%From DLANTZ@center.colgate.edu Thu Oct 13 10:53:29 1994

%From:	IN%"heinzer@math.purdue.edu"     3-AUG-1994 16:30
%Subj:	lbir28.tex

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\tolerance 1000
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\def\Z{{\Bbb Z}}
%\def\Z{\text{\bf Z}}
\def\htwo{{\Cal H}_2}
\def\Spec{\operatorname{Spec}}
\def\jS{j\text{-\nolinebreak}\operatorname{Spec}}
\def\G{\operatorname{G}}
\def\Le{\operatorname{L_e}}
\def\m{{\text{\bf m}}}
\def\n{{\text{\bf n}}}
\def\q{{\text{\bf q}}}
\def\p{{\text{\bf p}}}


\topmatter

\title Prime Ideals in Birational Extensions 
of Polynomial Rings II\endtitle
\author William J. Heinzer,  David Lantz and 
 Sylvia M. Wiegand \endauthor
\thanks{Heinzer and Wiegand gratefully acknowledge the support
of the National Science Foundation. Lantz and Wiegand thank
Purdue University for its hospitality while we were working on
this research.}
\endthanks
\address{Mathematics Department, Mathematical Sciences Building,
Purdue University, West Lafayette, IN 47907-1395}
\endaddress
\email heinzer\@math.purdue.edu
\endemail
\address{Mathematics Department, Colgate University, 13 Oak Drive,
Hamilton, NY 13346-1398}
\endaddress
\email dlantz\@center.colgate.edu
\endemail
\address{Mathematics Department, University of Nebraska, Lincoln, 
NE 68588-0323}
\endaddress
\email swiegand\@unlinfo.unl.edu
\endemail
\subjclass{13E05, 13F20, 13G05, 13H99, 13J15}
\endsubjclass
%\date {August 3, 1994}
%\enddate
\endtopmatter

\document
\baselineskip 15pt 
\heading 1. Introduction \endheading
Let $R$ be a semilocal Noetherian domain of dimension one, and let $x$ 
be an indeterminate over $R$. By a {\it birational extension} of the
polynomial ring $R[x]$ we mean an integral domain $B$ containing $R[x]$
and contained in the fraction field of $R[x]$.  We are interested in
the prime ideal structure of birational extensions of $R[x]$.
A general objective, of which the present paper is a part, is to
determine which partially ordered sets, or equivalently topological
spaces, can occur as
the prime spectrum, $\Spec(B)$, of a finitely generated birational
extension $B$ of $R[x]$. A related objective is to determine
necessary and sufficient conditions on two finitely generated
birational extensions $B_1$ and $B_2$ of $R[x]$ in order that
$\Spec(B_1)$ and $\Spec(B_2)$ be order-isomorphic as partially
ordered sets.

Let $\m_1,\m_2, \dots,\m_n$ denote the maximal ideals of $R$.  In order
to obtain precise results, we restrict
to extensions of the form $B= R[x, \{ g_i/f_i\}_{i=1}^t]$,
where $t$ is a positive integer, each
$f_i\in R[x] -\bigcup^n_{j=1}\m_j[x]$, and $(f_i,g_i)R[x]$
is not contained in any height-one prime ideal of $R[x]$.
For ease of reference we refer to a finitely generated birational
extension of this special form as a {\it ($t$-generated) sfb-extension}
of $R[x]$. We often restrict to the case where $R$ is the
localization $k[y]_{(y)}$ of the polynomial ring in an indeterminate
$y$ over a countable algebraically closed field $k$. This
article is a continuation of our work in \cite{Heinzer et al. 
(1994b)} and also relies heavily on results and notation in 
\cite{Heinzer and Wiegand (1989)} and \cite{Heinzer et al. (1994a)}.

Let $D$ be a two-dimensional Noetherian domain with infinitely
many height-two maximal ideals and only finitely many height-one prime
ideals that are contained in infinitely many maximal ideals.
We refer to the height-one primes of $D$ that are
contained in infinitely many maximal ideals as the {\it special
primes } of $D$. Note that a height-one prime ideal $P$ of $D$ is
special if and only if it is nonmaximal and is an intersection of
maximal ideals.  It is
easy to see that $\Spec(D)$ as a partially ordered set is
determined by the maximal prime ideals of $D$ and the 
containments between the nonmaximal height-one primes and the 
height-two maximal ideals. This
amounts to considering two things:
\roster
\item The  $j$-spectrum of $D$, $\jS(D)$, (a prime $P \in \Spec(D)$ 
is a $j$-prime if $P$ is an intersection of maximal ideals, and
 $\jS(D)$ is the partially ordered set of $j$-primes in $\Spec(D)$) and
\item The ``inverse $j$-radicals" of finite sets $T$ of height-two
maximal ideals of $\Spec(D)$, that
is, the set of all height-one prime ideals that are contained in every
element of $T$ but are in no other maximal ideals.
\endroster
%In keeping with our previous notation for partially ordered sets,
%we refer to the sets in item (2) as the ``exactly-less-than" sets,
%$$
%\Le(T)=\{ w\in\Spec(D)\, \vert \, w<t \iff t \in T\}.
%$$

\noindent
In \cite{Heinzer et al. (1994b)}, restricting our attention to a 
cyclic (i.e., one-generated) sfb-extension $B$ of $R[x]$:
\roster
\item we characterize $\jS(B), $
\item we show that the inverse $j$-radical 
%(exactly-less-than set) 
of every singleton set $T$ of height-two maximals of $B$ is 
infinite, and
\item for certain domains $R$ we obtain complete results on the 
%sets $\Le(T)$, 
inverse $j$-radicals of finite sets of height-two maximals,
thus describing the entire spectrum of $B$.
\endroster

In Section~2 of
the present article, we prove that if $R$ is 
of finite type over a countable
field and $B$ is a cyclic sfb-extension of $R$, then $\Spec(B)$ is
uniquely determined by the type of $\jS(B)$. This extends the result
in \cite{Heinzer et al. (1994b)} mentioned in (3). For the rest 
of the paper, we allow
noncyclic sfb-extensions and show that the $j$-spectrum 
of a noncyclic sfb-extension can be much different from the cyclic 
case. A transient in an sfb-extension $B=R[x,g_1/f_1,\dots,g_n/f_n]$
is a height-one nonmaximal $j$-prime that does not survive in 
$R[x,1/(f_1\dots f_n)]$. In Section~3, we provide explicit examples
to show that the $j$-spectra of noncyclic sfb-extensions may 
differ from the cyclic case, in that two transients need not be
comaximal, and in fact their sum may be contained in more than
one maximal ideal. In Section 4, we formulate an axiom on
partially ordered sets that is satisfied by the $j$-spectra of
sfb-extensions of $R[x]$ for $R$ local. We describe a 
procedure that we believe will realize a given ``feasible''
partially ordered set satisfying this axiom as the $j$-spectrum
of an sfb-extension.

All rings we consider are commutative and contain a multiplicative
identity. The terms ``local" and ``semilocal" as used in this paper
include a Noetherian hypothesis on the ring.

\demo{1.1 Notation} For $U$ a partially ordered set of
finite dimension, elements $u, w$ of $U$, and $T$ a 
finite subset of $U$, we set
$$\align
\G(u) &= \{ w \in U \ \vert\  w>u  \} \qquad\text{and} \\
\Le(T)&= \{ w \in U \ \vert\ w<t  \iff t \in T \} \\
      &= \{ w \in U \ \vert\ \G(w) = T \} \quad.\\
\endalign$$
And we denote by ${\Cal H}_i(U)$ the set of elements of $U$ of height $i$.
\enddemo

\demo{Note}  We say that a list of properties {\it characterizes} a given
partially ordered set if every partially ordered set with these
properties is order-isomorphic to the given partially ordered set.
\enddemo

Let $R$ be a countable semilocal domain of dimension one with exactly
$n$ maximal ideals.  As Theorem 1.2 below indicates, $\jS(R[x])$ is
then uniquely determined, and there are at most two possibilities for
$\Spec(R[x])$. For notational purposes
we introduce the {\it $j\text{-\nolinebreak}$subset},
$j\text{-\nolinebreak}V$, of a partially ordered set
$V$ of finite dimension, which we define recursively as follows:
$u \in j\text{-\nolinebreak}V$
iff either (i)~$u$ is maximal in $V$, or (ii)~there are infinitely
many covers of $u$ in $j\text{-\nolinebreak}V$, where an element $w$
of $V$ is called a {\it cover} of $u$ if $u<w$ and there is no
element $v$ of $V$ for which $u<v<w$.

\proclaim{1.2 Theorem}{\cite{Heinzer et al. (1994b), Theorem~1.3}} 
Let $R$ be a countable semilocal domain of dimension one with 
exactly $n$ maximal ideals, $x$ an indeterminate, and 
$U=\Spec(R[x])$. Then:

\noindent
(1) There are
at most two possibilities for the partially ordered set $U$, and
they are distinguished by the properties (P6) and (P6$\,'$) below:

\noindent
In case $R$ is not Henselian, $U$ satisfies

\itemitem{(P6)} For each nonempty finite subset $T$ of $\htwo(U)$,
$\Le(T)$ is infinite.

