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\title{Extending a valuation centered in a local domain to the formal
  completion.}
\author{F. J. Herrera Govantes\\
Departamento de \'Algebra\\
Facultad de Matem\'aticas\\
Avda. Reina Mercedes, s/n\\
Universidad de Sevilla\\
41012 Sevilla, Spain \and
M. A. Olalla Acosta\\
Departamento de \'Algebra\\
Facultad de Matem\'aticas\\
Avda. Reina Mercedes, s/n\\
Universidad de Sevilla\\
41012 Sevilla, Spain \and
M. Spivakovsky\\
Laboratoire de Math\'ematiques E. Picard\\
Universit\'e Paul Sabatier\\
118, route de Narbonne\\
31062 Toulouse cedex 4, France. \and
B. Teissier\\
Equipe `` G\'eom\'etrie et Dynamique'',\\
Institut math\'ematique de Jussieu\\
175 Rue du Chevaleret\\
F-75013 Paris, France.}

\date{}

\begin{document}
\maketitle

\noi\textbf{Version of \today}
\medskip

\section{Introduction}
\label{In}

All the rings in this paper will be commutative with 1.

Let $(R,m,k)$ be a local noetherian domain with field of fractions $K$ and
$\nu:K^*\rightarrow\Gamma$ a valuation of $K$, centered at $R$. Let $\hat
R$ denote the $m$-adic completion of $R$ (which, of course, need not in
general be a domain). In the applications of valuation theory to
commutative algebra and the study of singularities, one is often induced to
replace $R$ by its $m$-adic completion $\hat R$ and $\nu$ by a suitable
extension $\hat\nu$ to $\frac{\hat R}P$ for a suitably chosen prime ideal
$P$, such that $P\cap R=(0)$ (below, we will mention two specific
applications we have in mind). It is well known and not hard to prove that
such extensions $\hat\nu$ exist for some minimal prime ideals $P$ of $\hat
R$. In general, such a $\hat\nu$ is far from being unique. The purpose of
this paper is to give, assuming that $R$ is a G-ring (see the definition
below), a systematic description of all such extensions $\hat\nu$ and to
identify certain classes of extensions which are of particular interest for
applications.

When studying the extensions of $\nu$ to the completion of $R$, one is led to
the study of its extensions to the henselization $\tilde R$ of $R$ as a
natural first step. We therefore start out by letting $\sigma:R\rightarrow
R^\dag$ denote one of the two operations of henselization or completion:
\begin{eqnarray}
R^\dag&=&\hat R\mbox{ or}\label{eq:hatR}\\
R^\dag&=&\tilde R,\label{eq:tildeR}
\end{eqnarray}
with $R$ a G-ring.

Let $r$ denote the (real) rank of $\nu$. Let $(0)=\Delta_r\supsetneqq
\Delta_{r-1}\supsetneqq\dots\supsetneqq\Delta_0=\Gamma$ be the isolated
subgroups of $\Gamma$ and $P_0=(0)\subset P_1\subset\dots\subset P_r=m$ the
prime valuation ideals of $R$, which need not, in general, be distinct. In
this paper, we will assume that $R$ is a G-ring. Under this assumption, we
will canonically associate to $\nu$ a chain $H_1\subset H_3\subset\dots\subset
H_{2r+1}=mR^\dag$ of ideals of $R^\dag$, numbered by odd integers from 1 to $2r+1$,
such that $H_{2\ell+1}\cap R=P_\ell$ for $0\le \ell\le r$. We will show that all the
ideals $H_{2\ell+1}$ are prime. We will define $H_{2\ell}$ to be the unique minimal
prime ideal of $P_\ell R^\dag$, contained in $H_{2\ell+1}$ (that such a minimal prime is
unique follows from the regularity of the homomorphism $\sigma$; if
$R^\dag=\tilde R$ then necessarily $H_{2\ell}=H_{2\ell+1}$).

We will thus obtain, in the cases (\ref{eq:hatR}) and (\ref{eq:tildeR}), a
chain of $2r+1$ prime ideals
$$
H_0\subset H_1\subset\dots\subset H_{2r}=H_{2r+1}=mR^\dag,
$$
satisfying $H_{2\ell}\cap R=H_{2\ell+1}\cap R=P_\ell$ and such that $H_{2\ell}$ is a
minimal prime of $P_\ell R^\dag$ for $0\le \ell\le r$. Moreover, if $R^\dag=\tilde
R$, then $H_{2\ell}=H_{2\ell+1}$. We call $H_\ell$ the $\ell${\bf -th implicit prime
  ideal} of $R^\dag$, associated to $R$ and $\nu$. The ideals $H_\ell$ behave
well under local blowing ups along $\nu$ (that is, birational local
homomorphisms $R\rightarrow R'$, essentially of finite type, such that $\nu$
is centered at $R'$). This means that given any local blowing up $R\rightarrow
R'$ along $\nu$ the $l$-th implicit prime ideal $H'_\ell$ of ${R'}^\dag$ has the
property that $H'_\ell\cap\hat R=H_\ell$.

For a prime ideal $P$ in a ring $R$, $\kappa(P)$ will denote the residue
field $\frac{R_P}{PR_P}$.

Let
$(0)=\mathbf{P}_0\subsetneqq\mathbf{P}_1\subsetneqq\dots\subsetneqq
\mathbf{P}_r=m_\nu$ be the prime ideals of the valuation ring $R_\nu$. By
definitions, our valuation $\nu$ is a composition of $r$ rank one
valuations $\nu=\nu_1\circ\nu_2\dots\circ\nu_\ell$, where $\nu_\ell$ is a
valuation of the field $\kappa(\mathbf{P}_{\ell -1})$, centered at
$\frac{(R_\nu)_{\mathbf{P}_\ell}}{\mathbf{P}_{\ell -1}}$.

