\titlea{}{Preface}
\leftrunningtitle{Specialization of Quadratic and Symmetric Bilinear
Forms}
\rightrunningtitle{Preface}
\noi
{\it \hskip5.5cm A Mathematician Said Who

\noi
\hskip 5,5cm Can Quote Me a Theorem that's True?

\noi
\hskip 5.5cm For the ones that I Know

\noi
\hskip 5.5cm Are Simply not So,

\noi
\hskip 5.5cm When the Characteristic is Two!}


\vskip1,0cm\noi
This  pretty limerick first came to my ears in May 1998 during a
talk by T.Y.~Lam on field invariants from the theory of
quadratic forms.\fonote{``Some reflections on
quadratic invariants of fields'', 3 May 1998 in Notre Dame (Indiana)
on the occasion of O.T. O'Meara's 70th birthday.}
It is -- poetic exaggeration allowed -- a suitable motto
for this monograph.

What is it about? In the beginning of the seventies I drew up a
specialization theory of quadratic and symmetric bilinear
forms over fields~[K$_4$].
Let $\lambda \colon K \to L \cup \infty$ be a place. Then one can
assign  a form $\lambda_* (\varphi)$
to a form $\varphi$ over $K$  in a meaningful way if $\varphi$ has
``good reduction'' with respect to  $\lambda$ (see~\S1). The basic idea is
to simply apply the place $\lambda$ to the coefficients of
$\varphi$ which therefore of course have to be in the valuation
ring of $\lambda$.

The specialization theory of that time was satisfactory as long as
the field~$L$, and therefore also $K$, had characteristic
$\not=2$. It served me in the first place as the foundation for a
theory of generic splitting of quadratic forms~[K$_5$], [K$_6$].
After a very modest beginning, this theory is now in full bloom.
It became important for the understanding of quadratic forms over
fields, as can be seen from
the book [IKKV] of Izhboldin-Kahn-Karpenko-Vishik for instance.
One should note that there exists a
theory of (partial) generic splitting of central simple algebras
and reductive algebraic groups, parallel to the theory of generic
splitting of quadratic forms (see [Ke R] and the literature cited
there).

In this book I would like to present a specialization theory
of quadratic and symmetric bilinear forms with respect to a place
$\lambda \colon K \to L \cup \infty$, without the assumption that
$\Char L\not=2$. This is where complications arise. We have to
make a distinction between bilinear and quadratic forms and study
them both over fields and valuation rings. From the viewpoint of
reductive algebraic groups, the so-called regular quadratic forms
(see below) are the natural objects. But, even if we are only
interested in such forms, we have to know a bit about
specialization of
nondegenerate symmetric bilinear forms, since they occur as
``multipliers'' of quadratic forms: if $\varphi$ is such a
bilinear form and $\psi$ is a regular quadratic form, then we
can form a tensor product $\varphi \otimes \psi$, see~\S5. This is
a quadratic form, which is again regular when $\psi$ has even
dimension ($\dim\psi=$ number of variables occurring in $\psi$).
However -- and here already we run into trouble --
when $\dim\psi$ is odd, $\varphi \otimes \psi$ is not
necessarily regular.

Even if we only want to understand quadratic forms over a field
$K$ of characteristic zero, it might be necessary to look at
specializations with respect to places from $K$ to fields of
characteristic~$2$, especially in arithmetic investigations. When
$K$ itself has characteristic~$2$, an often more complicated
situation may occur, for which we are not prepared by the
available literature. Surely, fields of characteristic~$2$ were
already allowed in my work on specializations in 1973~[K$_4$], but
from today's point of view   satisfactory results were only
obtained for symmetric bilinear forms. For quadratic forms there
are gaping holes. We have to study quadratic forms over a
valuation ring in which $2$ is not a unit. Even the beautiful and
extensive book of Ricardo Baeza~[Ba] doesn't give us enough for
the theory of specializations, although Baeza even allows
semilocal rings instead of valuation rings. He only studies
quadratic forms whose associated bilinear forms are nondegenerate.
This forces those forms to have even dimension.
\medskip

Let me now discuss the contents of this book. After an introduction
to the problem in~\S1, which can be understood without any
previous knowledge of quadratic and  bilinear forms, the
specialization theory of symmetric bilinear forms is presented
in~\S2~-~\S3. There are good, generally accessible sources available for the
foundations of the algebraic theory of symmetric bilinear forms.
Therefore many results are presented without a proof, but with a reference
to the literature instead. As an important application, the outlines of
the theory of generic splitting
in characteristic~$\not=2$ are sketched in~\S4, nearly without
proofs.

