, there exists a rank 3 quadratic bundle 
 $(V_T, q_T)$ induced from a rank 3 bilinear form bundle $(V_T, b_T)$ 
 on $T$ such that $C_0(V_T, q_T)\cong A_T$ as algebra bundles. 
 (this last conclusion holds for any affine scheme $T$ over $X$, 
even if ${\cal O}_X.w$ were not 
 an ${\cal O}_X$-direct summand of $W$). 
\end{description}
 \end{theorem}


 Notice that under the hypotheses (1) and (2) of Theorem \ref{surjectivity-limited}, there exists 
 by (1), Theorem \ref{structure-of-specialisation},  
 a rank 3 bundle $ V$ on $X$ such that $\Lambda^{even}({\cal V})= 
{\cal O}_X\oplus\Lambda^2({\cal V})\cong{\cal A}$ with ${\cal O}_X\cong 
 {\cal O}_X.1_{\cal A}.$ 
 In this situation the `if' part of Theorem \ref{surjectivity-limited}
 as well as the latter part of (e) of Theorem
 \ref{surjectivity-affine-limited} 
 are consequences of the next one 
 which describes the specialisations as bilinear forms. 
 We continue with the notations introduced before
 Theorem \ref{relations-with-self-duality}.  
 If 
 $\hbox{\rm Stab}_{w}\subset \hbox{\rm GL}_{W}$
 is the stabiliser subgroupscheme of $w$,  recall from Theorems 
 3.4 and 3.8, Part A, \cite{tevb-paper1}, that there exists a canonical 
 action of $\hbox{\rm Stab}_{w}$ on $\hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_W$ such that the 
 natural inclusions \label{clubsuit} 
 $$(\clubsuit)\hspace*{1cm}\hbox{\rm Id-}w\hbox{\rm -Azu}_W\hookrightarrow \hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_W \hookrightarrow \hbox{\rm Id-}w\hbox{\rm -Assoc-Alg}_W$$
 are all $\hbox{\rm Stab}_{w}$-equivariant.
  Now let
  $ V$ be a rank 3 vector bundle on the scheme $X$ and 
 $\hbox{\rm Bil}_{V}$
 be the associated  rank 9 vector bundle of 
 bilinear forms on $ V$ with values in ${\cal O}_X.$ 
 Let 
$\hbox{\rm Bil}^{sr}_{V}\hookrightarrow
 \hbox{\rm Bil}_{V}$
 correspond to the 
 open subscheme of semiregular bilinear forms---we say that a bilinear 
 form is semiregular if the quadratic form it induces is semiregular (though
 it  may turn out that a semiregular bilinear form may be degenerate).
 Let $ W:=\Lambda^{even}( V)$ and let $ w\in\Gamma(X, 
 W)$ be the nowhere-vanishing global section corresponding to the 
 unit for the natural multiplication in the even-exterior algebra bundle.
There is an obvious
  natural action of $\hbox{\rm GL}_{V}$ on $\hbox{\rm Bil}_{V}.$ 
 There is 
also a natural morphism of groupschemes  $\hbox{\rm GL}_{V}\longrightarrow
\hbox{\rm Stab}_{w}$ given on valued points
 by $g\mapsto \Lambda^{even}(g)$ and therefore  
 the natural inclusions marked by  $(\clubsuit)$ above are  
 $\hbox{\rm GL}_{V}$-equivariant. Finally, note that there is an obvious 
 involution $\Sigma$ on $\hbox{\rm Id-}w\hbox{\rm -Assoc-Alg}_W$
 given by $A\mapsto \hbox{\rm opposite}(A)$ which leaves the open 
 subscheme $\hbox{\rm Id-}w\hbox{\rm -Azu}_W$ invariant. 

\begin{theorem}\label{bilinear-forms-as-specialisations}\
\begin{description}
\item[(1)]
 Let $V$ be a rank 3 vector bundle on the scheme $X$,
 $ W:=\Lambda^{even}( V)$ and  $ w\in\Gamma(X, 
 W)$ correspond to 1 in the even-exterior algebra bundle. 
 There is a natural $\hbox{\rm GL}_{V}$-equivariant 
 morphism of $X$--schemes 
$\Upsilon'=\Upsilon'_X: \hbox{\rm Bil}_{V}\longrightarrow
\hbox{\rm Id-}w\hbox{\rm -Assoc}_W$ whose 
 schematic image is precisely the scheme of specialisations 
$\hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_W.$ Further 
if  $\Upsilon'$ factors canonically through 
$\Upsilon=\Upsilon_X: \hbox{\rm Bil}_{V}\longrightarrow
\hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_W$, 
 then $\Upsilon$ is a $\hbox{\rm GL}_{V}$-equivariant
  isomorphism and it maps the $\hbox{\rm GL}_{V}$-stable 
 open subscheme $ \hbox{\rm Bil}^{sr}_{V}$
 isomorphically onto the $\hbox{\rm GL}_{V}$-stable open 
 subscheme $\hbox{\rm Id-}w\hbox{\rm -Azu}_W.$
\item[(2)] The involution 
 $\Sigma$ of  $\hbox{\rm Id-}w\hbox{\rm -Assoc-Alg}_W$ 
 defines a unique involution (also denoted by $\Sigma$) on 
 the scheme of specialisations $\hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_W$
 leaving the open subscheme $\hbox{\rm Id-}w\hbox{\rm -Azu}_W$
 invariant, and therefore via the isomorphism $\Upsilon$, it defines 
 an involution on $\hbox{\rm Bil}_{V}.$ This involution is none other
 than the one on valued points given by $B\mapsto \hbox{\rm transpose}(-B).$  
\item[(3)]
For an $X$-scheme $T$, let $V_T$ (resp.$\thinspace W_T$) denote the 
 pullback of $V$ (resp.$\thinspace W$) to $T$,
  and let $w_T$ be the global section of $W_T$ induced by 
 $w.$ Then the base-changes of $\Upsilon'_X$ and $\Upsilon_X$ to $T$, 
 namely  $\Upsilon'_X\times_X T: \hbox{\rm Bil}_{V}\times_X T\longrightarrow
\hbox{\rm Id-}w\hbox{\rm -Assoc}_W\times_X T$ and 
 $\Upsilon_X\times_X T: \hbox{\rm Bil}_{V}\times_X T\cong
\hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_W\times_X T$ may be  
 canonically identified with the corresponding ones over $T$ namely 
 $\Upsilon'_T: \hbox{\rm Bil}_{V_T}\longrightarrow
\hbox{\rm Id-}w_T\hbox{\rm -Assoc}_{W_T}$ and 
 $\Upsilon_T: \hbox{\rm Bil}_{V_T}\cong
\hbox{\rm Id-}w_T\hbox{\rm -Sp-Azu}_{W_T}.$
\end{description} 
\end{theorem}
 The explicit computation of the morphism $\Upsilon$ locally over $X$ is
 an important step in proving the above theorem. To describe this,  
 suppose that $V$ is free of rank 3 over $X$, so that 
 we may fix a basis $\{e_1, e_2, e_3 \}$ for $V$, which naturally gives rise 
 to a basis of $\hbox{\rm Bil}_{V}.$  For any $X$-scheme $T$, 
 a $T$-valued point $B$ of $\hbox{\rm Bil}_{V}$ is just   
 a global bilinear form with values in ${\cal O}_T$
 on the pull-back $V\otimes_X T$ of 
 $V$ to $T.$  Such a $B$ 
 is given uniquely 
 by a $(3\times 3)$-matrix
 $(b_{ij})$ with the $b_{ij}$ being global sections of the trivial 
 line bundle ${\mathbb A}^1_T$ (or equivalently, elements of 
 $\Gamma(T, {\cal O}_T)$).
 The chosen basis for $V$ also gives rise to 
 the basis $\{\epsilon_0:=w=1\hspace*{1mm};\hspace*{1mm}
\epsilon_1:=e_1\wedge e_2\hspace*{1mm},\hspace*{1mm}
\epsilon_2:=e_2\wedge e_3\hspace*{1mm},\hspace*{1mm}
\epsilon_3:=e_3\wedge e_1\}$ 
of $W=\Lambda^{even}(V).$ A $T$-valued point $A$ of 
$\hbox{\rm Id-}w\hbox{\rm -Assoc-Alg}_W$ is just a
 $w_T:=(w\otimes_X T)$-unital 
 associative algebra structure on the bundle $W_T:=W\otimes_X T.$ 
 Let $\cdot_A$ denote the multiplication in the algebra bundle $A$, 
 and for ease of notation, let $s^{\circ}$ denote the section 
 $s\otimes_X T$ induced from a section $s$ (for example, $(w\otimes_X T)=
 w^{\circ}, \epsilon_i\otimes_X T=\epsilon_i^\circ$ etc).

\begin{theorem}\label{multiplication-table-wrt-bilinear}
In addition to the hypothesis of
 Theorem \ref{bilinear-forms-as-specialisations}, assume that $V$ is 
 free of rank 3. Then fixing a basis for $V$ and adopting 
 the notations above, 
 the map $\Upsilon(T)$ takes $B=(b_{ij})$ to 
$(A, 1_A, \cdot)=(W_T, w_T=w^\circ , \cdot_A)$
 with multiplication given as follows, where $M_{ij}(B)$ is the 
 determinant of the minor of the element $b_{ij}$ in $B:$
\begin{enumerate}
\item[$\bullet$] 
$\epsilon_1^{\circ}\cdot_A\epsilon_1^{\circ}=-M_{33}(B)w^\circ + (b_{21}-b_{12})\epsilon_1^{\circ}$
\item[$\bullet$] 
$\epsilon_2^{\circ}\cdot_A\epsilon_2^{\circ}=-M_{11}(B)w^\circ + (b_{32}-b_{23})\epsilon_2^{\circ}$
\item[$\bullet$] 
$\epsilon_3^{\circ}\cdot_A\epsilon_3^{\circ}=-M_{22}(B)w^\circ + (b_{13}-b_{31})\epsilon_3^{\circ}$
\item[$\bullet$] 
$\epsilon_1^{\circ}\cdot_A\epsilon_2^{\circ}=-M_{31}(B)w^\circ -b_{23}\epsilon_1^{\circ} -b_{12}\epsilon_2^{\circ} 
-b_{22}\epsilon_3^{\circ}$
\item[$\bullet$] 
$\epsilon_2^{\circ}\cdot_A\epsilon_3^{\circ}=+M_{12}(B)w^\circ -b_{33}\epsilon_1^{\circ} -b_{31}\epsilon_2^{\circ} 
-b_{23}\epsilon_3^{\circ}$
\item[$\bullet$] 
$\epsilon_3^{\circ}\cdot_A\epsilon_1^{\circ}=+M_{23}(B)w^\circ -b_{31}\epsilon_1^{\circ} -b_{11}\epsilon_2^{\circ} 
-b_{12}\epsilon_3^{\circ}$
\item[$\bullet$] 
$\epsilon_1^{\circ}\cdot_A\epsilon_3^{\circ}=+M_{32}(B)w^\circ +b_{13}\epsilon_1^{\circ} +b_{11}\epsilon_2^{\circ} 
+b_{21}\epsilon_3^{\circ}$
\item[$\bullet$] 
$\epsilon_2^{\circ}\cdot_A\epsilon_1^{\circ}=-M_{13}(B)w^\circ +b_{32}\epsilon_1^{\circ} +b_{21}\epsilon_2^{\circ} 
+b_{22}\epsilon_3^{\circ}$
\item[$\bullet$] 
$\epsilon_3^{\circ}\cdot_A\epsilon_2^{\circ}=-M_{21}(B)w^\circ +b_{33}\epsilon_1^{\circ} +b_{13}\epsilon_2^{\circ} 
+b_{32}\epsilon_3^{\circ}$
\end{enumerate}
\end{theorem}

 The key to the proofs of Theorems \ref{injectivity}, \ref{lifting-of-isomorphisms} and \ref{lifting-of-automorphisms} lies in a deeper study of a different 
 identification 
 of the scheme of specialisations, 
 namely one related to the scheme of quadratic 
 forms on a trivial rank 3 bundle in the special situation when $W$ is free 
 and $w$ part of a global basis. Without loss of 
 generality we may in this situation therefore 
 take $V$ to be a free rank 3 vector bundle on $X$ and 
 $(W, w)=(\Lambda^{even}(V), 1)$, so that we are in the situation of 
 Theorem \ref{multiplication-table-wrt-bilinear} above. This relationship with 
 quadratic forms  was shown in Theorem 5.3, Part A, \cite{tevb-paper1}, 
  which we briefly recall next. Let $\hbox{\rm Quad}_{V}$ denote the 
 bundle of quadratic forms on $V$ (with values in ${\cal O}_X$) and 
 $\hbox{\rm Quad}^{sr}_{V}$ the open subscheme of semiregular quadratic forms. 
 Let $A_0$ denote the algebra bundle structure (with unit $w=1$) on
 $W=\Lambda^{even}(V)$ given by $\Lambda^{even}(V)$ itself. Fix a basis for 
 $V$ and adopt the notations preceding Theorem \ref{multiplication-table-wrt-bilinear} above. Then $\hbox{\rm Stab}_{w}$ is the semidirect product of a 
 commutative 3-dimensional subgroupscheme $\hbox{\sf L}_w\cong ({\mathbb A}^3_X, +)$
 with the stabiliser subgroupscheme $\hbox{\rm Stab}_{A_0}$ of $A_0$ in 
 $\hbox{\rm Stab}_{w}$ (Lemma 5.1, Part A, \cite{tevb-paper1}).  

\begin{theorem}[Definition 5.2 \& Theorem 5.3, Part A, \cite{tevb-paper1}]
\label{Thetaisom}
There is a natural isomorphism $\Theta: \hbox{\rm Quad}_{V}\times_X \hbox{\sf L}_w \cong
 \hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_W$ which maps the open subscheme 
 $\hbox{\rm Quad}^{sr}_{V}\times_X \hbox{\sf L}_w$ isomorphically onto the open subscheme 
 $\hbox{\rm Id-}w\hbox{\rm -Azu}_W.$  
\end{theorem}
Section \ref{sec3} is essentially devoted to studying $\Theta.$
 There we compute $\Theta$ explicitly and
 in Theorem \ref{mult-table-wrt-Theta} we write out the 
  multiplication table of every specialised algebra structure on any fixed
  free rank 4 vector bundle with fixed unit that is part of a global basis.
It turns out that $\Theta$ is not equivariant 
 with respect to $\hbox{\rm GL}_{V}$, but nevertheless satisfies a `twisted' 
 form of equivariance (Theorem \ref{tequiv}). 
A $T$-valued point $q$ of $\hbox{\rm Quad}_{V}\cong{\mathbb A}^6_X$ 
may be identified uniquely with a 6-tuple $(\lambda_1,\lambda_2,\lambda_3,
\lambda_{12},\lambda_{13},\lambda_{23})$ corresponding to the 
 quadratic form $(x_1,x_2,x_3)\mapsto \Sigma_i \lambda_i x_i^2 + \Sigma_{i<j}\lambda_{ij}x_ix_j.$ A $T$-valued point $\underline{t}$ 
  of $\hbox{\sf L}_w\cong({\mathbb A}^3_X, +)$
 may be identified uniquely with a 3-tuple $(t_1, t_2, t_3)$ which 
 corresponds to the valued point of $\hbox{\rm Stab}_{w}$ given by the 
 $(4\times 4)$-matrix
 \begin{displaymath}
 \left(\begin{array}{c|ccc}
1 & t_1 & t_2 & t_3 \\
\hline
0 & & I_3 & 
\end{array}
\right)
 \end{displaymath} 
where $I_3$ is the $(3\times 3)$-identity matrix. 
 With these notations, 
the identification of Theorems \ref{bilinear-forms-as-specialisations} and 
 \ref{multiplication-table-wrt-bilinear} may be compared with that of 
 the above Theorem \ref{Thetaisom} as follows.

\begin{theorem}\label{Thetaisom-Upsilonisom}
The isomorphism $\Upsilon^{-1}\circ\Theta
:\hbox{\rm Quad}_{V}\times_X \hbox{\sf L}_w
\cong \hbox{\rm Bil}_{V}$ takes the valued point 
$(\hspace*{2mm}q,\hspace*{2mm} \underline{t}\hspace*{2mm})=(\hspace*{2mm}(\lambda_1,\lambda_2,\lambda_3,
\lambda_{12},\lambda_{13},\lambda_{23}),\hspace*{2mm} (t_1, t_2, t_3)\hspace*{2mm})$
to the valued point $B=(b_{ij})$ given by 
\begin{displaymath}
B=\left(\begin{array}{ccc}
\lambda_1 & t_1 & \lambda_{13}-t_3 \\
\lambda_{12}-t_1 & \lambda_2 & t_2 \\
t_3 & \lambda_{23}-t_2 & \lambda_3
\end{array}
\right)
 \end{displaymath} 
Moreover, under this identification, the involution $B\mapsto {(-B)}^t$ on 
  $\hbox{\rm Bil}_{V}$ (induced from the isomorphism 
 $\Upsilon$ of Theorem \ref{bilinear-forms-as-specialisations})
 translates into the involution on 
 $\hbox{\rm Quad}_{V}\times_X \hbox{\sf L}_w$ given by 
 $$(\hspace*{2mm}q,\hspace*{2mm} (t_1, t_2, t_3)\hspace*{2mm})\mapsto (\hspace*{2mm}-q,\hspace*{2mm} (t_1-\lambda_{12}, t_2-\lambda_{23}, t_3-\lambda_{13})\hspace*{2mm}).$$ 
\end{theorem}


 We next make some comments relevant to the question of the 
 surjectivity of $\hbox{\sf Witt}(X):{\mathscr Q}_3(X)
\longrightarrow  {\mathscr A}_4^{Sp-Azu}(X).$   
 Given a specialised algebra structure $A$ (on a rank 4 vector bundle on $X$) 
which is itself an Azumaya algebra, we first indicate briefly how 
 to naturally  retrieve  
a rank 3 quadratic bundle $\vq$ such that $\covq\cong A.$ 
 Let $R$ be a unital commutative ring and $A$ a unital associative $R$-algebra.
  Given any involution $\sigma$ on $A$, we may 
 define the  associated trace $tr_\sigma:x\mapsto 
 x+\sigma(x)$ and the associated norm $n_{\sigma}: x\mapsto x.\sigma(x)$;  
 in para.1.3, Chap.I, \cite{Knus}, Knus calls $\sigma$
 {\em standard}  if 
 $\sigma$ fixes $R.1_A$ and both $tr_\sigma$ and $n_{\sigma}$ take 
 values in $R.1_A.$ In  Prop.1.3.4 of the same chapter, he proves that
  a standard  involution is unique if it exists, provided the $R$-module
 underlying  $A$  is finitely generated projective and faithful.
 Thereafter, in para.1.3.7,  Knus defines  
 $A$ to be 
 a {\em quaternion} algebra if $A$ is a projective $R$-module of rank 4 and 
 $A$ has a standard involution. Thus we may define a rank 4 algebra bundle on 
 a scheme $X$ to be a quaternion algebra bundle if it is locally 
 (in the Zariski topology) a quaternion algebra in Knus' sense, and  
 it would follow that the local standard involutions glue to define a
 unique global standard involution on the bundle. 
 
 \begin{proposition}\label{spln-is-quaternion}  
Any specialised algebra bundle is a quaternion algebra bundle.
\end{proposition}
 This result 
 can be deduced from the following two facts:
\begin{description}
\item[(1)] Any specialised algebra is 
 locally (in the Zariski topology) the even Clifford algebra of a rank 
 3 quadratic bundle (Theorems 3.8 and  5.3, Part A, \cite{tevb-paper1}).
\item[(2)] The even Clifford algebra of a quadratic 
  module of rank 3 over a commutative ring
 has a standard involution which is none other than the restriction of the `standard' involution on the full 
 Clifford algebra (Prop.3.1.1, Chap.V, \cite{Knus}). {\bf Q.E.D.}
\end{description}

\begin{theorem}\label{Witt-bijective-on-q3sr}
For each scheme $X$, the map $\hbox{\sf Witt}(X):{\mathscr Q}_3^{sr}(X)
\longrightarrow 
 {\mathscr A}_4^{Azu}(X)$ is bijective. 
\end{theorem}
The proof of the above Theorem 
  follows from Prop.3.2.3 and Prop.3.2.4, Chap.V, \cite{Knus}
 generalised to the scheme-theoretic setting.
 We recall how the surjectivity is established. 
 Let $A$ be a specialised algebra bundle on the scheme $X$. 
 By the results just quoted if $A$ is Azumaya, or more generally by
 Prop.\ref{spln-is-quaternion}, we have the existence of a unique 
 standard involution $\sigma_A$  on $A$, to which are associated the 
 norm $n_{\sigma_A}:A\longrightarrow {\mathbb A}^1_X$ 
given on sections by $x\mapsto x.\sigma_A(x)$ and 
 the trace $tr_{\sigma_A}:A\longrightarrow {\mathbb A}^1_X $
 given on sections by $x\mapsto x+\sigma_A(x).$ 
 Let $A':=\hbox{\rm kernel}(tr_{\sigma_A})\hookrightarrow A$ 
be the subsheaf of  trace zero elements.
As the calculations in para.3.2, Chap.V, \cite{Knus} show,  
the trace map is surjective if 
 $A$ is itself an Azumaya algebra; if this is the case, then 
 it is further shown there that
 the rank 3 quadratic bundle $\vq:=(A', n_{\sigma_A}|A')$ is   
 semiregular and its even Clifford algebra  $\covq\cong A.$

 However the above method of retrieving a canonical 
 rank 3 quadratic bundle fails badly for specialised non-Azumaya algebras. 
 Consider even the case of $X=\hbox{\rm Spec}(k)$ where $k$
 is a field of characteristic 
 two and the Clifford algebra $A=C_0(V,q)$ 
 of a quadratic form $q$ on $V=k^{\oplus 3}$
 which is a perfect 
 square (i.e., a square of a linear form or equivalently a sum of squares). 
 In this case an easy computation shows that 
 the subspace $A'$ of trace zero elements is the full space $A.$ 
 However, the underlying module of $A$ trivially satisfies the hypotheses of 
 Theorem \ref{surjectivity-limited}, and so 
 the existence of $\vq$ follows from that Theorem.

 
 The good algebraic properties of Azumaya algebras
 are reflected as good geometric properties of the scheme of 
 Azumaya algebra structures on a fixed vector bundle: this scheme is 
 separated, of finite type and smooth relative to the base scheme (over 
 which the vector bundle is fixed) and also base-changes well 
 relative to the base scheme (Theorem 3.4, Part A, 
 \cite{tevb-paper1}). When the vector bundle is of rank 4, 
  the nice thing that happens is that all these good 
 properties also pass over to the limit i.e., to the 
  scheme of specialisations, defined 
 to be the schematic image of the scheme of Azumaya algebra structures 
 (Theorem 3.8, Part A, \cite{tevb-paper1}). 
 If $A$ is Azumaya over $X$, then as seen above, 
 $A\cong \covq$ with $\vq$ semiregular. So by (b), Theorem 
 \ref{structure-of-specialisation}, we have that 
 $\hbox{\rm det}(A)\in 2.\hbox{\rm Pic}(X).$ On the other hand, 
 by the same Theorem, this is a necessary condition for a specialisation 
 to arise as $C_0(V,q)$ for a global $q.$ Therefore the  
 author expects this condition to hold in general (the assertions in
 (e), Theorem \ref{surjectivity-affine-limited}, arose after 
 an attempt to investigate this viewpoint). 
 Of course, if this 
 holds and also if ${\cal O}_X.1_A$ is an ${\cal O}_X$-direct 
 summand of $A$, then by 
 Theorem \ref{surjectivity-limited} we 
 get something more, i.e., even a bilinear form bundle $(V,b)$ 
 such that $C_0(V,q_b)\cong A.$ But the latter hypothesis 
 is seemingly strong for non-affine $X.$
The author hopes that  
the surjectivity of $\hbox{\sf Witt}(X)$ is
 valid for general $X$, and believes that this may follow 
 from a deeper understanding of the geometry of the scheme of specialisations.
 We next state some results on  
 rank 3 quadratic forms and specialised algebras in certain particular cases. 

\begin{proposition}\label{semilocal-unique-semireg}
Let $S$ be a commutative semilocal ring that is 2-perfect i.e., such that  
the square map $S\longrightarrow S: s\mapsto s^2$ is surjective, and 
 $V$ a free rank 3 $S$-module. Then the set of semiregular quadratic $S$-forms 
 on $V$ forms a single $\hbox{\rm GL}(V)$--orbit; in other words, upto 
 isometry, $\exists$ only one semiregular quadratic $S$-module structure 
 on $V.$ 
\end{proposition} 

\begin{corollary}\label{local-unique-azumaya}
Let $S$ be a commutative local ring that is 2-perfect. Then any two rank 4 
 Azumaya $S$-algebras are isomorphic. If $S$ were only semilocal, the 
 conclusion still holds provided the identity elements for multiplication 
 for each of the two Azumaya $S$-algebras can be completed to an $S$-basis.
\end{corollary}
 
Since $\hbox{\sf Witt}(X):{\mathscr Q}_3^{sr}(X)\cong {\mathscr A}_4^{Azu}(X)$ 
 is bijective, taking $X=\hbox{\rm Spec}(S)$ with $S$ as in the Prop.\ref{semilocal-unique-semireg} proves the first assertion of the above corollary.
 The second may be deduced by an application of Theorem \ref{Thetaisom} 
 alongwith Prop.\ref{semilocal-unique-semireg}.