If $R^\dag=\tilde R$, we will prove that there is a unique
extension $\tilde\nu$ of $\nu$ to $\frac{R^\dag}{H_0}$. If
$R^\dag=\hat R$, the situation is more complicated. We will show
that if $\nu$ extends to a valuation of $\hat\nu$ centered at
$\frac{\hat R}P$ with $P\cap R=(0)$ then the prime $P$ must
contain the minimal prime $H_0$ of $\hat R$. We will then show
that specifying an extension $\hat\nu$ of $\nu$ as above is
equivalent to specifying a chain of prime valuation ideals $\tilde
H_0\subset\tilde H_1\subset\dots\subset\tilde H_{2r}=m\hat R$ such
that $H_\ell\subset\tilde H_\ell$ for all $l\in\{0,\dots,2r\}$, and
valuations $\hat\nu_1,\hat\nu_2,\dots,\hat\nu_{2r}$, where
$\hat\nu_j$ is a valuation of the field
$\kappa\left(\hat{\mathbf{P}}_{j-1}\right)$, arbitrary when $j$ is
odd and satisfying certain conditions, coming from the valuation
$\nu_{\frac j2}$, when $j$ is even. Here $\hat{\mathbf{P}}_{j-1}$
denotes the $(j-1)$-st prime ideal of the valuation ring
$R_{\hat\nu}$; its intersection with $\hat R$ is $\tilde H_{j-1}$.

The prime ideals $H_j$ are defined recursively in $j$. Namely, we define
\begin{equation}
H_{2\ell-1}
=\bigcap\limits_{\beta\in\Delta_{\ell -1}}\left(
\left(\lim\limits_{\overset\rightarrow{R'}}
P'_\beta {R'}^\dag\right)\bigcap R^\dag\right),\label{eq:defin}
\end{equation}
where $R'$ ranges over all the local blowings up of $R$ along
$\nu$. The ideal $H_{2\ell-2}$ is, by definition, the unique minimal
prime of $P_{\ell -1}\hat R$, contained in $H_{2\ell-2}$ (that such a
minimal prime is unique follows easily from the fact that $R$ is a
G-ring).

The questions answered in this paper originally arose from our work on the
Local Uniformization Theorem, where passage to completion is required in
both the approaches of \cite{Spi2} and \cite{Te}. In \cite{Te}, one really
needs to pass to completion for valuations of arbitrary rank. One of the
main applications of the theory of implicit prime ideals, proved in the
present paper, is the following result, announced in \cite{Te}.
\begin{thm}\label{teissier} With the above notation, assume that the Local
  Uniformization Theorem holds for all the local rings of dimension at most $\dim\ R-1$, which are birational and
  essentially of finite type over $\frac{\hat R}P$, where $P$ is some prime ideal of $\hat
  R$. Then there exists a prime ideal $H$ of $\hat R$ with $H\cap R=(0)$ and a
  valuation $\hat\nu$, centered at $\frac{\hat R}H$, such that
  $\gr_{\hat\nu}\frac{\hat R}H$ is scalewise birational to $\gr_\nu R$ (the
definition of scalewise birational will be recalled below).
\end{thm}
As observed in \cite{Te}, there is no danger of circular reasoning when
applying this result to the Local Uniformization Theorem, since one can
proceed by induction on $\dim\ R$. In the above Theorem, local uniformization
is assumed up to dimension $\dim\ R-1$ and it is used for local uniformization
in dimension $\dim\ R$.

The approach to the Local Uniformization Theorem taken in \cite{Spi2} is to
reduce the problem to the case of rank 1 valuations. The theory of implicit
prime ideals is much simpler for valuations of rank 1.

Another recent application of these results on extending valuations to the
formal completion is the work \cite{LMSS} on the Pierce-Birkhoff
conjecture. This conjecture was reduced by J. Madden to studying the
separating ideal $<\alpha,\beta>$ of two points $\alpha$ and $\beta$ in the
real spectrum of $A$ (of course $A$ must be of characteristic 0 in order for
$\sper\ A$ to be non-empty). It is easy to reduce the problem to the case when
$A$ is local. However, for various technical reasons one is induced to pass to
the formal completion $\hat A$ of $A$. At this point, we need to know that there
exist points $\hat\alpha$ and $\hat\beta$ in the real spectrum of $\hat A$
such that $<\hat\alpha,\hat\beta>\cap A=<\alpha,\beta>$. This fact is deduced
readily from \ref{teissier}.

The paper is organized as follows. In \S\ref{basics} we define the
odd-numbered implicit ideals $H_{2\ell+1}$ and prove that $H_{2\ell+1}\cap
R=P_\ell$. We observe that by the very definition, the ideals $H_{2\ell+1}$ behave
well under local blowings up along $\nu$. Proving that $H_{2\ell+1}$ is indeed
prime is postponed until \S\ref{prime}, since the proof of this for $\hat R$
uses the uniqueness of the extension $\tilde\nu$ of $\nu$ to $\tilde R$ modulo
its first implicit prime ideal. We also give two examples to motivate the
definition \ref{eq:defin} -- one to explain the need to localize at
$H_{2\ell+1}$, the other the need of taking the limit with respect to $R'$.

In \S\ref{henselization} we prove that $H_{2\ell+1}$ is prime for the
henselization. We study the extension of $\nu$ to $\tilde R$ modulo its first
implicit prime ideal and prove that such an extension is unique.

In \S\ref{prime} we prove that $H_{2\ell+1}$ are indeed prime for the
completion. We defined $H_{2\ell}$ to be the unique minimal prime ideal of
$P_\ell\hat R$ and that $H_{2\ell}\cap R=H_{2\ell+1}\cap R=P_\ell$.

It follows from the noetherianity of $R^\dag$, that there exists a specific
$R'$ for which the limit is achieved. In \S\ref{henselization} and
\S\ref{prime} we explain in more detail what that $R'$ is; it turns out that
the $R'$ which has this property for the henselization, automatically has it
for the completion.

In \S\ref{extensions} we describe the set of extensions $\hat\nu$ to
$\frac{\hat R}P$, where $P$ is a prime ideal such that $P\cap R=(0)$. We
describe one class of such extensions, which we call ``natural'', for which
$\tilde H_j=H_j$ for all $j$ and the valuation $\hat\nu_{2\ell+1}$ is completely
determined by $\nu_\ell$ for each $l$. Up until this point in the paper the Local
Uniformization Theorem has not yet appeared. Finally, \ref{locuni} is devoted
to the part of the theory which uses the Local Uniformization Theorem, notably
Theorem \ref{teissier}. We
would like to acknowledge the paper \cite{HeSa} by Bill Heinzer and Judith
Sally on this subject which inspired one of the authors to continue thinking
about the passage to formal completion.