\noindent
In case $R$ is Henselian (and then $n=1$), $U$ satisfies

\itemitem{(P6$\,'$)} For each finite subset $T$ of $\htwo(U)$
of cardinality greater than one, $\Le(T)$ is empty. For each element
$t$ of $\htwo(U)$, $\Le(\{t\})$ is infinite.

\noindent
(2) Let $V_1,\,V_2$ be countable partially ordered sets of dimension
two with unique minimal elements such that

\itemitem{(i)} There exists an order-isomorphism $\varphi$:
$j\text{-\nolinebreak}V_1\to j\text{-\nolinebreak}V_2$.

\itemitem{(ii)} For $i=1,2$, the height of each element of
$j\text{-\nolinebreak}V_i$ agrees with its height in $V_i$, and

\itemitem{(iii)} For each finite subset $T$ of $\htwo(V_1)$,
$\Le(T)$ and $\Le(\varphi(T))$ are either both empty or both 
infinite.

\noindent
Then $V_1$ is order-isomorphic to $V_2$ by an extension of $\varphi$.

\noindent
(3) Let $V$ be a countable partially ordered set of dimension two with
unique minimal element such that:

\itemitem{(i)} The subset $j\text{-\nolinebreak}V$ has the 
properties (P0)--(P4) below:

{\leftskip 1 true in\parindent 0 pt

(P0)  $j$-$V$  is countable.

(P1) $j$-$V$ has a unique minimal element $u_0$.

(P2) $j$-$V$ has dimension $2$.

(P3) $ j$-$V$ has infinitely many
height-one maximal elements.

(P4) $ j$-$V$ has exactly $n$ special 
elements. We denote these elements $u_1,u_2,\dots u_n$. They satisfy:}

{\leftskip 1.5 true in

(a) $\G(u_1) \cup \dots \cup \G(u_n) = \htwo(j\text{-\nolinebreak}V)$,

(b) $\G(u_i) \cap \G(u_j) = \emptyset$ for $i\ne j$, and

(c) $\G(u_i)$ is infinite for each $i$, $1\le i\le n$.

}

\itemitem{(ii)} The height of each element of $j\text{-\nolinebreak}V$ 
agrees with its height in $V$, and

\itemitem{(iii)} $V$ satisfies (P6) or (P6$\,'$)  of 1.2(1) 
(whichever $U$ satisfies).

\noindent
Then $V$ is order-isomorphic to $U$.
\endproclaim

\demo{Note} When we stipulate the $j\text{-\nolinebreak}$spectrum
and either (P6) or (P6$'$) of 1.2, we have described all possibilities for
$\Spec(R[x])$ given that $R$ is a countable semilocal domain of
dimension one
with exactly $n$ maximal ideals. However, condition~(ii) is necessary in
(2) and (3), because conditions~(i) and (iii) together do not uniquely
determine a partially ordered set $V$. For example,
%$j\text{-\nolinebreak}V$ might have dimension two while $V$ has 
%dimension three, or 
there could be maximal elements of $V$ of height two in $V$ but only 
of height one in $j\text{-\nolinebreak}V$.
\enddemo

(1.3)  Let $(R,\m_1,\dots,\m_n)$ be a semilocal domain of dimension 
one, $x$ an indeterminate, and let $B$ be a {\it $t$-generated 
sfb-extension} of $R[x]$,
i.e., $B= R[x, \{ g_i/f_i\}_{i=1}^t]$, where $t$ is a positive
integer, each $f_i\in R[x] -\bigcup^n_{j=1}\m_j[x]$ and each 
$(f_i,g_i)R[x] = R[x]$ is not contained in any height-one prime
ideal of $R[x]$. It follows from the dimension formula \cite{M2,
page~119} that for each height-two maximal ideal $M$ of $B$,
$M\cap R =\m_j$ for some $j$ and $B/M$ is a finite algebraic 
extension of $R/\m_j$.
Let $f=\prod_{i=1}^t f_i$.  If $f \in P$, a prime ideal of $R[x]$,
then $f \in Q \subseteq P$ for some height-one prime $Q$ of $R[x]$.
Now $f \in Q$ implies $f_i \in Q$ for some $i$. But then
$g_i \notin Q$ and so $g_i/f_i \notin R[x]_Q \supseteq R[x]_P$.
Hence $B \nsubseteq R[x]_P$.  Thus a prime ideal
$P$ of $R[x]$  does not contain $f$ if and only if 
$B \subseteq R[x]_P$.  
Therefore the radical of $fR[x]$ is uniquely
determined by $B$.  For convenience we use the following
terms from \cite{Heinzer et al. (1994b)} for the {\it special
primes } of $B$ --- the nonmaximal height-one $j$-primes.
\roster
\item  A special prime ideal $P$ of $B$ is
called a {\it survivor} if it survives in $B[1/f]$, that is,
if $PB[1/f]\subsetneq B[1/f]$, or, equivalently, if $f \not\in P$.
(There is exactly one survivor $j$-prime of $B$ contracting to
each maximal ideal $\m_i$ of $R$, namely $\m_i R[x,1/f] \cap B$.)
\item A special prime ideal $P$
of $B$ is called a {\it transient} if it does not
survive in $B[1/f]$, that is, if $PB[1/f] = B[1/f]$,
or, equivalently, if $f \in P$.
\endroster

\medskip
(1.3.1) The special prime ideals  of $B$ are
precisely the minimal primes of the $\m_i B$. To see this, 
we may suppose that $(R,\m)$ is local and consider
$P$ a special prime of $B$; then $P$ is the
intersection of maximal ideals of $B$ of height two. Each
of these maximal ideals contains $\m$, so $\m \subseteq P$
and $P$ is a minimal prime of $\m B$. On the other hand,
if $P$ is a minimal prime of $\m B$, then $P$ is a minimal
prime of a principal ideal, so $P$ is of height one. Again by the
dimension formula \cite{M2, page~119}, $B/P$ is of
transcendence degree one over $R/\m$. Since $B/P$ is
finitely generated as an $R/\m$-algebra, it follows from Hilbert's
Nullstellensatz (cf. \cite{M2, page~33}) that
$P$ is not maximal and is an intersection of maximal ideals
of $B$. Thus $P$ is a special prime of $B$.

(1.3.2) In \cite{Heinzer et al. (1994a)} and 
\cite{Heinzer et al. (1994b)}, we consider only
extensions $B=R[x,g/f]$ in which $f,g$ is an $R[x]$-sequence. This
restriction allows us to use the explicit description 
$(fu-g)R[x,u]$ for the kernel of the homomorphism of 
$R[x]$-algebras from the polynomial ring $R[x,u]$ in the 
indeterminates $x,u$ onto $B$ determined by sending $u$ to
$g/f$; and it rules out the case $R[x,g/f] = R[x,1/f]$, a
ring of fractions of $R[x]$, of which the spectrum is well
understood. Since $R$ and hence $R[x]$ are Cohen-Macaulay rings,
to say that $f,g$ is an $R[x]$-sequence is equivalent to saying
that $(f,g)R[x]$ is not contained in any height-one prime of
$R[x]$ \underbar{and} that $(f,g)R[x] \neq R[x]$. In continuing
our study of sfb-extensions of $R[x]$, we 
now allow as generators fractions $g_i/f_i$ for which 
$(f_i,g_i)R[x] = R[x]$. We have two reasons for this. First,
even if $(f_1,g_1)R[x]$ and $(f_2,g_2)R[x]$ are proper ideals
of $R[x]$, there is no description, analogous to the case of
one generator, for the kernel of $R[x,u,v]\to 
R[x,g_1/f_1,g_2/f_2]:u\mapsto g_1/f_1,v\mapsto g_2/f_2$. (We
use the computer program MACAULAY to compute such a kernel.
See Section~3.) Second, it seems useful to us to be able to
say, if $R[x]\subseteq A \subseteq B$ with $B$ an sfb-extension
and $A$ a finitely generated extension of $R[x]$, that $A$
is also an sfb-extension. In fact, we do not know that in
general; cf. (1.3.3). But if we had required that $(f_i,g_i)R[x]
\neq R[x]$ for each $i$, the statement would have failed even
for a cyclic sfb-extension of 
$R[x] = k[y]_{(y)}[x]$. For example, consider 
$B = R[x][y/(x(yx + 1))]$ and $A = R[x][1/(yx+1)]$.

(1.3.3) If $B = R[x,g_1/f_1,\dots,g_t/f_t]$ is an
sfb-extension of $R[x]$ and if the irreducible factors of 
the $f_i$ are prime elements of $R[x]$, then every
finitely generated $R[x]$-subalgebra of $B$ is also an 
sfb-extension of $R[x]$.  In particular this holds if 
$R$ is a PID. 