 Let $W$ be a rank 4 vector bundle on a scheme $X$, $w\in\Gamma(X,W)$ 
 a nowhere-vanishing global section and $\hbox{\rm Stab}_{w}\subset \hbox{\rm 
 GL}_{W}$ the stabiliser subgroupscheme of $w.$ Recall that the 
 natural inclusion $\azuWw\hookrightarrow\spazuWw$ is 
 $\hbox{\rm Stab}_{w}$-equivariant 
 (see ($\clubsuit$), page \pageref{clubsuit}).  
 When $X=\hbox{\rm Spec}(k)$ where $k$ is an algebraically closed field,
 there is a canonical $\hbox{\rm Stab}_{w}$-stratification of the 
 $k$-variety underlying $\hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_W$ as follows. 

\begin{theorem}\label{stratification}\ 
\begin{description}
\item[(1)] Let $k$ be a quadratically closed field and $X=\hbox{\rm Spec}(k).$ 
Then ${\mathscr Q}_3(X)$ has 4 elements which correspond 
 to (a) semiregular quadratic modules; (b) rank 2 quadratic modules i.e., 
 those that are not semiregular but which are regular on a two-dimensional 
 subspace; (c) nonzero perfect squares and (d) the zero form. If $V$ is a 3-dimensional
 vector space over $k$ and $\{e_1, e_2, e_3\}$ a $k$-basis for $V$, then 
 representatives for these 4 $\hbox{\rm GL}_{V}$-orbits in the space $ \hbox{\rm Quad}_{V}$ 
 of quadratic forms on $V$ can respectively be taken to be: 
 $\hbox{ (a) } q^{(1)}(\Sigma_{i=1}^3x_ie_i)=x_1x_2+x_3^2\hspace*{0.25cm};\hspace*{0.25cm}
 \hbox{ (b) } q^{(2)}(\Sigma_{i=1}^3x_ie_i)=x_1x_2\hspace*{0.25cm};\hspace*{0.25cm}
 \hbox{ (c) }q^{(3)}(\Sigma_{i=1}^3x_ie_i)=x_3^2\hspace*{0.25cm};\hspace*{0.25cm}
 \hbox{ (d) } q^{(4)}=0.$
\item[(2)] In addition to the hypotheses and notations of (1) above, 
 assume that $k$ is an algebraically closed field. Then the 
 four orbits $\hbox{\rm Quad}_{V}^{(i)}:=\hbox{\rm GL}_{V}\cdot q^{(i)}$ for 
 $1\leq i\leq 4$ form a stratification of the $k$-variety $\hbox{\rm Quad}_{V}$ 
 in the sense that we have  
 $$\overline{\hbox{\rm Quad}_{V}^{(1)}}=\hbox{\rm Quad}_{V}\hspace*{0.25cm}and\hspace*{0.25cm}
  \overline{\hbox{\rm Quad}_{V}^{(i+1)}}=\overline{\hbox{\rm Quad}_{V}^{(i)}}\hspace*{2mm}\backslash\hspace*{2mm}
\hbox{\rm Quad}_{V}^{(i)}\hspace*{0.25cm}for\hspace*{0.25cm}1\leq i\leq 3$$  and further we also have  
 $$\hbox{\rm Sing}(\overline{\hbox{\rm Quad}_{V}^{(i+1)}})=\overline{\hbox{\rm Quad}_{V}^{(i+1)}}\hspace*{2mm}\backslash\hspace*{2mm}
\hbox{\rm Quad}_{V}^{(i+1)}\hspace*{0.25cm}for\hspace*{0.25cm}1\leq i\leq 2$$
 unless the characteristic of 
 $k$ is 2 in which case $\overline{\hbox{\rm Quad}_{V}^{(3)}}$ is itself
 smooth  
 (the notation $\overline{T}$
 denotes the orbit closure and $\hbox{\rm Sing}(T)$ denotes the subset of 
 singular (non-smooth) points of $T$, each given the canonical reduced 
 induced closed subscheme structure).  
\item[(3)] Continuing with the notations and hypotheses of (2) above, 
 set $(W, w):=(\Lambda^{even}(V), 1).$ For 
 ease of notation denote $\hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_W$ by $\hbox{\rm SpAzu}$ and 
 $\hbox{\rm Stab}_{w}$ by $H.$
 Then the four orbits  $\hbox{\rm SpAzu}^{(i)}:=H\cdot\Theta(q^{(i)}, I_4)$
 for 
 $1\leq i\leq 4$ form a stratification of the $k$-variety $\hbox{\rm SpAzu}$ 
 in the sense that we have 
 $$\overline{\hbox{\rm SpAzu}^{(1)}}=\hbox{\rm SpAzu}\hspace*{0.25cm}and\hspace*{0.25cm}
  \overline{\hbox{\rm SpAzu}^{(i+1)}}=\overline{\hbox{\rm SpAzu}^{(i)}}\hspace*{2mm}\backslash\hspace*{2mm}
\hbox{\rm SpAzu}^{(i)}\hspace*{0.25cm}for\hspace*{0.25cm}1\leq i\leq 3$$  and further we also have  
 $$\hbox{\rm Sing}(\overline{\hbox{\rm SpAzu}^{(i+1)}})=\overline{\hbox{\rm SpAzu}^{(i+1)}}\hspace*{2mm}\backslash\hspace*{2mm}
\hbox{\rm SpAzu}^{(i+1)}\hspace*{0.25cm}for\hspace*{0.25cm}1\leq i\leq 2$$
 unless the characteristic of 
 $k$ is 2 in which case $\overline{\hbox{\rm SpAzu}^{(3)}}$ is itself smooth. 
\end{description}
\end{theorem}

%Shree Maataaravindaarpanam

\section{\label{sec2} Reduction of Theorems \ref{injectivity} \& \ref{lifting-of-isomorphisms} to the Free Case}
In this section, we prove Prop.\ref{transfer-to-lambda2} and 
 reduce the proof of Theorem \ref{injectivity} to 
 Theorem \ref{lifting-of-isomorphisms}. Thereafter we reduce the proof of 
 Theorem \ref{lifting-of-isomorphisms} to the case when $V$ is free.
 Our means towards this end are 
 several standard results which we recall below. 
 Bourbaki's tensor operations are described first, followed by  
 basic facts on tensoring quadratic bundles 
 with discriminant bundles, and thereafter by 
some facts about similitudes between
 quadratic bundles.   
 On the way we justify 
 the definitions of the functors ${\mathscr Q}_3^{sr}\hookrightarrow
 {\mathscr Q}_3$, 
 and the natural transformation $\hbox{\sf Witt}$
 introduced in \S \ref{sec1} above.
\begin{theorem}[Bourbaki's Tensor Operations, \S 9, Chap.9, \cite{Bourbaki};
 para.1.7, Chap.IV, \cite{Knus}]\label{Bourbaki} In the following, let
 $R$ be a commutative ring (with 1) and
 $V$ an $R$-module.
\begin{description}
\item[(1)]
Let $q:V\longrightarrow R$ be a quadratic form on $V$ and $f\in V^\vee:=\hbox{\rm Hom}_R(V, R)$ be a 
 functional on $V.$ Then there exists an $R$-linear endomorphism $t_f$ of 
 the tensor algebra $TV$ which is unique with respect to the first three
 of the following properties it satisfies:
\begin{description}
\item[(a)] $t_f(1)=0;$ 
\item[(b)]$t_f(x\otimes y)=f(x).y-x\otimes t_f(y)$ for every $x\in V$ and every $y\in TV;$
\item[(c)]If $J(q)$ is the two-sided ideal of $TV$ generated by the set 
 $\{x\otimes x-q(x).1\thinspace|\thinspace x\in V\}$, then $t_f(J(q))\subset J(q);$
\item[(d)]$t_f$ is homogeneous of degree -1 (for elements it does not 
 annihilate);
\item[(e)](Recall that the Clifford algebra of $q$ is $C(V,q):=TV/J(q)$).
 By (c) above, $t_f$ 
induces a $(\mathbb Z/2.\mathbb Z)$-antigraded endomorphism 
 $d_f^q:C(V,q)\longrightarrow C(V,q);$
\item[(f)]$t_f\circ t_f=0;$
\item[(g)]if $g\in V^\vee$ is also a functional, then $t_f\circ t_g + t_g\circ t_f=0;$
\item[(h)]if $\alpha\in \hbox{\rm End}_R(V)$, then  $t_f\circ T(\alpha)=T(\alpha)\circ t_{\alpha^*f}$ where $\alpha^*f\in \hbox{\rm End}_R(V)$ is defined by 
 $x\mapsto f(\alpha(x));$
\item[(i)]$t_f\equiv 0$ on the subalgebra of $TV$ generated by $\hbox{\rm kernel}(f).$ 
\end{description} 
\item[(2)] Let $q, q':V\longrightarrow R$ be two quadratic forms whose
 difference is the quadratic form $q_b$ induced by 
 a bilinear form $b\in\hbox{\rm Bil}_R(V):=\hbox{\rm Hom}_R(V\otimes_R V, R)$ 
i.e., $q'(x)-q(x)=q_b(x):=b(x,x)\thinspace \forall x\in V.$ Further, for any $x\in V$ denote by $b_x$ the functional on $V$ given by 
 $y\mapsto b(x,y).$ Then there exists an $R$-linear automorphism 
 $\Psi_b$ of 
 $TV$ which is unique with respect to the first three of the following 
 properties it satisfies: 
\begin{description}
\item[(a)]$\Psi_b(1)=1;$
\item[(b)]$\Psi_b(x\otimes y)=x\otimes\Psi_b(y)+t_{b_x}(\Psi_b(y))$ 
 for any $x\in V$ and any $y\in TV;$ 
\item[(c)]$\Psi_b(J(q'))\subset J(q);$ 
\item[(d)]by the previous property, $\Psi_b$ induces an isomorphism of $(\mathbb Z/2.\mathbb Z)$-graded 
 $R$-modules $$\psi_b:C(V,q')\cong C(V,q);$$ in particular, given a 
 quadratic form $q_1:V\longrightarrow R$, we may take a bilinear form $b_1$
 that 
 induces $q_1$ (i.e., such that $q_1(x)=q_{b_1}(x):=b_1(x,x)
\thinspace \forall x\in V$), and get a 
 $(\mathbb Z/2.\mathbb Z)$-graded linear isomorphism $\psi_{b_1}:
C(V,q_1=q_{b_1})\cong 
 C(V,0)=\Lambda(V);$ 
\item[(e)]$\Psi_b(T^{2n}V)\subset \oplus_{(i\leq n)}T^{2i}V$ and 
 $\Psi_b(T^{2n+1}V)\subset \oplus_{(\hbox{\rm\tiny odd }i\leq 2n+1)}T^{i}V;$
\item[(f)]in particular, for $x,x'\in V$, $\Psi_b(x\otimes x')
 =x\otimes x' + b(x,x').1_{TV}$ 
 so that for $\psi_b:C_0(V,q_b)\cong C_0(V,0)=\Lambda^{even}(V)$ we have 
 $\psi_b(x.x')=x\wedge x'+ b(x,x').1$ where $x.x'$ denotes the 
 product in $C(V,q_b).$ 
\item[(g)]if $f\in V^\vee$ and $t_f$ is given by (1) above,
 then $\Psi_b\circ t_f=t_f\circ \Psi_b;$
\item[(h)]if $b_i$ are bilinear forms on $V$, then $\Psi_{b_1+b_2}=\Psi_{b_1}\circ 
 \Psi_{b_2}$ and $\Psi_0=\hbox{\rm Identity on } {TV};$ 
\item[(i)]for any $\alpha\in\hbox{\rm End}_R(V)$, $\Psi_b\circ T(\alpha)=T(\alpha)\circ \Psi_{(b.\alpha)}$ where $(b.\alpha)(x,x'):=b(\alpha(x),\alpha(x'))\thinspace\forall x,x'\in V;$ 
\item[(j)]by property (h), 
one has a homomorphism of groups $(\hbox{\rm Bil}_R(V), +)
\longrightarrow (\hbox{\rm Aut}_R(TV), \circ):b\mapsto \Psi_b;$ the associative
 unital monoid $(\hbox{\rm End}_R(V), \circ)$ acts on $\hbox{\rm Bil}_R(V)$ 
 on the right by $b'\leadsto b'.\alpha$ and acts on the 
 left (resp. on the right) of $\hbox{\rm End}_R(TV)$ by $\alpha.\Phi:=T(\alpha)\circ\Phi$ (resp. by $\Phi.\alpha:=\Phi\circ T(\alpha)$), and the 
 homomorphism
 $b\mapsto \Psi_b$ satisfies $\alpha.\Psi_{(b.\alpha)}=\Psi_b.\alpha;$ 
 the group $\hbox{\rm Aut}_R(V)=\hbox{\rm GL}_R(V)$ acts on
  the left of $\hbox{\rm Bil}_R(V)$ by 
 $g.b: (x,x')\mapsto b(g^{-1}(x), g^{-1}(x'))$ and on the left of 
 $\hbox{\rm Aut}_R(TV)$ by conjugation via the natural 
 group homomorphism $\hbox{\rm GL}_R(V)\longrightarrow \hbox{\rm Aut}_R(TV): 
 g\mapsto T(g)$ i.e., $g.\Phi:=T(g)\circ \Phi\circ T(g^{-1})$, and the 
 homomorphism $b\mapsto \Psi_b$ is $\hbox{\rm GL}_R(V)$-equivariant: 
 $\Psi_{g.b}=g.\Psi_b.$ 
\end{description} 
\item[(3)] For a commutative $R$-algebra $S$ (with 1), let 
 $(q\otimes_R S), (q'\otimes_R S): (V\otimes_R S=:V_S)\longrightarrow S$ 
 be the quadratic $S$-forms induced from 
 the quadratic forms $q,q'$ of (2) above
 and $(b\otimes_R S)\in\hbox{\rm Bil}_S(V_S)$ the bilinear $S$-form induced 
 from the bilinear form $b$ of (2) above. 
 Then as a result  of the uniqueness properties (2a)--(2c) satisfied 
 by $\Psi_b$ and $\Psi_{(b\otimes_R S)}$, the $S$-linear automorphisms
 $(\Psi_b\otimes_R S)$ and $\Psi_{(b\otimes_R S)}$ 
may be canonically  identified. In particular,
 the $(\mathbb Z/2.\mathbb Z)$-graded $S$-linear isomorphism 
 $(\psi_{b}\otimes_R S):C(V_S,(q'\otimes_R\thinspace S))\cong C(V_S,(q\otimes_R S))$ 
 induced from $\psi_b$ of
 (2d) above may be canonically identified with $\psi_{(b\otimes_R S)}.$ 
\end{description}
\end{theorem}

\paragraph{Tensoring by Discriminant Bundles.} We recall that a bilinear form (resp. alternating form, resp. quadratic form) with values in ${\cal O}_X$ on a vector bundle $V$ over an open set $U\hookrightarrow X$ of the scheme $X$ is by 
 definition a section over $U$ of the vector bundle $\hbox{\rm Bil}_{V}$ (resp. of $\hbox{\rm Alt}^2_{V}$, resp. of $\hbox{\rm Quad}_{V}$), or equivalently, an 
 element of $\Gamma\left(U, \hbox{\rm Bil}_{\cal V}:=(T^2_{{\cal O}_X}({\cal V}))^\vee\right)$ (resp. 
 of $\Gamma\left(U, \hbox{\rm Alt}^2_{\cal V}:=(\Lambda^2_{{\cal O}_X}({\cal V}))^\vee\right)$, resp. 
 of   $\Gamma\left(U, \hbox{\rm Quad}_{\cal V}\right)$ ), where the sheaf 
 $\hbox{\rm Quad}_{\cal V}$---the (coherent locally-free) sheaf of ${\cal O}_X$-modules corresponding to the bundle $\hbox{\rm Quad}_{V}$ of quadratic forms 
 on $V$---is defined by the exactness of the following sequence: 
$$0\longrightarrow \hbox{\rm Alt}^2_{\cal V} \longrightarrow \hbox{\rm Bil}_{\cal V} \longrightarrow \hbox{\rm Quad}_{\cal V}\longrightarrow 0.$$
 In terms of the corresponding (geometric) vector bundles over $X$, the above 
 translates into  the following 
 sequence of morphisms of vector bundles, with the first one a closed immersion and the second one a Zariski locally-trivial principal 
$\hbox{\rm Alt}^2_V$-bundle: 
$$\hbox{\rm Alt}^2_{V} \hookrightarrow \hbox{\rm Bil}_{V} \twoheadrightarrow \hbox{\rm Quad}_{V}.$$ 
Given a quadratic form $q\in\Gamma(U, \hbox{\rm Quad}_{\cal V})$, recall 
 that  the usual 
 `associated' bilinear form $b_q\in\Gamma(U, \hbox{\rm Bil}_{\cal V})$ 
 is given on sections (over open subsets of $U$) by $v\otimes v'\mapsto q(v+v')-q(v)-q(v').$ This association in general does not lead to 
  a bijective correspondence
 between quadratic forms and symmetric bilinear forms (which is nevertheless 
 correct when $2\in \Gamma(X, {\cal O}_X^*)$  or equivalently when 
 the residue field of each point of $X$ is of characteristic $\neq 2$). 
 Given a (not-necessarily symmetric!) bilinear form
 $b$, we also have the induced quadratic form $q_b$ given on sections by 
 $v\mapsto b(v\otimes v).$ Since 
 a surjection of sheaves does 
 not necessarily imply a surjection on global sections,  
 a global quadratic form may not be induced 
 from a global bilinear form (unless we assume something more, for e.g., 
 that the scheme is affine, or more 
 generally  that the sheaf cohomology group $\hbox{\rm H}^1(X, \hbox{\rm Alt}^2_{\cal V})=0$).   


\begin{proposition}\label{tensoring-with-sym-bil-module}
Let $\vq$ be a quadratic bundle on $X$ and $(M,b)$ be a symmetric 
bilinear form bundle on $X$ (i.e., $M$ is a vector bundle on $X$ and $b$ is a global symmetric bilinear form on $M$ with values in ${\cal O}_X$). 
\begin{description}
\item[(1)] Then we can tensor $\vq$ with $(M,b)$ to get a unique 
 quadratic bundle $(V\otimes M, 
 q\otimes b).$ The quadratic form on $V\otimes M$ is given on 
 sections by $v\otimes m\mapsto q(v).b(m\otimes m)$
 and has associated bilinear 
 form $b_{q\otimes b}=b_q\otimes b.$
\item[(2)] When the rank of $M$ is 1
 i.e., $M$ is a line bundle, $(M,b)$ is regular iff $(M, q_b)$  is 
 semiregular iff $b:{\cal M}\otimes {\cal M}\cong {\cal O}_X$ is an 
 isomorphism (i.e., iff $(M,b)$ is a discriminant bundle). 
\item[(3)] Let $V$ be of odd rank and $(M, b)$ a discriminant bundle.
 Then $\vq$ is semiregular iff $\vq\otimes(M,b)=(V\otimes M, q\otimes b)$ is semiregular. 
\end{description}
\end{proposition} 

By taking an affine open cover for $X$ (which always exists by 
 definition since $X$ is a scheme),  
the proof of the assertions in the above proposition may be reduced
 to the case when $X$ is itself an affine scheme. In this case we may 
 further assume that $V$ and $M$ are themselves free. Then (1) 
 follows from para.8.4, Chap.I, \cite{Knus}, while (2) and (3) are 
 easy consequences of the definition and basic properties of semiregularity
 (\S 3, Chap.IV, \cite{Knus}). Note that statments (1) and (3)
 above justify respectively the definition 
 of the functor ${\mathscr Q}_3$ and its subfunctor ${\mathscr Q}_3^{sr}$ 
 introduced in \S \ref{sec1}. 

\begin{proposition}\label{tensoring-with-disc-module}
Let $V$ and $V'$ be vector bundles of the same rank on the scheme $X$, with 
 associated locally-free sheaves $\cal V$ and $\cal V'$ respectively. 
Let $\alpha:V'\cong V\otimes L$ be an isomorphism of bundles where 
 $(L,h)$ is a discriminant bundle on $X.$
\begin{description}
\item[(1)] Over any open subset
 $U\hookrightarrow X$, given a bilinear form 
 $b\in\Gamma\left(U, \hbox{\rm Bil}_{\cal V}\right)$, we can define a 
 bilinear form $b'\in\Gamma\left(U, \hbox{\rm Bil}_{\cal V'}\right)$ 
 using $\alpha$ and $h$ as follows: we let $b':=b\circ \zeta_{(\alpha, h)}$
 where $\zeta_{(\alpha,h)}:{\cal V'}\otimes{\cal V'}\cong {\cal V}\otimes{\cal V}$ is the ${\cal O}_X$-module isomorphism given
 by the composition of the following natural morphisms:
$${\cal V'}\otimes{\cal V'}\stackrel{\alpha\otimes\alpha\thinspace(\cong)}{\longrightarrow}{\cal V}\otimes{\cal L}\otimes{\cal V}\otimes{\cal L}\stackrel{\hbox{\rm\tiny SWAP}(2,3)\thinspace(\equiv)}{\longrightarrow}{\cal V}\otimes{\cal V}\otimes{\cal L}\otimes{\cal L}\stackrel{\hbox{\rm\tiny Id}\otimes h\thinspace(\cong)}{\longrightarrow}{\cal V}\otimes{\cal V}\otimes{\cal O}_X\stackrel{\hbox{\rm\tiny CANON}\thinspace(\equiv)}{\longrightarrow}{\cal V}\otimes{\cal V}.$$
Then the association $b\mapsto b'$ induces ${\cal O}_X$-linear isomorphisms 
 shown by vertical downward arrows in the following diagram (with 
 exact rows) making it commutative: 
$$\begin{CD}
0 @>>> \hbox{\rm Alt}^2_{\cal V} @>>> \hbox{\rm Bil}_{\cal V} @>>> \hbox{\rm Quad}_{\cal V} @>>> 0\\
 & & @V{\cong}VV  @V{\cong}VV  @V{\cong}VV & & \\
0 @>>> \hbox{\rm Alt}^2_{\cal V'} @>>> \hbox{\rm Bil}_{\cal V'} @>>> \hbox{\rm Quad}_{\cal V'} @>>> 0
\end{CD}$$ 
Therefore one also has the following commutative diagram of vector bundle 
 morphisms with the vertical downward arrows being isomorphisms: 
$$\begin{CD}
 \hbox{\rm Alt}^2_{V} @>{\hbox{\rm closed}}>{\hbox{\rm immersion}}> \hbox{\rm Bil}_{V} @>{\hbox{\rm locally}}>{\hbox{\rm trivial}}> \hbox{\rm Quad}_{V} \\
 @V{\cong}VV  @V{\cong}VV @V{\cong}VV \\
\hbox{\rm Alt}^2_{V'} @>{\hbox{\rm closed}}>{\hbox{\rm immersion }}> \hbox{\rm Bil}_{V'} @>{\hbox{\rm locally}}>{\hbox{\rm trivial}}> \hbox{\rm Quad}_{V'} 
\end{CD}$$ 
\item[(2)]
 Let $b\in\Gamma\left(X, \hbox{\rm Bil}_{\cal V}\right)$ be a global 
 bilinear form and let it induce $b'\in\Gamma\left(X, \hbox{\rm Bil}_{\cal V'}\right)$ via $\alpha$ and $h$  as defined in (1) above. Let $\Psi_b\in\hbox{\rm Aut}_{{\cal O}_X}(T{\cal V})$ (resp. $\Psi_{b'}\in\hbox{\rm Aut}_{{\cal O}_X}(T{\cal V'})$) be the $(\mathbb Z/2.\mathbb Z)$-graded linear  
 automorphism of the tensor algebra (with even elements given degree 0 and 
 odd elements given degree 1) induced by 
 $b$ (resp. by $b'$) defined locally (and hence globally) 
 as in (2e) of Theorem \ref{Bourbaki} above. Let $Z_{(\alpha, h)}:T^{even}_{{\cal O}_X}({\cal V'})\cong  T^{even}_{{\cal O}_X}({\cal V})$ be the ${\cal O}_X$-algebra isomorphism induced via the isomorphism $\zeta_{(\alpha,h)}:T^2_{{\cal O}_X}({\cal V'})\cong T^2_{{\cal O}_X}({\cal V})$ defined in (1) above. Then 
 the following diagram commutes:
$$\begin{CD}
T^{even}_{{\cal O}_X}({\cal V'}) @>{Z_{(\alpha, h)}}>{\cong}>T^{even}_{{\cal O}_X}({\cal V})\\
@V{\Psi_{b'}}V{\cong}V  @V{\cong}V{\Psi_b}V \\
T^{even}_{{\cal O}_X}({\cal V'}) @>{\cong}>{Z_{(\alpha, h)}}>T^{even}_{{\cal O}_X}({\cal V})
\end{CD}$$ 
thereby inducing (see (2d), Theorem \ref{Bourbaki}) the following commutative diagram of ${\cal O}_X$-linear 
 isomorphisms
$$\begin{CD}
C_0({\cal V'}, q_{b'}) @>{\hbox{\rm via }Z_{(\alpha, h)}}>{\cong}>C_0({\cal V}, q_b)\\
@V{\psi_{b'}}V{\cong}V @V{\cong}V{\psi_b}V \\
\Lambda^{even}_{{\cal O}_X}({\cal V'}) @>{\cong}>{\hbox{\rm via }Z_{(\alpha, h)}}>\Lambda^{even}_{{\cal O}_X}({\cal V})
\end{CD}$$ 
\item[(3)]
 Let $b$ and $b'$ be as in (2) above. Then $\alpha:V'\cong V\otimes L$ 
 induces an isometry of bilinear form bundles $\alpha:(V',b')\cong(V,b)\otimes(L,h)$ and also an isometry of the induced 
 quadratic bundles $\alpha: (V',q_{b'})\cong 
 (V, q_b)\otimes (L,h).$ Moreover, if we are just given a global quadratic 
 form $q$ on $V$ (resp. $q'$ on $V'$), then we may define the global quadratic 
 form $q'$ on $V'$ (resp. $q$ on $V$) via $q':=(q\otimes h)\circ \alpha$
 (resp. via $q:=(q'\otimes{(h^\vee)}^{-1})\circ{(\alpha\otimes L^{-1})}^{-1}$) 
 and  
 again $\alpha:(V',q')\cong (V,q)\otimes(L,h)$ becomes an
 isometry of quadratic 
 bundles.   
\end{description} 
\end{proposition}  
 

\paragraph{\bf Similitudes.} 
Let $(V,q)$ and $\vprqpr$ be quadratic bundles on the scheme $X.$ We recall 
 that a morphism of vector bundles $g:V\longrightarrow V'$ is called 
 a similitude (or similarity) 
 with multiplier $l\in\Gamma(X, {\cal O}_X^*)$, and we write 
 $g:\vq\thinspace{\cong}_{l}\thinspace\vprqpr$, if 
 $g$ is an isomorphism of vector bundles and the following diagram involving
 their  associated locally-free sheaves commutes (where $q$ and $q'$ are 
 considered as morphisms of sheaves of {\em sets}): 
$$\begin{CD}
{\cal V} @>{g}>{\cong}> {\cal V'}\\
@V{q}VV @VV{q'}V\\
{\cal O}_X @>{\cong}>{\hbox{\rm\tiny MULT. BY }l}> {\cal O}_X
\end{CD}$$
Moreover, when the multiplier $l=1$, we write 
 $g:\vq\cong\vprqpr$ and call $g$ an isometry. 
\begin{proposition}\label{simil-induces-iso-of-even-cliff}
Let $g:\vq\thinspace{\cong}_{l}\thinspace\vprqpr$
 be a similarity with multiplier
 $l\in\Gamma(X, {\cal O}_X^*).$ 
\begin{description}
\item[(1)] There exists a unique isomorphism of ${\cal O}_X$-algebra bundles
$C_0(g,l):C_0(V,q)\cong C_0(V',q')$ such that  
 $C_0(g,l)(v.v')=l^{-1}g(v).g(v')$ on sections.
\item[(2)] There exists a unique vector bundle isomorphism 
$C_1(g,l):C_1(V,q)\cong C_1(V', q')$ such that
\begin{description}
\item[(a)] 
 $C_1(g,l)(v.c)=g(v).C_0(g,l)(c)$ and 
\item[(b)] $C_1(g,l)(c.v)=C_0(g,l)(c).g(v)$ 
\end{description} 
for any section $v$ of $V$ and any section $c$ of $C_0(V,q).$
 Thus $C_1(g,l)$ is $C_0(g,l)$-semilinear.  
\item[(3)]
 If $g_1:\vprqpr\thinspace{\cong}_{l_1}\thinspace(V'',q'')$ is another 
  similarity with multiplier $l_1$, then the composition 
 $g_1\circ g:\vq\thinspace{\cong}_{ll_1}\thinspace(V'',q'')$ is 
 also a similarity with multiplier given by the product of 
 the multipliers. Further $C_i(g_1\circ g,ll_1)=
 C_i(g_1,l_1)\circ C_i(g,l)$ for $i=0,1.$ 
\end{description}
\end{proposition}
\noindent 
The proof of the above follows from the case of an affine scheme which is 
 established in Prop.7.1.1, Chap.IV, \cite{Knus}. 
 A local computation 
 shows that tensoring by a discriminant bundle amounts to (locally) applying 
 a similarity. In this case also one gets a global
  isomorphism of even Clifford 
 algebras: 

\begin{proposition}\label{isom-covq-covlqh}
Let $\vq$ be a quadratic bundle on a scheme $X$ and $(L,h)$ be a 
 discriminant bundle. There exists a unique isomorphism of algebra bundles 
 $\gamma_{(L,h)}:C_0\left(\vq\otimes(L,h)\right)\cong\covq$ 
 given by $\gamma_{(L,h)}\left((v\otimes\lambda).(v'\otimes \lambda')\right)=h(\lambda\otimes \lambda')v.v'$ for any  sections $v,v'$ of $V$ and $\lambda,\lambda'$ of $L.$  
\end{proposition}
\noindent 
The proof of the above may be reduced to the case of an affine scheme, 
 in  which case, we may further assume that $L$ is free, so that tensoring
 by $(L,h)$ is the same as applying a similarity. In this case, the 
 result follows from Prop.\ref{simil-induces-iso-of-even-cliff}. 
 The above result justifies the definition of the natural transformation 
 $\hbox{\sf Witt}$ introduced in \S \ref{sec1}. 