\section{Definition and first properties of the implicit ideals.}
\label{basics}

Let the notation be as above. We define our main object of study, the $j$-th
implicit prime ideal $H_j$ recursively in the descening order of $j$, as
follows. To start the recursion, put
$$
H_{2r}=H_{2r+1}=mR^\dag.
$$
Now let $l$ be a strictly positive integer, $l\le r$, and assume that
$H_{2\ell+1}$ is already defined. Put
\begin{equation}
H_{2\ell-1}=
\bigcap\limits_{\beta\in\Delta_{\ell -1}}\left(
\left(\lim\limits_{\overset\rightarrow{R'}}
P'_\beta {R'}^\dag\right)\bigcap R^\dag\right),\label{eq:defin1}
\end{equation}
where $R'$ ranges over all the local blowings up of $R$ along $\nu$.

The following two examples illustrate the need for taking the limit over the
local blowings up $R'$.

\noi\textit{Example.}

\noi\textit{Example.} Let
$R=\frac{k[x,y,z]_{(x,y,z}}{(z^2-y^2(1+x))}$. Let $\Gamma=\Z^2$
with the lexicographical ordering. Let $t$ be an independent
variable and let $\nu$ be the valuation, centered in $R$, induced
by the $t$-adic valuation of $k[[t^\Gamma]]$ under the injective
homomorphism $\iota:R\hookrightarrow k[[t^\Gamma]]$, defined by
$\iota(x)=t^{(0,1)}$, $\iota(y)=t^{(1,0)}$ and
$\iota(z)=t^{(1,0)}\sqrt{1+t^{(0,1)}}$. Then we have
$\bigcap\limits_{\beta\in\Phi}P_\beta\hat R=(0)$ and
$\bigcap\limits_{\beta\in\Phi\cap\Delta_1}P_\beta\hat R=(y,z)\hat
R=P_1\hat R$, where the ideal $(0)$ is not prime in $\hat R$. Now,
let $R'=R\left[\frac zy\right]_{M'}$, where $M'=\left(x,y,\frac
zy-1\right)$ is the center of $\nu$ in $R\left[\frac zy\right]$.
We have $z-y\sqrt{1+x}\in P'_\beta\hat R'\setminus P_{(2,0)\hat
R}$; in particular,
$$
z-y\sqrt{1+x}\in\bigcap\limits_{\beta\in\Phi}P'_\beta\hat
R'\setminus P_{(2,0)\hat R}.
$$
Thus this example also shows that
the ideals $P_\beta\hat R$ and
$\bigcap\limits_{\beta\in\Phi}P_\beta\hat R$ do not behave well
under blowing up.
\bigskip

Note that both of these examples occur not only for the completion $\hat
R$ but also for the henselization $\tilde R$.
\begin{prop}\label{contracts} We have $H_{2\ell-1}\cap R=P_{\ell -1}$.
\end{prop}
\noi\textit{Proof:} To be filled in. $\Box$
\begin{prop}\label{behavewell} The ideals $H'_{2\ell-1}$ behave well under local
  blowings up along $\nu$. In other words, let $R\rightarrow R'$ be a local
  blowing up along $\nu$ and let $H'_{2\ell-1}$ denote the $(2l-1)$-st implicit
  prime ideal of $\hat R'$. Then $H_{2\ell-1}=H'_{2\ell-1}\cap R^\dag$.
\end{prop}
\noi\textit{Proof:} Immediate from the definitions. $\Box$
\smallskip

To study the ideals $H_{2\ell+1}$, we need to understand more explicitly the
nature of the limit appearing in (\ref{eq:defin1}). To study the relationship
between the ideals $P_\beta R^{\dag}$ and $P'_\beta {R'}^\dag\bigcap
R^\dag$, it is useful to factor the natural map $R^\dag\rightarrow{R'}^\dag$
as $R^\dag\rightarrow R^\dag\otimes_RR'\overset\phi\rightarrow{R'}^\dag$. In
general, the ring $R^\dag\otimes_RR'$ is not local (see the above examples),
but it has one distinguished maximal ideal $M'$, namely, the ideal generated by
$mR^\dag\otimes1$ and $1\otimes m'$, where $m'$ denotes the maximal ideal of
$R'$. The map $\phi$ factors through the local ring
$\left(R^\dag\otimes_RR'\right)_{M'}$ and the resulting map
$\left(R^\dag\otimes_RR'\right)_{M'}\rightarrow{R'}^\dag$ is either
the formal completion or the henselization; in either case, it is faithfully
flat. Thus $P'_\beta{R'}^\dag\cap\left(R^\dag\otimes_RR'\right)_{M'}= P'_\beta
\left(R^\dag\otimes_RR'\right)_{M'}$. This shows that we may replace
${R'}^\dag$ by $\left(R^\dag\otimes_RR'\right)_{M'}$ in (\ref{eq:defin1})
without affecting the result, that is,
\begin{equation}
H_{2\ell-1}=
\bigcap\limits_{\beta\in\Delta_{\ell -1}}\left(
\left(\lim\limits_{\overset\rightarrow{R'}}
P'_\beta\left(R^\dag\otimes_RR'\right)_{M'}\right)\bigcap
R^\dag\right).\label{eq:defin3}
\end{equation}
From now on, we will use (\ref{eq:defin3}) as our working definition of the
implicit prime ideals. One advantage of the expression (\ref{eq:defin3}) is
that it makes sense in a situation more general than the completion and the
henselization. Namely, to study the case of the henselization $\tilde R$, we
will need to consider \'etale extensions $R^\dag$ of $R$, which are contained
in $\tilde R$ (particularly, those which are essentially of finite type). The
definition (\ref{eq:defin3}) of the implicit prime ideals makes sense also in
that case.

\section{Extending a valuation centered in a local G-domain to its
  henselization.}
\label{henselization}

Let $\tilde R$ denote the henselization of $R$, as above. The completion
homomorphism $R\rightarrow\hat R$ factors through the henselization:
$R\rightarrow\tilde R\rightarrow\hat R$. In this section, we will show that
$H_1$ a minimal prime of $\tilde R$, that $\nu$ extends uniquely to a
valuation $\tilde\nu$ of rank $r$ centered at $\frac{\tilde R}{H_1}$, and that
$H_1$ is the unique prime ideal $P$ of $\tilde R$ such that $\nu$ extends to a
valuation of $\frac{\tilde R}P$. Furthermore, we will prove that $H_{2\ell+1}$ is
a minimal prime of $P_\ell\tilde R$ for all $l$ and that these are precisely the
prime $\tilde\nu$-ideals of $\tilde R$.

