\titlea{\S4}{Generic Splitting in Characteristic $\not=2$}
\leftrunningtitle{Chapter I. Fundamentals of Specialization Theory}

%\leftrunningtitle{Specialization of Quadratic and Symmetric
%Bilinear Forms}
\rightrunningtitle{\S4 Generic Splitting in
Characteristic $\not=2$}


In this section we outline an important application area of the
specialization theory developed in \S3, namely the theory of
generic splitting of bilinear forms. Many proofs will only
be given in \S7, after also having developed a specialization
theory of quadratic forms.

Let $k$ be a field and let $\varphi$ be a {\it form} over $k$,
which is just as before understood to be a nondegenerate symmetric
bilinear form over $k$. Our starting point is the following simple

\begobservation{} Let $K$ and $L$ be fields, containing $k$, and
let $\lambda \colon K \to L \cup \infty$ be a place {\it over}
$k$, i.e. with $\lambda (c) = c$ for all $c \in k$.
\smallskip
\setitemindent{(a)}
\item{(a)} Then $\varphi \otimes K$ has good reduction with
respect to $\lambda$ and
$\lambda_* (\varphi \otimes K) \approx \varphi
\otimes L$.\fonote{$\varphi \otimes K$ is the form $\varphi$,
considered over $K$ instead of over  $k$. If ${E}$ is a bilinear
space  over ${k}$ corresponding to ${\varphi}$, then
${K \otimes_k E}$ -- as described in \S3 -- is a bilinear space
corresponding to ${\varphi \otimes K}$.}
Indeed, if $\varphi \cong (a_{ij})$ with $a_{ij} \in
k$, $\det (a_{ij}) \ne 0$, then also $\varphi \otimes K \cong
(a_{ij})$, and this is a unimodular representation of $\varphi$
with respect to $\lambda$, since $k$ is contained in the valuation
ring ${\nam o}$ of $\lambda$.
So $\lambda_* (\varphi
\otimes K) \approx (\lambda (a_{ij})) = (a_{ij})$ and since this
symmetric matrix is now considered over $L$, we conclude that
$\lambda_*
(\varphi \otimes K) \approx \varphi \otimes L$.
\smallskip
\item{(b)} Suppose that $\varphi \otimes K$ has kernel form
$\varphi_1$ and Witt index $r_1$, i.e.
$$\varphi \otimes K \,\approx \, \varphi_1 \perp r_1 \times H,$$
where $H$ denotes from now on the hyperbolic ``plane''\fonote{We do not
make a notational distinction between the forms ${\left( {0 \atop 1} {1 \atop 0
} \right)}$ over the different fields occurring here.}
$\left( {0 \atop 1} {1 \atop 0} \right)$.
According to the corollary of Theorem~4 in \S3, the form
$\varphi_1$ has good reduction with respect to $\lambda$.
Therefore, applying $\lambda_*$ yields
$$\varphi \otimes L \,\approx \, \lambda_* (\varphi_1) \perp r_1
\times H.$$
Hence we conclude that $\ind (\varphi \otimes L) \geq
\ind (\varphi \otimes K)$ and that
$$\lambda_* (\ker (\varphi \otimes K)) \,\sim \, \ker (\varphi
\otimes L).$$
(Recall the terminology of \S2, Definition~1.)
\endobservation

\begdefinition{1.} We call two fields $K \supset k$, $L \supset k$
over $k$ {\it specialization equivalent}, or just {\it
equivalent}, if there exist places $\lambda \colon K \to L \cup
\infty$  and  $\mu \colon L \to K \cup \infty$ over $k$. We then
write $K \sim_k L$.
\enddefinition

The following conclusions  can be drawn immediately from our
observation:

\begremark{1.} If $K \sim_k L$, then every form $\varphi$ over
$k$ satisfies:
\smallskip
\setitemindent{(2)}
\item{(1)} $\ind (\varphi \otimes K) = \ind (\varphi \otimes L)$.
\smallskip
\item{(2)} $\ker (\varphi \otimes K)$ has good reduction with
respect to every place $\lambda$ from $K$ to $L$ and $\lambda_* (\ker
(\varphi \otimes K)) \cong \ker (\varphi \otimes L)$. (Note that
``$\cong$'' holds, not just ``$\approx$''!)
\endremark