\paragraph{Proof of Prop.\ref{transfer-to-lambda2}:}
Start with an isomorphism of algebra-bundles $\phi:\covq\cong\covprqpr.$
Let ${\{U_i\}}_{i\in I}$ be an affine open covering of $X$ (which may 
also be chosen so as to trivialise
 the rank 3 vector bundles $V$ and $V'$ if needed). Choose bilinear forms 
 $b_i\in\Gamma(U_i, \hbox{\rm Bil}_{V})$ and $b'_i\in\Gamma(U_i, \hbox{\rm Bil}_{V'})$ such that $q|U_i=q_{b_i}$ and $q'|U_i=q_{b'_i}$ for each $i\in I.$ 
 By (2d), Theorem \ref{Bourbaki}, we have  isomorphisms of vector bundles 
$\psi_{b_i}$ and 
 $\psi_{b'_i}$, which preserve 1 by (2a) of the same Theorem, 
  and we define the isomorphism of vector bundles $\phi_{\Lambda_i^{ev}}$ 
 so as to make the following diagram commute: 
$$\begin{CD}
C_0(V|U_i, q|U_i) @>{\phi|U_i}>{\cong}> C_0(V'|U_i, q'|U_i)\\
@V{\psi_{b_i}}V{\cong}V @V{\cong}V{\psi_{b'_i}}V\\
\Lambda^{even}(V|U_i) @>{\cong}>{\phi_{\Lambda_i^{ev}}}> \Lambda^{even}(V'|U_i) 
\end{CD}$$
The linear isomorphism $\phi_{\Lambda_i^{ev}}$ preserves 1  and 
 therefore it induces a linear isomorphism from $\Lambda^2(V|U_i)$ to 
 $\Lambda^2(V'|U_i)$, which we denote by  ${({\phi_{\Lambda^2}})}_i.$ 
Observe that ${({\phi_{\Lambda^2}})}_i$ is independent of the choice of 
 the bilinear forms $b_i$ and $b'_i.$ For, replacing these respectively 
 by $\widehat{b_i}$ and $\widehat{b'_i}$, it follows from (2f),
 Theorem \ref{Bourbaki}, that $\psi_{b_i}\circ{(\psi_{\widehat{b_i}})}^{-1}$ 
 (resp. $\psi_{b'_i}\circ{(\psi_{\widehat{b'_i}})}^{-1}$) followed by the 
 canonical projection onto $\Lambda^2(V|U_i)$ (resp. onto $\Lambda^2(V'|U_i)$) 
 is the same as the projection itself. By this observation, it is also 
 clear that the isomorphisms ${\{{({\phi_{\Lambda^2}})}_i\}}_{i\in I}$
 agree on (any open affine subscheme of, and hence on all of)
 any intersection $U_i\cap U_j.$ Therefore they glue to give a global 
 isomorphism of vector bundles $\phi_{\Lambda^2}:\Lambda^2(V)\cong\Lambda^2(V').$ {\bf Q.E.D, Prop.\ref{transfer-to-lambda2}.}

 \paragraph{Reduction of Proof of Theorem \ref{injectivity} to 
 Theorem \ref{lifting-of-isomorphisms}.} We start with an isomorphism of algebra-bundles $\phi:\covq\cong\covprqpr$, construct the isomorphism of vector
 bundles $\phi_{\Lambda^2}:\Lambda^2(V)\cong\Lambda^2(V')$ and keep the notations introduced in the proof of Prop.\ref{transfer-to-lambda2}. 
 Firstly we deduce a linear isomorphism 
  $\hbox{\rm det}{({(\phi_{\Lambda^2})}^\vee)}^{-1}:
\hbox{\rm det}({(\Lambda^2(V))}^\vee)\cong
\hbox{\rm det}({(\Lambda^2(V'))}^\vee).$  Since $V$ and 
 $V'$ are of rank 3,  
  there are canonical isomorphisms $\eta:\Lambda^2(V)\equiv V^\vee\otimes\hbox{\rm det}(V)$ and $\eta':\Lambda^2(V')\equiv (V')^\vee\otimes\hbox{\rm det}(V').$
It follows therefore that if we set $L:=\hbox{\rm det}(V')\otimes{(\hbox{\rm det}(V))}^{-1}$ then we get a discriminant line bundle $(L,h)$ and a vector 
 bundle isomorphism $\alpha:V'\cong V\otimes L.$ Now for each $i\in I$, 
 the bilinear form 
 $b_i\in\Gamma(U_i, \hbox{\rm Bil}_{V})$ induces, via $\alpha|U_i$ and $(L|U_i,h|U_i)$ and (1), Prop.\ref{tensoring-with-disc-module}, a bilinear form 
 $b''_i\in\Gamma(U_i, \hbox{\rm Bil}_{V'}).$ By (3) of the same Proposition, 
 over each $U_i$  we get an isometry  of bilinear form bundles 
$\alpha|U_i:(V'|U_i, b''_i)\cong (V|U_i, b_i)\otimes (L|U_i, h|U_i)$ 
 and also an isometry of quadratic bundles
$\alpha|U_i:(V'|U_i, q_{b''_i})\cong (V|U_i, q_{b_i}=q|U_i)\otimes (L|U_i, h|U_i).$ On the other hand, by an assertion in (3), Prop.\ref{tensoring-with-disc-module}, we could also 
 define the global
 quadratic bundle $(V',q'')$ using $(V,q)$, $\alpha$ and $(L,h)$, 
 so that we have an isometry of quadratic bundles 
$\alpha:(V',q'')\cong(V,q)\otimes(L,h).$ It follows therefore that 
 the $q_{b''_i}$ glue to give $q''.$ 
Notice that in general 
 the $b''_i$ (resp. the $b_i$) need not glue to give a global bilinear 
 form $b''$ (resp. $b$) such that $q_{b''}=q''$ (resp. $q_b=q$). 
 By (1), Prop.\ref{simil-induces-iso-of-even-cliff}, there exists 
 a  unique isomorphism of ${\cal O}_X$-algebra bundles
$C_0(\alpha,1):C_0(V',q'')\cong C_0\left((V,q)\otimes(L,h)\right)$ and by 
 Prop.\ref{isom-covq-covlqh} 
 we have a unique isomorphism of algebra bundles 
 $\gamma_{(L,h)}:C_0\left(\vq\otimes(L,h)\right)\cong\covq.$
 Therefore the composition 
 of the following sequence of isomorphisms of algebra bundles on X
$$C_0(V',q'')\stackrel{C_0(\alpha,1)\thinspace{(\cong)}}{\longrightarrow}
C_0\left((V,q)\otimes(L,h)\right)\stackrel{\gamma_{(L,h)}\thinspace{(\cong)}}{\longrightarrow}
\covq\stackrel{\phi\thinspace{(\cong)}}{\longrightarrow}\covprqpr$$
is an element of $\hbox{\rm Iso}[C_0(V',q''),C_0(V',q')]$, which, granting 
 Theorem \ref{lifting-of-isomorphisms}, is induced by a similarity in  
 $\hbox{\rm Sim}[(V',q''),(V',q')].$ Therefore, we would have that 
 $(V',q'')$ and $(V',q')$ are globally similar (which means that 
 they differ by a discriminant bundle with underlying line bundle being trivial), and 
 this combined with the fact that 
   $(V,q)$ and $(V',q'')$ are isometric 
 upto the discriminant bundle $(L,h)$ (by the construction above),
 we would have that $(V,q)$ and $(V',q')$ also differ by a discriminant 
 bundle. Therefore the proof of Theorem \ref{injectivity} reduces 
 to the proof of 
 Theorem \ref{lifting-of-isomorphisms}. 

\paragraph{Reduction of Theorem \ref{lifting-of-isomorphisms} to the Free Case:}\label{reduction-to-free-case} For a similarity $g$ with multiplier $l$, we have $C_0(g,l)$
 given by (1), Prop.\ref{simil-induces-iso-of-even-cliff}, so that the map $\simvqvqpr \longrightarrow \isomcovqcovqpr$ mentioned in 
 the statement is the natural $g\mapsto C_0(g,l).$
  We shall  
 show (locally in the Zariski topology and hence globally)
 the equality $\hbox{\rm det}\left({(C_0(g,l))}_{\Lambda^2}\right)
=l^{-3}\hbox{\rm det}^2(g)$
 so that $\isomvqvqpr$ and $\sisomvqvqpr$ are respectively mapped 
 into $\isomcovqcovqprprime$ and $\sisomcovqcovqpr$ as claimed.
 This equality will be 
verified in Lemma \ref{locally-det-coglambda-equals-detgbylcube}
 following  the proof of 
the surjectivity of $\simvqvqpr \longrightarrow \isomcovqcovqpr.$ 
  We start with an isomorphism of algebra-bundles $\phi:\covq\cong\covqpr$,
 which by the above discussion leads to the automorphism of vector
 bundles $\phi_{\Lambda^2}:\Lambda^2(V)\cong\Lambda^2(V).$
Firstly, define the global bundle automorphism  $g'\in\hbox{\rm GL}\left(V\otimes{(\hbox{\rm det}(V))}^{-1}\right)$ so that the following diagram commutes:
$$\begin{CD}
{(\Lambda^2(V))}^\vee @>{({({\phi_{\Lambda^2}})}^\vee)}^{-1}>{\cong}> 
{(\Lambda^2(V))}^\vee\\
@V{(\eta^\vee)}^{-1}V{\equiv}V @V{\equiv}V{(\eta^\vee)}^{-1}V\\
V\otimes{(\hbox{\rm det}(V))}^{-1} @>{\cong}>g'> V\otimes{(\hbox{\rm det}(V))}^{-1}
\end{CD}$$ where 
 $\eta:\Lambda^2(V)\equiv V^\vee\otimes\hbox{\rm det}(V)$
 is the canonical isomorphism (since $V$ is of rank 3). Now let 
 $g\in\hbox{\rm GL}(V)\stackrel{\cong}{\longleftarrow}\hbox{\rm GL}
(V\otimes{(\hbox{\rm det}(V))}^{-1})$ be the image of $g'$ i.e., 
 the image of $g'\otimes \hbox{\rm det}(V)$ under the canonical identification 
 $\hbox{\rm GL}(V\otimes{(\hbox{\rm det}(V))}^{-1}\otimes\hbox{\rm det}(V))
\equiv \hbox{\rm GL}(V).$ 
 Next, let $l\in\Gamma(X, {\cal O}_X^*)$ be a global section such that 
 $\gamma(l):=(l^3).\hbox{\rm det}(\phi_{\Lambda^2})$ has a square root in 
 $\Gamma(X, {\cal O}_X^*).$ For example, we have the following special 
 cases when this is true: 
\begin{description}
\item[Case 1.]  \label{cases}
 If $\hbox{\rm det}(\phi_{\Lambda^2})$ is itself a square, 
set $l:=1.$ If further $\hbox{\rm det}(\phi_{\Lambda^2})=1$, set 
 $\sqrt{\gamma(l)}=1$, otherwise let $\sqrt{\gamma(l)}$ denote any 
 fixed square root of $\hbox{\rm det}(\phi_{\Lambda^2}).$
\item[Case 2.] If $\hbox{\rm det}(\phi_{\Lambda^2})$ is not 
 a square, given an integer $k$,
 take $l={(\hbox{\rm det}(\phi_{\Lambda^2}))}^{2k+1}$ and
 let $\sqrt{\gamma(l)}$ denote any fixed square root of 
 ${(\hbox{\rm det}(\phi_{\Lambda^2}))}^{6k+4}.$
\end{description}
 For each integer 
 $k$, we now associate to $\phi$ the element \label{glphi-def}
 $g_l^\phi:=(l^{-1}\sqrt{\gamma(l)}\thinspace)g$ with 
 $g$ as defined above.  We shall 
 show the following locally for the Zariski toplogy on $X$  
 (more precisely, for each open subscheme of $X$ over which 
 $V$ is free): 
\begin{description}
\item[(1)]  
 that $g_l^\phi$ is a 
 similitude from $q$ to $q'$ with multiplier $l$ (Lemma
 \ref{glphi-simil-q-qpr-with-mult-l}); 
\item[(2)] that $g_l^\phi$  
 induces $\phi$ i.e., with the notations of (1), 
Prop.\ref{simil-induces-iso-of-even-cliff}, 
 that $C_0(g_l^\phi, l)=\phi$ (Lemma 
 \ref{glphi-induces-phi}); 
\item[(3)] that $\hbox{\rm det}(g_l^\phi)=\sqrt{\gamma(l)}$
 so that 
 $\hbox{\rm det}^2(g_l^\phi)
=\hbox{\rm det}(\phi_{\Lambda^2})$ 
when $\hbox{\rm det}(\phi_{\Lambda^2})$ is itself a square
 (Lemma \ref{B-versus-Bglphi}) and 
\item[(4)]
 that the  map $\sisomvqvqpr\longrightarrow\sisomcovqcovqpr$ is injective (Lemma \ref{bijection-of-special-isomorphisms}).
\end{description}
 It would follow then that these statements are also true globally. The 
 maps $s_{2k+1}:\phi\mapsto g_l^\phi$ with $l$ as in Case 2 and 
 $s': \phi\mapsto g_l^\phi$ with $l$ as in Case 1 will then give the 
 sections to the maps (which would imply their surjectivities) 
as mentioned in Theorem \ref{lifting-of-isomorphisms}.
 But these maps 
  are  not necessarily multiplicative since a computation reveals that 
 if $\phi_i\in\hbox{\rm Iso}[C_0(V,q_i),C_0(V,q_{i+1})]$ 
is associated to $g_{l_i}^{\phi_i}\in\hbox{\rm Sim}[(V,q_i),(V,q_{i+1})]$, 
 and $\phi_2\circ\phi_1$ to $g_{l_{21}}^{\phi_2\circ\phi_1}$, then 
 $g_{l_{21}}^{\phi_2\circ\phi_1}=\delta g_{l_2}^{\phi_2}\circ 
g_{l_1}^{\phi_1}$ for 
 $\delta\in\mu_2(\Gamma(X, {\cal O}_X^*))$ because of the ambiguity in 
 the initial global choices of square roots for 
 $\gamma(l_i)$ and $\gamma(l_{21}).$  However 
 this can be remedied as follows. For any given 
 $\phi\in\hbox{\rm Iso}[C_0(V,q),C_0(V,q')]$, irrespective of whether or not
 $\hbox{\rm det}(\phi_{\Lambda^2})$ is a square,
 take 
$$l={(\hbox{\rm det}(\phi_{\Lambda^2}))}^{2k+1}, 
 \gamma(l)=l^3\hbox{\rm det}(\phi_{\Lambda^2}),
  \sqrt{\gamma(l)}:={(\hbox{\rm det}(\phi_{\Lambda^2}))}^{3k+2}\hbox{ and }
 s_{2k+1}^{+}(\phi):=g_l^\phi=\left(l^{-1}\sqrt{\gamma(l)}\right)g.$$
 Then it is clear that each $s_{2k+1}^{+}$ is multiplicative with the 
 properties as claimed in the statement.
 We thus reduce the proof of Theorem \ref{lifting-of-isomorphisms} 
 to the case when the rank 3 vector bundle 
 $V$ is free. This will be taken up next in \S \ref{sec3}. 



%Shree Maataaravindaarpanam

\section{\label{sec3} Investigation of $\Theta$\thinspace; Proof of Theorem 
\ref{lifting-of-automorphisms}}
In this section, we conclude the proofs of Theorems \ref{injectivity} and  
\ref{lifting-of-isomorphisms} which were begun in \S \ref{sec2}  and  also 
 prove Theorem \ref{lifting-of-automorphisms}. As means to these ends, 
 we carry out two explicit computations. Firstly we compute 
the isomorphism $\Theta$ of Theorem \ref{Thetaisom}. This provides us with
 the multiplication table of every specialised algebra structure on any fixed
  free rank 4 vector bundle with fixed unit which is part of a global basis (Theorem \ref{mult-table-wrt-Theta} below). 
 This result  will also be used in \S \ref{sec4}   
 in the proof of Theorem \ref{Thetaisom-Upsilonisom}.
It turns out that $\Theta$ is not equivariant 
 with respect to $\hbox{\rm GL}_{V}$, but nevertheless satisfies a `twisted' 
 form of equivariance (Theorem \ref{tequiv}). 
 Secondly, we explicitly  compute the algebra bundle 
 isomorphism $C_0(g,l):C_0(V,q)\cong C_0(V,q')$ of 
 (1), Prop.\ref{simil-induces-iso-of-even-cliff} induced by a similarity $g:\vq\thinspace{\cong}_{l}\thinspace
\vqpr$
with multiplier $l\in\Gamma(X, {\cal O}_X^*)$ in the case when 
 $V$ is free of rank 3 (Theorem \ref{b-co-phi-lambda-indep-of-q}).


\paragraph{The Action of $\hbox{\rm GL}_{V}.$}
Let $V$ be a vector bundle over a scheme $X$ with associated locally-free 
 sheaf $\cal V.$ The $X$-smooth $X$-groupscheme $\hbox{\rm GL}_{V}$ acts
 naturally on the left on  
 the sheaves $\hbox{\rm Alt}^2_{\cal V}$, $\hbox{\rm Bil}_{\cal V}$ and 
 $\hbox{\rm Quad}_{\cal V}$ of alternating, bilinear and quadratic forms on 
 $\cal V.$ Namely, for $U\hookrightarrow X$ an open subscheme, and for 
 $b\in\Gamma(U, \hbox{\rm Bil}_{\cal V})$ (resp. $a\in\Gamma(U, \hbox{\rm Alt}^2_{\cal V})$, resp. $q\in\Gamma(U, \hbox{\rm Quad}_{\cal V})$), and for 
 $g\in\Gamma(U,\hbox{\rm GL}_{V})=\hbox{\rm GL}(V|U)$, the corresponding 
 form of the same type $g.b$ (resp. $g.a$, resp. $g.q$) is defined on 
 sections (over open subsets of $U$) by $(g.b)(v,v'):=b(g^{-1}(v), g^{-1}(v'))$
 (resp.$(g.a)(v,v'):=a(g^{-1}(v), g^{-1}(v'))$, resp. 
 $(g.q)(v):=q(g^{-1}(v))$). 
 It is immediate that the following short-exact-sequence of sheaves, 
 indicated in the discussion before Prop.\ref{tensoring-with-sym-bil-module},
 is  equivariant with respect to this action:
$$\label{spadesuit}(\spadesuit)\hspace*{8mm} 0\longrightarrow \hbox{\rm Alt}^2_{\cal V} \longrightarrow \hbox{\rm Bil}_{\cal V} \longrightarrow \hbox{\rm Quad}_{\cal V}\longrightarrow 0.$$
Equivalently, the $X$-groupscheme $\hbox{\rm GL}_{V}$  acts on the 
 corresponding geometric vector bundles such that  both of the $X$-morphisms 
 of $X$-vector bundles in the  following sequence are
  $\hbox{\rm GL}_{V}$-equivariant: 
 $$\hbox{\rm Alt}^2_{V} \hookrightarrow \hbox{\rm Bil}_{V} \twoheadrightarrow \hbox{\rm Quad}_{V}.$$ Notice that it is one and the same thing to require that 
  $\hbox{\rm GL}(V|U)\ni g:(V|U, q)\thinspace{\cong}_{l}
\thinspace(V|U, q')$ be a 
 similitude with multiplier $l\in\Gamma(U, {\cal O}_X)$, and to 
 require that $g.q={l}^{-1}q'.$ 

\paragraph{Definition of the isomorphism $\Theta$.}\label{thetadef}
We briefly recall the definition of $\Theta$
 from Part A of \cite{tevb-paper1}. We keep the notations introduced just 
 before Theorem \ref{Thetaisom}; for ease of notation, 
 the pullback of a section $s$ (of a vector bundle or its associated sheaf)
 is denoted by $s^\circ.$  Since $V$ is free of rank 3 on $X$, we 
  choose an identification ${\cal V}\equiv
{\cal O}_X.e_1\oplus{\cal O}_X.e_2\oplus{\cal O}_X.e_3.$
This gives the identification of the dual bundle as ${\cal V}^\vee\equiv
{\cal O}_X.f_1\oplus{\cal O}_X.f_2\oplus{\cal O}_X.f_3$ (defined uniquely 
 by $f_i(e_j)=\delta_{ij}$, the Kronecker delta). Therefore the dual of 
 the sheaf 
 of quadratic forms on $V$, which is 
${(\hbox{\rm Quad}_{\cal V})}^\vee:={(\hbox{\rm Bil}_{\cal V}/\hbox{\rm Alt}^2_{\cal V})}^\vee={({(T^2{\cal V})}^\vee/{(\Lambda^2{\cal V})}^\vee)}^\vee$
 has global ${\cal O}_X$-basis
given by $\{e_i\otimes e_i; (e_i\otimes e_j+e_j\otimes e_i)\}.$
 This leads to 
 an identification of the associated sheaf of 
 symmetric algebras $\hbox{\rm Sym}_{{\cal O}_X}(\hbox{\rm Quad}_{\cal V}^\vee)\equiv
 {\cal O}_X[Y_1, Y_2, Y_3, Y_{12}, Y_{13}, Y_{23}]$, where 
 $e_i\otimes e_i\equiv Y_i$ and $e_i\otimes e_j+e_j\otimes e_i\equiv Y_{ij}$,
 and therefore $\hbox{\rm Quad}_{V}:=
 \hbox{\bf Spec}\left(\hbox{\rm Sym}_{{\cal O}_X}(\hbox{\rm Quad}_{\cal V}^\vee)\right)\equiv
 \hbox{\bf Spec}({\cal O}_X[Y_1, Y_2, Y_3, Y_{12}, Y_{13}, Y_{23}])={\mathbb A}^6_X.$  Consider the 
 universal quadratic bundle  $({\mathbf V},{\mathbf q})$ where 
 $\mathbf V$ is the pullback of $V$ by $\hbox{\rm Quad}_{V}\longrightarrow X.$
 The universal quadratic form $\mathbf q$ is given by 
 $(x_1, x_2, x_3)\mapsto \Sigma_i Y_i.(x_i)^2 + \Sigma_{i<j} Y_{ij}.x_i.x_j$
 and moreover the global bilinear form on $\mathbf V$ given by 
 $${\mathbf b}({\mathbf q}): \left((x_1, x_2, x_3), (x'_1, x'_2, x'_3)\right)\mapsto
  \Sigma_i Y_i.x_i.x'_i + Y_{12}.x_2.x'_1+Y_{23}.x_3.x'_2+Y_{13}.x_1.x'_3$$
 induces 
 $\mathbf q$ (the bilinear form
 `associated in the usual sense' to $\mathbf q$, viz. $b_{\mathbf q}$ is 
 not $\mathbf b({\mathbf q})$ but in fact its symmetrisation). 
  Therefore, by (2d), Theorem \ref{Bourbaki}, we get an 
 isomorphism of vector bundles $\psi_{{\mathbf b}({\mathbf q})}:C_0({\mathbf V}, {\mathbf q}=q_{{\mathbf b}({\mathbf q})})\cong \Lambda^{even}({\mathbf V})=:{\mathbf W}$ which, according 
 to (2a) and (2f) of the same Theorem,  carries  the ordered
 Poincar\'e-Birkhoff-Witt basis $\{1;e^\circ_1.e^\circ_2, e^\circ_2.e^\circ_3, e^\circ_3.e^\circ_1\}$  onto 
 the corresponding ordered basis of the even exterior algebra (=even 
 Clifford algebra of the zero quadratic form on $\mathbf V$) given by 
 $\{w^\circ=1^\circ=1;e^\circ_1\wedge e^\circ_2, e^\circ_2\wedge e^\circ_3, e^\circ_3\wedge e^\circ_1\}.$ The choices $e^\circ_3.e^\circ_1$ and $e^\circ_3\wedge e^\circ_1$ instead of the usual $e^\circ_1.e^\circ_3$ and $e^\circ_1\wedge e^\circ_3$ are deliberate---for example, $\psi_{{\mathbf b}({\mathbf q})}$ would 
 carry $\{1;e^\circ_1.e^\circ_2, e^\circ_2.e^\circ_3, e^\circ_1.e^\circ_3\}$
 onto  $\{w^\circ=1^\circ=1;e^\circ_1\wedge e^\circ_2, e^\circ_2\wedge e^\circ_3, e^\circ_1\wedge e^\circ_3 + Y_{13}.w^\circ\}$ which depends 
 on $Y_{13}.$
 Thus 
 the even Clifford algebra bundle $C_0({\mathbf V},
 {\mathbf q}=q_{{\mathbf b}({\mathbf q})})$
 induces  via $\psi_{{\mathbf b}({\mathbf q})}$ a
 $w^\circ$-unital algebra structure on the pullback bundle $\mathbf W$
 of $W:=\Lambda^{even}(V)$ (where $w$ corresponds to 1
 in ${\Lambda^{even}(V)}$).
 But by definition, this algebra structure corresponds precisely
 to an $X$-morphism $\theta:\hbox{\rm Quad}_{V}
\longrightarrow \hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_W.$ The isomorphism 
 $\Theta$ is now given by the composition of the following $X$-morphisms (cf.
Def.5.2, Part A, \cite{tevb-paper1}):
$$\hbox{\rm Quad}_{V}\times_X\hbox{\sf L}_w\stackrel{\theta\times\hbox{\rm\tiny ID}}{\longrightarrow}\hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_W\times_X\hbox{\sf L}_w\stackrel{\hbox{\rm\tiny SWAP}\thinspace(\cong)}{\longrightarrow}
\hbox{\sf L}_w\times_X\hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_W\stackrel{\hbox{\rm\tiny ACTION}}{\longrightarrow}\hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_W.$$ 
 The association of $\mathbf q$ with ${\mathbf b}({\mathbf q})$
 also defines a splitting
 of the exact sequence  $(\spadesuit)$  above,
 so that more generally,
 given a valued point $q\in(\hbox{\rm Quad}_{V})(T)$, we may 
 associate uniquely a valued point $b(q)\in(\hbox{\rm Bil}_{V})(T)$
 which induces it. That this association is not $\hbox{\rm GL}_V$-equivariant 
 is reflected in the lack of equivariance of the isomorphism $\Theta$
 (Theorem \ref{tequiv}). 