Studying the implicit prime ideals of $\tilde R$ and the extension of $\nu$ to
$\tilde R$ is a logical intermediate step before attacking the formal
completion, for the following reason. As we will show in the next section, if
$R$ is already henselian in (\ref{eq:defin1}) then $P'_\beta\hat
R'_{H'_{2\ell+1}}\cap\hat R=P_\beta\hat R$ for all $\beta$ and $R'$ and thus we have that
$H_{2\ell-1}=\bigcap\limits_{\beta\in\Gamma+}\left(P_\beta\hat R\right)$.

We state the main result of this section. In the case when $R^\dag$ is an
\'etale extension of $R$, contained in $\tilde R$, we use (\ref{eq:defin3}) as
our definition of the implicit prime ideals.
\begin{thm}\label{hensel0} Let $R^\dag$ be a local \'etale extension of $R$, contained in $\tilde R$. Then:

(1) The ideal $H_{2\ell+1}$ is prime for $0\le l\le r$; it is a minimal prime of
$P_\ell$. In particular, $H_1$ is a minimal prime of $R^\dag$.

(2) The ideal $H_1$ is the unique prime $P$ of $R^\dag$ such that there
    exists an extension $\nu^\dag$ of $\nu$ to $\frac{R^\dag}{H_1}$; the extension $\nu^\dag$ is unique. The graded algebra
    $\gr_{\nu^\dag}\frac{R^\dag}{H_1}$ is
 scalewise birational to $\gr_\nu R$; in particular, $rk\ \nu^\dag=r$.


(3) The ideals $H_{2\ell+1}$ are precisely the prime $\nu^\dag$-ideals of
    $R^\dag$.
\end{thm}
\noi\textit{Proof:} By assumption, the ring $R^\dag$ is a direct limit of
local, strict \'etale extension of $R$ which are essentially of finite
type. All the assertions (1)-(3) behave well under taking direct limits, so it
is sufficient to prove the Theorem in the case when $R^\dag$ is essentially of
finite type over $R$. From now on, we will restrict attention to this case.

The next step is to characterize explicitly those local blowings up
$R\rightarrow R'$ for which the limit in (\ref{eq:defin3}) is attained.

Since $R^\dag$ is algebraic, essentially of finite type over $R$,
the ring $\kappa(P_\ell)\otimes_RR^\dag$ is finite over
$\kappa(P_\ell)$; this ring is reduced but it may contain zero
divisors. In fact, it is known that the minimal primes of
$\frac{\tilde R}{P_\ell\tilde R}$ (and of $\frac{\hat
  R}{P_\ell\hat R}$) correspond one-to-one to maximal ideals of the normalization
of $\frac R{P_\ell}$ (here we use the fact that $R$ is a G-ring). Let $\Lambda$
denote the set of minimal primes of $\frac{\tilde R}{P_\ell\tilde R}$. Since we
have strict \'etale maps $\frac
R{P_\ell}\rightarrow\frac{R^\dag}{P_\ell R^\dag}\rightarrow\frac{\tilde R}{P_\ell\tilde
  R}$, there exists a partition $\Lambda=\coprod\limits_{i=1}^s\Lambda_i$ such
that the sets $\Lambda_i$ are in one-to-one correspondence with the minimal
primes of $\frac{R^\dag}{P_\ell R^\dag}$. Now, let $N$ denote the center of the
valuation induced by $\nu$ on $\frac R{P_\ell}$ in the normalization $S$ of
$\frac R{P_\ell}$. Since $N$ lies over $\frac m{P_\ell}$, it is a maximal ideal of
$S$. The ring $S_N\otimes_RR^\dag$ is contained in the henselization of the
normal local G-ring $S_N$ and is therefore an integral domain. Thus the zero
ideal of $S_N\otimes_RR^\dag$ contracts to a certain minimal prime of
$\frac{R^\dag}{P_\ell R^\dag}$; let us denote this minimal prime by $Q_0$. Since
the ring $\kappa(P_\ell)\otimes_RR^\dag$ is reduced and finite over
$\kappa(P_\ell)$, it is a direct product of fields, each of which is finite over
$\kappa(P_\ell)$. These fields are precisely the fields of fractions of the rings
of the form $\frac{R^\dag}{P_\ell R^\dag}/Q$, where $Q$ ranges over the minimal
primes of $\frac{R^\dag}{P_\ell R^\dag}$. Let $L=L_0$ be the field of fractions of
$\frac{R^\dag}{P_\ell R^\dag}/Q_0$; $L$ is a finite extension of $\kappa(P_\ell)$.

