\titlea{\S12}{Unified Theory of Generic Splitting}
\leftrunningtitle{Chapter II. Generic Splitting Theory}
%\leftrunningtitle{Specialization of Quadratic and Symmetric
%Bilinear Forms}
\rightrunningtitle{\S12 Unified Theory of Generic
Splitting}


Just as in \S 9 and \S 10 we will not use geometric language and
talk of quadratic forms instead of quadratic spaces in this
section. The definitions concerning \qs s over fields, encountered
in \S 11, will be faithfully  adopted in the language of forms.
{\it In what follows, a ``form'' will always be understood to be a
quadratic form.}

If $\gamma \colon k \to L \cup \infty$ is a place and $\varphi$ a
{\it regular} form over $k$ which has GR \wrt\ $\gamma$, then the
splitting behaviour of $\gamma_* (\varphi)$ under extensions of
the field $L$ is controlled by a given generic splitting tower
$(K_r\; \vert \; 0 \leq r \leq h)$
of $\varphi$, as seen in \S 4 and \S 9 (\S 9, Theorem~4 and
generalization of \S 4, Scholium~3). According to \S 10, something
similar happens for a place $\gamma$ and a
form $\vf$ over $k$ having GR \wrt\ $\gamma$ for which
 $\gamma_* (\varphi)$ is
nondegenerate, provided some care is exercised.
Now we want to extend these results to the case
where $\vf$  has just FR \wrt\ $\gamma$.

At the same time we want to unite the results of \S 9 and \S 10
under one roof, starting with the following definition.

\begdefinition{1.} Let $\vf$ be a \nd\ form over a field $K$. We
call a field extension
$K \hookrightarrow L$
$\vf$-{\it conservative} (or: {\it conservative for} $\vf$)\Idx{fi-conservative field extension@$\vf$-conservative field extension} 
if the form
$\varphi \otimes L$
is again \nd, i.e. if the quasilinear part $QL(\vf)$ of $\vf$
remains anisotropic under a field extension from $K$ to $L$.
\enddefinition

If $\vf$ is regular, then $\dim QL(\varphi) \leq 1$ and so every
field extension of $K$ is $\vf$-conservative. This is true in
particular when $\Char K\not=2$. If $\Char K=2$, then every
separable field extension $K \hookrightarrow L$ is
$\vf$-conservative by \S 10, Theorem~1.

\begtheorem{1.} Let $\lambda \colon K \to L \cup \infty$ be a
place, $K'\supset K$ a field extension and $\mu \colon K' \to L \cup
\infty$ an extension of $\lambda$. Let $\vf$ be a $($\nd$)$ form over
$K$, having \FR\ $($resp. \GR$)$ \wrtl, and let $\lambda_* (\varphi)$ be
\nd.

Claim: the extension $K \hookrightarrow K'$ is $\vf$-conservative.
The form $\varphi \otimes K'$ has \FR\ $($resp. \GR$)$ \wrt\ $\mu$ and
$\mu_* (\varphi \otimes K') \cong \lambda_* (\varphi)$.
\endtheorem

\begproof{} In the case of good reduction, this is Theorem~2 from
\S 10. The proof in the case of fair reduction is similar and goes
as follows.