\begin{theorem}\label{mult-table-wrt-Theta} Let $T$ be an $X$-scheme. Let $q$
 be
a $T$-valued point of $\hbox{\rm Quad}_{V}\equiv{\mathbb A}^6_X$ which is
identified uniquely with a 6-tuple $(\lambda_1,\lambda_2,\lambda_3,
\lambda_{12},\lambda_{13},\lambda_{23})$ corresponding to the 
 quadratic form $(x_1,x_2,x_3)\mapsto \Sigma_i \lambda_i x_i^2 + \Sigma_{i<j}\lambda_{ij}x_ix_j.$ Let $\underline{t}$ be a  $T$-valued point  
  of $\hbox{\sf L}_w\equiv({\mathbb A}^3_X, +)$ which 
 is identified uniquely with a 3-tuple $(t_1, t_2, t_3)$ that 
 corresponds to the $T$-valued point of $\hbox{\rm Stab}_{w}$ given by the 
 $(4\times 4)$-matrix
 \begin{displaymath}
 \left(\begin{array}{c|ccc}
1 & t_1 & t_2 & t_3 \\
\hline
0 & & I_3 & 
\end{array}
\right)
 \end{displaymath} 
where $I_3$ is the $(3\times 3)$-identity matrix. Then the multiplication 
 table for the specialised algebra structure $\Theta(q,\underline{t})=
\underline{t}.\theta(q)$ on the pullback bundle $W_T$ with 
 unit $w^\circ=w_T$ is given as follows (in terms of the global basis 
 $\{w^\circ=1^\circ=1\hspace*{1mm};\hspace*{1mm}
\epsilon^\circ_1:=e^\circ_1\wedge e^\circ_2\hspace*{1mm},\hspace*{1mm}
\epsilon^\circ_2:=e^\circ_2\wedge e^\circ_3\hspace*{1mm},\hspace*{1mm}
\epsilon^\circ_3:=e^\circ_3\wedge e^\circ_1\}$ induced from that 
of $W=\Lambda^{even}(V):$)  
\begin{enumerate}
\item[$\bullet$] $\epsilon^\circ_1.\epsilon^\circ_1= (t_1\lambda_{12}-\lambda_1\lambda_2-t_1^2).w^\circ+(\lambda_{12}-2t_1).\epsilon^\circ_1\thinspace;$
\item[$\bullet$] $\epsilon^\circ_2.\epsilon^\circ_2= (t_2\lambda_{23}-\lambda_2\lambda_3-t_2^2).w^\circ+(\lambda_{23}-2t_2).\epsilon^\circ_2\thinspace;$
\item[$\bullet$] $\epsilon^\circ_3.\epsilon^\circ_3= (t_3\lambda_{13}-\lambda_1\lambda_3-t_3^2).w^\circ+(\lambda_{13}-2t_3).\epsilon^\circ_3\thinspace;$
\item[$\bullet$] $\epsilon^\circ_1.\epsilon^\circ_2=(\lambda_2\lambda_{13}-\lambda_2t_3-t_1t_2).w^\circ -t_2\epsilon^\circ_1-t_1\epsilon^\circ_2-\lambda_2\epsilon^\circ_3
\thinspace;$
\item[$\bullet$] $\epsilon^\circ_2.\epsilon^\circ_3=(\lambda_3\lambda_{12}-\lambda_3t_1-t_2t_3).w^\circ -\lambda_3\epsilon^\circ_1-t_3\epsilon^\circ_2-t_2\epsilon^\circ_3
\thinspace;$
\item[$\bullet$] $\epsilon^\circ_3.\epsilon^\circ_1=(\lambda_1\lambda_{23}-\lambda_1t_2-t_1t_3).w^\circ -t_3\epsilon^\circ_1-\lambda_1\epsilon^\circ_2-t_1\epsilon^\circ_3
\thinspace;$
\item[$\bullet$] $\epsilon^\circ_2.\epsilon^\circ_1=(\lambda_2t_3-(\lambda_{12}-t_1)(\lambda_{23}-t_2)).w^\circ +(\lambda_{23}-t_2)\epsilon^\circ_1+(\lambda_{12}-t_1)\epsilon^\circ_2+\lambda_2\epsilon^\circ_3\thinspace;$
\item[$\bullet$] $\epsilon^\circ_3.\epsilon^\circ_2=(\lambda_3t_1-(\lambda_{13}-t_3)(\lambda_{23}-t_2)).w^\circ +\lambda_3\epsilon^\circ_1+(\lambda_{13}-t_3)\epsilon^\circ_2+(\lambda_{23}-t_2)\epsilon^\circ_3\thinspace;$
\item[$\bullet$] $\epsilon^\circ_1.\epsilon^\circ_3=(\lambda_1t_2-(\lambda_{12}-t_1)(\lambda_{13}-t_3)).w^\circ +(\lambda_{13}-t_3)\epsilon^\circ_1+\lambda_1\epsilon^\circ_2+(\lambda_{12}-t_1)\epsilon^\circ_3$.
\end{enumerate}
\end{theorem}

\paragraph{Proof of Theorem \ref{mult-table-wrt-Theta}:} 
 For clarity, let $*_q$ denote the multiplication in $C_0(V_T,q)$, 
 and for uniformity, let $\epsilon_0:=w.$  Since $q=q_{b(q)}$, we have by 
 (2d), Theorem \ref{Bourbaki}, 
 the isomorphism $\psi_{b(q)}:C_0(V_T,q)\cong \Lambda^{even}(V_T)=W_T.$ 
 Let $*_{b(q)}$ denote the 
 product in the algebra structure $\theta(q)$ thus induced on $W_T.$  
 Since 
 the $\epsilon^\circ_i$ are a basis
for $W_T$,    
 it is  enough to compute the products  
$\epsilon^\circ_i*_{b(q)} \epsilon^\circ_j$ for $1\leq i,j \leq 3.$
 For example, consider 
the product $\epsilon^\circ_2*_{b(q)} \epsilon^\circ_1.$ 
Using the properties of the multiplication in $C(V_T, q)$, and the 
 properties of the isomorphism $\psi_{b(q)}$ from (2), Theorem \ref{Bourbaki},
 we get the following: 
\begin{equation}
\begin{split}
\epsilon^\circ_2*_{b(q)} \epsilon^\circ_1 
& =\psi_{b(q)}\left(
\{ \psi_{b(q)}^{-1}( e_2^{\circ}\wedge e_3^{\circ} ) \} *_q
\{ \psi_{b(q)}^{-1}( e_1^{\circ}\wedge e_2^{\circ} ) \} \right) \notag \\
& =\psi_{b(q)}\left(
  (e_2^{\circ}*_q e_3^{\circ})   *_q
  (e_1^{\circ}*_q e_2^{\circ})   \right) \notag \\
& =\psi_{b(q)}\left(
 (\lambda_{23}(1^{\circ})-e_3^{\circ}*_q e_2^{\circ})*_q
 (\lambda_{12}(1^{\circ})-e_2^{\circ}*_q e_1^{\circ}) \right) \\ \notag 
& =\psi_{b(q)}\left(
\lambda_{23}\lambda_{12}(1^{\circ})-\lambda_{23}e_2^{\circ}*_q
 e_1^{\circ}-\lambda_{12}e_3^{\circ}*_q e_2^{\circ}
  + (e_3^{\circ}*_q
 e_2^{\circ})*_q(e_2^{\circ}*_q
 e_1^{\circ})\right) \\ \notag 
 & =\psi_{b(q)}\left(
\lambda_{23}\lambda_{12}(1^{\circ})-\lambda_{23}(\lambda_{12}(1^{\circ})-e_1^{\circ}*_q
 e_2^{\circ})
 -\lambda_{12}(\lambda_{23}(1^{\circ})-e_2^{\circ}*_q e_3^{\circ}) 
+ e_3^{\circ}*_q
 (e_2^{\circ}*_q e_2^{\circ})*_q
 e_1^{\circ}\right) \\ \notag
 & = (-\lambda_{12}\lambda_{23})
w^{\circ}+\lambda_{23}\epsilon^\circ_1
+\lambda_{12}\epsilon^\circ_2 +\lambda_{2}\epsilon^\circ_3. \notag 
\end{split}\end{equation}
In a similar fashion, the other products may be computed; this amounts to
 computing $\theta$ on $T$-valued points.  
The following result is needed to compute $\Theta$ from $\theta.$ 
\begin{lemma}\label{compute-Theta-from-theta}
Let  $*_{(b(q),\underline{t})}$ denote the multiplication in the algebra 
 $\Theta(q,\underline{t})=\underline{t}.\theta(q)$
 and as before, $*_{b(q)}$ denote 
 the multiplication in $\theta(q).$  
 Then we have 
\begin{description}
\item[1.]
 $\underline{t}(\epsilon^\circ_i)=t_{i}w^{\circ}+\epsilon^\circ_i$ for
 $1\leq i \leq 3;$
\item[2.]
  $(\underline{t})^{-1}(\epsilon^\circ_i)=-t_{i}w^{\circ}+\epsilon^\circ_i$ for
 $1\leq i \leq 3;$
\item[3.]
 $\epsilon^\circ_i*_{(b(q),\underline{t})} \epsilon^\circ_j=\underline{t}(\epsilon^\circ_i*_{b(q)} \epsilon^\circ_j) -t_{j}\epsilon^\circ_i-t_{i}\epsilon^\circ_j-t_it_j w^{\circ}.$
\end{description}
\end{lemma}
\noindent While the first two of the above
formulae follow easily by direct computation, 
the third follows by using the first two  
 alongwith the following: 
$$\epsilon^\circ_i*_{(b(q),\underline{t})}\epsilon^\circ_j=\underline{t}
 \left(\ 
(\underline{t}^{-1}(\epsilon^\circ_i))*_{b(q)}
(\underline{t}^{-1}(\epsilon^\circ_j))\ 
 \right).$$
\noindent We may now 
compute 
the multiplication in
the algebra $\Theta(q,\underline{t})=\underline{t}.\theta(q)$ by making use of
the formulas listed in the above lemma and the expressions for 
the products of the form
 $\epsilon^\circ_i*_{b(q)} \epsilon^\circ_j$ whose computation had
already been illustrated before the lemma. {\bf Q. E. D., Theorem \ref{mult-table-wrt-Theta}.}
\paragraph{Computation of $C_0(g, l)$ in the free case.}
We continue with the notations introduced above. In the following 
 we study the lack of equivariance of the isomorphism $\Theta$ relative to 
 $\hbox{\rm GL}_{V}$ and show  
 that it satisfies a curious
 `twisted' version of equivariance. Firstly we consider the 
 morphism of $X$-groupschemes
  $\Lambda^{even}:\hbox{\rm GL}_{V}\longrightarrow\hbox{\rm Stab}_{w}$
given on valued points by $g\mapsto \Lambda^{even}(g).$
Recall that $\Lambda^{even}(V)=:A_0\in\hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_W(X)$ is the even graded part of the Clifford algebra of the zero quadratic form
 on $V.$ A simple computation reveals the following result.
 \begin{lemma}\label{bgdef} For each $X$-scheme $T$, 
define the map 
$$\hbox{\rm GL}(V_T)\longrightarrow 
\hbox{\rm Stab}({(A_0)}_T): 
g\mapsto \begin{pmatrix}1 & {\bf 0}\\{\bf 0}& B(g)\end{pmatrix}\in\hbox{\rm Stab}(w_T)$$
where $B(g):=\hbox{\rm det}(g)\left(E_{12}E_{23}(g^{-1})^t E_{23}E_{12}\right)$
with 
$$E_{12}=\begin{pmatrix}0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1\end{pmatrix}
\hbox{ and }  
E_{23}=\begin{pmatrix}1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0\end{pmatrix}.$$
Then the  above maps define a morphism of $X$-groupschemes 
 which in fact is  none other than $\Lambda^{even}:\hbox{\rm GL}_{V}\longrightarrow \hbox{\rm Stab}_{w}$;  
  in other 
 words: $B(g)=\Lambda^2(g).$ 
\end{lemma}
Recall from Lemma 5.1, Part A, \cite{tevb-paper1}, that 
$\hbox{\rm Stab}_{w}$
 is the semidirect product of $\hbox{\rm Stab}_{A_0}$ and $\hbox{\sf L}_w$, 
 so that $\hbox{\rm Stab}_{A_0}$ naturally acts on $\hbox{\sf L}_w$ by ``outer conjugation''. Let $\hbox{\rm GL}_{V}$ act on $\hbox{\sf L}_w$
 via the homomorphism  $\Lambda^{even}$ i.e., 
 for $g\in\hbox{\rm GL}(V_T)$ and $\underline{t}\in\hbox{\sf L}_w(T)$, 
$$g.\underline{t}:=\Lambda^{even}(g).\underline{t}:= \Lambda^{even}(g)\thinspace\underline{t}\thinspace\Lambda^{even}(g^{-1}).$$
Any element $h\in\hbox{\rm Stab}(w_T)$ \label{hshl-decomp}
 can be uniquely written as $h=h_sh_l=
h_l'h_s$ where $h_s\in \hbox{\rm Stab}({(A_0)}_T)$ and $h_l,\thinspace
h_l'\in\hbox{\sf L}_w(T).$
 Then the relation between $h_l$ and $h_l'$ can be written
 as $h_l'=h_s.h_l$ or $h_l=h_s^{-1}.h_l'$ where ``$.$''
 stands for the action of $\hbox{\rm Stab}_{A_0}$ on $\hbox{\sf L}_w.$
Thus one has a $\hbox{\rm GL}_{V}$-action
 on $\hbox{\rm Quad}_{V}\times_X \hbox{\sf L}_w$ induced by the 
diagonal embedding $\hbox{\rm GL}_{V}\stackrel{\Delta}{\hookrightarrow}\hbox{\rm GL}_{V}\times_X\hbox{\rm GL}_{V}.$
Since $\hbox{\rm Id-}w\hbox{-Sp-Azu}_W$ comes with 
 a canonical action of $\hbox{\rm Stab}_{w}$ on it, we let 
 $\hbox{\rm GL}_{V}$ act on $\hbox{\rm Id-}w\hbox{-Sp-Azu}_W$ via 
 $\Lambda^{even}.$ 
The following result describes the lack of $\hbox{\rm GL}_{V}$-equivariance 
of the isomorphism $\Theta.$

\begin{theorem}\label{tequiv} Let $T$ be an $X$-scheme. 
For $T$-valued points $g, q, \underline{t}$ respectively of $\hbox{\rm GL}_{V}$, $\hbox{\rm Quad}_{V}$, and $\hbox{\sf L}_w$, there exists
 a unique $T$-valued point of $\hbox{\sf L}_w$ given by an 
  isomorphism $h_l'(g,q)$ of ${\cal O}_T$-algebra bundles 
$$h_l'(g,q): g.\Theta(q,\underline{t}) \stackrel{\cong}{\longrightarrow}
 \Theta(g.q,g.\underline{t}).$$
 Further, $h_l'(g,q)$ satisfies the formula
$$h_l'(gg',q)=h_l'(g,g'.q)(g.h_l'(g',q)).$$ 
\end{theorem}
Therefore $\Theta$ satisfies a `twisted' version of
 $\hbox{\rm GL}_{V}$-equivariance. The next theorem,
 which was originally motivated by the proof of this `twisted equivariance',
 will be of central importance to us for the rest of
 this section.
\begin{theorem}\label{b-co-phi-lambda-indep-of-q}
Given  a 
 similarity  $g:(V_T, q)\thinspace\cong_l\thinspace(V_T,q')$ with multiplier 
 $l\in\Gamma(T, {\cal O}_T^*)$,  let $h(g,l,q,q')$ be the 
 automorphism of $(W_T, w_T)$ given by the composition of the 
 following isomorphisms:
$$W_T\stackrel{{(\psi_{b(q)})}^{-1}\thinspace(\cong)}{\longrightarrow}
C_0(V_T,q)\stackrel{C_0(g,l)\thinspace{(\cong)}}{\longrightarrow}
C_0(V_T,q')\stackrel{\psi_{b(q')}\thinspace(\cong)}{\longrightarrow} W_T$$
where the algebra bundle isomorphism $C_0(g,l)$ comes from (1), 
 Prop.\ref{simil-induces-iso-of-even-cliff} and the linear isomorphisms 
 $\psi_{b(q)}$ and  $\psi_{b(q')}$ come from (2d), Theorem \ref{Bourbaki}.
In terms of actions, this means that $h(g,l,q,q').\theta(q)=\theta(q').$ 
Write $h(g,l,q,q')\in\hbox{\rm Stab}(w_T)$ 
 uniquely as a product 
$$h(g,l,q,q')=h_s(g,l,q,q')h_l(g,l,q,q')$$ with 
 the first factor in $\hbox{\rm Stab}_{A_0}(T)$ and 
 the second in $\hbox{\sf L}_w(T)$ as explained earlier. 
 Then $h_s(g,l,q,q')$ depends only on $g$ and $l$ and 
 not on $q$ or $q'.$ In fact, one has 
$$h_s(g,l,q_1,q_2)=h_s(g,l):=\begin{pmatrix}1 & {\bf 0}\\{\bf 0}& l^{-1}\Lambda^2(g)\end{pmatrix} \forall q_1,q_2\in\hbox{\rm Quad}(V_T).$$
\end{theorem}

\paragraph{\sc Proof:} We directly compute the ${\cal O}_T$-linear
 automorphism
 $h(g,l,q,q')$ of $W_T$ as follows.
 Of course, this automorphism fixes $w^\circ=w_T.$
 So we need to only compute the images of the three remaining basis elements  
 $\epsilon^\circ_1=e^\circ_1\wedge e^\circ_2$, $\epsilon^\circ_2=e^\circ_2\wedge e^\circ_3$ and $\epsilon^\circ_3=e^\circ_3\wedge e^\circ_1$ 
 in terms of the basis elements $w^\circ$ and $\epsilon^\circ_i.$
 Let $q$ and  $l(g.q)=q'$ respectively
 correspond to the 6-tuples 
$(\mu_1,\mu_2,\mu_3,
\mu_{12},\mu_{13},\mu_{23})$  and   
$(\mu'_1,\mu'_2,\mu'_3,
\mu'_{12},\mu'_{13},\mu'_{23})
\in \Gamma(T, {{\cal O}_T}^{\oplus 6}).$
(We caution the reader that 
 $l(g.q)\neq(lg).q=l^{-2}(g.q)$!)
 Let 
 $g\in\hbox{\rm GL}(V_T)\equiv\hbox{\rm GL}_3(\Gamma(T,{\cal O}_T))$
 be given by the matrix $(g_{ij}).$ 
 Observe that the $\mu'$ are polynomials in the $\mu$ and $g_{ij}.$
In the following computation, for the sake of clarity, we denote the 
 product in $C(V_T,q)$ by $*_q.$
For example, we have
\begin{equation}
\begin{split} 
h(g,l,q,q')\epsilon_1
&= \psi_{b(q')}\circ C_0(g,l)\left(
{(\psi_{b(q)})}^{-1}(e^\circ_1 \wedge e^\circ_2)\right)
=\psi_{b(q')}\left( C_0(g,l)\left(e^\circ_1*_qe^\circ_2\right)\right) 
\hbox{(by (2f), Theorem \ref{Bourbaki})} \notag\\
&=\psi_{b(q')}\left(l^{-1}(g(e^\circ_1)*_{q'}g(e^\circ_2))\right)
 \hbox{\hspace*{5mm}(by (1), Prop.\ref{simil-induces-iso-of-even-cliff})} \notag \\
&=l^{-1}\psi_{b(q')}\left((g_{11}e^\circ_1+g_{21}e^\circ_2+g_{31}e^\circ_3)*_{q'} (g_{12}e^\circ_1+g_{22}e^\circ_2+g_{32}e^\circ_3)\right) \notag \\
&=l^{-1}\psi_{b(q')}\left(\right.(g_{11}g_{12}\mu'_1+g_{21}g_{22}\mu'_2+g_{31}g_{32}\mu'_3+ g_{21}g_{12}\mu'_{12}+g_{31}g_{22}\mu'_{23}+g_{11}g_{32}\mu'_{13})w^\circ +\notag \\
&\quad +(g_{11}g_{22}-g_{21}g_{12}) e^\circ_1*_{q'}e^\circ_2 +
  (g_{21}g_{32}-g_{31}g_{22}) e^\circ_2*_{q'}e^\circ_3
+(g_{31}g_{12}-g_{11}g_{32}) e^\circ_3*_{q'}e^\circ_1\left.\right) \notag \\
&=l^{-1}\left(P_1(g,l,q,q')w^\circ+C_{33}(g)\epsilon^\circ_1+C_{13}(g)\epsilon^\circ_2+C_{23}(g)\epsilon^\circ_3\right)
\hbox{\hspace*{5mm}(by (2f), Theorem \ref{Bourbaki})} \notag
\end{split}\end{equation}
where $P_1(g,l,q,q')$ is the polynomial in the $\mu'$ and $g_{ij}$ 
 (as computed in the previous step) and where $C_{ij}(g)$ represents 
 the cofactor determinant of the element $g_{ij}$ of the matrix $g=(g_{ij}).$
 Similarly one computes the values of $h(g,l,q,q')\epsilon_2$ and $h(g,l,q,q')\epsilon_3.$
 Then the matrix of $h(g,l,q,q')$ is given by 
$$h(g,l,q,q')=
\begin{bmatrix}
1 & l^{-1}P_1(g,l,q,q') & l^{-1}P_2(g,l,q,q') & l^{-1}P_3(g,l,q,q') \\
0 & l^{-1}C_{33}(g) &  l^{-1}C_{31}(g) &  l^{-1}C_{32}(g) \\
0 & l^{-1}C_{13}(g) &  l^{-1}C_{11}(g) &  l^{-1}C_{12}(g) \\
0 & l^{-1}C_{23}(g) &  l^{-1}C_{21}(g) &  l^{-1}C_{22}(g) 
\end{bmatrix}$$ 
which implies that 
$$h_s(g,l,q,q')=
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & l^{-1}C_{33}(g) &  l^{-1}C_{31}(g) &  l^{-1}C_{32}(g) \\
0 & l^{-1}C_{13}(g) &  l^{-1}C_{11}(g) &  l^{-1}C_{12}(g) \\
0 & l^{-1}C_{23}(g) &  l^{-1}C_{21}(g) &  l^{-1}C_{22}(g) 
\end{bmatrix}\hbox{ depends only on }g\hbox{ and }l.$$
Next define the matrix
$$\widehat{g}=
\begin{bmatrix}
 g_{33} &  g_{13} &  g_{23} \\
 g_{31} &  g_{11} &  g_{21} \\
 g_{32} &  g_{12} &  g_{22} 
\end{bmatrix}\hbox{ so that }h_s(g,l,q,q')=
\begin{bmatrix}
1 & {\bf 0}\\
{\bf 0} & l^{-1}{C(\widehat{g})}^t
\end{bmatrix}$$
where $C(\widehat{g})$ is the cofactor matrix of $\widehat{g}.$
Now if $E_{12}$ and $E_{23}$ are the matrices defined in Lemma \ref{bgdef}
 above, premultiplying by $E_{ij}$ has the effect of 
 interchanging the $i$th and $j$th rows, while postmultiplying has a 
similar effect on the columns.
 Thus we get $\widehat{g}=E_{12}E_{23}(g^t)E_{23}E_{12}$
 from which it follows that 
\begin{equation}
{C(\widehat{g})}^t = \hbox{ Adjoint }(\widehat{g}) 
 = \hbox{ det }(\widehat{g}).(\widehat{g})^{-1}
 = \hbox{ det }(g).\left(E_{12}E_{23}{(g^{-1})}^t E_{23}E_{12}\right),\notag  
\end{equation} 
showing that ${C(\widehat{g})}^t=\Lambda^2(g)$ by Lemma \ref{bgdef}. {\bf Q.E.D., Theorem 
 \ref{b-co-phi-lambda-indep-of-q}.}