Now let $\pi:R\rightarrow R'$ be a local blowing up of $R$ along
$\nu$. Let $\kappa(P_\ell)\rightarrow\kappa(P'_\ell)$ be the field
extension induced by $\pi$. Let $d$ be the greatest integer such
that for some local blowing up $\pi$, $L$ contains an extension of
$\kappa(P_\ell)$ of degree $d$, isomorphic to some extension of
$\kappa(P_\ell)$, contained in $\kappa(P'_\ell)$.
\smallskip
\begin{lem}\label{anirred1} Fix an integer $l\in\{1,\dots,r\}$. There exists a
  local blowing up $R\rightarrow R'$ along $\nu$ having the following
  property. Let $P'_\ell$ denote the $\ell$-th prime $\nu$-ideal of $R'$. Then
  the ring $\frac{R'}{P'_\ell}$ is analytically irreducible; in particular, $\frac{R'}{P'_\ell}\otimes R^\dag$ is an integral
  domain.
\end{lem}
\noi\textit{Proof of Lemma \ref{anirred1}:} Since $R$ is a G-ring, every
homomorphic image of $R$ is Nagata \cite{Mat}. Let $\pi:\frac
R{P_\ell}\rightarrow S$ be the normalization of $\frac R{P_\ell}$ and let
$\rho:R\rightarrow R''$ be a birational homomorphism such that $\spec\ S$ is
the (scheme-theoretic) inverse image of $\frac R{P_\ell}$ in
$\spec\ R''$. Explicitly, such a map $\rho$ can be described as follows. Write
$S$ as $S=\frac R{P_\ell}\left[\frac{\bar a_1}{\bar b_1},\dots,\frac{\bar
    a_n}{\bar b_n}\right]$, where $\bar a_i,\bar b_i$ are elements of $\frac
R{P_\ell}$. Let $a_i$ be a representative of $\bar a_i$ in $R$, and $b_i$ a
representative of $\bar b_i$. Since $\bar b_i\ne0$, we have $b_i\notin
P_\ell R''$. Put $R''=R\left[\frac{a_1}{b_1},\dots,\frac{a_n}{b_n}\right]$. Now
let $R'$ be the localization of $R''$ at the center of $\nu$ and let $P'_\ell$ be
the $l$-th prime $\nu$-ideal of $R'$. On the one hand,
\begin{equation}
P'_\ell\supset P_\ell R';\label{eq:inclusion}
\end{equation}
on the other, the ring $\frac{R'}{P_\ell R'}$ is a localization of $S$
at a prime ideal; in particular, $\frac{R'}{P_\ell R'}$ is a domain.
Thus $P_\ell R'$ is prime. Now, for any element $x\in P'_\ell$, we have
$xb^N_1\dots b^N_n\in P_\ell R'$ for $N$ sufficiently large. Since
$b_1\dots b_n\notin P_\ell R'$ and $P_\ell R'$ is prime, we have $x\in
P_\ell R'$. Together with (\ref{eq:inclusion}), this shows that
$P_\ell R'=P'_\ell$. The ring $\frac{R'}{P'_\ell}\cong\frac{R'}{P_\ell R'}$ is a
localization of $S$ at a prime ideal, hence it is a normal G-ring.
In particular, it is analytically irreducible {}, as desired.
$\Box$
\medskip

Now, take a local blowing up $\pi$ such that $\kappa(P'_\ell)$
contains an extension of $\kappa(P_\ell)$ of degree $d$, isomorphic
to some extension of $\kappa(P_\ell)$, contained in $L$. Applying
Lemma \ref{anirred1} to the result, we obtain an $R'$ such that,
in addition, $\frac{R'}{P'_\ell}\otimes R^\dag$ is an integral
domain.
\smallskip

\noi\textit{Claim.} The limit in (\ref{eq:defin3}) is attained for this $R'$.
\smallskip

\noi\textit{Proof of Claim:} Replacing $R$ by $R'$, we may
rephrase the Claim as follows. Assume that for any local blowing
up $\pi:R\rightarrow R'$, no non-trivial extension of
$\kappa(P_\ell)$, contained in $\kappa(P'_\ell)$, is isomorphic to an
extension of $\kappa(P_\ell)$, contained in $L$. For an element
$\beta\in\Phi$, let $\beta_{\ell -1}=\min\{\gamma\in\Phi\ |\
\beta-\gamma\in\Delta_\ell\}$. We must show that for any local
blowing up $\pi$ and any $\beta\in\Delta_{\ell -1}\cap\Phi$, we have
\begin{equation}
P'_{\beta_{\ell -1}}\left(R^\dag\otimes_RR'\right)_{M'}\bigcap R^\dag=
P_{\beta_{\ell -1}}R^\dag.\label{eq:stab}
\end{equation}
One inclusion in (\ref{eq:stab}) is trivial; we must show that
\begin{equation}
P'_{\beta_{\ell -1}}\left(R^\dag\otimes_RR'\right)_{M'}\bigcap
R^\dag\subset P_{\beta_{\ell -1}}R^\dag.\label{stab1}
\end{equation}
This is the same as proving the injectivity of the map
\begin{equation}
\bar\pi:\frac{
R_{P_\ell}}{P_{\beta_{\ell -1}}}\otimes_RR^\dag\rightarrow\left(\frac{
R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}\otimes_RR^\dag\right)_T\label{eq:inclusion2}
\end{equation}
induced by $\pi$, where $T$ denotes the image of the
multiplicative set $R'\otimes_RR^\dag\setminus M'$ in $\frac{
R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}\otimes_RR^\dag$ under the natural map
$R'\otimes_RR^\dag\rightarrow\frac{
R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}\otimes_RR^\dag$. To prove the
injectivity of $\bar\pi$, we start with the field extension
$\kappa(P_\ell)\hookrightarrow\kappa(P'_\ell)$ induced by $\pi$. Since
$R^\dag$ is flat over $R$, the induced map
$\pi_1:\kappa(P_\ell)\otimes_RR^\dag\rightarrow\kappa(P'_\ell)\otimes_RR^\dag$
is also injective. Now, $\kappa(P_\ell)\otimes_RR^\dag$ is algebraic
and finitely generated over the field $\kappa(P_\ell)$; moreover, it
is a domain by the assumptions on $R$. Thus
$\kappa(P_\ell)\otimes_RR^\dag$ is a finite field extension of
$\kappa(P_\ell)$.

Also by the assumptions on $R$, there is no non-trivial extension
$L$ of $\kappa(P_\ell)$ which embeds into both $\kappa(P'_\ell)$ and
$\kappa(P_\ell)\otimes_RR^\dag$. Hence the finitely generated
algebraic extension
$$
\kappa(P'_\ell)\otimes_RR^\dag\cong\kappa(P'_\ell)\otimes_{\kappa(P_\ell)}R^\dag
$$
of $\kappa(P'_\ell)$ is an integral domain and hence a field. Thus
$\kappa(P'_\ell)\otimes_RR^\dag$ is a finite field extension of
$\kappa(P'_\ell)$.

Next, we observe that
\begin{equation}
\kappa(P'_\ell)\otimes_RR^\dag=\left(\frac{
R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}\otimes_RR^\dag\right)_{red}.\label{eq:reduced}
\end{equation}
Since $\kappa(P'_\ell)\otimes_RR^\dag$ is a field, (\ref{eq:reduced})
shows that $\frac{R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}\otimes_RR^\dag$ is
a local artinian ring. The multuplicative system $T$ is disjoint
from the nilradical of
$\frac{R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}\otimes_RR^\dag$, hence
\begin{equation}
\left(\frac{R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}\otimes_RR^\dag\right)_T=
\frac{R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}\otimes_RR^\dag.\label{eq:localiz}
\end{equation}
Now, the map $\frac{R_{P_\ell}}{P_{\beta_{\ell -1}}}\hookrightarrow\frac{
R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}$ is injective, hence so is $\frac{
R_{P_\ell}}{P_{\beta_{\ell -1}}}\otimes_RR^\dag\rightarrow\frac{
R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}\otimes_RR^\dag$ by the flatness of
$R^\dag$ over $R$. Combining this with (\ref{eq:localiz}), we
obtain (\ref{eq:inclusion2}). This completes the proof of Claim.