(1.3.4) Let $B = R[x,g_1/f_1,\dots,g_t/f_t]$ be an sfb-extension
of $R[x]$, let  $P_0$ be a survivor prime and $P_1$ a transient
prime of $B$ with respect to $R[x]$. We show that $P_0 + P_1$ is
contained in at most one maximal ideal of $B$. By considering
their respective extensions to the ring of fractions 
$R[x,1/(f_1\dots f_t)]$ of $B$, we see that the contraction
$P_0 \cap R[x]$ has the form $\m R[x]$ for a maximal ideal $\m$ of
$R$, while $P_1 \cap R[x] = Q$ is a height-two maximal ideal.
If $\m R[x] \nsubseteq Q$, then $P_0$ and $P_1$ are comaximal. If
$\m R[x] \subseteq Q$, the maximal ideals of $B$ containing both
$P_0$ and $P_1$ correspond to the maximal ideals of 
$(R[x] - Q)^{-1}(B/P_0)$ that contain the image of $P_1$. But
$(R[x] - Q)^{-1}(B/P_0)$ is a birational extension of
$((R/\m)[x])_Q$, a DVR, so $(R[x]-Q)^{-1}(B/P_0)$ is that DVR. 
Therefore, a survivor and a transient are contained in at most one
(height-two) maximal ideal of $B$. Moreover, if $P_0$ is a
survivor, if $P_1$ and $P_2$ are two transients with
the same center on $R[x]$, and if $M_1$ is a maximal ideal
containing $P_0 + P_1$ and $M_2$ is a maximal ideal
containing $P_0 + P_2$, then $M_1 = M_2$. 
In other words, $P_1$ and $P_2$ cannot be contained in different maximal
ideals of $B$ that both contain a survivor.

It is noted in \cite{Heinzer et al. (1994b)} that, 
when $R$ is local and $B$ is
a cyclic sfb-extension, the survivor is often distinguished in the
partially ordered set $\jS(B)$ from other special
primes, in that it is the only special prime of $B$ which might 
be contained in two maximal ideals which contain other special 
primes. Specifically, in Diagram~1.4.1 below, for each of the 
transients $Q_j$, the survivor $P$ and $Q_j$ are both
contained in one maximal ideal, while $P$ is comaximal with each
of the transients $H_i$. 

\proclaim{1.4 Theorem}\cite{Heinzer et al. (1994b), Theorems 2.6, 2.7}
(1) Let $R$ be a local domain of dimension one, and let $x$ 
be an indeterminate. Then
every cyclic sfb-extension $B$ of $R[x]$ has
$\jS(B)$ like Diagram~1.4.1  below, for some choice of positive
integers $m$ and $n$, not both zero.

(2) Moreover, for every countable partially ordered set $U$ of the form of
Diagram~1.4.1, with $m$ and $n$ not both zero, there exists a
cyclic sfb-extension $B$ of $R[x]$ with
$j\text{-\nolinebreak}$spectrum order-isomorphic to $U$.

(3) If $R$ and $B$ are as in (1) \underbar{except} that $R$ has 
$r$ maximal ideals, where $r>1$, then $\jS(B)$ is a union of 
$\jS((R-\m)^{-1}B$, over all maximal
ideals $\m$ of $R$---the union is disjoint except that the zeros 
and the height-one maximal ideals are identified, and no new
inclusions are added.
\endproclaim

\vskip 18 pt

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\box4
\vskip 12 pt
\centerline{Diagram 1.4.1}
\vskip 18 pt

The paper \cite{Heinzer et al. (1994b)} contains the following four 
results on $\Spec(B)$ for cyclic sfb-extensions of $R[x]$:

\proclaim{1.5 Theorem} \cite{Heinzer et al. (1994b), Theorem 3.1} 
Suppose $R$ is a semilocal
domain of dimension one, $x$ is an indeterminate, and $B$ is a cyclic
sfb-extension of $R[x]$. Then for each height-two maximal ideal
$N$ of $B$, there exist infinitely many height-one primes $P$ such that
$N$ is the only height-two maximal ideal containing $P$. That is, (P6)
or (P6$\,'$) of Theorem~1.2(1) holds for singleton subsets $T$ of the set 
of all height-two elements of Spec($B$).
\endproclaim

\proclaim{1.6 Theorem} \cite{Heinzer et al. (1994b), Theorem 4.1} 
Let $R$ be a Henselian local domain
of dimension one, and let $B$ be a cyclic sfb-extension of $R[x]$.
Then $\Spec(B)$ satisfies (P6$\,'$) of Theorem~1.2(1). If $R$ is 
countable, then $\Spec(B)$ is uniquely determined by the partially 
ordered set $\jS(B)$.
\endproclaim

\proclaim{1.7 Theorem} \cite{Heinzer et al. (1994b), Theorem 4.5} 
Let $R$ be a semilocal domain of dimension one, $x$ an indeterminate 
over $R$, and $B=R[x,g/f]$ a cyclic sfb-extension of $R[x]$. Suppose
there exists a one-dimensional Noetherian domain $D \subset R$  
such that $R$ is a localization of $D$ and such 
that $U=\Spec(D[x,g/f])$ satisfies (W):

\itemitem{(W)} Let $S$ be a finite set of height-one
elements of $U$ and $T$ a finite subset of $\htwo(U)$. Then there exist
infinitely many height-one elements $w\in U$ such that $\G(s)\cap
\G(w)\subseteq T\subseteq \G(w)$, for all $s\in S$.

\noindent
Then $\Spec(B)$ satisfies (P6) of 1.2.
\endproclaim

The condition (W) is similar to an axiom described by Roger Wiegand in 
\cite{R. Wiegand (1978)} and \cite{R. Wiegand (1986)}. 
Condition (W) is satisfied by the prime
spectrum of a polynomial ring over the integers, $\Spec(\Z[x])$ and
by the spectrum of every domain of dimension two that is finitely
generated as a $k$-algebra, if $k$ is a field contained in the algebraic
closure of a finite field \cite{R. Wiegand (1986)}, 
\cite{Heinzer et al. (1994b)}. Thus the
following corollary is obtained in \cite{Heinzer et al. (1994b)}.

\proclaim{1.8 Corollary} \cite{Heinzer et al. (1994b), Corollary 4.6} 
Suppose $k$ is a field 
algebraic over a finite field.  Let $z$ and $x$ be indeterminates
over $k$, $\{ \p_i \ \vert\  1 \le i\le n \}$ a finite set of nonzero
prime ideals of $k[z]$, $S = k[z] - \bigcup^n_{i=1} \p_i$,
and $R = S^{-1}k[z]$. Let $B = R[x, g/f]$ be a cyclic
sfb-extension of $R[x]$. Then $\Spec(B)$ satisfies (P6) of
Theorem~1.2 and so is uniquely determined by the isomorphism type 
of $\jS(B)$.
\endproclaim

\medskip

\heading{2. Further results on cyclic sfb-extensions of $R[x]$}
\endheading

In Theorem 2.2  we extend  Corollary 1.8  to the case where
$R$ is a one-dimensional semilocal domain which is a ring
of fractions of a finitely generated algebra 
over an arbitrary ground field $k$.

\medskip

\noindent
{\bf 2.1 Remarks.} (1) Corollary 1.8 stated above holds 
more generally if $R$ is a semilocal ring of fractions of
a one-dimensional affine domain over a field $k$ which is
algebraic over a finite field and  
$B$ is any  sfb-extension of $R[x]$ (not necessarily cyclic). 
To prove this more general
statement, modify the original proof in \cite{Heinzer et al. (1994b)} by 
replacing ``$ g/f$''  by ``$g_1/f_1,\dots,g_t/f_t$''.
(The crucial tool used in the proof of (1.8) is Theorem 2 of
\cite{R. Wiegand (1986)} which holds for all two-dimensional affine
domains over a field algebraic over a finite field.)

(2) If $R$ is a one-dimensional semilocal domain that is a
localization of an affine domain $D$ over a field $k$, then
$R$ is also a localization of a one-dimensional affine domain
over a possibly larger field. This can be seen using
(14.G) and (14.F) of Matsumura (1980).

\medskip

The $j\text{-\nolinebreak}$spectrum of a cyclic sfb-extension 
$B$ of $R[x]$ is described in Theorem~1.4. In view of 
Theorem 1.2(2), in order to determine the entire spectrum of $B$, 
we consider, as in \cite{Heinzer et al. (1994b)}, the
exactly-less-than sets $\Le({\Cal T})$, where ${\Cal T}$ is a finite subset
of $\htwo(\Spec(B))$. 

\proclaim{2.2 Theorem} Let $R$ be a semilocal (Noetherian)
domain of dimension one, and let $x$ be an indeterminate over $R$. 
Suppose that $R$ is a ring of fractions of an 
affine domain over a field and $B$ is a cyclic
sfb-extension of $R[x]$. Then $\Spec(B)$ satisfies (P6) of 
Theorem~1.2. In particular, if $R$ is countable, then $\Spec(B)$
is uniquely determined by the type of $\jS(B)$. 
\endproclaim

\demo{Proof} By (2.1.2) we may assume that $R$ is a localization of a
one-dimensional affine domain $D = k[d_1,\dots,d_m]$ over a field $k$.
Let $B= R[x,g/f]$, where $f$ is
outside all the extensions to $R[x]$ of the maximal ideals of $R$
and either $(f,g)$ is an $R[x]$-sequence or $(f,g)R[x] = R[x]$. 
We first give the proof in the case where $k$ is an
algebraically closed field.

By localizing $D$ at a unit of $R$, we may assume that
$f,g \in D[x]$ and $(f,g)D[x]$ is either of height two or all
of $D[x]$. It follows that $A = D[x,g/f]$ is of the form
a polynomial ring $D[x,y]$ modulo a principal ideal and
hence is Cohen-Macaulay.