From a technical point of view, it is a good idea to treat the
following special case of Definition~1 separately:

\begdefinition{2.} We call a field extension $K \supset k$ {\it
inessential} if there exists a place $\lambda \colon K \to k
\cup \infty$ over $k$.
\enddefinition

Obviously this just means that $K$ is equivalent to $k$ over $k$.
In this case, Remark~1 becomes:

\begremark{2.} If $K$ is an inessential extension of $k$, then
every form $\varphi$ over $k$ satisfies:
\smallskip
\setitemindent{(2)}
\item{(1)} $\ind (\varphi \otimes K) = \ind (\varphi)$,
\smallskip
\item{(2)} $\ker (\varphi \otimes K)=\ker(\varphi)\otimes K$.
\endremark

We will  see that
the forms $\varphi$ and $\varphi \otimes K$
exhibit the ``same'' splitting  behaviour
with respect to an inessential extension $K/k$ in a broader sense
(see Scholium~4 below).

Let us return to the general observation above. It gives rise
to the following

\begproblem{}
Let $\varphi$ be a form over $k$, $\dim \varphi =n \geq 2$, and
let $t$ be an integer with $1 \leq t \leq \frac{n}{2}$. Does there
exist a field extension $K \supset k$ which is ``generic with
respect to
splitting off $t$ hyperbolic planes''?, i.e. with the following
properties:
\smallskip
\setitemindent{(b)}
\item{(a)} $\ind (\varphi \otimes K) \geq t.$
\smallskip
\item{(b)}If $L$ is a field extension of $k$ with $\ind
(\varphi \otimes L) \geq t$, then there exists a place from  $K$ to $L$ over $k$.
\endproblem

We first  address this problem for the case $t=1$. As before, let
$\varphi$ be a form over a field $k$.

\begdefinition{3.} An extension field $K \supset k$ is called a {\it
generic zero field of} $\varphi$ if the following conditions
hold:
\smallskip
\setitemindent{(a)}
\item{(a)} $\varphi \otimes K$ is isotropic.
\item{(b)} For every field extension $k \hookrightarrow L$ with
$\varphi \otimes L$ isotropic, there exists a place
$\lambda \colon K \to L \cup \infty$ over $k$.
\enddefinition

Note that if $\varphi$ is isotropic, then the field $k$ is of course
itself  a generic zero field of $\varphi$.

{\it In the rest of this section, we assume that $\Char k\not=2$.}
Now $\varphi$ can also be viewed as a quadratic form,\fonote{In
keeping with our earlier conventions, it would perhaps be better
to write $\varphi (x) = \frac{1}{2} \varphi (x, x)$. However, the factor
$\frac{1}{2}$ is not important for now.}
$\varphi (x) \colon = \varphi (x, x)$.
We define a field extension $k (\varphi)$ of $k$, which is a
priori suspected to be a generic zero field of $\varphi$.

\begdefinition{4.} Let $\dim \varphi \geq 3$ or let
$\dim \varphi = 2$ and $\varphi\not\cong H$. Let $k (\varphi)$
denote the function field of the affine quadric
$\varphi(X_1,~\dots,~X_n)~=~0$  (where $n =\dim \varphi$), i.e.
the quotient field of the ring $A \colon =
k [X_1, \dots, X_n ]/(\varphi (X_1, \dots, X_n))$ with
indeterminates $X_1, \dots, X_n$.
\enddefinition

Observe that the polynomial $\varphi (X_1, \dots, X_n)$ is
irreducible. To prove this we may suppose that $\varphi$ is
diagonalised, $\varphi = \langle a_1, a_2, \dots, a_n
\rangle$. The polynomial $a_1 X_1^2 + a_2 X_2^2 + \dots + a_n
X_n^2$ is clearly not a product of two linear forms (since
$\Char k \ne 2$).

If $\varphi = H$, then $\varphi (X_1, X_2) = X_1 X_2$. On formal
grounds we then set $k(\varphi) = k(t)$ for an indeterminate $t$.