From \S5 onwards we address the theory of quadratic forms. 
In characteristic~$2$ fewer results can be found 
in the literature for such forms than for bilinear forms, even at the basic level.
%
%When
%characteristic~$2$ is allowed, one can find in principle much less about this
%in the literature than about bilinear forms. 
Therefore we present
most of the proofs. We also concern ourselves with the so-called
``weak specialization'' (see~\S1) and get into  areas which may
seem strange even to specialists in the theory of quadratic forms.
In particular we have to require  a quadratic form over
$K$ to be ``obedient'' in order to weakly specialize it
with respect to a place
$\lambda \colon K \to L \cup \infty$  (see~\S7). I have never encountered such a thing anywhere  in the literature.

At the end of Chapter~I we reach a level in the specialization theory of quadratic forms
that facilitates a generic splitting theory, useful for many applications. 
In the first two sections (\S9, \S10) of Chapter~II we produce such a generic splitting theory in two versions, both of which deserve interest in their own right.

%In \S9 and \S10 we then develop, as a first
%application of the specialization theory, a theory of generic
%splitting, which is not yet available in the literature in this
%form. The theory of~\S9 includes the theory of \S4, and so the missing
%proofs of \S4 are subsequently filled in.

We call a quadratic form $\varphi$ over a field $k$ {\it
nondegenerate} when its quasilinear part (cf. Arf~[A]), which we
denote by $QL(\varphi)$, is anisotropic. We further call -- deviating
from Arf~[A] -- $\varphi$
{\it regular} when $QL(\varphi)$ is at most one-dimensional and
{\it strictly regular} when $QL(\varphi)=0$ (cf. \S6, Definition~3).
When $k$ has characteristic $\not=2$, every nondegenerate form is
strictly regular, but in characteristic $2$ the quasilinear part
causes complications. For in this case $\varphi$ can become
degenerate under a field extension $L\supset k$. Only in the
regular case  is this impossible.

In \S9 we study the splitting behaviour of a regular quadratic form
$\varphi$ over $k$ under field extensions, while in
\S10 any nondegenerate form $\varphi$, but only separable
extensions of $k$ are allowed. The theory of~\S9 incorporates the theory of \S4, and so the missing
proofs of \S4 are subsequently filled in.

Until  the end of \S10 our specialization theory is based on an
obvious  ``canonical'' concept of {\it good reduction}
of a form $\varphi$ over a field $K$ (quadratic or symmetric
bilinear) to a valuation ring ${\nam o}$ of $K$, similar to what
is known under this name in other areas of mathematics
(e.g. abelian varieties). There is nothing wrong with this theory, however for
many applications it is too limited.

This is particularly clear
when studying specializations with respect to a place
$\lambda \colon K \to L \cup \infty$ with $\Char K=0$, $\Char
L=2$.
If $\varphi$ is a nondegenerate quadratic form over $K$ with good
reduction with respect to $\lambda$, then the specialization  $\lambda_* (\varphi)$ is
automatically strictly regular. However, we would like to have a
more general specialization concept, in which  forms with
quasilinear part $\not=0$ can arise over $L$. Conversely, if the place
$\lambda$ is surjective, i.e. $\lambda (K) = L \cup \infty$, we
would like to ``lift'' {\it every} nondegenerate quadratic form $\psi$ over
$L$ with respect to $\lambda$ to a form $\varphi$ over $K$, i.e.
to find a form $\varphi$ over $K$ which specializes to $\psi$
with respect to $\lambda$. Then we could use the theory of forms
over $K$ to make statements about $\psi$.

We present such a general specialization theory  in \S11.  It is
based on the concept of ``{\it fair reduction}'', which is less
orthodox than good reduction, but which possesses nevertheless 
quite satisfying properties.

Next, in \S12, we present a theory of generic splitting, which
unites the theories of \S4, \S9 and \S10 under one roof and which
incorporates fair reduction. This theory is  deepened in \S13 and
\S14 through the study of generic splitting towers and so we reach the end of Chapter~II.

Chapter~III (\S15 - \S27) is a long chapter in which we present a panorama of results about quadratic forms over fields  for which specialization and generic splitting of forms play an important role.  This only scratches the surface of 
applications of the specialization theory of Chapters~I and II.
Certainly many more results can be unearthed.  

We return to the foundations of specialization theory in the final short Chapter~IV (\S28 - \S32). Quadratic and bilinear forms over a field can be specialized with respect to a more general ``quadratic place''  $\Lambda \colon K \to L \cup
\infty$ (defined in \S 28)
instead of  a usual place $\lambda \colon K \to L \cup \infty$.
This represents a considerable broadening of the specialization theory of Chapters~I and II.
Of course we require again ``obedience'' from a quadratic form $q$ over $K$ in order for its specialization  $\Lambda_* (q)$ to reasonably exist.
It then turns out that the generic splitting behaviour of $\Lambda_* (q)$ is governed by the splitting behaviour of $q$ and $\Lambda$, in so far   good or fair reduction  is present in a weak sense, as elucidated for ordinary places in Chapter~II.  