Let ${\Cal T}$ be a finite set of height-two maximal ideals of $B$.
The proof of Theorem~1.5 given in \cite{Heinzer et al. (1994b), 
Theorem~3.1} shows that, for each height-two maximal
ideal $N$ of $B$, there is an element of $B$ which is
contained in $N$ and in no other height-two maximal ideal of
$B$; and clearly we can choose that element to be in $A$.
So by multiplication we obtain an element $h$ of $A$ that is
contained in all the elements of ${\Cal T}$ and in no other
height-two maximal ideal of $B$. If we choose from $D$ an element
$r$ in the intersection of the maximal ideals of $R$ and outside all
the minimal primes of $hA$, then $h,r$ is an $A$-sequence:
Since $h$ and $r$ are both contained in the elements of
${\Cal T}$, they do not generate the unit ideal even in $B$,
much less in $A$; and our choice of $r$ assures that $(h,r)A$
has height greater than 1.

With $A = k[d_1,\dots,d_m, x, g/f]$ a two-dimensional 
affine domain over an algebraically closed field $k$,  
by Lemma~4 of \cite{R. Wiegand (1978)}, there is a nonempty 
open subset $U$ of $(k)^{m+3}$ such that, for each point 
$\text{\bf p} = (\alpha,\beta_1,\ldots,\beta_m,\gamma,\delta)$ 
in $U$, the radical of the principal ideal in $A$ generated by 
the element
$$
a_{\text{\bf p}} =
    h + (\alpha + \beta_1 d_1 + \cdots + \beta_m d_m + \gamma x
    + \delta (g/f)) r
$$
is a (height-one) prime ideal $P$. Every height-two maximal
ideal $N$ of $B$ contains a maximal ideal of $R$ and hence
$r$, so if $N$ contains $P$, then it also contains $h$ and hence
is one of the elements of ${\Cal T}$; and clearly the elements of
${\Cal T}$ contain $P$, that is, $P \in \Le({\Cal T})$.

We want to show that varying the elements
$a_{\text{\bf p}}$ over $U$ will give rise to infinitely many such
height-one primes: Select one point {\bf p} in $U$; then
by leaving fixed the values of the $\beta_i$'s, $\gamma$ and
$\delta$, and varying the value of $\alpha$, we get infinitely many
other points in $U$. Assume by way of contradiction that two of
these {\bf p} in $U$, differing only in the first component
$\alpha$, yield elements $a_{\text{\bf p}}$ with the same prime
radical $P$; then the difference between the two $a_{\text{\bf p}}$
is in $P$, and so $r\in P$. But since $P$ is
contained in only finitely many of the height-two maximals that
survive in $B$, it intersects $R$ in $(0)$, the desired
contradiction. This completes the proof in the case where
$k$ is algebraically closed.

Now let us consider the case where $k$ is not algebraically closed
and observe how the proof follows in this case. Again 
by localizing $D$ at a unit of $R$ we may assume that $f,g \in D[x]$.
It follows that $A = D[x, g/f]$ is of the form the polynomial
ring $D[x,y]$ modulo a principal ideal and hence is Cohen-Macaulay.
Let $k^*$
denote an algebraic closure of $k$. Let $D^*$ denote a domain quotient
of $k^*\otimes_kD$ obtained by factoring out a minimal prime, and let
$R^*$ denote the result of localizing $D^*$ at the same
multiplicative system at which $D$ was localized
to get $R$, $A = D[x,g/f]$, $A^* = D^*[x,g/f]$, and $B^* = R^*[x,g/f]$.

(Notes: (1)~Though
$k^*$ is usually an infinite algebraic extension of $k$, 
the domain  $D^*$ is a one-dimensional Noetherian domain of
finite type over $k^*$ and an integral extension of $D$. Hence  
there are only finitely many primes of $D^*$ lying over a prime in
$D$ --- namely the minimal primes of the extension to $D^*$ of the
prime in $D$ --- so $R^*$ is still semilocal, as well as an integral
extension of $R$. (2)~ Our hypothesis on $(f,g)$ implies
$A^*$ is still a domain.)

Let ${\Cal T}$ be a finite set of height-two maximal ideals of $B$,
and let ${\Cal T}^*$ denote the set of all primes of $B^*$ lying
over elements of ${\Cal T}$.
As above, we obtain an element $h$ of $A$ that is contained
in all the elements of ${\Cal T}$ and in no other height-two
maximal ideals of $B$ and an element $r \in D$ in the intersection of
the maximal ideals of $R$ and outside all of the minimal primes
of $hA$. 

Let $U$ be a nonempty open subset of $(k^*)^{m+3}$ obtained
via an application of Lemma~4 of \cite{R. Wiegand (1978)} where
$R^*$, $A^*$, $B^*$ and ${\Cal T}^*$now play the roles previously 
played by $R$, $A$, $B$ and ${\Cal T}$.
We consider the intersections with $R$ of the extensions to $R^*$ of
all the (prime) radicals $P^*$ of the principal ideals generated by
the elements $a_{\text{\bf p}}$ as {\bf p} varies over $U$. Since only
finitely many primes of $R^*$ lie over a single prime in $R$, there
are infinitely many different such intersections, and it is clear that
each is a prime contained in every element of ${\Cal T}$. By the Going
Up Theorem and the fact that each $P^*$ is contained only in elements
of ${\Cal T}^*$, each such intersection is contained in no other
height-two maximal ideal of $B$. Therefore, $\Spec(B)$ satisfies
(P6).~\qed
\enddemo

\medskip

\noindent
{\bf 2.3 Remark.} It would be interesting to know whether 
Theorem~2.2 also holds for noncyclic sfb-extensions. The 
proof given here does not extend to noncyclic sfb-extensions 
because it relies on Lemma~3.5 of \cite{Heinzer et al. (1994b)} 
to deduce that for
each height-two maximal ideal $N$ of $B$ there is an element of $B$
which is contained in $N$ and in no other height-two maximal ideal
of $B$. We show in Example~2.4 that this need not hold for 
$B$ a noncyclic sfb-extension. 

\medskip
\noindent
{\bf 2.4 Example.} Let $k$ be a field of characteristic zero, 
let $R$ be the DVR $k[y]_{(y)}$, and consider the sfb-extension
$B = R[x, (y/x)^2, (y/x)^3]$ of $R[x]$. Then 
$B \subseteq R[x, y/x] = C$ and $xC \cap B = P$ is a transient
height-one prime of $B$ such that $B/P  \cong k[t^2, t^3]$, 
where $t$ is an indeterminate over $k$. Since $k$ is of
characteristic zero, for any nonzero $a \in k$, the maximal
ideal $(t-a)k[t] \cap k[t^2,t^3]$ is not the radical of a principal
ideal. Hence if $N$ is a height-two maximal ideal of $B$ for which 
$N/P \cong (t-a)k[t] \cap k[t^2,t^3]$, then there exists no element
of $B$ which is contained in $N$ and in no other height-two 
maximal ideal of $B$. 


\heading{3. Examples of $j\text{-\nolinebreak}$spectra  of finite
birational extensions}
\endheading

This section displays two examples showing that the 
$j$-spectra of noncyclic sfb-extensions can be more 
complicated that the cyclic case. In both cases, $k$ is
the field $\Z_{31991}$ of integers modulo the prime 31991.

\medskip
\noindent
{\bf 3.1 Example.} An example of a 2-generated sfb-extension
of $k[a]_{(a)}[x]$ (where $a,x$ are indeterminates over $k$)
in which two transient $j$-primes are
not comaximal. This shows that if an sfb-estension is not cyclic,
a picture different from Diagram~1.4.1 can occur.

The example is the ring 
$$
B = k[a]_{(a)}[x,x/(x-a),(x-a)^2/x]\ .
$$
To compute the kernel $P$ of the $k[a]_{(a)}[x]$-homomorphism 
from the polynomial ring $k[a]_{(a)}[x,u,v]$ onto $B$ determined by
sending $u$ to $x/(x-a)$ and $v$ to $(x-a)^2/x$, we used the
computer algebra program MACAULAY, written by David Bayer and
Michael Stillman. But since MACAULAY is written for 
computations with quasihomogeneous polynomial rings, we added a
homogenizing indeterminate $d$; and to be sure that the program found
the entire kernel, and not just the piece generated by
$(x-a)u-x$ and $xv-(x-a)^2$, we added a indeterminate $w$ to
map to $1/x(x-a)$. Thus, we really began with
the polynomial ring $C = k[a,x,u,v,d][w]$ (the
isolation of $w$ is accomplished by placing it first and specifying
the monomial order $1\ 5$) and its ideal $I$ generated
by $(x-a)u-x$, $xv-(x-a)^2$ and $x(x-a)w-d^3$. The ideal
is described by its generators, with the MACAULAY convention that
an integer after a variable is interpreted as an exponent.
(The example is a bit unusual in that we did not have to multiply 
some terms of the numerators and denominators of the generators of 
our sfb-extension by powers of $d$ to ``homogenize'' them, and that
after we had given weight 1 to each of
$w,a,x,d$, the weights we had to assign to $u,v$ to make the 
generators into quasihomogeneous polynomials were also 1.) 
Then we replaced the given generating set for $I$ with a
``standard'' (i.e., Gr\"obner) basis, ``eliminated'' $w$ (i.e, 
found the intersection $J$ of $I$ with the coefficient ring
$k[a,x,u,v,d]$), and found a standard basis for $J$, of which we
then displayed a minimal generating set.  Here is the slightly
edited ``monitor file'' of our computer run; the characters
we entered are in ``typewriter'' typeface:

\vskip 6 pt

{\obeylines
\% {\tt ring C}
! characteristic (if not 31991)       ? 
! number of variables                 ? {\tt 6}
!   6 variables, please               ? {\tt waxuvd}
! variable weights (if not all 1)     ? {\tt 1 1 1 1 1 1}
! monomial order (if not rev. lex.)   ? {\tt 1 5}
;   largest degree of a monomial        : 512 217 

\% {\tt ideal I}
! number of generators ? {\tt 3}
! (1,1) ? {\tt (x-a)u-xd}
! (1,2) ? {\tt xv-(x-a)2}
! (1,3) ? {\tt x(x-a)w-d3}

\% {\tt std I I}

\% {\tt elim I J}

\% {\tt std J J}

\% {\tt type J}
; au-xu+xd a2-2ax+x2-xv uvd3+ad4-xd4 
}

\vskip 6 pt
\noindent
Setting $d=1$, we obtain a generating
set for the desired kernel $P$: $(x-a)u-x$, $xv-(x-a)^2$, and
$uv-(x-a)$. (The first two, of course, are not new, and the third
is not surprising but is not in the ideal generated by the
first two.) Now since all the height-two and special primes in $B$ 
contain the extension of the maximal ideal $\m$ of $k[a]_{(a)}$, 
we can take their preimages in $k[a]_{(a)}[x,u,v]$ and factor out 
the extension of $\m$ to this ring, i.e., set
$a=0$ to get $k[x,u,v]$. The resulting generators of the image 
of $P$ are $x(u-1)$, $x(v-x)$ and $uv-x$. The image in $k[x,u,v]$
of the preimage in $k[a]_{(a)}[x,u,v]$ of the
survivor prime in $B$ is the minimal prime of the image of $P$  
that does not contain $x$ and hence is generated by 
$u-1$ and $v-x$. So the preimage of the survivor in
$k[a]_{(a)}[x,u,v]$ is $(a,u-1,v-x)$. The image of the preimage
of a transient prime contains $x$ and hence contains either $u$ or 
$v$; so the preimages of the transients are $(a,x,u)$ and 
$(a,x,v)$. These primes are clearly contained in the maximal ideal 
$(a,x,u,v)$, so their images in $B$, the transients, are not 
comaximal. Moreover, the preimages of the second transient and the 
survivor are both contained in the maximal ideal $(a,x,u-1,v)$.
Here is a diagram of the poset $\jS(B)$:

\vskip 18 pt

\setbox4=\vbox{\hbox{%
   \rlap{\kern.5in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern1.4in\lower0in\hbox to .3in{\hss$(a,x,u,v)$\hss}}%
   \rlap{\kern2.3in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern3.2in\lower0in\hbox to .3in{\hss$(a,x,u-1,v)$\hss}}%
   \rlap{\kern4.1in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern.7in\lower1in\hbox to .3in{\hss$(a,x,u)$\hss}}%
   \rlap{\kern1.5in\lower1in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern2.3in\lower1in\hbox to .3in{\hss$(a,x,v)$\hss}}%
   \rlap{\kern3.9in\lower1in\hbox to .3in{\hss$(a,u-1,v-x)$\hss}}%
   \rlap{\kern2.3in\lower1.8in\hbox to .3in{\hss$(0)$\hss}}%
   \rlap{\special{pa 550 80} \special{pa 750 870} \special{fp}}%1
   \rlap{\special{pa 950 80} \special{pa 750 870} \special{fp}}%4
   \rlap{\special{pa 950 80} \special{pa 2350 870} \special{fp}}%4
   \rlap{\special{pa 2350 80} \special{pa 2350 870} \special{fp}}%4
   \rlap{\special{pa 3250 80} \special{pa 2350 870} \special{fp}}%7
   \rlap{\special{pa 3250 80} \special{pa 3950 870} \special{fp}}%7
   \rlap{\special{pa 4150 80} \special{pa 3950 870} \special{fp}}%8
   \rlap{\special{pa 750 1080} \special{pa 2350 1670} \special{fp}}%10
   \rlap{\special{pa 1550 1080} \special{pa 2350 1670} \special{fp}}%10
   \rlap{\special{pa 2350 1080} \special{pa 2350 1670} \special{fp}}%11
   \rlap{\special{pa 3950 1080} \special{pa 2350 1670} \special{fp}}%14
}}

\box4
\vskip 12 pt
\centerline{Diagram 3.1.1}
\vskip 18 pt

\medskip

\noindent
{\bf 3.2 Example.}
An example of a 2-generated sfb-extension of $k[x-y]_{(x-y)}[x]$,
where $x,y$ are indeterminates over the field $k$, in 
which two transients are contained 
in two maximal ideals. This cannot happen in a cyclic sfb-extension.

Let
$$
B = k[x-y]_{(x-y)}[x,(x^2+2y^3)/xy,\,(x^2+2y^3-2xy)/y^2] \ .
$$
By using MACAULAY, we find that the kernel of the map from
$k[x-y]_{(x-y)}[x,u,v]$ onto $B$ determined by sending $u$
to $(x^2+2y^3)/xy$ and $v$ to $(x^2+2y^3-2xy)/y^2$ is generated
by
$$\gather x^2 (v+1-2x)\ ,\qquad x^2(u - 1 - 2x)\ ,\qquad
x(u -2 - v)\ , \\
x(u(v-2x) - v + 4x)\ ,\qquad v(u^2 - 2u - v) - 2x(u-2)^2\ .
\endgather$$
In this case, factoring out the extension of $\m$ means replacing $y$
with $x$ in the generators, and we find that the preimage of
the survivor is $(x-y,u-v-2,2x-u+1)$ and those of the transients are 
$(x,y,v)$ and $(x,y,u^2-2u-v)$. Thus, the preimages of the transients
are contained in both $(x,y,u,v)$ and $(x,y,u-2,v)$, while the
preimages of the survivor and the first transient are contained
in $(x,y,u-1,v+1)$. The survivor and the other transient are
comaximal.

\vskip 18 pt
\setbox4=\vbox{\hbox{%
   \rlap{\kern0in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern1in\lower0in\hbox to .3in{\hss$(x,y,u,v)$\hss}}%
   \rlap{\kern1.9in\lower0in\hbox to .3in{\hss$(x,y,u-2,v)$\hss}}%
   \rlap{\kern2.8in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern3.8in\lower0in\hbox to .3in{\hss$(x,y,u-1,v+1)$\hss}}%
   \rlap{\kern4.6in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern.1in\lower1in\hbox to .3in{\hss$(x,y,v)$\hss}}%
   \rlap{\kern1in\lower1in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern2.3in\lower1in\hbox to .3in{\hss$(x,y,u^2-2u-v)$\hss}}%
   \rlap{\kern4.0in\lower1in\hbox to .3in{\hss$(x-y,2x-u+1,u-v-2)$\hss}}%
   \rlap{\kern2.3in\lower1.8in\hbox to .3in{\hss$(0)$\hss}}%
   \rlap{\special{pa 50 80} \special{pa 150 870} \special{fp}}%1
   \rlap{\special{pa 1050 80} \special{pa 150 870} \special{fp}}%4
   \rlap{\special{pa 1050 80} \special{pa 2350 870} \special{fp}}%4
   \rlap{\special{pa 1950 80} \special{pa 150 870} \special{fp}}%4
   \rlap{\special{pa 1950 80} \special{pa 2350 870} \special{fp}}%4
   \rlap{\special{pa 2850 80} \special{pa 2350 870} \special{fp}}%4
   \rlap{\special{pa 3850 80} \special{pa 2350 870} \special{fp}}%7
   \rlap{\special{pa 3850 80} \special{pa 4050 870} \special{fp}}%7
   \rlap{\special{pa 4650 80} \special{pa 4050 870} \special{fp}}%8
   \rlap{\special{pa 150 1080} \special{pa 2350 1670} \special{fp}}%10
   \rlap{\special{pa 1050 1080} \special{pa 2350 1670} \special{fp}}%11
   \rlap{\special{pa 2350 1080} \special{pa 2350 1670} \special{fp}}%11
   \rlap{\special{pa 4050 1080} \special{pa 2350 1670} \special{fp}}%14
}}