Let $R'$ be a local blowing up of $R$ which satisfies the
conclusion of the Claim. Once Theorem \ref{hensel0} is proved for
$R'$, the same results for $R$ will follow easily by intersecting
all the ideals of ${R'}^\dag$ in sight with $R^\dag$. Therefore
from now on we will replace $R$ by $R'$, that is, we will assume
that (\ref{eq:stab}) holds for any local blowing up $R'$ of $R$.

We are now in the position to prove (1) of Theorem \ref{hensel0}.

Let $\Phi=\nu(R\setminus\{0\})$. Since $R$ is noetherian, $\Phi$
is well ordered. For an element $\beta\in\Phi$, let $\beta+$
denote the immediate successor of $\beta$ in $\Phi$ and
$\beta(l-1,+)$ the smallest element of $\Phi$ greater than $\beta$
such that $\beta(l-1,+)-\beta\notin\Delta(l-1)$.

Take any elements $x,y\in R^\dag\setminus H_{2\ell-1}$. Then there
exist elements $\beta,\gamma\in\Phi\cap\Delta_{\ell -1}$ such that
\begin{equation}
x\in P_{\beta}R^\dag\setminus P_{\beta(\ell-1,+)}R^\dag\label{eq:xinbeta}
\end{equation}
and
\begin{equation}
y\in P_{\gamma}R^\dag\setminus P_{\gamma(\ell-1,+)}R^\dag.\label{eq:yingamma}
\end{equation}
To prove (1), it is sufficient to prove that
\begin{equation}
xy\notin P_{\beta+\gamma(\ell-1,+)}R^\dag.\label{eq:xynotin}
\end{equation}
Let $(a_1,...,a_n)$ be a set of generators of $P_\beta$ and
$(b_1,...,b_s)$ a set of generators of $P_\gamma$, with
$\nu(a_1)=\beta$ and $\nu(b_1)=\gamma$. Let $R'$ be a local
blowing up along $\nu$ such that $R'$ contains all the fractions
$\frac{a_i}{a_1}$ and $\frac{b_j}{b_1}$. Moreover, choose $R'$ so
that $P'_\ell{R'}^\dag$ is a prime ideal; this is possible by Lemma
\ref{anirred1}. Then $a_1\ |\ x$ and $b_1\ |\ y$ in ${R'}^\dag$.
Write $x=za_1$ and $y=wb_1$ in ${R'}^\dag$. The equality
(\ref{eq:stab}) implies that $z,w\notin P'_\ell{R'}^\dag $, hence
\begin{equation}
zw\notin P'_\ell{R'}^\dag.\label{eq:zwnotin}
\end{equation}
We obtain $xy=a_1b_1zw$. Since $\nu$ is a valuation on $R'$, we
have $\left(P'_{\beta+\gamma(l-1,+)}:(a_1b_1)R'\right)\subset
P'_\ell$. By faithful flatness of ${R'}^\dag$ over $R'$ we obtain
\begin{eqnarray}
\left(P'_{\beta+\gamma(\ell-1,+)}{R'}^\dag:(a_1b_1){R'}^\dag\right)&\subset&P'_\ell{R'}^\dag\mbox{
and}\label{eq:colon}\\
a_1b_1&\notin&P_{\beta+\gamma(\ell-1,+)}{R'}^\dag.
\end{eqnarray}
Combining this with (\ref{eq:zwnotin}), we obtain
(\ref{eq:xynotin}), as desired. This proves that the ideal
$H_{2\ell-1}$ is prime. By Proposition \ref{contracts}, $H_{2\ell-1}$
maps to $P_\ell$ under the map $\spec\ R^\dag\rightarrow\spec\ R$.
Since this map is \'etale, its fibers are zero-dimensional, which
shows that $H_{2\ell-1}$ is a minimal prime of $P_\ell$. This completes
the proof of (1) of Theorem \ref{hensel0}.

To describe the extension $\nu^\dag$ of $\nu$ to
$\frac{R^\dag}{H_1}$, we use induction on $r$. For $r=0$, $R$ is a
field, $R=R^\dag$ (since it is a strict \'etale extension) and
there is nothing to prove. Assume that the result is true for
valuations of rank $r-1$. For an element $x\in R^\dag\setminus
H_1$, there exists an element $\beta\in\Phi$ such that $z\in
P_\beta R^\dag\setminus P_{\beta(0,+)}R^\dag$ (where, of course,
we allow $\beta=0$). Let $(a_1,...,a_n)$ be a set of generators of
$P_\beta$ and let $R\rightarrow R'$ be a local blowing up along
$\nu$ such that $R'$ contains all the fractions $\frac{a_i}{a_1}$
and such that $P'_1{R'}^\dag$ is prime; the latter is possible by
Lemma \ref{anirred1}. Note that in view of (1) of Theorem
\ref{hensel0}, the primality of $P'_1{R'}^\dag$ is also equivalent
to saying that $P'_1{R'}^\dag=H'_3$. Write
\begin{equation}
x=za_1\label{eq:xza}
\end{equation}
in $R'$; we have $z\in{R'}^\dag\setminus P'_1{R'}^\dag$. Let
$\bar\nu$ denote the rank $r-1$ valuation of
$\kappa(\mathbf{P_1})$ such that $\nu=\nu_1\circ\bar\nu$. Let
$\bar\nu^\dag$ be the unique extension of $\bar\nu$ to
$\lim\limits_{\overset\rightarrow{R'}}\frac{{R'}^\dag}{H'_3}$. To
be precise, the restriction of $\bar\nu^\dag$ to each
$\frac{{R'}^\dag}{H'_3}$ is defined and is unique by the induction
assumption, and $\bar\nu$ is defined and is unique by passing to
the limit. Now put
\begin{equation}
\nu^\dag(x)=\nu(a_1)+\bar\nu^\dag(z).\label{eq:nudag}
\end{equation}
The uniqueness of $\nu^\dag$ is already obvious. It is also clear
from (\ref{eq:nudag}) that every element outside of $H_1$ has a
value in $\Gamma$, so for any prime ideal $P$ of $R^\dag$ such
that $\nu$ extends to a valuation of $\frac{R^\dag}P$, we have
$P\subset H_1$, hence $P=H_1$ by the minimality of $H_1$.