Let ${\nam o} \colon = {\nam o}_{\lambda}$ and ${\nam o}' \colon = {\nam
o}_{\mu}$. Let $E = (E, q)$ be a space corresponding to $\vf$ and $M$
a \rnd\ \qom\ with $E \cong K \otimes_{{\nam o}} M$. Then
$\lambda_* (\varphi)$ corresponds to the space $\lambda_* (E) \cong L
\otimes_{\overline{\lambda}} \overline{M}$, where
$\overline{\lambda} \colon
\kappa ({\nam o}) \hookrightarrow L$
is the field extension associated to $\lambda$. We have the
factorization $\overline{\lambda} = \overline{\mu} \circ j$,
featuring
the inclusion $j \colon \kappa ({\nam o}) \hookrightarrow \kappa ({\nam
o}')$ and the field homomorphism $\overline{\mu}
\colon \kappa ({\nam o}') \hookrightarrow L$, determined by $\mu$.
Let $\overline{\varphi}$ be the form over $\kappa ({\nam o})$,
associated to $\overline{M}$, i.e.
$\lambda_*(\varphi)=\overline{\varphi}\otimes_{\overline{\lambda}}L$.
Since
$\lambda_* (\varphi)$
is \nd\ by assumption, $\overline{\lambda} \colon
\kappa ({\nam o}) \hookrightarrow L$ is conservative for
$\overline{\varphi}$. Therefore $j \colon \kappa ({\nam o}) \hookrightarrow
\kappa ({\nam o}')$ is conservative for $\overline{\varphi}$ and
$\overline{\mu} \colon \kappa ({\nam o}') \hookrightarrow L$ is
conservative for $\overline{\varphi} \otimes \kappa
({\nam o}')$. Now $\overline{\varphi} \otimes \kappa ({\nam o}')$
belongs to the quadratic $\kappa ({\nam o}')$-module
$\overline{M'}$ with $M' \colon = {\nam o}' \otimes_{{\nam o}} M$.
Therefore $\overline{M'}$ is \nd, in other words $M'$ is \rnd. We
have $\overline{M'} = \kappa ({\nam
o}') \otimes_{\kappa({\nam o})}\overline{M}$. Furthermore,
$K' \otimes E \cong K' \otimes_{{\nam o}'} M'$. Thus $\varphi \otimes
K'$ has \FR\ \wrt\ $\mu$ and
$$\mu_* (\varphi \otimes K') \cong (\overline{\varphi} \otimes
\kappa ({\nam o}')) \otimes_{\overline{\mu}} L = \overline{\varphi}
\otimes_{\overline{\lambda}} L \cong \lambda_* (\varphi).$$
In particular,  $\varphi \otimes K'$ is \nd\ (cf. \S 11, the Remark
following Definition~3). We conclude that $K \hookrightarrow K'$
is $\vf$-conservative.\qed
\endproof

{\it In the following let $\vf$ be a \nd\ form over a field $k$.}
Over every field $K$ we denote the form $\left[ {0 \atop 1} \;
{1 \atop 0} \right]$ by $H$.

\begtheorem{2.} Let $K$ and $L$ be extensions of the field $k$ and
let $\lambda \colon K \to L \cup \infty$ be a place over $k$.
Suppose further that $L$ is $\vf$-conservative and let
$\varphi \otimes K \cong r \times H \perp \psi$
be the Witt decomposition of $\vf$.

Claim: $K$ is also a $\vf$-conservative extension of $k$. The form
$\psi$ has \GR\ \wrtl, and $\varphi \otimes L \cong r
\times H \perp \lambda_* (\psi)$. Thus $\ind (\varphi \otimes L) \geq
\ind (\varphi \otimes K)$.
\endtheorem

\begproof{} We apply Theorem~1 to the trivial place $\gamma \colon k
\hookrightarrow L$ and its extension $\lambda$. Now $\vf$ has \GR\
\wrt\ $\gamma$ and $\gamma_* (\varphi) = \varphi \otimes L$. By
Theorem~1, $K$ is conservative for $\vf$, in other words $\varphi \otimes
K$ is \nd. Again by this theorem, $\varphi \otimes K$ has \GR\
\wrtl\ and $\lambda_* (\varphi \otimes K) \cong \varphi \otimes
L$. Since $r \times H$ also has \GR\ \wrtl, it follows that $\psi$
has \GR\ \wrtl\ (\S 8, Theorem~2), and we have $\varphi \otimes L \cong r \times H
\perp \lambda_* (\psi)$.\qed
\endproof

\begcorollary{} Let $K$ and $L$ be specialization equivalent
extensions of the field $k$. Then $K$ is \fc\ if and only if $L$
is \fc. In this case we have $\ind (\varphi
\otimes K) = \ind (\varphi \otimes L)$. Furthermore, if $\lambda \colon K
\to L \cup \infty$ is a place over $k$, then  $\ker (\varphi \otimes
K)$ has \GR\ \wrtl\ and $\lambda_* (\ker (\varphi \otimes K)) \cong \ker (\varphi
\otimes L)$.\qed
\endcorollary

Next we define a {\it generic splitting tower}\Idx{generic splitting tower} 
$(K_r \; \vert
\; 0 \leq r \leq h)$ associated to $\vf$ just as in \S 4 for
$\Char k\not =2$ and in \S9 for regular forms, with {\it higher
kernel forms}\Idx{higher kernel form}\Idx{higher index}
 $\varphi_r$ and {\it higher indices} $i_r \; (0 \leq r \leq
h)$: $K_0$ is an inessential extension of $k$, $\varphi_r \colon =
\ker (\varphi \otimes K_r)$, $K_{r+1}
\sim_k K_r (\varphi_r)$, $i_{r+1} \colon = \ind (\varphi_r \otimes K_{r+1})$,
in case $r < h$. The construction stops with step~$h$ if $\dim
\varphi_h - \dim QL(\varphi_h) \leq 1$.