\box4
\vskip 12 pt
\centerline{Diagram 3.2.1}


\heading{4. $j$-Spectra of birational extensions} \endheading

Let $B$ be an sfb-extension of $R[x]$ where $R$ is local, so
that there is a unique survivor $P_0$ in $\jS(B)$. 
Recall that for $P$ a transient prime of $B$, $P \cap R[x]$ is
a maximal ideal (from the discussion in (1.3.4)). For $P,Q$
transient primes of $B$, we say $P \sim Q$ if and only if
$P \cap R[x] = Q \cap R[x]$; and for maximal ideals $M,N$ of 
$B$, we define $M \sim N$ if and only if $M \cap R[x] = N \cap R[x]$.
There are finitely many transients in $\jS(B)$ and therefore 
a finite number $s$ of equivalence classes, 
$\Cal T_1, \dots \Cal T_s$, of transient primes
under this relation. Considering just the maximal ideals of 
$B$ which contain transients, they also fit into $s$ equivalence
classes,
$\Cal M_1,\dots,\Cal M_s$, where
$$
\Cal M_i = \{\  M \in \htwo(\jS(B)) : M \cap R[x] = P \cap R[x], 
\ P \in \Cal T_i \ \} \ .
$$ 
For each $i$, the only maximal ideals of $B$ containing
elements of $\Cal T_i$ are elements of $\Cal M_i$.
If two transients $P$ and $Q$ of $B$ are not equivalent,
they must be comaximal in $B$. By (1.3.4)
there is at most one connection of the survivor prime
$P_0$ to each $\Cal T_i$ via $\Cal M_i$; that is, if there
is a maximal ideal $M$ containing $P_0$ and some 
$P \in \Cal T_i$, then $M \in \Cal M_i$ and for each
$Q \in \Cal T_i$ and $N \in \Cal M_i$ with 
$P_0 + Q \subseteq N$, we have $N = M$.
Consider the poset diagram
of $\jS(B)$; there is at most one line upward 
joining $P_0$ to each set $\Cal M_i$ and thus at most one
connection from $P_0$ to each set $\Cal T_i$.
In other words, suppose we remove from the poset diagram
of $\jS(B)$ the prime (0) and regard the rest as an undirected
graph. Then $P_0$ is adjacent to at most one element of 
each $\Cal M_i$; thus a path through $P_0$ cannot be closed
(without passing back through $P_0$).
Thus $U = \jS(B)$ satisfies
\medskip
\itemitem{(P7)} Let $V$ be the graph resulting from taking
the poset diagram of $U$ and removing the unique minimal
element. Among the special elements there is one, $u_0$,
for which there is no closed path containing $u_0$
(without repeated edges) in $V$.

\medskip

In Theorem 4.1, we describe a systematic procedure for
constructing an sfb-extension with $j$-spectrum a given
countable partially ordered set $U$ that is otherwise 
reasonable (dimension two, unique minimal element, infinitely
many height-one maximals, finitely many height-two points 
that are greater than two given height-one points, etc.)
and satisfies a strong version of (P7): 
the special element $u_0$ has degree one in
the graph $V$.
In (4.4), we construct an example in which $u_0$ has
degree two. We believe that this construction can be
generalized to yield an sfb-extension with $j$-spectrum
isomorphic to a given feasible $U$ satisfying (P7).
The local ring $R$ used in our constructions is the DVR $k[y]_{(y)}$
where $k$ is a countable algebraically closed field and
$y$ is an indeterminate over $k$. 


\proclaim{4.1 Theorem} 
Let $U$ be a countable two-dimensional partially ordered set with
a unique minimal element and precisely $n + 1$  height-one elements
$u_0,u_1,\dots,u_n$, which are not maximal. Assume that

\roster
\item The set of maximal height-one elements of $U$ is infinite,
\item For each $i,\,0\le i\le n$, $\G(u_i)$ is infinite,
\item $|G(u_0) \cap (\cup_{i > 0}G(u_i))| \le 1$,
\item For every $i\ne j, \,1\le i,j\le n$, 
$\G(u_i)\cap\G(u_j)$ is finite,
\endroster
Then, for $k$ a countable algebraically closed
field, $y,x$  indeterminates over $k$, and  $R=k[y]_{(y)}$,
there exists an sfb-extension $B$ of $R[x]$ for which
$U\cong\jS(B)$. Furthermore the order-isomorphism
$\varphi: U\to \jS(B)$
can be chosen so that $\varphi(u_i)\cap R[x]=(x,y)$,
for each $i>0$, and so that $\varphi(u_0)$ is the 
survivor prime of the sfb-extension $B$ of $R[x]$.
\endproclaim

We first prove a version for which the partially ordered set is
closer to a tree.

\proclaim{4.2 Lemma} 
Let $U$ be a countable two-dimensional partially ordered set
with a unique minimal element and precisely $n+1$ height-one elements
$u_0,u_1,\dots,u_n$, which are not maximal. Assume that
\roster
\item The set of maximal height-one elements of $U$ is infinite,
\item For each $i,\,0\le i\le n$, $\G(u_i)$ is infinite,
\item For the pair $u_0,u_1$, 
$|G(u_0) \cap G(u_1)| \le 1$, while for
every $i\ne j, \{ i,j\} \ne \{0,1\},
\,0\le i,j\le n$, $\G(u_i)\cap\G(u_j)=\emptyset$.
\endroster
Then, for $k$ a countable algebraically closed
field, $y,x$  indeterminates over $k$, and  $R=k[y]_{(y)}$,
there exists an sfb-extension $A$ of $R[x]$ so that
$U\cong \jS(A)$. Furthermore the order-isomorphism
$\varphi: U\to \jS(A)$ can be chosen so that
$\varphi(u_i)\cap R[x]=(x,y)$, for each $i>0$, and so that
$\varphi(u_0)$ is the survivor prime of the sfb-extension $A$ of 
$R[x]$.  Also if $M$ is a maximal ideal of $A$, then $A/M \cong k$.
\endproclaim

\demo{Proof of Lemma 4.2} Choose distinct elements $a_1,\dots,a_n$ of $k$.
If $|G(u_0) \cap G(u_1)| = 1$, take $a_1=0$,  while for
$|G(u_0) \cap G(u_1)| =0$, take all $a_i\ne0$.
For $1\le i\le n,$ let $V_i\supset R[x][y/x]_{(x,(y/x)-a_i)}$ be the
valuation domain  of the ord valuation $v_i$ (determined by
the powers of the ideal $(x,(y/x)-a_i)\ )$; let
$D=R[x,1/x]$. 
The center of $V_i$ on the regular ring $R[x,y/x]$ is 
$(x,(y/x) - a_i)R[x,y/x]$ and the residue field  of  $V_i$ is a simple
transcendental extension of $k$ with the image of $((y/x) - a_i)/x$ 
as a field generator for this residue field over $k$.
Set $A=D\cap V_1\dots\cap V_n$.

Let $w = (\prod_{i=1}^n((y/x) - a_i))/x$; we show that
$A = R[x, y/x, w]$. It is easily seen that $R[x,y/x, w] \subseteq A$,
so we prove the reverse inclusion. Since $R[x,y/x]$ is regular,
the extension $R[x,y/x][w]$ is locally a complete intersection and
hence Cohen-Macaulay. Thus to show $A \subseteq R[x,y/x,w]$, it
suffices to show for each height-one prime $Q$ of $R[x,y/x,w]$
that $A \subseteq R[x,y/x,w]_Q$. If $x \notin Q$, then 
$R[x,y/x,w]_Q$ contains $D$ and hence $A$. Suppose $x \in Q$.
Then $\prod_{i=1}^n((y/x) - a_i) = xw \in Q$, so one factor
$(y/x) - a_i$ is in $Q$ and the rest are not because the
pairwise differences of the factors are units. Thus,
one element of $R[x,y/x,w]_Q$ is 
$$
\frac{w}{\prod_{j \ne i}((y/x)-a_j)} = \frac{(y/x) - a_i}{x}\ .
$$
It follows that $R[x, y/x, w]_Q$ contains 
$R[x,y/x]_{(x,(y/x) - a_i)}[((y/x) - a_i)/x]$,
of which $V_i$ is a localization, i.e., $R[x,y/x,w]_Q = V_i$.
This completes the proof that $A = R[x,y/x,w]$.
In particular, $A$ is an sfb-extension of $R[x]$.

Setting $P_0 = yD \cap A$, we have  $x\notin P_0$, and 
$P_0$ is the survivor prime with respect to the
extension $R[x] \subset A \subset R[x,1/x] = D$, i.e., we
have $P_0 = P_0D \cap A$, a height-one prime of $A$.
For $i \ge 1$, let $P_i$ be the center of $V_i$ on $A$;
then the $P_i$ are height-one primes of $A$, and $A_{P_i} = V_i$. 
Moreover the $P_i$ have the property that $P_i\cap R[x]=(x,y)R[x]$. 

By (1.3.1), $P_0, P_1,\dots,P_n$ are the special elements of $A$.
Define $\varphi$ on the $u_i$ by setting $\varphi(u_i)=P_i$ for $i\ge 0$.
Then $\G(\varphi(u_i))$ is infinite.  
Note that $j$-Spec($A$) has a unique minimal element and 
infinitely many height-one  maximal elements (since $R[x,1/x]$ does).
Thus to see that $\varphi$ can be extended to an 
order-isomorphism on all of $U$, it suffices to show that (3) holds 
with the $u_i$ replaced by $P_i$. This is done in the following claim: 

\proclaim{Claim} $P_0 + P_1 \ne A$ iff $a_1 = 0$. If
$a_1 = 0$, then $P_0+P_1$ is contained in a unique maximal ideal of $A$.
All pairs $P_i,\,P_j$ with $i\ne j, \{ i,j\}\ne \{ 0,1\}$
are comaximal.
\endproclaim

\demo{Proof of claim} 
We have $P_0$ contains $y/x$, $P_i $ contains $y/x-a_i$ for
$i>0$,  and  the $a_i$ are distinct. Thus 
the last  sentence of the claim is clear, and  if $a_1 \ne 0$, then
$P_0 + P_1 = A$. 