Let $x_1, \dots, x_n$ be the images of the indeterminates
$X_1, \dots, X_n$ in $A$. Then $\varphi (x_1, \dots, x_n) = 0$.
Hence $\varphi \otimes k (\varphi)$ is isotropic. (This is also
true for $\varphi \cong H$.) On top of that we have the following
important

\begtheorem{1.} Let $\dim \varphi \geq 2$. Then $k (\varphi)$ is a
generic zero field of $\varphi$.
\endtheorem

Note that the assumption $\dim \varphi \geq 2$ is necessary for
the existence of a generic zero field, since forms of dimension
$\leq 1$ are never isotropic.

Theorem~1 can  already be found in  [K$_5$]. We will wait until
\S9 to prove it  in the framework of a generic
splitting theory of quadratic forms.

It is clear from  above that every other generic zero field $K$ of
$\varphi$ over $k$ is equivalent  to $k (\varphi)$. It is unknown
which bilinear forms possess generic zero fields in characteristic
$2$.

Given an arbitrary form $\varphi$ over $k$, we now construct a
tower of fields $(K_r \; \vert \; 0 \leq r \leq h)$ together with
anisotropic forms $\varphi_r$ over $K_r$ and numbers
$i_r \in {\Bbb N}_0$ as follows: choose $K_0$ to be any inessential
extension of the field $k$.\fonote{In earlier works (especially
[K$_5$]) $K_0$ was always chosen to be $k$. From a technical point
of view it is favourable to allow $K_0$ to be an inessential
extension of $k$, just as in [KR].}
Let $\varphi_0$ be the kernel form of
$\varphi\otimes K_0$ and  $i_0$ the Witt index of $\varphi$.
Then
$$\varphi\otimes K_0 \, \cong \, \varphi_0 \perp i_0 \times H.$$
If $\dim \varphi_0 \leq 1$, then Stop! Otherwise choose a generic
zero field $K_1 \supset K_0$ of $\varphi_0$. Let $\varphi_1$ be
the kernel form of $\varphi_0 \otimes K_1$ and $i_1$ the Witt
index of $\varphi_0 \otimes K_1$. Then
$$\varphi_0 \otimes K_1 \, \cong \, \varphi_1 \perp i_1 \times
H.$$
If $\dim \varphi_1 \leq 1$, then Stop! Otherwise choose a generic
zero field $K_2 \supset K_1$ of $\varphi_1$. Let $\varphi_2$ be
the kernel form of $\varphi_1 \otimes K_2$ and $i_2$ the Witt
index of $\varphi_1 \otimes K_2$. Then
$$\varphi_1 \otimes K_2 \, \cong \, \varphi_2 \perp i_2 \times
H,$$
etc.\vfill\eject

The construction halts after $h \leq \left[ \frac{\dim \varphi}{2}
\right]$ steps with a Witt decomposition
$$\varphi_{h-1} \otimes K_h \, \cong \, \varphi_h \perp i_h
\times H,$$
$\dim \varphi_h \leq 1$.

\begdefinition{5.} $(K_r \; \vert \; 0 \leq r \leq h)$ is called
a {\it generic splitting tower of} $\varphi$. The number $h$ is
called the {\it height} of $\varphi$, denoted  $h = h(\varphi)$.
The number $i_r$ is called the $r$-th {\it higher index} of
$\varphi$ and the form $\varphi_r$  the $r$-th {\it
higher kernel form} of $\varphi$.
\enddefinition

\begremark{} Obviously  $\varphi_r$ is the kernel form of $\varphi \otimes
K_r$ and $\ind (\varphi \otimes K_r) = i_0 + \dots + i_r$. Note
that
$h = 0$  iff the form $\varphi$ ``splits'', i.e. iff its kernel
form is zero or one-dimensional.
\endremark

We will see that the number $h$ and the sequence
$(i_0, \dots,i_h)$ are independent of the choice of generic
splitting tower and also that the forms $\varphi_r$ are determined
uniquely by $\varphi$ in a precise way. For this the following
theorem is useful.