All this makes sense, since an inductive argument shows that $QL(\varphi_r) =
QL(\varphi) \otimes K_r$ is anisotropic for every $r \in \{ 0, \dots h
\}$. Indeed, the field extension $K_r (\varphi_r)/K_r$ is
separable for $r<h$. Hence we can conclude by Theorem~2 that
$K_{r+1}/K_r$ is $\vf_r$-conservative once we already know that
$QL(\varphi_r)$ is anisotropic, i.e. that $\varphi_r$ is \nd. The
extension $K_h/k$ is therefore \fc.

\begnote{} If $\Char k\not=2$, then $\dim \varphi_h \leq 1$. If
$\Char k=2$, then $\varphi_h = QL(\varphi_h)$.
\endnote{}

We now have to convince ourselves that the field tower
$(K_r \; \vert \; 0 \leq r \leq h)$
really accomplishes what we expect from a generic splitting tower.
As before in more special situations, it suffices for this purpose,
to prove a theorem of the following sort.

\begtheorem{3}{\rm (Extension of \S9, Theorem~4 and \S10, Theorem
3).}
Let $\gamma \colon k \to L \cup \infty$ be a place. Suppose that
the form $\vf$ has \FR\ \wrt\ $\gamma$ and that $\gamma_*
(\varphi)$ is \nd. Then $\gamma_* (\varphi)$ is isotropic \iff\
$\gamma$ can be extended to a place $\lambda \colon k(\varphi) \to L \cup \infty$.
\endtheorem

\begproof{} We have $\varphi \otimes k(\varphi) \cong H \perp
\psi$ for a form $\psi$ over $k(\varphi)$. If there exists a place
$\lambda \colon k (\varphi) \to L \cup \infty$ then, by Theorem~1,
$\varphi \otimes k (\varphi)$ has \FR\ \wrtl\ and
$\lambda_* (\varphi \otimes k (\varphi)) \cong \gamma_*
(\varphi)$. Then it follows from \S11, Theorem~4, that $\psi$ has
\FR\ \wrtl\ and that $\gamma_* (\varphi) \cong H \perp \lambda_*
(\psi)$. Hence $\gamma_* (\varphi)$ is isotropic.

Conversely, suppose that the \nd\ form
$\overline{\varphi} \colon = \gamma_* (\varphi)$
is isotropic. Since $H <
\overline{\varphi}$ we have $\dim \overline{\varphi} - \dim
QL(\overline{\varphi}) \geq 2$.
Now, by earlier work, we see that $\gamma$ can be
extended to a place $\tilde{\gamma} \colon k (\varphi)
\to L (\overline{\varphi}) \cup \infty$ (look again at the proof
of \S9, Theorem~3). $\{$Note: here we do not need the assumption
that $\overline{\varphi}$ is isotropic, but only that
$\dim \overline{\varphi} -\dim QL(\overline{\varphi}) \geq 2$.$\}$
Since $\overline{\varphi}$ is isotropic, $L(\overline{\varphi})$
is a purely transcendental extension of $L$ (\S9, Lemma~1). Thus
there exists a place $\rho \colon L (\overline{\varphi}) \to L
\cup \infty$ over $L$. The place $\lambda \colon = \rho
\circ \tilde{\gamma} \colon \; k (\varphi) \to L \cup \infty$
is an extension of $\gamma$.\qed
\endproof

In what follows, let $(K_r \; \vert \; 0 \leq r \leq h)$ be a
generic splitting tower associated to $\vf$ with higher kernel
forms $\varphi_r$ and indices $i_r$.

\begtheorem{4}{\rm (Extension of \S4, Theorem~3 and \S10,
Theorem~4).} Let $\gamma \colon k \to L \cup \infty$ be a place
\wrt\ which $\vf$ has \FR\ $($resp. \GR$)$. Suppose that the form $\gamma_*
(\varphi)$ is \nd. Finally, suppose that $\lambda \colon K_m \to L \cup
\infty$ is a place for some $m \in \{ 0,\dots, h \}$, which
extends $\gamma$ and which cannot be extended to $K_{m+1}$ in case
$m<h$. Then $\varphi_m$ has \FR\ $($resp. \GR$)$ \wrtl. The form $\gamma_*
(\varphi)$ has kernel form $\lambda_* (\varphi_m)$ and Witt index
$i_0 + \dots + i_m$.
\endtheorem