To finish the proof of (2), it remains to show that the valuation
$\nu^\dag$ is well defined by (\ref{eq:nudag}). In other words, we
must check that given another element $\gamma\in\Phi$ with
$\gamma_0=\beta_0$ and a factorization
\begin{equation}
x=wb_1\label{eq:xwb}
\end{equation}
with $w\in{R'}^\dag\setminus P'_1{R'}^\dag$, we have
\begin{equation}
\nu(a_1)+\bar\nu^\dag(z)=\nu(b_1)+\bar\nu^\dag(w).\label{eq:welldefined}
\end{equation}
Without loss of generality, we may assume that
$\nu(a_1)\le\nu(b_1)$. Enlarging $R'$, if necessary, we may assume
that $b_1=a_1t$ for some $t\in R'$. Then $tw=z$, hence $t\notin
P'_1{R'}^\dag$ and
$\bar\nu^\dag(z)=\bar\nu^\dag(t)+\bar\nu^\dag(w)$. Combining this
with (\ref{eq:xza}), (\ref{eq:xwb}) and \ref{eq:nudag}), we obtain
$$
\nu(b_1)+\bar\nu^\dag(w)=\nu(a_1)+\nu(t)+\bar\nu^\dag(w)=
)=\nu(a_1)+\bar\nu^\dag(t)+\bar\nu^\dag(w)=\nu(a_1)+\bar\nu^\dag(z),
$$
as desired. This completes the proof of (2). (3) of Theorem
\ref{hensel0} is now immediate from definitions. Theorem
\ref{hensel0} is proved. $\Box$
\medskip

We note the following corollary of the proof of (2) of Theorem
\ref{hensel0}. Let $\Phi^\dag=\nu^\dag(R^\dag\setminus\{0\})$ and
take an element $\beta\in\Phi^\dag$ and let $P^\dag_\beta$ denote
the $\nu^\dag$-ideal of of $R^\dag$ of value $\beta$.
\begin{cor}\label{blup} Take an element $x\in P^\dag_\beta$. There
exists a local blowing up $R\rightarrow R'$ such that
$\beta\in\nu(R')\setminus\{0\}$ and $x\in P'_\beta{R'}^\dag$.
\end{cor}

\section{The Main Theorem: the primality of the implicit ideals.}
\label{prime}

In this section we study the ideals $H_j$ for $\hat R$ instead of $\tilde R$. The main result of this section is
\begin{thm}\label{primality} The ideal $H_{2\ell-1}$ is prime.
\end{thm}
\noi\textit{Proof:} For the purposes of this proof, let $H_{2\ell-1}$
denote the implicit ideals of $\hat R$ and $\tilde H_{2\ell-1}$ the
implicit prime ideals of the henselization $\tilde R$ of $R$. Any
henselian local domain is analytically irreducible, hence $\tilde
H_{2\ell-1}\hat R$ is prime for all $\ell\in\{1,\dots,r+1\}$. Let
$\tilde\nu$ denote the unique extension of $\nu$ to $\frac{\tilde
R}{\tilde H_1}$, constructed in the previous section. Let
$H^*_{2\ell-1}\subset\frac{\tilde R}{\tilde H_1}$ denote the implicit
ideals associated to the henselian ring $\frac{\tilde R}{\tilde
H_1}$ and the valuation $\tilde\nu$.
\medskip

\noi\textit{Claim.} We have $H^*_{2\ell-1}=\frac{H_{2\ell-1}}{\tilde H_1}$.
\medskip

\noi\textit{Proof of Claim:} For $\beta\in\Gamma$, let $\tilde
P_\beta$ denote the $\tilde\nu$-ideal of $\frac{\tilde R}{\tilde
H_1}$ of value $\beta$. For all $\beta$, we have
$\frac{P_\beta}{\tilde H_1}\subset\tilde P_\beta$, and the same
inclusion holds for all the local blowings up of $R$, hence
$\frac{H_{2\ell-1}}{\tilde H_1}\subset H^*_{2\ell-1}$. To prove the
opposite inclusion, we may replace $\tilde R$ by a finitely
generated strict \'etale extension $R^\dag$ of $R$. Now let
$\Phi^\dag=\nu^\dag(R^\dag\setminus\{0\})$ and take an element
$\beta\in\Phi^\dag\cap\Delta_{\ell -1}$. By Corollary \ref{blup},
there exists a local blowing up $R\rightarrow R'$ such that $x\in
P'_\beta{R'}^\dag$. Letting $\beta$ vary over
$\Phi^\dag\cap\Delta_{\ell -1}$, we obtain that if $x\in H^*_{2\ell-1}$
then $x\in\frac{H_{2\ell-1}}{\tilde H_1}$, as desired. This completes
the proof of Claim.
\medskip

The Claim shows that replacing $R$ by $\tilde R$ in Theorem
\ref{primality} does not change the problem. In other words, we
may assume that $R$ is henselian.

The rest of the proof of Theorem \ref{primality} follows closely
the proof of Claim in the proof of Theorem \ref{hensel0}. Namely,
since $R$ is a henselian G-ring, it is algebraically closed in
$\hat R$; of course, the same holds for $\frac R{P_\ell}$ for all
$l$.