\begtheorem{2.} Let $\psi$ be a form over a field $K$. Let
$\lambda \colon K \to L \cup \infty$ be a place, such that $\psi$
has good reduction with respect to $\lambda$. Suppose that $L$
$($and therefore $K$$)$ has characteristic $\ne 2$. Then $\lambda_*
(\psi)$ is isotropic if and only if $\lambda$ can be extended to a place
$\mu \colon K (\psi) \to L \cup \infty$.
\endtheorem

If $\lambda$ is a trivial place, i.e. a field extension, then this
theorem says once more that $K(\psi)$ is a generic zero field of
$\psi$ (Theorem~1).

One direction of the assertion is trivial, just as for Theorem~1:
if $\lambda$ can be extended to a place $\mu \colon K (\psi) \to L \cup
\infty$, then $\psi \otimes K(\psi)$ has good reduction with
respect to $\mu$ and $\mu_* (\psi \otimes K
(\psi)) \cong \lambda_* (\psi)$. Since $\psi \otimes K (\psi)$ is
isotropic,  $\lambda_* (\psi)$ is also isotropic.

The other direction will be established in \S9. For a shorter and
simpler proof, see [K$_5$] and the books [S], [KS].

\begtheorem{3}{\rm (Corollary of Theorem~2).}
Let $\varphi$ be a form over a field $k$. Let
$(K_r \; \vert \; 0 \leq r \leq h)$ be a generic splitting tower
of $\varphi$ with associated higher kernel forms $\varphi_r$ and
indices $i_r$. Suppose that $\varphi$ has good reduction with respect
to some place $\gamma \colon k \to L \cup \infty$.
Suppose that $L$ $($and hence $k$$)$ has characteristic $\ne 2$.
Finally let $\lambda \colon K_m \to L \cup \infty$ be a place for
some $m, \; 0 \leq m \leq h$ which extends $\gamma$ and which
$\underline{\hbox{cannot}}$ be extended to $K_{m+1}$ if $m < h$. Then $\varphi_m$
has good reduction with respect to $\lambda$. The form $\gamma_*
(\varphi)$ has kernel form $\lambda_* (\varphi_m)$ and Witt index
$i_0 + \dots + i_m$.
\endtheorem

\begproof{} There is an isometry
$$\varphi \otimes K_m \, \cong \, \varphi_m \perp (i_0 + \dots +
i_m) \times H. \leqno(1)$$
The form $\varphi \otimes K_m$ has good reduction with respect to $\lambda$ and $\lambda_*
(\varphi \otimes K_m) = \gamma_* (\varphi)$. Using Theorem~4 and
Theorem~3 from \S3, (1) implies that $\varphi_m$ has good
reduction with respect to $\lambda$ and
$$\gamma_* (\varphi) \, \cong \, \lambda_* (\varphi_m) \perp (i_0 +
\dots + i_m) \times H \leqno(2)$$
(cf. the observation at the beginning of this section). If $\lambda_*
(\varphi_m)$ were isotropic, then we would have $m < h$, since
$\dim\varphi_h \leq 1$. It would then follow from Theorem~2 that
$\lambda$ can be extended to a place from $K_{m+1}$ to $L$,
contradicting our assumption. Therefore $\lambda_*(\varphi_m)$ is
anisotropic and so (2) is the Witt decomposition of
$\gamma_* (\varphi)$. \qed
\endproof

Note that this theorem implies in particular, that any attempt to
successively extend the given place $\lambda \colon k \to L \cup \infty$
to a  ``storey'' $K_m$ of the generic splitting
tower, which is as high as possible, always ends at the same number~$m$.

Theorem~3 gives rise to a number of interesting
statements about the splitting behaviour of forms and the
extensibility of places.