\begproof{} We give the proof for \FR. The case of \GR\ can be
treated in an analogous way. We have an isometry
$$\varphi \otimes K_m \cong \varphi_m \perp (i_0 + \dots + i_m) \times H.
\leqno(1)$$
By Theorem~1, $\varphi \otimes K_m$ has \FR\ \wrtl\ and $\lambda_*
(\varphi \otimes K_m) \cong \gamma_* (\varphi)$. From $(1)$ we
get, according to \S11, Theorem~4, that $\varphi_m$ has \FR\
\wrtl\ and that
$$\gamma_* (\varphi) \cong \lambda_* (\varphi_m) \perp (i_0 + \dots + i_m)
\times H. \leqno(2)$$
In particular,  $\lambda_* (\varphi_m)$ is \nd. If $\lambda_*
(\varphi_m)$ were isotropic, then we would surely have that
$\dim \varphi_m \geq \dim QL(\varphi_m) + 2$, i.e. $m<h$. But then
it follows from Theorem~3 that $\lambda$ can be extended to a
place $\mu \colon K_{m+1} \to L \cup \infty$, contradiction!
Therefore $\lambda_* (\varphi_m)$ is anisotropic and $(2)$ is the
Witt decomposition of $\gamma_* (\varphi_m)$.\qed
\endproof

By applying this theorem to the trivial place $\gamma$, we obtain

\begscholium{1}{\rm (Extension of \S4, Scholium~1).} Let $L \supset
k$ be a \fc\ field extension.
\medskip
\setitemindent{(1)}
\item{(1)} Let $\lambda \colon K_m \to L \cup \infty$ be a place
over $k$ for some $m\in \{ 0, \dots, h \}$, which {\it cannot} be
extended to a place from $K_{m+1}$ to $L$ in case $m < h$. Then
$\varphi_m$ has good reduction \wrtl\ and $\lambda_* (\varphi_m)$
is the kernel form of $\varphi\otimes L$.
\medskip
\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
$\mu \colon K_m \to L \cup \infty$.
\medskip
\item{(3)} For a given  number $t$ with
$$0 \leq t \leq \left[ \frac{\dim \varphi - \dim
QL(\varphi)}{2} \right] = i_0 + \dots + i_h$$
and  $m \in {\Bbb N}_0$ minimal  such that $t \leq i_0 + \dots + i_m$,
$K_m$ is a generic \fc\ field extension of $k$ for the
splitting off of $t$ hyperbolic planes of $\vf$
 (cf. the problem posed at the beginning of \S4).\qed
\endscholium

Thus, the field tower
$(K_r \; \vert \; 0 \leq r \leq h)$
rightfully merits the name ``generic splitting tower for $\vf$'' also  in the
current general situation.

\begdefinition{2.} As before we call $h$ the {\it height}\Idx{height} 
of the
form $\vf$ and write $h=h(\vf)$.\Idx{111hfi@$h(\varphi)$} 
Further, we define the {\it
splitting pattern} $\SP(\varphi)$ as the set of all indices $\ind (\varphi \otimes
L)$, where $L$ runs through all \fc\ field extensions of $k$.
By the scholium we have
$$\SP (\varphi) = \{ i_0 + \dots + i_r \; \vert \; 0 \leq r \leq h \}.$$
\enddefinition

\begdefinition{3.} We call any field extension $E \supset k$
which is specialization equivalent to $k(\varphi)$ over $k$ a {\it
generic zero field of}\Idx{generic zero field} 
$\vf$. Further, for $r \in \{
0, \dots, h \}$,
we call any
field extension $F \supset k$ which is specialization equivalent
to $K_r$ over $k$ a {\it partial generic splitting field
of}\Idx{partial generic splitting field}\Idx{generic splitting field of level r@generic splitting field of level $r$} 
$\vf$ or, more precisely, a {\it generic splitting field of
level} $r$.
This signifies that $F$ is generic
for the splitting off of as many hyperbolic planes
as the $(r+1)$-th number $i_0 + \dots + i_r$ in $\SP (\varphi)$
indicates.
In case $r=h$, we speak of a {\it total generic splitting field}.\Idx{total generic splitting field}
\enddefinition

This terminology generalizes notions from \S4 and \S9, but
one has to exercise some caution; the extensions $E \supset k$ and $F \supset k$
have the mentioned generic properties -- as far as we know --
{\it only within the class
of \fc\ field extensions of $k$}.
\bigskip