\paragraph{Proof of Theorem \ref{tequiv}.}
Note that $g$ is an isometry from $(V_T,q)$ to $(V_T, g.q)$ and hence 
according to (1),
 Prop.\ref{simil-induces-iso-of-even-cliff}, 
 induces the algebra isomorphism
 $C_0(g,l=1):C_0(V_T,q)\cong C_0(V_T,g.q).$  Let 
 $h(g,q):=h(g,1,q,g.q)$ where $h(g,l,q,q')$ was 
 defined in Theorem \ref{b-co-phi-lambda-indep-of-q} above.
 As explained in page \pageref{hshl-decomp}, 
 there are two canonical decompositions of $\hbox{\rm Stab}(w_T)$, one
  leading to 
 the unique (ordered) decomposition of $h(g,q)$ 
 as $h'_l(g,q)h_s(g,q)$, and the other leading to the  unique ordered 
  decomposition  
  $h_s(g,q)h_l(g,q).$ 
  By Theorem
 \ref{b-co-phi-lambda-indep-of-q} above,  
 $h_s(g,q_1)=h_s(g,q_2)=h_s(g,1)=\Lambda^{even}(g)=:h_s(g)\ \forall q_1,q_2\in\hbox{\rm Quad}(V_T)$ and hence we get:
\begin{equation}
\begin{split}
\Theta(g.q, g.\underline{t}) &:= (g.\underline{t}).\theta(g.q) \notag \\
 &= (\Lambda^{even}(g)\underline{t}\Lambda^{even}(g^{-1})).(h(g,1,q,g.q).\theta(q)) \notag \\
 &= (h_s(g)\underline{t}h_s^{-1}(g)).(h(g,q).\theta(q)) \notag \\
 &= ((h_s(g)\underline{t}h_s^{-1}(g))(h_s(g,q)h_l(g,q))).\theta(q) \notag \\
 &= (h_s(g)\underline{t}h_l(g,q)).\theta(q) \notag \\
 &= (h_s(g)h_l(g,q)).(\underline{t}.\theta(q)) \notag \\
 &= (h_s(g,q)h_l(g,q)).(\Theta(q,\underline{t})) \notag \\
 &= (h'_l(g,q)h_s(g,q)).(\Theta(q,\underline{t})) \notag \\
 &= h'_l(g,q).(h_s(g,q).\Theta(q,\underline{t})) \notag \\
 &= h'_l(g,q).(g.\Theta(q,\underline{t})). \notag 
\end{split}
\end{equation}
Note that $h_l(g,q)$ was explicitly computed in the proof of 
 Theorem \ref{b-co-phi-lambda-indep-of-q} above to be 
$$h_l(g,q)=
\begin{bmatrix}
1 & P_1(g,1,q,g.q) & P_2(g,1,q,g.q) & P_3(g,1,q,g.q) \\
{\bf 0} &  & I_3 & 
\end{bmatrix}\in\hbox{\sf L}_w(T).$$ 
The formula for $h'_l(g_1g_2,q)$ stated in the theorem is gotten thus:
\begin{equation}
\begin{split}
h'_l(g_1g_2,q) &=  (h'_l(g_1g_2,q)h_s(g_1g_2)) h_s^{-1}(g_1g_2) \notag \\
 &= h(g_1g_2,q)h_s^{-1}(g_1g_2). \notag    
\end{split}
\end{equation}
Now by (3) of Prop.\ref{simil-induces-iso-of-even-cliff} it follows that 
\begin{equation}
\begin{split}
h'_l(g_1g_2,q) &=  (h(g_1,g_2. q)h(g_2,q))h_s^{-1}(g_1g_2) \notag \\
 &= h'_l(g_1,g_2. q)h_s(g_1,g_2. q)h_s(g_2,q)
h_l(g_2,q)h_s^{-1}(g_1g_2)\notag  \\
 &= h'_l(g_1,g_2. q)h_s(g_1)h_s(g_2)
h_l(g_2,q)h_s^{-1}(g_1g_2)\notag  \\
 &= h'_l(g_1,g_2. q)h_s(g_1g_2)
h_l(g_2,q)h_s^{-1}(g_1g_2)\notag  \\
 &= h'_l(g_1,g_2. q)((g_1g_2). h_l(g_2,q)).\notag    
\end{split}
\end{equation}
But $g_2^{-1}.h'_l(g_2,q)=h^{-1}_s(g_2)(h'_l(g_2,q)h_s(g_2))=h^{-1}_s(g_2)(h_s(g_2)h_l(g_2,q))=h_l(g_2,q)$ and  therefore  
\begin{equation}
\begin{split}
h'_l(g_1g_2,q) &= h'_l(g_1,g_2. q).((g_1g_2)
.(g_2^{-1}. h'_l(g_2,q))) \notag \\
 &= h'_l(g_1,g_2. q).(g_1
. h'_l(g_2,q)). \notag    
\end{split}
\end{equation}
Finally, one has to show the uniqueness of $h'_l(g,q)\in\hbox{\sf L}_w(T).$
 Suppose $h_l\in\hbox{\sf L}_w(T)$ is also an algebra isomorphism 
$h_l: g.\Theta(q,\underline{t}) \stackrel{\cong}{\longrightarrow}
 \Theta(g.q, g.\underline{t})$, i.e., $h_l.( g. \Theta(q,\underline{t})) = \Theta(g. q,g.\underline{t}).$
Notice that while showing the `twisted' equivariance of $\Theta$ above, 
 we have also proved that $\Theta(g.q,g.\underline{t})=h(g,q)\Theta(q,\underline{t})$. Therefore we get 
\begin{equation}
\begin{split}
(h_lh_s(g)).(\underline{t}.\theta(q)) &= 
 h(g,q).\Theta(q,\underline{t}) \notag \\
\Rightarrow (h_lh_s(g)).(\underline{t}.\theta(q)) &= 
 (h_s(g)h_l(g,q)).\Theta(q, \underline{t}) \notag \\
\Rightarrow (h^{-1}_s(g)h_lh_s(g)).(\underline{t}.\theta(q)) &= 
 (h_l(g,q)\underline{t}).\theta(q) \notag \\
\Rightarrow \Theta(q, (g^{-1}. h_l)\underline{t}) &= \Theta(q, h_l(g,q)\underline{t}). 
\notag 
\end{split}
\end{equation}
But since $\Theta$ is an  isomorphism by Theorem~\ref{Thetaisom}, this 
 implies that $(g^{-1}. h_l)\underline{t}=h_l(g,q)\underline{t}$ which gives
$$h_l=h_s(g)h_l(g,q)h^{-1}_s(g) = h'_l(g,q).$${\bf Q.E.D., Theorem \ref{tequiv}.}

\paragraph{Proofs of Theorems \ref{lifting-of-isomorphisms} and \ref{injectivity}:}
We remind the reader that towards the end of \S \ref{sec2}, we had 
 reduced the proof of Theorem \ref{injectivity} to that of Theorem \ref{lifting-of-isomorphisms}, and had indicated in page \pageref{reduction-to-free-case}
 that it would be enough to prove the latter in 
 the case when $V$ is free---which has been the case in this section so far. 
 Starting with an isomorphism of algebra bundles $\phi:C_0(V,q)\cong C_0(V,q')$
 we arrive at the element $g_l^\phi\in\hbox{\rm GL}(V)$ as defined in page 
 \pageref{glphi-def}; to briefly recall this, firstly $g\in\hbox{\rm GL}(V)$
 was defined by the following commuting diagram: 
$$\begin{CD}
{(C_0(V,q))}^\vee @>{{({(\psi_{b(q)})}^\vee)}^{-1}}>{\cong}> 
{(\Lambda^{even}(V))}^\vee @>{\hbox{\rm\tiny surjection}}>> 
{(\Lambda^2(V))}^\vee @>{{(\eta^\vee)}^{-1}}>{\equiv}> 
V\otimes{{(\hbox{\rm det}(V))}^{-1}} 
@>{\otimes{\hbox{\rm det}(V)}}>{\equiv}> V\\
@A{{\phi^\vee}}A{\cong}A @A{{{(\phi_{\Lambda^{ev}})}^\vee}}A{\cong}A
  @A{{{(\phi_{\Lambda^2})}^\vee}}A{\cong}A @A{{(g')}^{-1}}A{\cong}A @A{g^{-1}}A{\cong}A\\
{(C_0(V,q'))}^\vee @>{{({(\psi_{b(q')})}^\vee)}^{-1}}>{\cong}> 
{(\Lambda^{even}(V))}^\vee @<{\hbox{\rm\tiny inclusion}}<< 
{(\Lambda^2(V))}^\vee @>{{(\eta^\vee)}^{-1}}>{\equiv}> 
V\otimes{{(\hbox{\rm det}(V))}^{-1}} 
@>{\otimes{\hbox{\rm det}(V)}}>{\equiv}> V
\end{CD}$$
Secondly, we had defined $g_l^\phi:=(l^{-1}\sqrt{\gamma(l)}\thinspace)g.$
Our current 
 special choices of bilinear forms $b(q)$ and $b(q')$ that induce $q$ and 
 $q'$ respectively do 
 not affect the generality,
 as was observed in the proof of Prop.\ref{transfer-to-lambda2}. We shall 
 now show that $g_l^\phi$ is a 
 similitude from $(V,q)$ to $(V,q')$ with multiplier $l$ and that 
 this similitude induces $\phi$ i.e., with the notations of (1), Prop. 
\ref{simil-induces-iso-of-even-cliff},  that $C_0(g_l^\phi, l)=\phi.$
 We shall also check that $\hbox{\rm det}^2(g_l^\phi)=\hbox{\rm det}
(\phi_{\Lambda^2})$ when $\hbox{\rm det}(\phi_{\Lambda^2})$ is itself a square.
We proceed with the proof which will follow from  several lemmas. 
\begin{lemma}\label{glphi-in-matrix-form}
Write the element $h:=\phi_{\Lambda^{ev}}\in(\hbox{\rm Stab}_{w})(X)$ 
 uniquely as an ordered product $h=h_sh_l=h'_lh_s$ where 
 $h_l,h'_l\in(\hbox{\sf L}_w)(X)$ and $h_s\in(\hbox{\rm Stab}_{A_0})(X)$ 
 as explained in page \pageref{hshl-decomp};
 let $B$ be the matrix corresponding to
 $\phi_{\Lambda^2}$, and let the matrices $E_{ij}$ be as defined 
 in Lemma \ref{bgdef}. Then we have matrix representations: 
$$h_s=\begin{pmatrix}1 & {\bf 0}\\{\bf 0}& B
\end{pmatrix} \hbox{ and }g_l^\phi=\left(l^{-1}\sqrt{\gamma(l)}\right)E_{23}E_{12}{((B)^t)}^{-1}E_{12}E_{23}$$
\end{lemma}
The proof of the above lemma follows from the fact that the matrix of 
 the canonical isomorphism $\eta:\Lambda^2(V)\equiv V^\vee\otimes\hbox{\rm det}(V)$ is given by $E_{23}E_{12}$, which can be verified by a simple computation.
\begin{lemma}\label{mult-in-covqpr-and-l-inv-covqpr}
$h_s(\hbox{\rm Identity}, l^{-1}, q', l^{-1}q')
=\begin{pmatrix}1 & {\bf 0}\\{\bf 0}& l\times I_3
\end{pmatrix} \hbox{\rm and }h_s(\hbox{\rm Identity}, l^{-1}, q', l^{-1}q').\theta(q')=
\theta(l^{-1}q').$
\end{lemma}
The identity map on $V$ is obviously a similarity with multiplier 
 $l^{-1}$ from $(V,q')$ to $(V,l^{-1}q').$ 
Hence the above lemma follows
 by taking $T=X$, $g=\hbox{Identity}$, and the $l^{-1}$
 and the $q'$ at hand for the $l$ and the  
$q$ of Theorem 
 \ref{b-co-phi-lambda-indep-of-q} (caution:  
 the $q'$ there would have to be replaced by $l^{-1}q'\thinspace$!).
 This can also be seen directly from 
 the multiplication tables for $\theta(l^{-1}q')=\Theta(l^{-1}q',I_4)$ and 
 $\theta(q')=\Theta(q',I_4)$
 written out in Theorem \ref{mult-table-wrt-Theta}, 
 where we must take $T=X$ and $\underline{t}=I_4$
 i.e., $t_i=0\thinspace\forall\thinspace i.$ 
 We observe from the multiplication table that each of the coefficients of 
 $\epsilon_i$ for $1\leq i\leq 3$ is a single $\lambda$, whereas each 
 coefficient of $w=1=\epsilon_0$ is a product of two $\lambda\thinspace$s, 
 and this observation implies the lemma above. 
As the reader might have noticed, there are two crucial facts
 about the identifications in this section; namely, firstly,  for 
 any $X$-scheme $T$, each of 
 the maps $\psi_{b(q)}$ (for different $q$) 
 identify $(C_0(V_T,q),1)$ with the same 
 $(W_T,w_T)$ and secondly, relative to the bases chosen, all these identifying 
 maps have trivial determinant. The latter is also true of the identification 
 $\eta$, since it is given by the matrix $E_{23}E_{12}$ 
 (as was noted after Lemma \ref{glphi-in-matrix-form}).
  It therefore follows that 
 $\hbox{\rm det}(\phi)=\hbox{\rm det}(\phi_{\Lambda^{ev}})=\hbox{\rm det}(\phi_{\Lambda^2})=\hbox{\rm det}(g')=\hbox{\rm det}(g)=\hbox{\rm det}(B^{-1}).$
 But we had chosen $l\in\Gamma(X, {\cal O}_X^*)$ such that $\gamma(l):=(l^3).\hbox{\rm det}(\phi_{\Lambda^2})=l^3\hbox{\rm det}(B).$ 
 Using these facts alongwith Lemma \ref{glphi-in-matrix-form} above, 
 a straightforward computation gives the following.
\begin{lemma}\label{B-versus-Bglphi}
$\hbox{\rm det}(g_l^\phi)=\sqrt{\gamma(l)}$ from which
 it follows that $B(g_l^\phi)=l\times B$ where $B(g_l^\phi)$ and $B$ are
 as defined in Lemmas \ref{bgdef} and \ref{glphi-in-matrix-form} respectively.
In particular, $\hbox{\rm det}^2(g_l^\phi)=\hbox{\rm det}
(\phi_{\Lambda^2})$ when $\hbox{\rm det}(\phi_{\Lambda^2})$ is itself a square,
 since in this case we had chosen $l:=1.$ 
\end{lemma}
\begin{lemma}\label{glphi-simil-q-qpr-with-mult-l}
 $g_l^\phi$ is a 
 similitude from $(V,q)$ to $(V,q')$ with multiplier $l.$
\end{lemma}
 The hypothesis $\phi:C_0(V,q)\cong C_0(V,q')$ is an algebra isomorphism 
 translates in terms of actions into $h.\theta(q)=\theta(q')$ where 
$h=\phi_{\Lambda^{ev}}\in(\hbox{\rm Stab}_w)(X).$   
Let $h(g_l^\phi,q):=h(g_l^\phi,1,q,g_l^\phi.q)$ where $h(g,l,q,q')$ was 
 defined in Theorem \ref{b-co-phi-lambda-indep-of-q} above. Then we have
\begin{equation}
\begin{split}
\Theta(g_l^\phi.q,I_4)&:=\theta(g_l^\phi.q)
=h(g_l^\phi,q).\theta(q)=h(g_l^\phi,q).(h^{-1}.\theta(q')) 
=\left(h'_l(g_l^\phi,q)h_s(g_l^\phi,q){h_s}^{-1}{(h'_l)}^{-1}\right).\theta(q') \notag \\
&=\left(h'_l(g_l^\phi,q)\begin{pmatrix} 1 & {\bf 0}\\{\bf 0} & B(g_l^\phi)\end{pmatrix}
\begin{pmatrix} 1 &{\bf 0}\\ {\bf 0} & B^{-1}\end{pmatrix}
{(h'_l)}^{-1}\right).\theta(q') \hspace*{4mm}\hbox{\rm (by Theorem
 \ref{b-co-phi-lambda-indep-of-q}; 
 Lemmas \ref{bgdef} \& \ref{glphi-in-matrix-form})} \notag \\
&=\left(h'_l(g_l^\phi,q)\begin{pmatrix} 1 &{\bf 0}\\{\bf 0} & l\times I_3\end{pmatrix}
{(h'_l)}^{-1}\right).\theta(q')
\hspace*{6mm}\hbox{\rm (by Lemma \ref{B-versus-Bglphi})} \notag \\
&=\left(h'_l(g_l^\phi,q)\thinspace h_s(\hbox{\rm Identity}, l^{-1}, q', l^{-1}q')
\thinspace {(h'_l)}^{-1}\right).\theta(q') 
\hspace*{6mm}\hbox{\rm (by Lemma \ref{mult-in-covqpr-and-l-inv-covqpr})} \notag \\
&=(h'_l(g_l^\phi,q)h''_l).\left(h_s(\hbox{\rm Identity}, l^{-1}, q', l^{-1}q').\theta(q')\right)
\hspace*{6mm}\hbox{\rm (since $\hbox{\rm Stab}_{w}$ is a semidirect product)} 
 \notag \\
&=(h'_l(g_l^\phi,q)h''_l).\theta(l^{-1}q')) 
\hspace*{6mm}\hbox{\rm (by Lemma \ref{mult-in-covqpr-and-l-inv-covqpr})}
  \notag \\
&=:\Theta(l^{-1}q',(h'_l(g_l^\phi,q)h''_l)) \notag
\end{split}
\end{equation}
But since $\Theta$ is an isomorphism, this implies the claim of the above 
 Lemma i.e., that 
$g_l^\phi.q=l^{-1}q'$ (and further that
 $h'_l(g_l^\phi,q)={(h''_l)}^{-1}$).
\begin{lemma}\label{glphi-induces-phi}
 $g_l^\phi:(V,q){\cong}_l(V,q')$ induces $\phi$ i.e.,
 with the notations of (1), Prop.\ref{simil-induces-iso-of-even-cliff},
 $C_0(g_l^\phi, l)=\phi.$
\end{lemma}
We have $C_0(g_l^\phi, l)=\phi\iff 
h(g_l^\phi, l,q,q'):=
\psi_{b(q')}\thinspace\circ\thinspace C_0(g_l^\phi, l) \thinspace\circ\thinspace \psi_{b(q)}^{-1}
=\psi_{b(q')}\thinspace\circ\thinspace\phi \thinspace\circ\thinspace \psi_{b(q)}^{-1}=: \phi_{\Lambda^{ev}}=:h.$
Now using successively Theorem \ref{b-co-phi-lambda-indep-of-q},
Lemma \ref{bgdef}, Lemma \ref{B-versus-Bglphi} and Lemma \ref{glphi-in-matrix-form}, 
we get the following sequence of equalities: 
$$h_s(g_l^\phi,l,q,q')=\begin{pmatrix}1 & {\bf 0}\\{\bf 0}& l^{-1}\times\Lambda^2(g_l^\phi)\end{pmatrix}=\begin{pmatrix}1 & {\bf 0}\\{\bf 0}& l^{-1}\times B(g_l^\phi)\end{pmatrix}=\begin{pmatrix}1 & {\bf 0}\\{\bf 0}& B\end{pmatrix}=h_s$$
Therefore the present hypotheses translated in terms of actions give
\begin{equation}\begin{split}
h(g_l^\phi,l,q,q').\theta(q)=\theta(q')&=h.\theta(q)\notag \\
\Rightarrow h_s(g_l^\phi,l,q,q').(h_l(g_l^\phi,l,q,q').\theta(q))&=
h_s.(h_l.\theta(q))\notag \\
\Rightarrow\Theta(q,h_l(g_l^\phi,l,q,q'))&=\Theta(q,h_l).\notag
\end{split}\end{equation}
But $\Theta$ being an isomorphism, the last equality implies that 
 $h_l(g_l^\phi,l,q,q')=h_l$ which gives $h(g_l^\phi,l,q,q')=h$ as wanted.
\begin{lemma}\label{locally-det-coglambda-equals-detgbylcube}\
\begin{description}
\item[(1)]
For a similarity $g\in\simvqvqpr$
 with multiplier $l$ and the
 induced isomorphism $C_0(g,l)\in\isomcovqcovqpr$ 
 given by (1), Prop.\ref{simil-induces-iso-of-even-cliff},
 we have
 the equality $\hbox{\rm det}({(C_0(g,l))}_{\Lambda^2})
=l^{-3}\hbox{\rm det}^2(g).$
 Therefore the map $$\simvqvqpr \longrightarrow
 \isomcovqcovqpr:g\mapsto C_0(g,l)$$ 
maps the subset $\isomvqvqpr$  
 into the subset $\isomcovqcovqprprime$ and
 $\sisomvqvqpr$ into $\sisomcovqcovqpr.$
\item[(2)] In the case $q'=q$, if $C_0(g,l)$ is the identity on 
$\covq$, then 
$g=l^{-1}\hbox{\rm det}(g)\times\hbox{\rm Id}_V$,
 and 
 further if $g\in\ovq$ then 
$g=\hbox{\rm det}(g)\times\hbox{\rm Id}_V$ with $\hbox{\rm det}^2(g)=1.$
\end{description}
\end{lemma} 
By definition, 
${(C_0(g,l))}_{\Lambda^{ev}}=
\psi_{b(q')}\thinspace\circ\thinspace C_0(g,l)
\thinspace\circ\thinspace\psi_{b(q)}^{-1}$, and the latter isomorphism is 
$h(g,l,q,q')$ from Theorem \ref{b-co-phi-lambda-indep-of-q} which 
further gives a formula for    
$h_s(g,l,q,q').$ 
Now using the facts that 
 $\psi_{b(q)}$ and $\psi_{b(q')}$ have trivial determinant (as noted 
 before Lemma \ref{B-versus-Bglphi}) 
we get assertion (1): 
$$\hbox{\rm det}\left({(C_0(g,l))}_{\Lambda^2}\right)
=\hbox{\rm det}\left({(C_0(g,l))}_{\Lambda^{ev}}\right)
=\hbox{\rm det}(h(g,l,q,q'))=\hbox{\rm det}(h_s(g,l,q,q'))
=l^{-3}\hbox{\rm det}^2(g).$$
 If $q=q'$ and $C_0(g,l)$ is the identity, then
  the same argument in fact shows that 
$l^{-1}\Lambda^2(g)=I_3$ and by using the formula in Lemma \ref{bgdef} for 
 $B(g)=\Lambda^{2}(g)$, we get $g=l^{-1}\hbox{\rm det}(g)I_3$;  
 taking determinants in the last equality 
gives $\hbox{\rm det}^2(g)=l^3$, so that when $g\in\ovq$ i.e., $l=1$, 
 $\hbox{\rm det}^2(g)\in\mu_2(\Gamma(X, {\cal O}_X))$ and 
assertion (2) follows. 
\begin{lemma}\label{bijection-of-special-isomorphisms}
 The  map $\sisomvqvqpr\longrightarrow\sisomcovqcovqpr:g\mapsto C_0(g,1,q,q')$ 
 is a bijection. 
\end{lemma}
Given $\phi\in\sisomcovqcovqpr$, by the definition in 
 Prop.\ref{transfer-to-lambda2} we have 
$\hbox{\rm det}(\phi_{\Lambda^2})=1$, so by Lemma \ref{B-versus-Bglphi}
$\hbox{\rm det}(g_l^\phi)=\sqrt{\gamma(l)}:=1$ by our 
 earlier choice (see Case 1 on page \pageref{cases}). Therefore 
 $g_l^\phi\in\sisomvqvqpr$ and is, according to Lemma 
\ref{glphi-induces-phi}, such that    
 $C_0(g_l^\phi,l=1,q,q')=\phi$ which gives the surjectivity.
As for the injectivity, if
 $g_1,g_2:(V,q){\cong}_1(V,q')$ are isometries with determinant 1 such 
 that $C_0(g_1,1,q,q')=C_0(g_2,1,q,q')$, then we have 
 $h(g_1,1,q,q')=h(g_2,1,q,q')$ so that $h_s(g_1,1,q,q')=h_s(g_2,1,q,q')$ 
 whence by Theorem
 \ref{b-co-phi-lambda-indep-of-q} and 
Lemma \ref{bgdef} $B(g_1)=B(g_2)\Rightarrow g_1=g_2.$ 
\paragraph{To sum up the above discussion:} 
Let now $V$ be a not-necessarily globally-trivial 
  rank 3 vector bundle on the scheme $X$, and given $\phi$, 
 we define $l, \gamma(l), \sqrt{\gamma(l)}$ and $g_l^\phi$ globally as 
 before. 
 Then the statements of 
 Lemmas \ref{glphi-simil-q-qpr-with-mult-l}, \ref{glphi-induces-phi}, 
 \ref{locally-det-coglambda-equals-detgbylcube} and 
 \ref{bijection-of-special-isomorphisms} are again valid, for the objects 
 involved are defined globally, but the assertions  
 are of a local nature.  
Given an isomorphism of algebra bundles  
$\phi\in\isomcovqcovqpr$, 
when $\hbox{\rm det}(\phi_{\Lambda^2})\not\in{(\Gamma(X, {\cal O}_X^*))}^{2}$,
 we have thus lifted $\phi$ to a family 
 of similarities $s_{2k+1}(\phi):=g_l^\phi\in\simvqvqpr$
 with multipliers
 $l:={(\hbox{\rm det}(\phi_{\Lambda^2}))}^{2k+1}$ parametrised 
 by integers $k$; when
 $1\neq\hbox{\rm det}(\phi_{\Lambda^2})\in{(\Gamma(X, {\cal O}_X^*))}^{2}$, 
we have lifted $\phi$ to a similarity with multiplier $l=1$ i.e.,
 to an isometry $s'(\phi):=g_l^\phi\in\isomvqvqpr$ 
such that $\hbox{\rm det}(\phi_{\Lambda^2})=\hbox{\rm det}^2(g_l^\phi)$, 
 and moreover, when $\phi_{\Lambda^2}$ has determinant 1, 
 we have lifted it to a unique isometry $g_l^\phi$ with determinant 1. 
 Of course, $s_{2k+1}$ and $s'$ are multiplicative only upto 
 an element of $\mu_2(\Gamma(X, {\cal O}_X^*))$, but this can 
 be remedied by considering $s_{2k+1}^{+}$
 as noted at the end of  section \ref{sec2}. 
{\bf Q.E.D., Theorems \ref{lifting-of-isomorphisms} \& \ref{injectivity}.}

\paragraph{Proof of Theorem \ref{lifting-of-automorphisms}.} Taking 
 $q'=q$ in Theorem \ref{lifting-of-isomorphisms} gives the commutative 
 diagram of groups and homomorphisms as asserted in the statement of 
 the theorem. We continue with the notations used in the proof of 
 Theorem \ref{lifting-of-isomorphisms}. For $g\in\govq$ with multiplier $l$,
 the equality $\hbox{\rm det}(C_0(g,l))
=\hbox{\rm det}\left((C_0(g,l))_{\Lambda^2}\right)
=l^{-3}\hbox{\rm det}^2(g)$ was proved in (1),
 Lemma \ref{locally-det-coglambda-equals-detgbylcube}.
 Assertion (2) of the same Lemma gives 
 exactness at $\govq$ and at $\ovq.$ 

We proceed to prove assertion (b). 
 Let $\phi\in\autcovq$, and consider the self-similarity
 $s_{2k+1}^{+}(\phi)=g_l^\phi$ with multiplier
 $l=\hbox{\rm det}(\phi)^{2k+1}.$ 
 For the moment, assume that $V$ is trivial over 
 $X$ with global basis $\{e_1,e_2,e_3\}$, and set $e'_i=g_l^\phi(e_i).$
 It follows from Kneser's  definition 
 of the half-discriminant $d_0$---see formula (3.1.4), Chap.IV, 
\cite{Knus}---that $d_0(q, \{e_i\})
=d_0(q, \{e'_i\})\thinspace\hbox{\rm det}^2(g_l^\phi).$ 
 Since we have 
 $g_l^\phi.q=l^{-1}q$, a simple computation shows that $d_0(q, \{e'_i\})=
l^{3}d_0(q,\{e_i\}).$ The hypothesis that $q\otimes\kappa(x)$ is 
 semiregular means that the image of the 
 element $d_0(q,\{e_i\})\in\Gamma(X,{\cal O}_X)$ in $\kappa(x)$ is nonzero. 
 Since $X$ is integral, we therefore deduce that 
$\hbox{\rm det}^2(g_l^\phi)=l^{-3}.$ On the other hand, we know that 
 $\hbox{\rm det}^2(g_l^\phi)l^{-3}=\hbox{\rm det}(\phi).$ It follows 
 that $\hbox{\rm det}^{12k+7}(\phi)
=1\thinspace\forall\thinspace k\in\mathbb Z$,
 which forces $\hbox{\rm det}(\phi)=1$ as claimed. 
{\bf Q.E.D., Theorem \ref{lifting-of-automorphisms}.}
 

%Shree Maataaravindaarpanam

\section{\label{sec4} Specialisations as Bilinear Forms:  
 Theorems \ref{bilinear-forms-as-specialisations}--\ref{Thetaisom-Upsilonisom}} 
In this section we reduce the proof of Theorem \ref{bilinear-forms-as-specialisations} to Theorem \ref{multiplication-table-wrt-bilinear}. We 
 prove the latter and using it  
  alongwith Theorem \ref{mult-table-wrt-Theta},  
 deduce Theorem \ref{Thetaisom-Upsilonisom}.