Why are quadratic places of interest, compared to ordinary places? To answer this question we observe the following. If a form $q$ over $K$ has bad reduction with respect to a place 
 $\lambda \colon K \to L \cup \infty$ , it often happens that $\lambda$ can be ``enlarged'' to a quadratic place $\Lambda \colon K \to L \cup \infty$ such that $q$ has good or fair reduction with respect to $\Lambda$ in a weak sense, and the splitting properties of $q$ are handed down to
$\Lambda_* (q)$ while there is no form $\lambda_* (q)$ available for which this would be the case.
The details of such a notion of reduction are much more tricky compared to what happens in Chapters~I and II. The central term which renders possible a unified theory of generic splitting of quadratic forms is called ``stably conservative reduction'', see \S31.



%From \S15 on more
%applications of the specialization theory of quadratic forms
%follow, which use  fair reduction significantly, and also
%applications of the specialization theory of bilinear forms from
%\S3 over fields of characteristic~$2$.


One has to get used to the fact
that for bilinear forms there is in general no Witt cancellation
rule, in contrast to quadratic forms.
Nevertheless the specialization theory  is in
many respects easier  for bilinear forms than for quadratic forms.

On the other hand we do not have any theory of generic splitting
for symmetric bilinear forms over fields of characteristic $2$.
Such a theory might not even be
 possible in a meaningful way.
This may well be connected to the fact that the automorphism
groups of such forms can be very far from being reductive groups
(which may also account for the absence of a good
cancellation rule).
\medskip

This book is intended for  audiences with different
interests. For a mathematician with perhaps only a little knowledge
of quadratic or symmetric bilinear forms, who just wants to get
an impression of specialization theory,
it suffices to read \S1~-~\S4. The theory of generic
splitting in characteristic $\not=2$ will acquaint him with an important application area.

From \S5 onwards the book is intended for scholars, working in the
algebraic theory of quadratic forms, and also for specialists in
the area of algebraic groups. They have always been given
something to look at by the theory of quadratic forms.


%. They often benefit from quadratic form theory for illustrative
%purposes.

%The theory of quadratic forms has
%always provided them with good examples.

When a reader has reached \S10 of the book, he can lean back in
his chair and take a well-deserved break. He has then
learned about the  specialization theory which is based  on the
concept of good reduction and has gained
a certain perspective on specific phenomena
in characteristic $2$. Furthermore he has been introduced to
the foundations of generic splitting and so has seen the
specialization theory in action. Admittedly he has not yet seen
independent applications of the weak specialization theory (\S3,
\S7), for this theory has only appeared up to then  as an auxiliary
one.

The remaining sections \S11 - \S14 of Chapter~II develop the specialization theory sufficiently far to allow an understanding of the classical algebraic theory of  quadratic forms (as presented in the books  of Lam [L], [L$'$] and Scharlau [S]) without the usual restriction that the characteristic should be different from $2$. Precisely this happens in Chapter~III where the reader will also obtain sufficient illustrations, enabling him to relieve other classical theorems from the characteristic $\not=2$ restriction, although this is often a nontrivial task. 

The final Chapter~IV is ultimately intended for the mathematician who wants to embark on a more daring expedition in the realm of quadratic forms over fields. It cannot be a mere coincidence that the specialization theory for quadratic places works just as well as the specialization theory for ordinary places. It is therefore a safe prediction that quadratic places will turn out to be generally useful and important in a future theory of quadratic forms over fields.
\bigskip

\hskip 7cm{\it Manfred Knebusch}
\smallskip

\hskip7cm{\it Regensburg, June 2007}
\bigskip
\bigskip
\bigskip
\bigskip

\noindent
{\bf Postscriptum} (October 2009)

\medskip\noindent
I had the very good luck to find a translator of the German text into English, who beside two languages
also could understand the mathematical content of this book in depth. I met Professor Thomas Unger within
the framework of the European network ``Linear Algebraic Groups, Algebraic K-theory, and Related Topics'',
and most of the translation and our collaboration has been done under the auspices of this network, which
we acknowledge gratefully.

\medskip\noindent
I further owe deep thanks to my former secretary Rosi Bonn, who typed the whole German text in various
versions and large parts of the English text.

\medskip\noindent
The German text, {\it Spezialisierung von quadratischen und symmetrischen bilinearen Formen}, can be found on my homepage
{\tt
http://www-nw.uni-regensburg.de/$\sim$.knm22087.mathematik.uni-regensburg.de}.