For an element $\beta\in\Phi$, let
$\beta_{\ell -1}=\min\{\gamma\in\Phi\ |\ \beta-\gamma\in\Delta_\ell\}$.
Let $M'$ denote the maximal ideal of $\hat R\otimes_RR'$,
generated by $m\hat R\otimes_RR$ and $\hat R\otimes_Rm'$. First,
we will show that for any local blowing up $\pi$ and any
$\beta\in\Delta_{\ell -1}\cap\Phi$, we have
\begin{equation}
P'_{\beta_{\ell -1}}\left(\hat R\otimes_RR'\right)_{M'}\bigcap\hat R=
P_{\beta_{\ell -1}}\hat R.\label{eq:stab2}
\end{equation}
One inclusion in (\ref{eq:stab2}) is trivial; we must show that
\begin{equation}
P'_{\beta_{\ell -1}}\left(\hat R\otimes_RR'\right)_{M'}\bigcap \hat
R\subset P_{\beta_{\ell -1}}\hat R.\label{eq:stab3}
\end{equation}
This is the same as proving the injectivity of the map
\begin{equation}
\bar\pi:\frac{ R_{P_\ell}}{P_{\beta_{\ell -1}}}\otimes_R\hat
R\rightarrow\left(\frac{ R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}\otimes_R\hat
R'\right)_T\label{eq:inclusion3}
\end{equation}
induced by $\pi$, where $T$ denotes the image of the
multiplicative set $R'\otimes_R\hat R\setminus M'$ in $\frac{
R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}\otimes_R\hat R$ under the natural map
$R'\otimes_R\hat R\rightarrow\frac{
R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}\otimes_R\hat R$. To prove the
injectivity of $\bar\pi$, we start with the field extension
$\kappa(P_\ell)\hookrightarrow\kappa(P'_\ell)$ induced by $\pi$. Since
$\hat R$ is flat over $R$, the induced map
$\pi_1:\kappa(P_\ell)\otimes_R\hat
R\rightarrow\kappa(P'_\ell)\otimes_R\hat R$ is also injective. Now,
since $\kappa(P_\ell)$ is algebraically closed in
$\kappa(P_\ell)\otimes_R\hat R$ by the assumptions on $R$, the ring
$\kappa(P'_\ell)\otimes_R\hat R$ is a domain.

Next, we observe that
\begin{equation}
\kappa(P'_\ell)\otimes_R\hat R=\left(\frac{
R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}\otimes_R\hat
R\right)_{red}.\label{eq:reduced1}
\end{equation}
Now, $\frac{R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}$ is a local artinian ring
and its only associated prime is its nilradical, the ideal
$P'_\ell\frac{R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}$; in particular, the $(0)$
ideal in this ring has no embedded components. Since $\hat R$ is
flat over $R$, $\frac{R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}\otimes_R\hat R$
is flat over $\frac{R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}$ by base change.
Hence the $(0)$ ideal of
$\frac{R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}\otimes_R\hat R$ has no
embedded components. In particular, since the multuplicative
system $T$ is disjoint from the nilradical of
$\frac{R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}\otimes_R\hat R$, the set $T$
contains no zero divisors, so localization by $T$ is injective.

Now, the map $\frac{R_{P_\ell}}{P_{\beta_{\ell -1}}}\hookrightarrow\frac{
R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}$ is injective, hence so is $\frac{
R_{P_\ell}}{P_{\beta_{\ell -1}}}\otimes_R\hat R\rightarrow\frac{
R'_{P'_\ell}}{P'_{\beta_{\ell -1}}}\otimes_R\hat R$ by the flatness of
$\hat R$ over $R$. Combining this with the injectivity of the
localization by $T$, we obtain (\ref{eq:inclusion3}). This
completes the proof of (\ref{eq:stab2}).

We now finish the proof of Theorem \ref{primality}.

Let $\Phi$ be as above; we will continue to use the notation
$\beta(l-1,+)$ for the smallest element of $\Phi$ greater than
$\beta$ such that $\beta(l-1,+)-\beta\notin\Delta(l-1)$.

Take any elements $x,y\in\hat R\setminus H_{2\ell-1}$. Then there
exist elements $\beta,\gamma\in\Phi\cap\Delta_{\ell -1}$ such that
\begin{equation}
x\in P_{\beta}\hat R\setminus P_{\beta(l-1,+)}\hat
R\label{eq:xinbeta1}
\end{equation}
and
\begin{equation}
y\in P_{\gamma}R^\dag\setminus P_{\gamma(l-1,+)}\hat
R.\label{eq:yingamma1}
\end{equation}
To prove (1), it is sufficient to prove that
\begin{equation}
xy\notin P_{\beta+\gamma(l-1,+)}R^\dag.\label{eq:xynotin1}
\end{equation}
Let $(a_1,...,a_n)$ be a set of generators of $P_\beta$ and
$(b_1,...,b_s)$ a set of generators of $P_\gamma$, with
$\nu(a_1)=\beta$ and $\nu(b_1)=\gamma$. Let $R'$ be a local
blowing up along $\nu$ such that $R'$ contains all the fractions
$\frac{a_i}{a_1}$ and $\frac{b_j}{b_1}$. Moreover, choose $R'$ so
that $P'_\ell\hat R'$ is a prime ideal; this is possible by Lemma
\ref{anirred1}. Then $a_1\ |\ x$ and $b_1\ |\ y$ in $\hat R'$.
Write $x=za_1$ and $y=wb_1$ in $\hat R'$. The equality
(\ref{eq:stab2}) implies that $z,w\notin P'_\ell\hat R'$, hence
\begin{equation}
zw\notin P'_\ell\hat R'.\label{eq:zwnotin1}
\end{equation}
We obtain $xy=a_1b_1zw$. Since $\nu$ is a valuation on $R'$, we
have $\left(P'_{\beta+\gamma(\ell-1,+)}:(a_1b_1)R'\right)\subset
P'_\ell$. By faithful flatness of $\hat R'$ over $R'$ we obtain
\begin{eqnarray}
\left(P'_{\beta+\gamma(\ell-1,+)}\hat R':(a_1b_1)\hat
R'\right)&\subset&P'_\ell\hat R'\mbox{ and}\label{eq:colon1}\\
a_1b_1&\notin&P_{\beta+\gamma(\ell-1,+)}\hat R'.
\end{eqnarray}
Combining this with (\ref{eq:zwnotin1}), we obtain
(\ref{eq:xynotin1}), as desired. This completes the proof of
Theorem \ref{primality}. $\Box$
\medskip

The ideal $H_{2\ell-2}$ is, by definition, the unique minimal prime of
$P_{\ell -1}\hat R$, contained in $H_{2\ell-2}$ (that such a minimal prime is unique
follows easily from the fact that $R$ is a G-ring).
\begin{prop} We have $H_{2\ell-2}\cap R=P_{\ell -1}$.
\end{prop}

\section{Extensions of $\nu$ to rings of the form $\frac{\hat R}P$.}
\label{extensions}

\section{Applications to Local Uniformization and the Pierce-Birkhoff
  conjecture.}
\label{locuni}

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\end{document}