\paragraph{Preliminaries on the notion of Schematic Image.}
\begin{definition}[Defs.6.10.1-2, Chap.I, EGA I \cite{ega1}]\rm\label{sch-img-def}
Let $f:X\longrightarrow Y$ be a morphism of schemes. If there exists 
a smallest closed subscheme $Y'\hookrightarrow Y$ such that the inverse 
 image scheme $f^{-1}(Y'):=Y'\times_Y {(}_fX)$ is equal to $X$, 
one calls $Y'$ the {\em schematic image} of $f$ (or of $X$ in $Y$ under $f$).
If $X$ were a subscheme of $Y$ and $f$ the canonical immersion, and 
 if $f$ has a schematic image $Y'$, then $Y'$ is called the {\em schematic 
 limit} or the {\em limiting scheme}
 of the subscheme $X\stackrel{f}{\hookrightarrow}Y.$ 
\end{definition}

\begin{proposition}[Prop.6.10.5, Chap.I, EGA I]\label{sch-img-existence}
The schematic image $Y'$ of $X$ by a morphism $f:X\longrightarrow Y$ exists 
 in the following two cases: 
(1) $f_*({\cal O}_X)$ is a quasi-coherent ${\cal O}_Y$-module, which 
 is for example the case when $f$ is quasi-compact and quasi-separated; (2) 
$X$ is reduced. 
\end{proposition}

\begin{proposition}\label{sch-img-props}
In each of the following statements whenever a schematic image is 
 mentioned, we assume that one of the two hypotheses  
 of the above Prop. is true so that the schematic image does exist. 
\begin{description}
\item[1.] Let $Y'$ be the schematic image of $X$ under a morphism 
 $f:X\longrightarrow Y$ and let $f$ factor as 
$X\stackrel{g}{\longrightarrow}Y'\stackrel{j}{\hookrightarrow}Y.$
Then $Y'$ is topologically the closure of $f(X)$ in $Y$, the morphism 
 $g$ is schematically dominant (i.e., $g^{\#}:{\cal O}_{Y'}\longrightarrow 
 g_*({\cal O}_X)$ is injective) and the 
 schematic image of $X$ in $Y'$ (under $g$) is $Y'$ itself. If $X$ is 
 reduced (respectively integral) then the same is true of $Y'.$ 
\item[2.] The schematic image of a closed subscheme under its canonical 
 closed immersion is itself.
\item[3.] (Transitivity of Schematic Image) Let there be given morphisms 
$X\stackrel{f}{\longrightarrow}Y\stackrel{g}{\longrightarrow}Z$, 
such that the schematic image $Y'$ of $X$ under $f$ exists, and further 
 such that if $g'$ is the restriction of $g$ to $Y'$, the schematic image $Z'$ 
 of $Y'$ by $g'$ exists. Then the schematic image of $X$ under $g\circ f$ 
 exists and equals $Z'.$   
\item[4.] Let $f:X\longrightarrow Y$ be a morphism which factors through 
 a closed subscheme $
Y_1$ of $Y$ by a morphism $f_1:X\longrightarrow Y_1.$
Then the scheme-theoretic image $Y'$ of $X$ in $Y$ is the same as the 
 scheme-theoretic image $Y'_1$ of $X$ in $Y_1$ considered canonically 
 as closed subscheme of $Y.$ 
\item[5.] If $f:X\longrightarrow Y$ has a schematic image $Y'$ then 
 $f$ is schematically dominant 
 iff $Y'=Y.$
\item[6.] The formation of schematic image commutes with flat base 
 change: i.e., if $f:X\longrightarrow Y$ is a morphism of $S$-schemes 
 which has a schematic image $Y'$ then for a flat morphism $S'\longrightarrow S$, one has that the induced $S'$-morphism
 $f\times_S S':X\times_S S'\longrightarrow Y\times_S S'$
 has a schematic image and it 
 may be canonically identified with $Y'\times_S{S'}.$ In particular this means 
 that the formation of schematic image is local over the base. 
\end{description}
\end{proposition}

Assertions (1) and (3) are respectively Prop.6.10.5 and Prop.6.10.3 
 in EGA I. The defining property of schematic image gives (2),
 while (4) can be deduced from the first three. As for (5), 
 from (1) it follows that $Y'=Y$ implies $f=g$ is schematically 
 dominant. For the other way around, one uses 
 the following characterisation of a schematically dominant morphism 
 in Prop.5.4.1 of EGA I: if $f:X\longrightarrow Y$ is a morphism of schemes, 
then $f$ is schematically dominant iff for every open subscheme $U$ of $Y$ 
 and every closed subscheme $Y_1$ of $U$ such that there exists a 
 factorisation 
$f^{-1}(U)\stackrel{g_1}{\longrightarrow}Y_1\stackrel{j_1}{\hookrightarrow}U$, 
of the restriction $f^{-1}(U)\longrightarrow U$ of $f$ (where $j_1$ is 
 the canonical closed immersion), one has $Y_1=U$---given $f$ is 
 schematically dominant, one just has to take $U=Y$, $Y_1=Y'$ and $g_1=g.$ 
Assertion (6) follows from statement (ii) (a) of Theorem 11.10.5 of EGA IV
 \cite{ega4}.


\paragraph{Preliminaries from Part A of \cite{tevb-paper1}.}
Until further notice we assume that $W$ is a vector bundle of fixed positive 
 rank on the scheme $X.$
Given any $X$-scheme $T$, 
  by a  $T$-algebra structure  
 on ${{W}}_T:={{W}}\times_X T$
 (also referred to as $T$-algebra bundle), we mean  
 a morphism 
 ${{W}}_T\times_T{{W}}_T\longrightarrow{{W}}_T$ 
 of vector bundles on $T$ arising from a morphism of the associated 
 locally-free sheaves. Given such a $T$-algebra structure and  $T'
\longrightarrow T$ an $X$-morphism, it is clear that one  
 gets by pullback (i.e., by base-change) a canonical $T'$-algebra 
 structure on ${{W}}_{T'}.$ Thus  
 one has a contravariant 
 ``functor of algebra structures on ${W}$'' from  
 $\{X-Schemes\}$ to $\{Sets\}$ 
 denoted $\falgW$ whose set of $T$-valued points 
 is the set of $T$-algebra structures on 
 ${{W}}_T$ viz. $\hbox{Hom}_{{\cal O}_T}\left({\cal W}_T\otimes{\cal W}_T,
 {\cal W}_T\right).$
By Prop.9.6.1, Chap.I of EGA I \cite{ega1}, it follows that the functor
 $\falgW$ is represented by the $X$-scheme
 $$\algW:=\spec{\sym{X}{{\cal W}}}.$$ 
 Hence
 $\algW$ is affine (hence separated),
  of finite presentation over $X$  and in fact smooth 
 of relative dimension $\hbox{rank}_X({{W}})^3$. If $X'\longrightarrow 
 X$ is an extension of base, then the construction $\algW$ base-changes 
 well i.e., one may canonically identify 
 $\algW\times_X X'$ with 
 $\hbox{Alg}_{{W}'}$ where ${{W}'}={{W}}\times_X X'$ 
 (cf. Prop.9.4.11, Chap.I, EGA I \cite{ega1}). We remark that 
 an algebra structure may fail to be associative and may fail 
 to have a (two-sided) identity element for multiplication. However, 
 a multiplicative identity for an 
 associative algebra structure must be a nowhere vanishing 
 section (Lemma 2.3, and (2)$\Rightarrow$(4) of Lemma 2.4, 
 Part A, \cite{tevb-paper1}).

The general linear groupscheme associated to ${W}$ viz $\gl{{{W}}}$ 
 naturally acts on $\algW$ on the left, so that for each $X$-scheme $T$, 
 $\algW(T)$ mod $\gl{{{W}}}(T)$ is the set of isomorphism classes of 
 $T$-algebra structures on ${{W}}\times_X T$.

 Let $w\in\Gamma(X, {\cal W})$ be a nowhere vanishing 
 section. 
For any  $X$-scheme $T$, let 
$\faaWw(T)$ denote the subset of $\algW(T)$ consisting of 
 associative algebra structures with multiplicative identity 
 the nowhere vanishing section $w_T$ over $T$ induced from $w$.
 Thus we obtain 
 a contravariant subfunctor 
$\faaWw$ of $\algW.$
Let $\stabw(T)\subset \gl{{{W}}}(T)$
 denote the stabiliser subgroup 
 of $w_T$, so that one gets a subfunctor in
 subgroups $\stabw\subset\gl{{{W}}}.$
 It is in fact represented by a closed subgroupscheme (also denoted
 by) $\stabw$ and further behaves well under base change relative 
 to $X$ i.e., $\stabw\times_X T$ can be canonically identified with
 $\hbox{\rm Stab}_{w_T}$ for any $X$-scheme $T.$
 These follow from para 9.6.6 of Chap.I, EGA I \cite{ega1}. 
 It is clear that the natural action of $\gl{{{W}}}$ on $\algW$ 
 induces one of $\stabw$ on $\faaWw.$
 It is easy to check
 (p.489, Part A, \cite{tevb-paper1})
 that the functor $\faaWw$ is a sheaf in the 
 big Zariski site over $X$ and further that this functor is 
 represented by a natural closed subscheme  $\aaWw\hookrightarrow
 \algW$ which is $\stabw$-invariant. 
 Further the construction $\aaWw$ behaves well with respect to 
 base-change (relative to $X$).
Consider the  subfunctor 
 $\fazuWw\hookrightarrow\aaWw$ 
 corresponding to Azumaya algebras.

\begin{theorem}[Theorem 3.4, Part A, \cite{tevb-paper1}]
\label{repbility-smoothness-azu}\ 
\begin{description}
\item[(1)] $\fazuWw$
 is represented by a $\stabw$-stable
 open subscheme $\azuWw\hookrightarrow\aaWw$ and the canonical open 
 immersion is an affine morphism.
\item[(2)] $\azuWw$ is affine (hence separated) 
 and of finite presentation over $X$, and 
 the construction
 $\azuWw$ behaves well with respect to base-change (relative to $X$). 
\item[(3)] Further, $\azuWw$ 
 is smooth of relative dimension $(m^2-1)^2$
 and geometrically irreducible over $X$, 
 where $m^2:=\hbox{rank}_X({\cal W})$. 
\end{description}
\end{theorem} 


 
\begin{theorem}[Theorem 3.8, Part A, \cite{tevb-paper1}]
\label{smoothness-spazu}\ 
\begin{description}
\item[(1)] The open immersion $\azuWw\hookrightarrow\aaWw$
 has a schematic image denoted 
 $\spazuWw$ which is affine (hence separated) 
  and of finite type over $X$ and is naturally a $\stabw$-stable
 closed subscheme of $\aaWw$,
 the action 
 extending the natural one on the open subscheme $\azuWw$.
 \item[(2)] When the rank of ${W}$ over 
 $X$ is 4, $\spazuWw$ is locally (over $X$) isomorphic to relative 
 9-dimensional affine space; in fact over every open affine subscheme $U$ of 
 $X$ where ${W}$ becomes trivial and $w$ becomes part of a global 
 basis, $\spazuWw|_U\cong{\mathbb A}^9_U.$
 Thus $\spazuWw$ is smooth of relative 
 dimension 9 and geometrically irreducible over $X.$ In particular, 
 it is of finite presentation over $X.$  
\item[(3)] When $\hbox{rank}_X({{W}})=4$,
 the construction 
 $\spazuWw\longrightarrow X$ base changes well. 
\end{description}
\end{theorem}

\paragraph{Definition of the Morphism $\Upsilon'$.}
We adopt the notations of Theorem \ref{bilinear-forms-as-specialisations}. Let $T$ be an $X$-scheme. Given a bilinear form $b\in\hbox{\rm Bil}_{V}(T)
=\hbox{\rm Bil}(V_T)$, consider the linear isomorphism $\psi_{b}:C_0(V_T, q_b)
\cong \Lambda^{even}(V_T)=W_T$ of 
 (2d), Theorem \ref{Bourbaki}. Let $A_b$ denote the algebra bundle structure
 on $W_T$ with unit $w_T=1$ induced via $\psi_b$ 
 from the even Clifford algebra $C_0(V_T, q_b)$. By definition, 
 $A_b\in\aaWw(T)$ and we get a map of $T$-valued points 
 $$\Upsilon'(T):\hbox{\rm Bil}_V(T)\longrightarrow \aaWw(T): b\mapsto A_b.$$
 This is functorial in $T$ because of (3), Theorem \ref{Bourbaki}, and 
 hence defines an $X$-morphism $\Upsilon':\hbox{\rm Bil}_V\longrightarrow 
 \aaWw.$ The morphism $\Upsilon'$ is $\hbox{\rm GL}_V$-equivariant 
 due to (2j), Theorem \ref{Bourbaki}.

\paragraph{Semiregular Bilinear forms.\label{semireg-bil-forms}}
 Fundamental problems in dealing with quadratic forms over
 arbitrary base schemes arise essentially from two abnormalities 
 in characteristic two: firstly,  
  the mapping that associates a quadratic form to its symmetric 
 bilinear form is not bijective and secondly, 
 there do not exist regular quadratic forms on any odd-rank bundle.
 The remedy for this is to consider  
 semiregular quadratic 
 forms, a concept due to M.Kneser \cite{Kneser} and 
 elaborated upon 
 by Knus in \cite{Knus}, which in fact works over an arbitrary base 
 scheme (and hence in a characteristic-free way) and further reduces to 
 the usual notion of regular form in characteristics $\neq 2.$
 Let $\hbox{\rm Spec}(R)=U\hookrightarrow X$ be an open affine subscheme 
 of $X$ such that $V|U$ is trivial.
 Consider a  quadratic form $q\in\Gamma(U, \hbox{\rm Quad}_V)$ on $V|U$ 
 and  its associated
 symmetric bilinear form $b_q$. The matrix of this bilinear form relative
 to any fixed basis is a symmetric matrix of odd rank and in particular, 
 if $U$ is of pure characteristic two (i.e., the residue field of
 any point of $U$ is 
 of characteristic two), then this matrix is also alternating
 and is  hence singular, immediately implying that $q$ cannot be regular.
 However, computing the 
 the determinant of such a matrix  in {\em formal variables} $\{\zeta_i,
\zeta_{ij}\}$
  shows that it is twice the 
 following  polynomial 
$$P_{3}(\zeta_{i},\zeta_{ij})
=4\zeta_{1}\zeta_{2}\zeta_{3}+\zeta_{12}\zeta_{13}\zeta_{23}
-(\zeta_{1}\zeta_{23}^{2}+\zeta_{2}\zeta_{13}^{2}+\zeta_{3}\zeta_{12}^{2}).$$
 The value $P_3(q(e_i),b_q(e_i,e_j))$ corresponding to a basis 
 $\{e_1,e_2,e_3\}$ is called the 
{\em half-discriminant} of $q$ relative to that basis,  
 and $q$ is said to be semiregular if its half-discriminant is 
 a unit. It turns out that this definition is independent of the 
 basis chosen (\S 3, Chap.IV, \cite{Knus}). Even if $V|U$ were 
 only projective (i.e., locally-free but not free), the semiregularity 
 of $q$ may be defined as the semiregularity of $q\otimes_R R_{\mathfrak m}$ 
 for each maximal ideal ${\mathfrak m}\subset R$, and it turns out that with 
this definition, the notion of a quadratic form
 being semiregular is local and is 
 well-behaved under base-change (Prop.3.1.5, Chap.IV, \cite{Knus}).
 We may thus define the subfunctor of $\hbox{\rm Quad}_V$
 of semiregular quadratic forms. This subfunctor is 
 represented by a $\hbox{\rm GL}_V$-invariant
  open subscheme $i:\hbox{\rm Quad}_V^{sr}\hookrightarrow 
 \hbox{\rm Quad}_V$ because, 
 over each affine open subscheme $U\hookrightarrow X$
 which trivialises $V$, it  
 corresponds to localisation by the non-zerodivisor $P_3$. Note that   
 this canonical open immersion 
 is affine and 
 schematically dominant as well.
 We next turn to semiregular bilinear forms. Recall that 
 we had defined a bilinear form $b$ to be semiregular iff its 
 induced quadratic form $q_b$ is semiregular.  
 Thus by definition, $\hbox{\rm Bil}_V^{sr}$ is the fiber product:
$$\begin{CD}
\hbox{\rm Bil}_V @>{p}>> \hbox{\rm Quad}_V\\
@A{i'}AA @AA{i}A\\
\hbox{\rm Bil}_V^{sr} @>>{p'}> \hbox{\rm Quad}_V^{sr}
\end{CD}$$
Since $p$ is a Zariski-locally-trivial principal $\hbox{\rm Alt}^2_V$-bundle,
 it is smooth and surjective (in particular faithfully flat). 
 It therefore follows that the affineness and 
 schematic dominance of $i$ imply those of 
 $i'.$ We record these facts below. 
\begin{proposition}\label{bilvsr-def}
The open immersion $\hbox{\rm Bil}_V^{sr}\hookrightarrow \hbox{\rm Bil}_V$
 is a $\hbox{\rm GL}_V$-equivariant schematically dominant affine morphism. 
 Further this open immersion behaves well under base-change (relative 
 to $X$).
\end{proposition}
\paragraph{Reduction of Proof of
 Theorem \ref{bilinear-forms-as-specialisations} 
to Theorem \ref{multiplication-table-wrt-bilinear}.}
 We first recall the following crucial fact 
(see (1), Prop.3.2.4, Chap.IV \cite{Knus}): 
{\em The even Clifford algebra of a semiregular quadratic form is an 
 Azumaya algebra.} 
 Using this fact and the definition of $\Upsilon'$, we see that
 the morphism $\Upsilon'$ restricted to $\hbox{\rm Bil}_V^{sr}$  
  factors through $\azuWw$ by a morphism $\Upsilon^{sr}$ such 
 that   the following  diagram is commutative  
$$\begin{CD}
\hbox{\rm Bil}_V @>{\Upsilon'}>> \aaWw \\
@AAA @AAA \\
\hbox{\rm Bil}_V^{sr} @>>{\Upsilon^{sr}}> \azuWw 
\end{CD}$$
 where the vertical arrows are the canonical open immersions. 
The above diagram base changes well in view of 
(2), Theorem \ref{repbility-smoothness-azu}, Prop.\ref{bilvsr-def} and 
 (3), Theorem \ref{Bourbaki}.  
Notice that since the structure morphism $\hbox{\rm Bil}_V\longrightarrow X$
 is an affine morphism, and since the same is true of $\aaWw\longrightarrow X$,
 it is also true of $\Upsilon'.$ 
 In particular, $\Upsilon'$ is quasi-compact and separated, and 
 therefore has a schematic image by case 
 (1) of Prop.\ref{sch-img-existence}. The same is true of
 each of the two vertical arrows and of 
 $\Upsilon^{sr}$ 
 in view of  
 Prop.\ref{bilvsr-def} and  (1) of Theorem 
\ref{repbility-smoothness-azu}.
 Further,  as noted in Prop.\ref{bilvsr-def},  
  $\hbox{\rm Bil}_V^{sr}\hookrightarrow\hbox{\rm Bil}_V$ is schematically 
 dominant and therefore by (5), Prop.\ref{sch-img-props},
 the limiting scheme of the former in the latter 
 is the latter itself. 
  So using the commutative diagram above, the transitivity 
 of the schematic image (assertion (3), Prop.\ref{sch-img-props}),
 and the definition of $\spazuWw$ (assertion (1), 
Theorem \ref{smoothness-spazu}), we see 
 that in order to prove (1), Theorem \ref{bilinear-forms-as-specialisations},
 it is enough to show that
\begin{description}
\item[(*)] $\Upsilon^{sr}$ is schematically dominant and surjective, and 
 $\Upsilon'$ is a closed immersion. 
\end{description}
We now claim that the above properties are equivalent to 
\begin{description}
\item[(**)] $\Upsilon^{sr}$ is proper and  $\Upsilon'$ is a closed immersion. 
\end{description}
 Suppose {\bf (**)} holds. To show {\bf (*)}, we only need 
 to show that $\Upsilon^{sr}$ is surjective and schematically dominant.
 From {\bf (**)} it follows 
 that  $\Upsilon_K^{sr}:=\Upsilon^{sr}\otimes_X K$ is functorially injective 
 and proper for each algebraically closed field $K$ with 
 an $X$-morphism $\hbox{\rm Spec}(K)\longrightarrow X.$
 
 That both the $K$-schemes $\hbox{\rm Bil}_V^{sr}\otimes_X K $ and 
 $\azuWw\otimes_X K$ are integral and 
 smooth of the same dimension follows from 
 the smoothness of relative dimension 9
  and geometric irreducibility $/X$ of 
 $\hbox{\rm Bil}_V^{sr}$ (which is obvious), and of 
  $\azuWw$  from (3), Theorem \ref{repbility-smoothness-azu}. 
 Since $\Upsilon_K^{sr}$ is differentially injective at each closed 
 point, it has 
 to be a smooth morphism by Theorem 17.11.1 of EGA IV \cite{ega4}
 and 
 thus has to be an open map. But by {\bf (**)} 
 it is also proper and hence a closed map. 
 Thus $\Upsilon_K^{sr}$ 
 is bijective etale, and hence an isomorphism. This also gives 
 that $\Upsilon^{sr}$ is surjective.  
 Now from Cor.11.3.11 of EGA IV and from the flatness of
 $\hbox{\rm Bil}_V^{sr}$ 
 over $X$,  
 it follows that $\Upsilon^{sr}$ is itself flat, and hence schematically 
 dominant since it is faithfully flat (being already surjective). 
 Therefore {\bf (**)}$\implies${\bf(*)}. 