\begscholium{1.} Let $\varphi$ be a form over $k$ and
$(K_r \; \vert \; 0 \leq r \leq h)$ a generic splitting tower
of $\varphi$ with kernel forms $\varphi_r$ and indices $i_r$.
Furthermore, let $k \subset L$ be a field extension. If we apply
Theorem~3 to the trivial place $\gamma \colon k \hookrightarrow
L$, we get:
\smallskip
\setitemindent{(3)}
\item{(1)} Let $\lambda \colon K_m \to L \cup \infty$ be a place
over $k \quad (0 \leq m \leq h)$, which {\it cannot} be extended
to $K_{m+1}$ in case $m < h$. Then $\varphi_m$ has good reduction
with respect to $\lambda$ and $\lambda_* (\varphi_m)$ is the
kernel form of $\varphi \otimes L$. The index of $\varphi \otimes
L$ is $i_0 + \dots + i_m$.
\smallskip
\item{(2)} If $\lambda' \colon K_r \to L \cup \infty$ is a place
over $k$, then $r \leq m$ and $\lambda'$ can be extended to a
place $\mu \colon K_m \to L \cup \infty$.
\smallskip
\item{(3)} Given a number $t$ with $1 \leq t \leq \left[ \frac{\dim
\varphi}{2} \right] = i_0 + \dots + i_h$, let $m \in {\Bbb N}_0$
be
minimal with $t \leq i_0 + \dots + i_m$. Then $K_m$ is a generic
field extension of $k$ with respect to splitting off $t$
hyperbolic planes of $\varphi$ (in the context of our problem
above).
\endscholium

\begdefinition{6.} Let $\varphi$ be a form over $k$. We call
the set of indices $\ind (\varphi \otimes L)$, where $L$ traverses
all field extensions of $k$, the {\it splitting pattern of}
$\varphi$, denoted by $\SP (\varphi)$.
\enddefinition

This definition also makes sense in characteristic $2$, and it is
a priori clear that $\SP (\varphi)$ consists of at most
$\left[ \frac{\dim \varphi}{2} \right] + 1$ elements. Usually the
elements of $\SP (\varphi)$ are listed in ascending order. If
$\Char k \ne 2$ and $(i_r \; \vert \; 0 \leq r \leq h)$ is the
sequence of higher indices of a generic splitting tower
$(K_r \; \vert \; 0 \leq r \leq h)$ of $\varphi$, then Scholium~1
shows:
$$\SP (\varphi) = (i_0, i_0 + i_1, i_0 + i_1 + i_2, \dots, i_0 +
i_1 + \dots + i_h).$$
Since the numbers $i_r$ with $r > 0$ are all positive, it is
evident that the height $h$ and the higher indices
$i_r \; (0 \leq r \leq h)$ are independent of the choice of
generic splitting tower of $\varphi$. Obviously is
$i_0 + \dots + i_h = \left[\frac{\dim \varphi}{2} \right]$.

\begscholium{2.} Let $(K_r \; \vert \; 0 \leq r \leq h)$ and
$(K'_r \; \vert \; 0 \leq r \leq h)$ be two generic splitting
towers of the form $\varphi$ over $k$. Applying Scholium~1 to the
field extensions $k \subset K_s$ and $k\subset K'_r$, yields: if
$\lambda \colon K_r\to K'_s \cup \infty$ is a place over $k$, then
$r \leq s$ and $\lambda$ can be extended to a place
$\mu \colon K_s \to K'_s\cup \infty$. The fields $K_s$ and
$K'_s$ are equivalent over $k$. For every place
$\mu \colon K_s \to K'_s\cup \infty$, the kernel form $\varphi_s$
of $\varphi\otimes K_s$ has good reduction with respect to $\mu$
and $\mu_*(\varphi_s)$ is the kernel form $\varphi'_s$ of
$\varphi \otimes K'_s$.
\endscholium