If we take $a_1 = 0$
and $n = 1$, so that $A = D \cap V_1$, then 
$A = R[x, y/x^2]$ by the argument above with $w = y/x^2$.
It follows that $R[x,y/x^2] = D \cap V_1 = A$, $P_0 = (y/x^2)A$
and $P_1 = xA$. Thus $P_0 + P_1 = (x,y/x^2)A$, a maximal ideal
of $A$ with residue field $k$. 

In the situation where $A = D \cap V_1 \cap \dots \cap V_n$,
 $n > 1$, and $a_1 = 0$, let $A_1 = D \cap V_1$. 
Then $A_1 = R[x,y/x^2],\  y/x \in P_0 \cap P_1$,
and  $(y/x) - a_i \in P_i$ for $i > 1$. Hence $P_0 \cap P_1$ and 
$P_i$ are comaximal for each $i > 1$. 
If $M$ is a maximal ideal of $A$ containing $P_0 + P_1$
and $S = A-M$, then $S$ meets each $P_i$ for $i > 1$,
so $S^{-1}A = S^{-1}A_1$.  It follows that $P_0 + P_1$
is contained in a  unique maximal ideal of $A$ which is 
$(x,y/x^2)A_1 \cap A$. This completes the proof of the claim.
\enddemo

As noted above, for $i > 0$,
$P_i \cap R[x] = (y,x)R[x]$ and $P_0$ is the survivor prime
of the sfb-extension $A$. 
If $M$ is a height-two maximal ideal of $A$, then $A/M$ is
an algebraic extension of $k$. Hence if $k$ is algebraically 
closed we have $A/M \cong k$, for each height-two maximal ideal $M$ of $A$.
Thus we have completed the proof of the lemma.~\qed
\enddemo

\demo{Proof of Theorem 4.1} We adapt the proof 
of \cite{S. Wiegand (1983), Theorem~1}: Suppose 
$$
H=\bigcup\{\G(u_i)\cap\G(u_j) : i,j = 1,\ldots,n,\ i\neq j\}
$$
has $t$ elements; use induction on $t$. If $t=0,$ 
then the theorem holds by the lemma
(possibly by renumbering). Thus we may assume that $t>0$ and that 
the theorem holds for every smaller number. 

Let $m\in H$. For convenience rearrange the $u_i$'s for $i\ge 1$ so
that $u_i<m$ if $1\le i\le r$ ($r\ge 2$), and $u_i\nless m$ if
$r<i\le n$.  Now let $W=\{ m_1,m_2,\dots,m_r\}$ be a set of $r$
elements disjoint from $U$, and let $U^*$ be the disjoint union
$(U-\{m\})\cup W$, with the additional relations: $u_i<m_i$ in $U^*$ 
for every $1\le i\le r$; $u_0 < m_1$  in $U^* \iff u_0 < m$ in $U$;
and no other new relations. As an illustration
of the process in the case $u_0 < m$, see the diagram below:

\vskip 18 pt

\setbox4=\vbox{\hbox{%
   \rlap{\kern.2in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern.4in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern.8in\lower0in\hbox to .3in{\hss$m$\hss}}%
   \rlap{\kern1.2in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern1.6in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern2.4in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern2.8in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern3.2in\lower0in\hbox to .3in{\hss$m_1$\hss}}%
   \rlap{\kern3.6in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern4.0in\lower0in\hbox to .3in{\hss$m_2$\hss}}%
   \rlap{\kern4.4in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern4.8in\lower0in\hbox to .3in{\hss$m_3$\hss}}%
   \rlap{\kern0in\lower1in\hbox to .3in{\hss$u_0$\hss}}%
   \rlap{\kern.4in\lower1in\hbox to .3in{\hss$u_1$\hss}}%
   \rlap{\kern.8in\lower1in\hbox to .3in{\hss$u_2$\hss}}%
   \rlap{\kern1.4in\lower1in\hbox to .3in{\hss$u_3$\hss}}%
   \rlap{\kern1.8in\lower1in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern2.4in\lower1in\hbox to .3in{\hss$u_0$\hss}}%
   \rlap{\kern2.8in\lower1in\hbox to .3in{\hss$u_1$\hss}}%
   \rlap{\kern3.6in\lower1in\hbox to .3in{\hss$u_2$\hss}}%
   \rlap{\kern4.4in\lower1in\hbox to .3in{\hss$u_3$\hss}}%
   \rlap{\kern4.8in\lower1in\hbox to .3in{\hss\boxed{\infty}\hss}}%
  \rlap{\kern.8in\lower1.8in\hbox to .3in{\hss$0$\hss}}%
  \rlap{\kern3.4in\lower1.8in\hbox to .3in{\hss$0$\hss}}%
  \rlap{\kern.8in\lower2.1in\hbox to .3in{\hss$U$\hss}}%
  \rlap{\kern3.4in\lower2.1in\hbox to .3in{\hss$U^*$\hss}}%
   \rlap{\special{pa 50 80} \special{pa 50 870} \special{fp}}%1
   \rlap{\special{pa 450 80} \special{pa 450 870} \special{fp}}%1
   \rlap{\special{pa 850 80} \special{pa 850 870} \special{fp}}%4
   \rlap{\special{pa 850 80} \special{pa 50 870} \special{fp}}%4
   \rlap{\special{pa 850 80} \special{pa 450 870} \special{fp}}%4
   \rlap{\special{pa 850 80} \special{pa 1450 870} \special{fp}}%4
   \rlap{\special{pa 1250 80} \special{pa 850 870} \special{fp}}%4
   \rlap{\special{pa 1650 80} \special{pa 1450 870} \special{fp}}%4
   \rlap{\special{pa 2450 80} \special{pa 2450 870} \special{fp}}%4
   \rlap{\special{pa 2850 80} \special{pa 2850 870} \special{fp}}%1
   \rlap{\special{pa 3250 80} \special{pa 2850 870} \special{fp}}%4
   \rlap{\special{pa 3250 80} \special{pa 2450 870} \special{fp}}%4
   \rlap{\special{pa 3650 80} \special{pa 3650 870} \special{fp}}%4
   \rlap{\special{pa 4050 80} \special{pa 3650 870} \special{fp}}%4
   \rlap{\special{pa 4450 80} \special{pa 4450 870} \special{fp}}%4
   \rlap{\special{pa 4850 80} \special{pa 4450 870} \special{fp}}%4
   \rlap{\special{pa 50 1080} \special{pa 850 1670} \special{fp}}%10
   \rlap{\special{pa 450 1080} \special{pa 850 1670} \special{fp}}%10
   \rlap{\special{pa 850 1080} \special{pa 850 1670} \special{fp}}%11
   \rlap{\special{pa 1450 1080} \special{pa 850 1670} \special{fp}}%14
   \rlap{\special{pa 1850 1080} \special{pa 850 1670} \special{fp}}%10
   \rlap{\special{pa 2450 1080} \special{pa 3450 1670} \special{fp}}%11
   \rlap{\special{pa 2850 1080} \special{pa 3450 1670} \special{fp}}%14
   \rlap{\special{pa 3650 1080} \special{pa 3450 1670} \special{fp}}%10
   \rlap{\special{pa 4450 1080} \special{pa 3450 1670} \special{fp}}%11
   \rlap{\special{pa 4850 1080} \special{pa 3450 1670} \special{fp}}%11
}}

\box4
\vskip 18 pt

\noindent
(If $u_0 \nless m$, delete the lines joining $u_0$ to
$m$ and $u_0$ to $m_1$.)

By the inductive hypothesis, there is an sfb-extension $B^*$ of
$R[x]$ and an order-isomorphism $\varphi: U^*\to \jS(B^*)$ so that
$\varphi(u_i)\cap R[x]=(x,y)R[x]$ for each $i>0$ and
$\varphi(u_0)$ is the survivor in $B^*$.

Now for each $i$, $M_i=\varphi(m_i)$ is a maximal ideal of $B^*$. 
Since $k$ is algebraically closed, 
$B^*/ \varphi(m_i)\cong k$ for each $i,\, 1\le i\le t$; let 
$\varepsilon_i:B^*\to k$ be the natural projection with kernel $M_i$.  
Let $B=\{b\in B^* : \varepsilon_1(b)=\dots =\varepsilon_t(b)\}$. 
Then by \cite{S. Wiegand (1983), Proposition~2}, the conductor of 
$B^*$ in $B$ is the maximal ideal $M=\bigcap_{i=1}^r M_i$ of $B$, and 
the spectrum of $B$ differs from the spectrum of $B^*$ only in that $M$ 
replaces $M_1,\dots,M_t$ which are the primes of $B^*$
lying over $M$. Furthermore, $B^*$ is a finitely 
generated $B$-module; therefore by the 
Artin-Tate Lemma \cite{Kunz (1985), Lemma 3.3, p.~16} and (1.3.3),
$B$ is an sfb-extension of $R[x]$. 
(Note that in order to insure that $R[x]\subseteq B$, the $M_i$'s must
intersect to the {\it same }maximal ideal of $R[x]$.)
It is clear that $\varphi: U^*\to \jS(B^*)$ can be adjusted to an
order-isomorphism $\varphi: U\to \jS(B)$ which has the desired properties
regarding maximal elements.~\qed

\enddemo

\noindent
{\bf 4.3 Remark.} The proofs of Theorem~4.1 and Lemma~4.2 also
apply under the assumption that $R$ is a countable excellent 
DVR with algebraically closed residue field. 