The property of a morphism being proper is
 local on the target (see for e.g., (f), Cor.4.8, Chap.IV, \cite{Hartshorne}) 
  and the same is true of the property of being a closed immersion. 
 Therefore, in verifying (**), we may assume that $V$ is free over $X$ 
 (so that $W=\Lambda^{even}(V)$ is also free over $X$ and 
  and $w$ is part of a global basis). We are now in the situation of 
 Theorem \ref{multiplication-table-wrt-bilinear}. Granting it, we see immediately 
 from the multiplication table that {\bf (**)} holds.
 For the table shows that the composition of the following $X$-morphisms
 $$\hbox{\rm Bil}_V\stackrel{\Upsilon'}{\longrightarrow}\aaWw\stackrel{\hbox{\rm\tiny CLOSED}}{\hookrightarrow}\algW$$ is a closed immersion, which 
 implies that $\Upsilon'$ is also a closed immersion. Further, 
  the multiplication 
 table also shows that both $\Upsilon'$ and $\Upsilon^{sr}$ satisfy the 
 valuative criterion for properness, and are therefore
 proper. Thus 
the conditions {\bf (**)} are verified.  So we have reduced  
 the proof of (1), Theorem \ref{bilinear-forms-as-specialisations}
 to Theorem \ref{multiplication-table-wrt-bilinear}. As for statement (2) 
 of Theorem \ref{bilinear-forms-as-specialisations}, firstly, 
  the  involution 
 $\Sigma$ of  $\hbox{\rm Id-}w\hbox{\rm -Assoc-Alg}_W$ 
 defines a unique involution (also denoted by $\Sigma$) on 
 the scheme of specialisations $\hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_W$
 (leaving the open subscheme $\hbox{\rm Id-}w\hbox{\rm -Azu}_W$
 invariant) because of   
  the defining property of the 
 schematic image involved; for we may verify  that an automorphism of 
 a scheme $T$ which leaves an open subscheme $U$ stable will also leave 
 stable the limiting scheme of $U$ in $T$ (of course we assume here that 
 the canonical open immersion $U\hookrightarrow T$ is a quasi-compact 
 open immersion, which ensures the existence of the limiting scheme). Secondly,
 a glance at the multiplication table of Theorem \ref{multiplication-table-wrt-bilinear}
 keeping in view the definition of opposite algebra shows that indeed 
 the induced $\Sigma\in\hbox{\rm Aut}_X\left(\hbox{\rm Bil}_V\right)$ takes the $T$-valued point 
$B=(b_{ij})$ to $\hbox{\rm transpose}(-B)=(-b_{ji}).$ Finally, assertion (3)
 of Theorem \ref{bilinear-forms-as-specialisations} is a consequence of 
 (1) taking into account (3), Theorem \ref{smoothness-spazu}.

\paragraph{\bf Proof of Theorem \ref{multiplication-table-wrt-bilinear}.}
Given $B=(b_{ij})\in\hbox{\rm Bil}_V(T)$, by our definition above, 
 $(\Upsilon'(T))(B)=A_B$ is the algebra structure induced from 
 the linear isomorphism $\psi_B:C_0(V_T, q_B)\cong \Lambda^{even}(V_T)$ 
 of (2d), Theorem \ref{Bourbaki}. 
The stated multiplication table for $A=A_B$
 is a consequence of straightforward calculation, keeping in 
 mind (2f), Theorem \ref{Bourbaki} and the standard properties 
 of the multiplication in the even Clifford algebra $C_0(V_T,q_B).$
{\bf Q.E.D., Theorems \ref{multiplication-table-wrt-bilinear} \& \ref{bilinear-forms-as-specialisations}.}

\paragraph{\bf Proof of Theorem \ref{Thetaisom-Upsilonisom}.}
The proof follows by comparing the multiplication table relative to 
 $\Theta$ as computed in Theorem \ref{mult-table-wrt-Theta} with 
 the multiplication table relative to $\Upsilon$ of Theorem  
\ref{multiplication-table-wrt-bilinear} computed above. 
{\bf Q.E.D., Theorem \ref{Thetaisom-Upsilonisom}.}


%Shree Maataaravindaarpanam

\section{\label{sec5} Properties of Specialisations/Self-duality: \ref{structure-of-specialisation}--\ref{surjectivity-affine-limited}}

 In this section, we prove Theorem \ref{structure-of-specialisation} and 
 use it alongwith 
 Theorem \ref{bilinear-forms-as-specialisations} to prove 
 Theorem \ref{surjectivity-limited}. Thereafter we investigate the 
 specialised algebras when the underlying bundle is self-dual and 
 prove Theorems \ref{relations-with-self-duality}
 and \ref{surjectivity-affine-limited}.


\paragraph{Proof of Theorem \ref{structure-of-specialisation}.}
 Let $W$ be the rank 4 vector bundle underlying 
 the specialised algebra $A$ and $w\in\Gamma(X, W)$
 be the global section corresponding to 
 $1_{\cal A}.$ We choose an affine open covering 
${\{U_i\}}_{i\in I}$ of $X$ such that $W|U_i$ is trivial and 
 $w|U_i$ is part of a global basis $\forall\thinspace i.$  
 Therefore on the one hand, 
for each $i\in I$, we can find a linear isomorphism
 $\zeta_i:\Lambda^{even}\left({\cal O}_X^{\oplus 3}|U_i\right)\cong 
 W|U_i$ taking $1_{\Lambda^{even}}$ onto $w|U_i.$ The $(w|U_i)$-unital
 algebra structure  $A|U_i$ induces via $\zeta_i$ an algebra structure $A_i$ on
 $\Lambda^{even}\left({\cal O}_X^{\oplus 3}|U_i\right)$ (so that $\zeta_i$
 becomes an algebra isomorphism). Since $A_i$ is also 
 a specialised algebra structure ((3), Theorem \ref{smoothness-spazu}), 
 by Theorem \ref{bilinear-forms-as-specialisations} we can on the other
 hand also
 find a quadratic form 
 $q_i$ on ${\cal O}_X^{\oplus 3}|U_i$ induced from a bilinear form 
 $b_i$ so that the algebra structure $A_i$ is precisely the one 
 induced by the linear isomorphism 
 $\psi_{b_i}:C_0\left({\cal O}_X^{\oplus 3}|U_i, q_i\right)
\cong\Lambda^{even}\left({\cal O}_X^{\oplus 3}|U_i\right)$ 
given by (2d) of Theorem \ref{Bourbaki}.  
 For each pair of indices $(i,j)\in I\times I$, let $\zeta_{ij}$ and 
 $\phi_{ij}$ be defined so that the following diagram commutes:
$$\begin{CD}
C_0({\cal O}_X^{\oplus 3}|U_{ij}, q_i|U_{ij}) 
@>{\psi_{b_i}|U_{ij}}>{\cong}> \Lambda^{ev}({\cal O}_X^{\oplus 3}|U_{ij})
@>{\zeta_i|U_{ij}}>{\cong}> A|U_{ij}\\
@V{\phi_{ij}}V{\cong}V @V{\zeta_{ij}}V{\cong}V @V{=}VV\\
C_0({\cal O}_X^{\oplus 3}|U_{ij}, q_j|U_{ij}) 
@>{\cong}>{\psi_{b_j}|U_{ij}}> \Lambda^{ev}({\cal O}_X^{\oplus 3}|U_{ij})
@>{\cong}>{\zeta_j|U_{ij}}> A|U_{ij}
\end{CD}$$
The above diagram means that the algebras $A_i$ glue along 
 $U_{ij}:=U_i\cap U_j$ via $\zeta_{ij}$ to give (an algebra bundle 
 isomorphic to) $A$, and in the same vein, the even Clifford algebras 
 $C_0({\cal O}_X^{\oplus 3}|U_i, q_i)$ glue along the $U_{ij}$ via 
 $\phi_{ij}$ to give $A$ as well.  
Now consider the similarity $g_{l_{ij}}^{\phi_{ij}}=s_{-1}^{+}(\phi_{ij}):({\cal O}_X^{\oplus 3}|U_{ij}, q_i|U_{ij})\thinspace\cong_{l_{ij}}\thinspace({\cal O}_X^{\oplus 3}|U_{ij}, q_j|U_{ij})$ with multiplier $l_{ij}:=\hbox{\rm det}(\phi_{ij})^{-1}$ given by (c), 
 Theorem \ref{lifting-of-isomorphisms}. Since $s_{-1}^{+}$ 
 is  multiplicative, and since $\phi_{ij}$ satisfy 
 the cocycle condition, it follows that $s_{-1}^{+}(\phi_{ij})$ also 
 satisfy the cocycle condition and therefore glue the
 ${\cal O}_X^{\oplus 3}|U_i$
 along the $U_{ij}$ to give a rank 3 vector bundle $V$ on $X.$ (Observe 
 that unfortunately the $q_i$ do not glue\thinspace!) We shall now 
 revert to the notations of Section \ref{sec3}. By Theorem \ref{b-co-phi-lambda-indep-of-q}, we have 
$$h_s(g_{l_{ij}}^{\phi_{ij}},l_{ij},q_i|U_{ij},q_j|U_{ij})=\begin{pmatrix}1 & {\bf 0}\\{\bf 0}& l_{ij}^{-1}\Lambda^2(g_{l_{ij}}^{\phi_{ij}})\end{pmatrix}$$
 which means that ${(\phi_{ij})}_{\Lambda^2}=\hbox{\rm det}(\phi_{ij})\times
 \Lambda^2(g_{l_{ij}}^{\phi_{ij}}).$ This immediately implies part (1) of
 assertion (a) of Theorem \ref{structure-of-specialisation}, from which 
 parts (2)---(4) can be deduced using the standard properties of the
  determinant  
 and the perfect 
 pairings between suitable exterior powers of a bundle. To prove (b), let 
 $A=C_0(V,q).$ Using a trivialisation ${\{U_i\}}_{i\in I}$
 for the 
 quadratic bundle $(V,q)$ with transition functions  
 $g_{ij}$, and remembering that by definition the $g_{ij}$ are
 isometries,  we have the following commutative diagram on the 
 intersections $U_{ij}$:
$$\begin{CD}
C_0({\cal O}_X^{\oplus 3}|U_{ij}, q_i|U_{ij}) 
@>{\psi_{b(q_i)}|U_{ij}}>{\cong}> \Lambda^{ev}({\cal O}_X^{\oplus 3}|U_{ij})
\\
@V{C_0(g_{ij},1,q_i|U_{ij},q_j|U_{ij})}V{\cong}V @V{\cong}V{h(g_{ij},1,q_i|U_{ij},q_j|U_{ij})}V \\
C_0({\cal O}_X^{\oplus 3}|U_{ij}, q_j|U_{ij}) 
@>{\cong}>{\psi_{b(q_j)}|U_{ij}}> \Lambda^{ev}({\cal O}_X^{\oplus 3}|U_{ij})
\end{CD}$$
But by the formula in Theorem \ref{b-co-phi-lambda-indep-of-q}, we have 
$$h_s(g_{ij},1,q_i|U_{ij},q_j|U_{ij})=\begin{pmatrix}1 & {\bf 0}\\{\bf 0}& 
\Lambda^2(g_{ij})\end{pmatrix}$$
which immediately implies that $C_0(V,q)/{\cal O}_X.1_{C_0}\cong \Lambda^2(V).$
{\bf Q.E.D., Theorem \ref{structure-of-specialisation}.}


\paragraph{Proof of `if' part of Theorem \ref{surjectivity-limited}.}
 Suppose that the hypotheses hold; by (1) 
 $\exists L\in\hbox{\rm Pic}(X)$ such that  $\hbox{\rm det}(A)\cong
L^{\otimes 2}$ and by (2), 
  we may choose a splitting $A\cong {\cal O}_X.1_A\oplus 
 (A/{\cal O}_X.1_A).$ Using assertion (1) of (a),
 Theorem \ref{structure-of-specialisation}, we see that there exists a
 rank 3 vector bundle $V'$ on $X$ such that
 $$A\cong {\cal O}_X.1_A\oplus (A/{\cal O}_X.1_A)\cong {\cal O}_X.1_A\oplus 
 (L^{\otimes 2}\otimes \Lambda^2(V'))\cong {\cal O}_X.1_A \oplus 
 \Lambda^2(V'\otimes L)\cong \Lambda^{even}(V)$$ where  
 $V:=V'\otimes L$ and the last isomorphism is chosen so as 
 to map ${\cal O}_X.1_A$ isomorphically onto ${\cal O}_X.1_{\Lambda^{ev}(V)}.$
 Therefore if $(W,w):=(\Lambda^{even}(V),1)$, then 
 by the above identification $A$ induces an element of $\spazuWw(X)$, 
 and since $\Upsilon:\hbox{\rm Bil}_V\cong \spazuWw$ is an $X$-isomorphism 
 by (1), Theorem \ref{bilinear-forms-as-specialisations}, it follows that 
 there exists a global quadratic form $q=q_b$ induced from a bilinear form 
 $b$ such that the algebra structure $\Upsilon(b)\cong A.$ To finish
 the proof, we only have to remember that
 $\Upsilon(b)$ is the algebra structure induced from the 
 linear isomorphism  $\psi_b:C_0(V,q=q_b)\cong \Lambda^{even}(V)=:W$ of 
 (2d), Theorem \ref{Bourbaki}, which preserves 1 by (2a) of the same 
 Theorem.
\paragraph{\bf Proof of `only if' part of Theorem \ref{surjectivity-limited}.}
Suppose $\exists\thinspace\vq$ such that $\covq\cong A$, 
 and that $q$ is induced from a bilinear form $b.$ Then by (2d), Theorem 
 \ref{Bourbaki}, we have a linear isomorphism
 $\psi_b:C_0(V,q=q_b)\cong \Lambda^{even}(V).$ 
 Since $\psi_b$ preserves 1 by (2a) of the same 
 Theorem, we get (2).
 Further (1) follows from assertion
 (b) of Theorem \ref{structure-of-specialisation}.
\paragraph{\bf Proofs of assertions on validity of hypotheses (1) \& (2).}
If (i) holds, the validity of (2) is a simple exercise in commutative 
 algebra. Since $A$ is a quaternion algebra bundle 
 by Prop.\ref{spln-is-quaternion}, we have the associated trace map
 $tr_{\sigma_A}:A\longrightarrow {\mathbb A}^1_X.$ Since (ii) is 
 equivalent to $2\in\Gamma(X, {\cal O}_X^*)$, when (ii) holds, 
the map $2^{-1}tr_{\sigma_A}$
 splits the canonical inclusion ${\cal O}_X.1_A\hookrightarrow A$, 
 showing that (2) holds. 
 If $A$ is Azumaya, then as we saw in the discussion following 
 Theorem \ref{Witt-bijective-on-q3sr},
 $A\cong \covq$ with $\vq$ semiregular. So (1) follows from (b), Theorem 
 \ref{structure-of-specialisation}. 
{\bf Q.E.D., Theorem \ref{surjectivity-limited}.}

\paragraph{Relations with Self-Duality.}
In the following we let $X$ be a scheme and $W$ a rank 4 vector bundle 
 over $X.$ We recall the following result: 
\begin{proposition}[Prop.3.3, Part A, \cite{tevb-paper1}]
\label{sch-azu-props}\ 
\begin{description}
\item[1.] 
Let $T$ be an $X$-scheme and $A$ an associative unital algebra 
 structure on $W_T:=W\otimes_X T.$ 
 Then the subset
$$U(T,A):=\{ t\in T\ |\ {\cal A}_t \hbox{ is an Azumaya }{\cal O}_{T,t}-\hbox{algebra}
\}$$ is an open (possibly empty) subset. When $U(T, A)$  is nonempty,
 denote by the same symbol the  
  canonical open subscheme structure. Then if 
 $f:T'\longrightarrow T$ is an $X$-morphism such that 
 the topological image intersects $U(T, A)$, then 
 $U(T', f^*(A)=A\otimes_T T')\cong U(T, A)\times_{T}T'$ as open 
 subschemes of $T'.$  Further  
  $U(T, A)\hookrightarrow  T$ is an affine morphism. 
\item[2.] $U(T,A)$ is the maximal open subset restricted to which 
 $A$ is  Azumaya. 
\item[3.] Further let $f:T'\longrightarrow T$ be a morphism of $X$-schemes 
 such that $f^*(A)$ is Azumaya. Then $f$ factors 
 through the open subscheme $U(T,A)$ defined above. 
\end{description}
\end{proposition}

\paragraph{Proof of Theorem \ref{relations-with-self-duality}.}
We first make certain observations when 
  $X$ is any reduced scheme and $W$ is a rank $n^2$ vector bundle 
 over $X$ with nowhere-vanishing global section $w.$ It is not hard to 
 see that the structure morphism $\azuWw\longrightarrow X$ is in fact a 
 morphism of finite presentation. Hence, in view of Prop.17.5.7, EGA IV 
 \cite{ega4}, assertion  (3) of Theorem 
\ref{repbility-smoothness-azu} implies that $\azuWw$ is reduced. 
 Since $\spazuWw$ is the schematic image of $\azuWw$, it follows from 
 (1), Prop.\ref{sch-img-props} that $\spazuWw$ is reduced as 
 well.  When  $X$ is integral, it is easy to see that 
 $\azuWw$ is integral as well. Consider any affine open subscheme 
 $U=\hbox{\rm Spec}(R)\hookrightarrow X$ 
such that $W|U$ is trivial and $w|U$ is part of a global 
 basis. There are $(w|U)$-unital Azumaya algebra structures on 
 $W|U$ which are isomorphic to the $(n\times n)$-matrix algebra over 
 $R.$ Consider the orbit morphism $\hbox{\rm Stab}_{w|U}\longrightarrow 
 \hbox{\rm Id-}(w|U)\hbox{\rm -Azu}_{W|U}$ corresponding to 
 one such algebra structure. Assertions (2) and   
 (3) of Theorem \ref{repbility-smoothness-azu} show that the topological 
 image of this morphism is dense.
 Further, $\hbox{\rm Stab}_{w|U}$ is integral since it is 
$\cong {\mathbb A}^{12}_U$ and since $U$ is integral. 
 Thus 
 $\hbox{\rm Id-}(w|U)\hbox{\rm -Azu}_{W|U}$ is integral.
Recall (e.g., Prop.2.1.6 \& Cor.2.1.7, Chap.0.2, EGA I \cite{ega1}) 
 that a nonempty topological space---whose set of irreducible components 
is locally finite---is locally irreducible iff each 
 irreducible component is open; further  
 it is irreducible iff it is locally irreducible and 
 connected. Now 
 since $\azuWw$ can be covered by irreducible open subschemes which 
 pairwise intersect (as $X$ is irreducible), 
 it follows that $\azuWw$ is integral as well. Therefore 
 (1), Prop.\ref{sch-img-props} implies that 
 $\spazuWw$ is also integral in the case when $X$ is integral. 
 Using again the facts that $\azuWw$ behaves well under base change and
 that the schematic image of an irreducible scheme is irreducible 
 (by (1), Prop.\ref{sch-img-props}), we may infer from the foregoing 
 that $X$ is irreducible iff $\azuWw$ is irreducible iff 
 $\spazuWw$ is irreducible. 


We next  observe that assertion
 (b) follows from (a) and from the following result: {\em The 
 vector bundle underlying a rank  4  Azumaya algebra bundle $A$ is self-dual.} 
 To see this, first remember from Prop.\ref{spln-is-quaternion}
 that $A$ comes with a standard 
 involution $\sigma=\sigma_A.$
 The associated norm $n_\sigma:x\mapsto x.\sigma(x)$
 is a (global) quadratic
 form with values in ${\cal O}$ 
 and its associated bilinear form $b_{n_\sigma}$ may be 
 given in terms of the associated 
 trace $tr_\sigma$ as $b_{n_\sigma}(x,x')=tr_\sigma(x\sigma(x'))=x\sigma(x')+
x'\sigma(x).$ Now $n_\sigma$ is nonsingular (=regular), as 
 may be verified locally over affine open subschemes---for
 details see para 7.3.5, Chap.I, \cite{Knus}.

As for assertion (a), we begin with a general remark: if $X'\longrightarrow X$
 is a surjective $k$-morphism, $k$ being the algebraically closed 
  base field, and if (a) holds for the pair $(X',A':=A\otimes_X X')$, 
 then it also holds for the pair $(X,A)$ where $A$ is an associative unital 
 algebra structure on $W.$ For, by (3), Prop.\ref{sch-azu-props}, 
 $U(X',A')=X'\Longrightarrow U(X,A)=X.$ Since the canonical morphism 
 $X_{RED}\hookrightarrow X$ is a surjective closed immersion, 
 we may therefore assume that $X$ is reduced. 
 Let $x\in U(X,A)\cap X_\alpha$ where $X_\alpha$ is an irreducible 
 component of $X$ given the canonical reduced induced closed subscheme 
 structure. If (a) holds for $(X_\alpha, A\otimes_X X_\alpha)$, then 
 by (1), Prop.\ref{sch-azu-props}, we have that $A\otimes_X X_\alpha$ 
 is Azumaya, and hence by (3) of the same Proposition,
 the canonical closed immersion $X_\alpha\hookrightarrow X$ factors through 
 $U(X, A).$ Now as  $X$ is connected, we observe that  
 (a) holds for $X$ if it holds for each irreducible component given
 the reduced induced closed subscheme structure (which is 
 also a proper scheme of finite type over $k$); thus we 
 may as well assume that $X$ is irreducible. By Chow's Lemma (Theorem 
 5.6.1, EGA II \cite{ega2}) there exists an integral projective scheme 
 $X'$ of finite type over $k$ alongwith a surjective projective 
 $k$-morphism $X'\longrightarrow X.$ Therefore we may further assume 
 that $X$ is projective. Finally, we may also assume that $X$ is normal, 
 for if $\widetilde{X}$ is the normalisation of $X$, then the 
 canonical morphism $\widetilde{X}\longrightarrow X$ is finite and 
 surjective, hence also projective. So we have reduced to the case 
 when $X$ is an integral normal projective scheme over $k.$ However, 
 the following proof works with the hypothesis of 
 $X$ being regular in codimension 1, which is satisfied when $X$ is 
 normal.


 Now  given an associative $w$-unital algebra structure $A$ on $W$
 such that the open subset $U(X,A)$ of Prop.\ref{sch-azu-props} is 
 nonempty, the restriction to the dense open subset $U(X,A)$ of 
 the section over $X$ corresponding to $A$ factors through $\azuWw$ as 
 a morphism (and not just topologically); further, since both 
 $X$ and $\spazuWw$ are reduced,  
 this section factors through $\spazuWw$ itself.
  Thus any $A$ such that $U(X,A)\neq\emptyset$
 corresponds to a specialised algebra structure. 
Let there be given an isomorphism of vector bundles  
 $\phi:W\cong W^\vee$ and an $A$ 
  such that $U(X,A)\neq\emptyset.$ To prove (a), we must 
 show that $U(X,A)=X.$ We proceed by contradiction: suppose 
 $D(X,A):=X\backslash U(X,A)$ is nonempty. Consider the composition of 
 the following morphisms of vector bundles: 
$$W\otimes_X W\equiv A\otimes_X A^{op}
\stackrel{(a\otimes b)\mapsto 
 (x\mapsto axb)}{\longrightarrow} \hbox{\rm End}_{\thinspace
 Linear\thinspace}(A)
\equiv W^\vee\otimes_X W
\stackrel{\phi\times Id}{\longrightarrow} W\otimes_X W$$ 
The above composite gives an endomorphism of the vector bundle $W\otimes W$ 
 which is an isomorphism precisely at the local rings of the points of 
 $U(X,A).$ Therefore, the induced element $s(\phi)\in\Gamma(X, {\cal O}_X)
\equiv \hbox{\rm H}^0(X, \hbox{\rm End}(\hbox{\rm det}(W\otimes W)))$ 
 goes into the maximal ideal of the local ring at each point of $D(X,A)$ and 
 to a unit of the local ring at each point of $U(X,A).$ It follows 
 that if $D(X,A)$ is the irredundant union of divisors
 $\{D_i|1\leq i\leq n\}$, then the divisor defined by $(s(\phi), {\cal O}_X)$ 
 namely $\Sigma_i n_iD_i$ is principal. Under the present hypotheses on 
 $X$, every principal divisor has degree zero (see for e.g., (d),
 Ex.6.2, Chap.II, \cite{Hartshorne}), so that 
 we get  
 $0=\Sigma_i n_i.(\hbox{\rm degree}(D_i)).$ This is a contradiction, since 
 $n_i\geq 1$ as it is the order of 
 vanishing of $s(\phi)$ on $D_i$ for each $i$, and since   
 the degree of each $D_i$ is positive (see for e.g., (a), Prop.7.6, Chap.I,  
 \cite{Hartshorne}). 
  {\bf Q.E.D., Theorem 
 \ref{relations-with-self-duality}.}

\paragraph{Proof of assertions (a)---(d) of 
 Theorem \ref{surjectivity-affine-limited}.}
The proofs  follow 
 essentially from the properties 
 of $\spazuWw\longrightarrow X$ and $\azuWw\longrightarrow X$
 as mentioned in Theorems \ref{smoothness-spazu} and
 \ref{repbility-smoothness-azu}. In this regard see also 
 the first paragraph of the proof of Theorem \ref{relations-with-self-duality}.
  We indicate  
 proofs for the not-so-obvious assertions,
 especially (c)  and for 
 the implication ($X$ irreducible
 $\Rightarrow$ $D_X$ irreducible) of (a). The 
 converse implication would follow from the fact that $D_X\longrightarrow X$ 
 is topologically surjective. Since the structure morphisms 
 $\spazuWw\longrightarrow X$ and $\azuWw\longrightarrow X$ are smooth, 
 they are faithfully flat (hence surjective).  
 Therefore in view of (b)---which is actually the result of good base-change 
 properties of $\spazuWw$ and $\azuWw$ relative to $X$, 
 the  irreducible components of $D_X$ and of $\spazuWw$ 
 are induced from those of $X$ by pullback, provided we check 
 the particular case when $X$ is irreducible and reduced. 
 