\noindent{\it Comment on the Theory so far.} If the form $\vf$
over $k$ is degenerate, but $\dim \varphi - \dim QL(\varphi) \geq
2$, then the function field $k(\varphi)$ can be constructed just
as in the \nd\ case. Thus we can formally define a ``generic
splitting tower'' $(K_r \; \vert\; 0~\leq~r~\leq~h)$ as above. We
may also inquire about the splitting behaviour of $\vf$ under
extensions $L \supset k$ of the field $k$, in other words about
the possibilities for the index $\ind (\varphi \otimes L)$ and the
kernel form $\ker (\varphi \otimes L)$ (see Definition~10 in \S 6).
However, can we also extend the theory so far to degenerate forms?

We have a decomposition $\varphi \cong \widehat{\varphi} \perp \delta
(\varphi)$ with $\delta (\varphi) \cong t \times [0]$ and $\widehat{\varphi}$
\nd, cf. \S6, Definition~9 ff. Clearly $k(\varphi)$ is a purely
transcendental extension of $k (\widehat{\varphi})$ of
transcendence degree $t$. Therefore $k(\varphi) \sim_k k
(\widehat{\varphi})$. Consequently, the above tower $(K_r \; \vert \; 0 \leq r \leq
h)$ is a generic splitting tower of $\widehat{\varphi}$. It is now
clear that our theorems so far all remain valid for degenerate
$\vf$, as long as we complete Definition~1 above as follows: for degenerate
$\vf$, a field extension $K \hookrightarrow L$ is called conservative for
$\varphi$ if it is conservative for $\widehat{\varphi}$.

This, however, furnishes us with a fairly bland extension of the theory so far to
degenerate forms. The obvious, difficult question seems to be: Let
$\vf$ be a \nd\ form over a field $k$ and $L \supset k$ a field
extension with $\varphi \otimes L$ degenerate. Is it so that the
splitting behaviour of $\varphi \otimes L$ under field extensions
of $L$ is controlled by a given generic splitting tower $(K_r \;
\vert \; 0 \leq r \leq h)$ of $\vf$ in a similar way as indicated
in Scholium~1(1) above? For example, is $\SP
(\varphi \otimes L) \subset \SP (\varphi)$?

Our theory does not give any information here. The
main problem seems to occur in Theorem~1 above. If we do not know
there that $\lambda_* (\varphi)$ is \nd, we can not conclude -- as
far as I can see -- that $(\varphi \otimes K')^{\wedge}$ has fair
reduction \wrt\ $\mu$.\qed
\bigskip

After this digression, we suppose again that $\vf$ is a {\it \nd}
form over $k$ and that $(K_r \; \vert \; 0 \leq r \leq h)$ is a
generic splitting tower of $\vf$ with higher kernel forms
$\varphi_r$ and indices $i_r$. From Theorem~4 above we immediately
obtain a literal repetition of \S4, Scholium~2 in the current,
more general situation,  if we bear in mind that for every generic
splitting tower $(K'_s \; \vert \; 0 \leq s \leq h)$ of $\vf$, all
extensions $K'_s$ of $k$ are conservative for $\vf$. Furthermore,
we obtain from Theorem~4 an extension of \S4, Scholium~3 in the
same way as we obtained  \S4, Scholium~3 from \S4, Theorem~3:

\begscholium{2.} Let $\gamma \colon k \to L \cup \infty$ be a
place \wrt\ which $\vf$ has \FR\ (resp. \GR). Suppose that $\gamma_*
(\varphi)$ is \nd. Then:
\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 $(L_s \; \vert \; 0 \leq s \leq
e)$ is a generic splitting tower of $\gamma_* (\varphi)$,
then $e \leq h$ and, for every $s$ with $0 \leq s\leq e$, we have
$$\ind (\gamma_* (\varphi) \otimes L_s) = i_0 + \dots + i_m$$
with $m \in \{ 0, \dots, h \}$. The number $m$ is the biggest
integer such that $\gamma$ can be extended to
$\lambda \colon K_m \to L_s\cup \infty$. The kernel form
$\varphi_m$ of $\varphi \otimes K_m$ has \FR\ (resp. \GR)
with
respect to every  extension $\lambda$ of this kind, and
$\lambda_* (\varphi_m)$ is the kernel form of
$\gamma_* (\varphi) \otimes L_s$.
\smallskip
\item{(3)} If $\rho \colon K_r \to L_s \cup \infty$ 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 $L_s$.\qed
\endscholium