\begscholium{3.} Let $\varphi$ be a form over $k$ and
$\gamma\colon k \to L \cup \infty$ a place such that
$\varphi$ has good reduction with respect to $\gamma$. Applying
Theorem~3 to the places $j\circ \gamma \colon k \to L \cup
\infty$, being the composites of $\gamma$ and trivial places
$j \colon L \hookrightarrow L'$, yields:
\smallskip
\setitemindent{(3)}
\item{(1)} $\SP (\gamma_* (\varphi)) \subset \SP (\varphi)$.
\smallskip
\item{(2)} The higher kernel forms of $\gamma_* (\varphi)$ arise
from certain higher kernel forms of $\varphi$ by means of
specialization. More precisely: if $(K_r \; \vert \; 0 \leq r \leq
h)$ is a generic splitting tower of $\varphi$ and
$(K'_s \; \vert \;0 \leq s \leq h')$ is a generic splitting tower
of $\gamma_* (\varphi)$, then $h' \leq h$ and for every $s$ with $0 \leq s\leq
h'$ we have
$$\ind (\gamma_* (\varphi) \otimes K'_s) = i_0 + \dots + i_m$$
for some $m \in \{ 0, \dots, h \}$. The number $m$ is the biggest
integer such that $\gamma$ can be extended to
$\lambda \colon K_m \to K'_s\cup \infty$. The kernel form
$\varphi_m$ of $\varphi \otimes K_m$ has good reduction with
respect to every  extension $\lambda$ of this kind, and
$\lambda_* (\varphi_m)$ is the kernel form of
$\gamma_* (\varphi) \otimes K'_s$.
\smallskip
\item{(3)} If $\rho \colon K_r \to K'_s \cup \infty$ is a place,
which extends $\gamma \colon k \to L \cup \infty$, then
$r \leq m$ and $\rho$ can be further extended to a place from
$K_m$ to $K'_s$.
\endscholium

\begscholium{4.} Let $L/k$ be an inessential field extension (see
Definition~2 above) and $\varphi$ again a form over $k$. Let
$(L_i \; \vert \; 0 \leq i \leq h)$ be a generic splitting tower
of $\varphi \otimes L$. It is then immediately clear from
Definition~5 that this is also a generic splitting tower of
$\varphi$. Hence $\SP(\varphi\otimes L) =\SP(\varphi)$, and the
higher kernel forms of $\varphi\otimes L$ can also be interpreted
as higher kernel forms of $\varphi$.
\endscholium

Which strictly increasing sequences $(0, j_1, j_2,\dots, j_n)$
occur as splitting patterns of anisotropic forms? What do
anisotropic forms of given height $h$ look like? These questions
are difficult and at the moment the object of intense research.
The first efforts towards answering them can be found in [K$_4$],
[K$_5$], while more recent ones can be found in [H$_2$], [HR$_1$], [HR$_2$],
[Ka] amongst others.

A complete answer is only known in the case $h = 1$. A form
$\langle 1, a_1 \rangle \otimes \dots \otimes \langle 1,
a_d \rangle \quad (d \geq 1, \, a_i \in k)$ is called a
{\it $d$-fold Pfister form}  over $k$.\fonote{The form $\langle
1\rangle$ also counts as a Pfister form, more precisely a $0$-fold
Pfister form.}
If $\tau$ is a
Pfister form of degree $d$, then the form $\tau'$, satisfying $\langle 1 \rangle \perp
\tau' = \tau$, is called the {\it pure part} of $\tau$.

\begtheorem{4.} An anisotropic form $\varphi$ over $k$ has height
$1$ if $\varphi \cong a \tau$ $($$\dim \varphi$ even$)$ or
$\varphi\cong a \tau'$ $($$\dim \varphi$ odd$)$, with $a \in k^*$ and
$\tau$ an anisotropic Pfister form of degree $d \geq 1$ in the
first case and $d \geq 2$ in the second case.
\endtheorem

Note that therefore $\SP (\varphi) = (0, 2^{d-1})$ when $\dim
\varphi$ is even and $\SP (\varphi) = (0, 2^{d-1} -1)$ when
$\dim \varphi$ is odd. Is $k = {\Bbb R}$ for example, then all
$d\geq 1$, resp. $d \geq 2$ occur. One can take for instance
$\varphi = 2^d \times \langle 1 \rangle$, resp.
$\varphi = (2^d -1) \times \langle 1 \rangle$.

A proof of Theorem~4 can be found in [K$_5$] and the books
[S], [KS]. In \S20 and \S21 we will prove two theorems for fields
of arbitrary characteristic, from which Theorem~4 can be obtained
in characteristic $\ne2$ (\S20, Theorem~5, \S21,
Theorem~$\blacksquare$).

Little is still known about forms of height~$2$, but the known
results are interesting and partly deep, see e.g.
[K$_6$], [F], [Ka], [H$_1$], [H$_3$].