So let $X$ be integral and first assume that $W$ is trivial and $w$ is
 part of a global basis. Without loss of generality,  
 we may take $(W,w)=(\Lambda^{even}(V), 1)$ for $V$ a rank 3 
trivial vector bundle over $X.$ After fixing a suitable 
 basis for $V$, we may define the morphism $\Theta$, which 
 by Theorem \ref{Thetaisom} is an isomorphism 
 that maps the closed subset 
 $(\hbox{\rm Quad}_V\times_X \hbox{\sf L}_w) \backslash(\hbox{\rm Quad}_V^{sr}\times_X\hbox{\sf L}_w)$ onto
 $D_0:=\spazuWw\backslash\azuWw.$ Therefore the irreducibility of 
 $D_0$ is equivalent to that of 
  the closed subset
 $\hbox{\rm Quad}_V\backslash\hbox{\rm Quad}_V^{sr}.$ Recall 
 from the discussion on semiregular forms (page \pageref{semireg-bil-forms},
 Section \ref{sec4}) that the open subscheme $\hbox{\rm Quad}_V^{sr}$ 
 corresponds to localisation by the polynomial $P_3.$ This 
 polynomial is irreducible as an element of $R[\zeta_i, \zeta_{ij}]\cong 
 R[\hbox{\rm Quad}_V]$ when $X=\hbox{\rm Spec}(R)$ is affine, though it is not 
 clear if it is a prime element (unless we assume something more 
 e.g., $R$ a UFD). The
 closed subset $\hbox{\rm Quad}_V\backslash\hbox{\rm Quad}_V^{sr}$ may be 
 given the canonical closed subscheme structure $Z(P_3)$ corresponding to 
 the vanishing of $P_3.$ Let $q^{(2)}$ be the global quadratic form 
 on $V$ given by $(x_1, x_2, x_3)\mapsto x_1x_2.$ It can be checked 
 that $q^{(2)}$ is not semiregular, but that its restriction to 
 the rank two (direct summand) vector subbundle generated by $\{e_1, e_2\}$
 is regular. Therefore the $X$-valued point corresponding to $q^{(2)}$ 
 lands topologically inside the closed subset underlying $Z(P_3).$
 Consider the orbit morphism $O(q^{(2)}):\hbox{\rm GL}_V\longrightarrow 
 \hbox{\rm Quad}_V$ corresponding to this $X$-valued point which also  
 lands topologically inside $Z(P_3).$ It will follow from assertion 
 (2), Theorem \ref{stratification}, that the topological image of 
 $O(q^{(2)})$ is dense in $Z(P_3).$ On the other hand this topological 
 image is irreducible, since $\hbox{\rm GL}_V\cong {\mathbb A}^9_X.$ 
  It follows that $\hbox{\rm Quad}_V\backslash
\hbox{\rm Quad}_V^{sr}$, and hence $D_0$,
  is irreducible in the case when $W$ is trivial and 
 $w$ is part of a global basis over $X.$ Since 
 the reduced closed subscheme structure on $Z(P_3)$ is given by 
 the radical of the ideal $(P_3)$, it follows that $\sqrt{(P_3)}$
 is the minimal prime divisor of $(P_3)$ and 
 by Krull's {\em Hauptidealsatz} this prime has height
 1. Therefore, the codimension of $D_0$ is also 1 in the present 
 case. 

 Now consider the case of a general $(W,w)$, choose an affine open covering 
  ${\{U_i=\hbox{\rm Spec}(R_i)\}}_{i\in I}$ 
 of $X$ such that $W_i:=W|U_i$ is trivial and $w_i:=w|U_i$ is part 
 of a global basis $\forall\thinspace i.$  The subset  
  $D_i:=\spazuWiwi\backslash\azuWiwi$ is irreducible for each $i$ by 
 the preceding paragraph. 
 On the other hand, by Theorems \ref{repbility-smoothness-azu}
 and  \ref{smoothness-spazu}, the subsets $D_i$ form 
 an open cover of $D:=\spazuWw\backslash\azuWw.$ 
 Since $X$ is irreducible, we
 have $D_i\cap D_j\neq\emptyset$  when $i\neq j.$
  Thus $D$ is 
 locally irreducible and connected, and hence irreducible (for e.g., by 
 Cor.2.1.7, Chap.0, EGA I \cite{ega1}).   
 Since $X$ is integral and noetherian, 
 assertion (2) of Theorem \ref{smoothness-spazu} implies that $\spazuWw$ is 
 integral and noetherian as well.
  Therefore the codimension of $D$ in 
 $\spazuWw$ is atleast 1. On the other hand (for e.g., 
 by Prop.14.2.3, Chap.0,  EGA IV, \cite{ega4})  
 this codimension is bounded above by $1=\hbox{\rm Codim}\thinspace
(D_i, \spazuWiwi)$
 for any $i$, since 
 $\spazuWiwi\hookrightarrow\spazuWw$
 is an open subset whose intersection with $D$ is precisely 
 $D_i.$ 
\paragraph{Proof of assertion (e) of 
 Theorem \ref{surjectivity-affine-limited}.} Let us remind the reader 
 that the pullbacks $W\otimes_X\spazuWw$ and $W\otimes_X\azuWw$ are naturally 
 endowed with $(w\otimes_X\spazuWw)$-unital associative
 ${\cal O}_{\spazuww}$-algebra structures with which they become respectively  
 the universal specialisation and universal Azumaya algebra relative 
 to the pair $(W,w)$---for details see the proof of Theorem 3.4, Part A,
 \cite{tevb-paper1}.  
By (c) and (d), the natural map 
$\hbox{\rm Pic}(\spazuWw)\longrightarrow \hbox{\rm Pic}(\azuWw)$ is 
 surjective and its kernel is generated by the image of ${\mathbb Z}.(D_X).$
 Given an isomorphism $\phi:W\cong W^\vee$, we consider its pullback 
 to $\spazuWw$, 
 and proceeding along the lines of the proof of 
Theorem \ref{relations-with-self-duality},
 with the triple $(\spazuWw, W\otimes_X\spazuWw, \phi\otimes_X\spazuWw)$
 replacing the triple  
 $(X,A,\phi)$ there, we infer that $n.(D_X)$ is principal for some $n\geq 1.$ 
Now $W\otimes_X\azuWw$ is an Azumaya algebra bundle,
 and as seen in the discussion following Theorem \ref{Witt-bijective-on-q3sr}, 
 it is isomorphic to the even Clifford algebra of a canonically 
 obtained rank 3 quadratic bundle  on $\azuWw.$ Therefore it follows 
 from (b), Theorem \ref{structure-of-specialisation} that 
 $\hbox{\rm det}(W\otimes_X\spazuWw)$ maps to an element of 
 $2.\hbox{\rm Pic}(\azuWw).$ The remaining assertions in (e) are 
 now consequences of Theorem \ref{surjectivity-limited}.  
{\bf Q.E.D., Theorem \ref{surjectivity-affine-limited}.}


%Shree Maataaravindaarpanam

\section{\label{sec6} Stratification of the Variety of Specialisations}
In this section, we prove Prop.\ref{semilocal-unique-semireg} and 
 Theorem \ref{stratification}. 

\paragraph{Proof of Prop.\ref{semilocal-unique-semireg}.} 
Fix an $S$-basis  $\{e_1, e_2, e_3\}$ for 
 $V$, and with respect to this basis, let $q^1$ denote the
 quadratic form given by $(x_1e_1+x_2e_2+x_3e_3)\longmapsto x_1x_2+x_3^2.$  
 It is easy to see that this quadratic form is semiregular. 
 We show that any 
 semiregular quadratic form $q$ can be moved to $q^1$ i.e., that $\exists\  
 g\in \hbox{\rm GL}(V)\hbox{ such that } g\cdot q=q^1.$ 
  
 By Prop.3.17, Chap.IV, \cite{Knus}, there exists 
 a basis  $\{e'_1, e'_2, e'_3\}$ for 
 $V$ such that $q$ restricted to the submodule
 generated by $e'_1$ and $e'_2$ is {\em regular} and further such that
 $q(e'_3)\in S^{*},\ 
b_q(e'_1, e'_2)=1 
\hbox{  and  }
 b_q(e'_1, e'_3)=0=b_q(e'_2, e'_3).$
  Let $g'\in \hbox{\rm GL}(V)$
 be the automorphism  that maps $e'_i$ onto the 
 $e_i$ for each $i$ and consider the quadratic form $q':=g'\cdot q.$ 
 Then by definition of the $\hbox{\rm GL}(V)$-action on the 
 set $\hbox{\rm Quad}(V)$ of quadratic $S$-forms on $V$ we have  
$q'(e_3)\in S^{*},\ 
b_{q'}(e_1, e_2)=1 
\hbox{  and  }
 b_{q'}(e_1, e_3)=0=b_{q'}(e_2, e_3).$
 So if we assume that
 $q'(e_i)=\lambda_i\ (\Leftrightarrow q(e'_i)=\lambda_i),$ 
then 
$q'(x_1e_1+x_2e_2+x_3e_3)=\lambda_1 x_1^2+\lambda_2x_2^2+
\lambda_3x_3^2+x_1x_2\ \forall\ x_i\in S.$
 Thus it is enough to show that $q'$ can be moved to $q^1.$ 
 We look for  an invertible matrix $g''=(u_{ij})\in
 \hbox{\rm GL}(V)$ such that $g''\cdot q'=q^1.$
 Writing this condition equivalently as
 $q'={(g'')}^{-1}\cdot q^1$ and comparing the polynomials in the $x_i$ gives 
  6 equations in terms of the $u_{ij}$ and the $\lambda_i$ which are to 
 be satisfied. We choose the $u_{ij}$ as follows. 
First set $u_{11}=u_{22}=0$ and  let $u_{12}\in S^{*}$ be a free parameter.  
Since every element of  $S$  has square roots in $S$, it makes sense to 
 choose 
 $u_{31}=\pm\sqrt{\lambda_1}$ and $u_{32}=\pm\sqrt{\lambda_2}.$
 We let 
 $\alpha=1+2u_{31}u_{32}$, $\beta=1-2u_{31}u_{32}$ and 
  $u_{21}=\beta/u_{12}.$ 
 Since $q'=g\cdot q$ and since $q$ is semiregular,  
  $q'$ is also semiregular. Its half-discriminant relative to the present 
 basis of $V$ is  (remembering that $\lambda_3\in S^{*}$)   
 $d_{q'}(e_1, e_2, e_3)=\lambda_3.(4\lambda_1\lambda_2-1)\in S^{*}.$
 This implies that 
 $\alpha\beta=1-4\lambda_1\lambda_2\in S^{*}\Longrightarrow \alpha,\beta
\in S^{*}.$ Therefore it makes sense to define 
$u_{33}=\pm\sqrt{(\beta\lambda_3)/\alpha}, 
 u_{13}=-2u_{31}u_{33}u_{12}/\beta\hbox{   and   } 
 u_{23}=-2u_{32}u_{33}/u_{12}.$
Note that $u_{33}\in S^{*}.$ 
A computation shows 
 that the determinant of the matrix $g''=(u_{ij})$ defined above
 is $-u_{33}\alpha\in S^{*}$ and hence $g''$ is invertible. 
It is also easily  checked that $g''\cdot q'=q^1.$  
{\bf Q.E.D., Prop.\ref{semilocal-unique-semireg}.}

 
\paragraph{Proof of assertion (2) of Theorem \ref{stratification}.}
(We shall not prove assertion (1) since it is well-known).
 Recall 
 from the discussion on semiregular forms (page \pageref{semireg-bil-forms},
 Section \ref{sec4}) that the open subscheme $\hbox{\rm Quad}_V^{sr}$ 
 corresponds to localisation by the polynomial $P_3$ and that this 
 polynomial is prime as an element of $k[\zeta_i, \zeta_{ij}]\cong 
 k[\hbox{\rm Quad}_V].$ Here a quadratic form $q$ corresponding to 
 the point $(\lambda_i,\lambda_{ij})\in{\mathbb A}^6_k$ is 
 given by $(x_1,x_2,x_3)\mapsto \Sigma_i \lambda_ix_i^2
+\Sigma_{i<j}\lambda_{ij}x_ix_j.$
 For 
 ease of readability (and typesetting !) let us denote the closure
 $\overline{T}$ of a subset $T$  
 (given the reduced closed subscheme structure)   
 by $\closure{T}$ in what follows.  
Since $\hbox{\rm Quad}_V^{(1)}$ is the same as the variety underlying 
the open subscheme $\hbox{\rm Quad}_V^{sr}$ of semiregular quadratic 
 forms, that $\closure{\hbox{\rm Quad}_V^{(1)}}={\hbox{\rm Quad}_V}$ follows 
 from the fact that ${\hbox{\rm Quad}_V}$ is irreducible. 

By assertion (1) of the present Theorem,
 ${\hbox{\rm Quad}_V}$ is the disjoint union of the $\hbox{\rm Quad}_V^{(i)}$; 
 therefore 
$\closure{\hbox{\rm Quad}_V^{(1)}}\backslash\hbox{\rm Quad}_V^{(1)}$
 is the disjoint union of $\{\hbox{\rm Quad}_V^{(i)}|2\leq i\leq 4\}$
 and also equals the closed subset $Z(P_3)$ defined by 
 the vanishing of $P_3.$ An explicit computation shows that 
the dimension of the  stabilizer of $q^{(2)}$ in $\hbox{\rm GL}(V)$ is 4.
Since $\hbox{\rm Quad}_V^{(2)}$ is an open dense subvariety of
 $\closure{\hbox{\rm Quad}_V^{(2)}}
\subset V(P_3)$, its closure is thus 5-dimensional. But since $P_3$ is an 
 irreducible polynomial, $Z(P_3)$ is also an irreducible 5-dimensional 
 subvariety. It  follows that $\closure{\hbox{\rm Quad}_V^{(2)}}
=\closure{\hbox{\rm Quad}_V^{(1)}}\backslash\hbox{\rm Quad}_V^{(1)}.$

 
Since $\hbox{\rm Quad}_V^{(2)}$ is smooth in its closure
 ($=Z(P_3)$ as seen above),  
 the singularities of its closure are contained in
 $\hbox{\rm Quad}_V^{(3)}\cup \hbox{\rm Quad}_V^{(4)}$
 which consists of quadratic forms that are perfect squares i.e., squares 
 of linear forms. 
 These singularities may be identified with 
  points $(\lambda_i, \lambda_{ij})\in {\mathbb A}^6_k
\cong\hbox{\rm Quad}_V$ at which 
 all the partial derivatives of $P_3$ vanish. A simple computation shows that 
 this set is the symmetric determinantal variety given by the vanishing of 
 the $(2\times 2)$-minors of the matrix of the symmetric bilinear form 
 associated to the generic quadratic form 
 given by $(x_1,x_2,x_3)\mapsto \Sigma_i \zeta_ix_i^2
+\Sigma_{i<j}\zeta_{ij}x_ix_j$ ; but it  can also be 
 shown that this set precisely corresponds to the  
 perfect squares. Therefore
 $\hbox{ Sing }(\closure{\hbox{\rm Quad}_V^{(2)}})
=\closure{\hbox{\rm Quad}_V^{(2)}}\backslash\hbox{\rm Quad}_V^{(2)}.$

That  
$\closure{\hbox{\rm Quad}_V^{(i+1)}}
=\closure{\hbox{\rm Quad}_V^{(i)}}\backslash
\hbox{\rm Quad}_V^{(i)}$ for $i=2$
follows from the above and the obvious fact that any quadratic form 
 can be specialised to the zero quadratic form. The case $i=3$ is trivial.

 
 To see that $\closure{\hbox{\rm Quad}_V^{(3)}}$ is smooth 
 if Char($k$)=2, we first note from the above and assertion (1) of the 
 present Theorem that the closure of 
$\hbox{\rm Quad}_V^{(3)}$ consists
 of perfect squares. Since char($k$)=2,
 under the identification ${\hbox{\rm Quad}_V}\cong{\mathbb A}^6_k$, 
 the perfect squares are seen to correspond to the copy of 
 ${\mathbb A}_k^3$ in ${\mathbb A}_k^6$
 given by the vanishing of the last three coordinates 
 $\lambda_{ij},\ 1\leq i<j\leq 3.$ 


 To see that the zero quadratic form is a 
 singularity of
 $\closure{\hbox{\rm Quad}_V^{(3)}}$ if char($k$) $\neq 2$, we note 
 from the above that 
 $\closure{\hbox{\rm Quad}_V^{(3)}}$ is defined by the same equations 
 that define the singularities of $Z(P_3)$ and
 is a certain symmetric determinantal variety.
 It is a well-known fact---an application of Standard Monomial 
 Theory (for e.g., see \cite{laxmi-css} or \cite{deconcini-procesi})---that the
 ideal defining these equations 
 is itself {\em reduced}, i.e., it is the {ideal of the variety}.  
 Checking the Jacobian criterion now shows that the zero quadratic 
 form is indeed a singular  point. 
  
\paragraph{Proof of assertion (3) of Theorem \ref{stratification}.} The 
 proof will follow from a series of observations. We remind the 
 reader of the $X$-morphism $\theta:\hbox{\rm Quad}_{V}
\longrightarrow \hbox{\rm Id-}w\hbox{\rm -Sp-Azu}_W$ 
 which was used to define the 
 isomorphism $\Theta$ (page \pageref{thetadef}, Section \ref{sec3}).\\[3mm] 
{\bf\em Claim 1: 
There are exactly 4 $H$-orbits in \hbox{\rm SpAzu}}.\\[2mm]
By Theorem \ref{Thetaisom},
 any point of \hbox{\rm SpAzu}
 is of the form $\underline{t}\cdot \theta(q).$ Its $H$-orbit is 
 $H\cdot \theta(q).$ There exists a unique $i$, $(1\leq i\leq 4)$,
 and some $g\in\hbox{\rm GL}(V)$ 
  such that $q=g\cdot q^{(i)}.$ Consider the algebra isomorphism
 $C_0(g,1)$ of (1), 
 Prop.\ref{simil-induces-iso-of-even-cliff} and 
 the induced isomorphism $h(g,1,q^{(i)},q)$ of
 Prop.\ref{b-co-phi-lambda-indep-of-q}. By definition,  
$\theta(q)=\theta(g\cdot q^{(i)})=h(g,1, q^{(i)},q)\cdot \theta(q^{(i)})
\Rightarrow H\cdot \theta(q)= H\cdot \theta(q^{(i)})$ so that 
 there are atmost 4 orbits. To see that there are atleast 4, we use  
 the following result. \\[3mm]
{\bf\em Claim 2. 
For $q\in \hbox{\rm Quad}_V$, we have
$\hbox{\sf L}_w\cdot\Theta((GL(V)\cdot q)\times\{I_4\})=\Theta((GL(V)\cdot q) \times \hbox{\sf L}_w)
=H\cdot \theta(q).$}\\[2mm]
On the one hand, we have 
\begin{equation}
\begin{split}
\Theta((GL(V)\cdot q) \times \hbox{\sf L}_w) &= \left\{ \underline{t}\cdot \theta( g\cdot q)
 \left|\right. \underline{t}\in \hbox{\sf L}_w \hbox{ and } g\in GL(V)\right\} \notag \\
 &= \left\{ \underline{t}\cdot(h(g,1,q,g\cdot q)\cdot \theta(q))
 \left|\right. \underline{t}\in \hbox{\sf L}_w \hbox{ and } g\in GL(V)\right\} \notag \\
 &\subset H\cdot \theta(q).\notag 
\end{split}
\end{equation}
Conversely take any  $h\cdot \theta(q)\in H\cdot \theta(q).$
 By Theorem~\ref{Thetaisom}, there exists a unique $\underline{t'}
\in\hbox{\sf L}_w$ and
 a unique $q'\in\hbox{\rm Quad}_V$ such that
 $h\cdot \theta(q)=\underline{t'}\cdot \theta(q').$
Therefore $({(\underline{t'})}^{-1}.h)\cdot \theta(q)=\theta(q').$
 Since $k$ is 
 algebraically- and hence quadratically closed, by (b), 
 Theorem \ref{lifting-of-isomorphisms},  
 there exists $g\in GL(V)$ such that 
 $q'=g\cdot q$ and ${(\underline{t'})}^{-1}.h=h(g,1,q,q').$ Hence 
$h\cdot \theta(q)=\underline{t'}\cdot \theta(g\cdot q)
=\Theta(g\cdot q, \underline{t'})
\Rightarrow H\cdot \theta(q)\subset\Theta((GL(V)\cdot q)
\times \hbox{\sf L}_w).$
This settles Claim 2.  As for Claim 1, 
if $q,q'\in\hbox{\rm Quad}_V$ are such that
 their $GL(V)$-orbits are distinct, then because 
 $\Theta$ is an isomorphism, Claim 2 shows that 
 $H\cdot \theta(q)$ is distinct from $H\cdot \theta(q').$ \\[3mm]
{\bf\em Claim 3. 
For each $i$, with $1\leq i\leq 4$,  
$ \closure{\hbox{\rm Quad}_V^{(i)}\times \hbox{\sf L}_w} 
= \closure{\hbox{\rm Quad}_V^{(i)}}\times \hbox{\sf L}_w.$}\\[2mm]
If $f:X\longrightarrow Y$ is a {\em  smooth 
 morphism} and $U\hookrightarrow Y$ is an open subset, then 
 $f^{-1}(\closure{U})=\closure{f^{-1}(U)}.$
 Since $\hbox{\sf L}_w\longrightarrow 
\hbox{Spec}(k)$ is smooth, so is the induced morphism 
$\closure{\hbox{\rm Quad}_V^{(i)}}\times\hbox{\sf L}_w\longrightarrow \closure{\hbox{\rm Quad}_V^{(i)}}.$
Taking $f$ to be this morphism and $U=\hbox{\rm Quad}_V^{(i)}$ gives Claim 3.\\[3mm] 
{\bf\em Claim 4.  The $GL(V)$-stratification of
 $\hbox{\rm Quad}_V$ 
 induces a $GL(V)$-stratification of
 $\hbox{\rm Quad}_V\times\hbox{\sf L}_w$ (the 
 $GL(V)$-action on $\hbox{\sf L}_w$ being taken to be trivial) with 
 strata given by 
$(\hbox{\rm Quad}_V\times\hbox{\sf L}_w)^{(i)}:= \hbox{\rm Quad}_V^{(i)}\times\hbox{\sf L}_w,\ (1\leq i\leq 4).$}\\[2mm]
To prove Claim 4, 
the only thing that needs to be checked is that 
 $$\closure{(\hbox{\rm Quad}_V\times\hbox{\sf L}_w)^{(i+1)}}
=\closure{(\hbox{\rm Quad}_V\times\hbox{\sf L}_w)^{(i)}}\backslash(\hbox{\rm Quad}_V\times\hbox{\sf L}_w)^{(i)}.$$
This follows by applying Claim 3 twice:
\begin{equation}
\begin{split}
\closure{(\hbox{\rm Quad}_V\times\hbox{\sf L}_w)^{(i+1)}} &= \closure{\hbox{\rm Quad}_V^{(i+1)}\times\hbox{\sf L}_w}
=  \closure{\hbox{\rm Quad}_V^{(i+1)}}\times\hbox{\sf L}_w 
=  (\closure{\hbox{\rm Quad}_V^{(i)}}\backslash\hbox{\rm Quad}_V^{(i)})\times\hbox{\sf L}_w \notag \\
 &=  \closure{\hbox{\rm Quad}_V^{(i)}}\times\hbox{\sf L}_w\backslash\hbox{\rm Quad}_V^{(i)}\times\hbox{\sf L}_w =  \closure{(\hbox{\rm Quad}_V\times\hbox{\sf L}_w)^{(i)}}\backslash(\hbox{\rm Quad}_V\times\hbox{\sf L}_w)^{(i)}. \notag 
\end{split}
\end{equation}
Now according to Claim 2, we have  
$\hbox{\rm SpAzu}^{(i)}=\Theta(GL(V)\cdot q^{(i)} \times \hbox{\sf L}_w)=\Theta((\hbox{\rm Quad}_V\times\hbox{\sf L}_w)^{(i)}).$ This combined with 
 Claim 4  and the fact that $\Theta$
 is an isomorphism completes  the proof of assertion (3) of Theorem 
 \ref{stratification}. {\bf Q.E.D., Theorem \ref{stratification}.}



%Shree Maaraaravindaarpanam
\begin{flushleft}
{\large \bf Acknowledgements}
\end{flushleft}

\addcontentsline{toc}{section}{Acknowledgements}
 The author gratefully acknowledges the Postdoctoral Fellowship 
 (September 2003--August 2004) of the {\em Graduiertenkolleg Gruppen
 und Geometrie} 
 under support from the {\em Deutschen Forschungsgemeinschaft}  and the 
 State of Niedersachsen at the Mathematisches Institut G\"ottingen 
 where this paper was written. The author also thanks the 
 National Board for Higher Mathematics, Department of Atomic Energy, 
 Government of India, for its Postdoctoral Fellowship (August 2002--August 2003) at the 
 Chennai Mathematical Institute, Chennai, India, during the latter half of 
 which certain special cases of the results of this work were obtained. 
 


%Shree Maataaravindaarpanam

\begin{thebibliography}{zzzzzz}

\addcontentsline{toc}{section}{References}

\bibitem{Kneser}
Kneser M, {\em Quadratische Formen, Ausarbeitung einer Vorlesung}, 
 Gottingen: Mathematisches Institut, 1974.

\bibitem{Knus}
Max-Albert Knus, Quadratic and Hermitian Forms over Rings, 
 {\em Grundlehren der mathematischen Wissenschaften} {\bf 294} (1991)
 (Springer-Verlag)

\bibitem{tevb-paper1}
Venkata Balaji T E, Limits of rank 4 Azumaya algebras and applications
 to desingularisation, Proc. Indian Acad. Sci. (Math. Sci.), Vol.112, 
 No.4, November 2002, pp.485--537

\bibitem{Bourbaki}
Bourbaki N, {\em \'El\'ements de Math\'ematique}  {\bf XXIV} Livre 2, 
 (1959) (Hermann, Paris)

\bibitem{phdthesis}
Venkata Balaji T E, On a Theorem of Seshadri on Limits of 
 $(2\times 2)$-Matrix Algebras with Applications to Desingularisation 
 and Quadratic Modules,  {\em Ph. D. Thesis, University of Madras, 
 June 2000}

\bibitem{ega1}
Grothendieck A, Dieudonn\'e J A, El\'ements de G\'eom\'etrie 
 Alg\'ebrique I, Die Grundlehren der mathematischen Wissenschaften 
 in Einzeldarstellungen, Band 166 (1971) (Springer-Verlag)

\bibitem{ega4}
Grothendieck A, Dieudonn\'e J A, El\'ements de G\'eom\'etrie 
 Alg\'ebrique IV, IHES Pub.Math.Nos.20, 24, 28, 32 (1964--1967)

\bibitem{ega2}
Grothendieck A, Dieudonn\'e J A, El\'ements de G\'eom\'etrie 
 Alg\'ebrique II, IHES Pub.Math.No.8 (1961)

\bibitem{Hartshorne}
Robin Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, 
 GTM 52 (Springer-Verlag, New York)

\bibitem{laxmi-css}
V. Lakshmibai and C. S. Seshadri, {\em Geometry of G/P -- II.}
 Proc. Ind. Acad. Sci. 87 A, 1978, pp. 1--54.


\bibitem{deconcini-procesi}
C. DeConcini and C. Procesi, {\em A Characteristic-free approach to 
 invariant theory.} Advances in Math. 21, 1976, pp. 330--354.


\end{thebibliography}


\end{document}