\medskip
\noindent
{\bf 4.4 Generalization.}
The construction in Theorem~4.1 and Lemma~4.2 can achieve only those
$j$-spectra with the survivor in at most one maximal ideal containing
another special prime. In order to produce the $j$-spectrum 
shown below, we outline a modification of this construction.

\vskip 18 pt
\setbox4=\vbox{\hbox{%
   \rlap{\kern.2in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern.8in\lower0in\hbox to .3in{\hss$\bullet$\hss}}%
   \rlap{\kern1.6in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern2.6in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%   
   \rlap{\kern3.6in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern4.15in\lower0in\hbox to .3in{\hss$\bullet$\hss}}%
   \rlap{\kern4.6in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern0in\lower1in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern.6in\lower1in\hbox to .3in{\hss$P_1$\hss}}%
   \rlap{\kern1.6in\lower1in\hbox to .3in{\hss$P_2$\hss}}%
   \rlap{\kern2.6in\lower1.0in\hbox to .3in{\hss$Q_1$\hss}}%
   \rlap{\kern3.6in\lower1.0in\hbox to .3in{\hss$Q_2$\hss}}%
   \rlap{\kern4.5in\lower1in\hbox to .3in{\hss$P_0$\hss}}%
   \rlap{\kern2.3in\lower1.8in\hbox to .3in{\hss$(0)$\hss}}%
   \rlap{\special{pa 250 80} \special{pa 650 870} \special{fp}}%1
   \rlap{\special{pa 850 40} \special{pa 650 870} \special{fp}}%4
   \rlap{\special{pa 850 40} \special{pa 4550 870} \special{fp}}%4
   \rlap{\special{pa 1650 80} \special{pa 1650 870} \special{fp}}%4
   \rlap{\special{pa 2650 80} \special{pa 2650 870} \special{fp}}%4
   \rlap{\special{pa 3650 80} \special{pa 3650 870} \special{fp}}%4
   \rlap{\special{pa 4200 40} \special{pa 2650 870} \special{fp}}%4
   \rlap{\special{pa 4200 40} \special{pa 4550 870} \special{fp}}%4
   \rlap{\special{pa 4650 80} \special{pa 4550 870} \special{fp}}%4
   \rlap{\special{pa 50 1080} \special{pa 2350 1670} \special{fp}}%10
   \rlap{\special{pa 650 1080} \special{pa 2350 1670} \special{fp}}%11
   \rlap{\special{pa 1650 1080} \special{pa 2350 1670} \special{fp}}%11
   \rlap{\special{pa 2650 1080} \special{pa 2350 1670} \special{fp}}%11
   \rlap{\special{pa 3650 1080} \special{pa 2350 1670} \special{fp}}%11
   \rlap{\special{pa 4550 1080} \special{pa 2350 1670} \special{fp}}%14
}}

\box4

\vskip 18 pt

Let $R = k[y]_{(y)}$, where
$k$ is a countable algebraically closed field, and let
$D = R[x, 1/x(x-1)]$. Set $z = y/(x(x-1))$ and let
$C = R[x,z]$. Then $C$ is a regular ring. For $a,b$ 
nonzero elements of $k$, 
consider the following valuations defined by ideals of $C$:

$v_1$ the ord valuation defined by powers of $(x,z)C$

$v_2$ the ord valuation defined by powers of $(x,z-a)C$

$w_1$ the ord valuation defined by powers of $(x-1,z)C$ and

$w_2$ the ord valuation defined by powers of $(x-1,z-b)C$;

\noindent
let $V_1, V_2, W_1, W_2$ be the associated valuation rings. 

Let $A=D\cap V_1\cap V_2\cap W_1\cap W_2$ and let 
$A_0 = C[(z(z-a))/x, (z(z-b))/(x-1)]$. We show that $A = A_0$.  
Let us first note that $A_0$ is Cohen-Macaulay.
To see this, observe that at each maximal ideal $Q$ of $C$,
$(C-Q)^{-1}A_0$  is a domain generated by one element over
the regular local ring $C_Q$ and hence the quotient of a
regular ring modulo a principal ideal. Thus $A_0$ is locally
a complete intersection and hence is Cohen-Macaulay.
Since  $A_0 \subseteq A$, to show $A_0 = A$ it suffices
to show that $A \subseteq (A_0)_P$ for each 
height-one prime $P$ of $A_0$.  If $x(x-1) \notin P$,
then $A \subseteq D \subseteq (A_0)_P$, so assume that
$x(x-1) \in P$. Suppose $x \in P$. Then $z$ or $z - a$ is
in $P$; say, $z \in P$. Set $Q = P \cap C$; then since
$z-a$ and $x-1$ are units in $C_Q$, we have 
$(C - Q)^{-1}A_0 = C_Q[z/x]$, so 
$$
PC_Q[z/x] = xC_Q[z/x] \qquad 
\text{ and }\qquad (A_0)_P = (C_Q[z/x])_{xC_Q[z/x]} = V_1.
$$
Similarly, if $x,z-a \in P$, then $(A_0)_P = V_2$;
if $x-1, z \in P$, then $(A_0)_P = W_1$; and if 
$x-1,z-b \in P$, then $(A_0)_P = W_2$. It follows that
$A_0 = A$ is an sfb-extension of $k[y]_{(y)}[x]$ with 
$j$-Spec($A$) as in the picture above. 

By an extension of this argument we can build examples that include
the examples in \cite{Heinzer et al. (1994b)} of cyclic sfb-extensions.

Using the gluing techniques from \cite{S. Wiegand (1983)} as in 
the proof of Theorem 4.1 
above, any maximal ideals above $P_1\cap P_2$ can be glued and any maximal 
ideals above $Q_1\cap Q_2$ can be glued together. However the technique does 
not permit gluing of pairs of maximals one above $P_i$ and one above $Q_j$ 
since the $P_i\cap R[x]$ and $Q_j\cap R[x]$ are comaximal.
                      
Thus we can get this picture:
\vskip 18 pt
\setbox4=\vbox{\hbox{%
   \rlap{\kern.2in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
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   \rlap{\kern.6in\lower1in\hbox to .3in{\hss$P_1$\hss}}%
   \rlap{\kern1.6in\lower1in\hbox to .3in{\hss$P_2$\hss}}%
   \rlap{\kern2.6in\lower1.0in\hbox to .3in{\hss$Q_1$\hss}}%
   \rlap{\kern3.6in\lower1.0in\hbox to .3in{\hss$Q_2$\hss}}%
   \rlap{\kern4.5in\lower1in\hbox to .3in{\hss$P_0$\hss}}%
   \rlap{\kern2.3in\lower1.8in\hbox to .3in{\hss$(0)$\hss}}%
   \rlap{\special{pa 250 80} \special{pa 650 870} \special{fp}}%1
   \rlap{\special{pa 850 40} \special{pa 650 870} \special{fp}}%4
   \rlap{\special{pa 850 40} \special{pa 1650 870} \special{fp}}%4
   \rlap{\special{pa 850 40} \special{pa 4550 870} \special{fp}}%4
   \rlap{\special{pa 1650 80} \special{pa 1650 870} \special{fp}}%4
   \rlap{\special{pa 2650 80} \special{pa 2650 870} \special{fp}}%4
   \rlap{\special{pa 3650 80} \special{pa 3650 870} \special{fp}}%4
   \rlap{\special{pa 4200 40} \special{pa 2650 870} \special{fp}}%4
   \rlap{\special{pa 4200 40} \special{pa 3650 870} \special{fp}}%4
   \rlap{\special{pa 4200 40} \special{pa 4550 870} \special{fp}}%4
   \rlap{\special{pa 4650 80} \special{pa 4550 870} \special{fp}}%4
   \rlap{\special{pa 50 1080} \special{pa 2350 1670} \special{fp}}%10
   \rlap{\special{pa 650 1080} \special{pa 2350 1670} \special{fp}}%11
   \rlap{\special{pa 1650 1080} \special{pa 2350 1670} \special{fp}}%11
   \rlap{\special{pa 2650 1080} \special{pa 2350 1670} \special{fp}}%11
   \rlap{\special{pa 3650 1080} \special{pa 2350 1670} \special{fp}}%11
   \rlap{\special{pa 4550 1080} \special{pa 2350 1670} \special{fp}}%14
}}

\box4

\vskip 18 pt
                      
But we can not get this picture:
\vskip 18 pt
\setbox4=\vbox{\hbox{%
   \rlap{\kern.2in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern.8in\lower0in\hbox to .3in{\hss$\bullet$\hss}}%
   \rlap{\kern1.6in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern2.1in\lower0in\hbox to .3in{\hss$\bullet$\hss}}%
   \rlap{\kern2.6in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%   
   \rlap{\kern3.6in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern4.15in\lower0in\hbox to .3in{\hss$\bullet$\hss}}%
   \rlap{\kern4.6in\lower0in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern0in\lower1in\hbox to .3in{\hss\boxed{\infty}\hss}}%
   \rlap{\kern.6in\lower1in\hbox to .3in{\hss$P_1$\hss}}%
   \rlap{\kern1.6in\lower1in\hbox to .3in{\hss$P_2$\hss}}%
   \rlap{\kern2.6in\lower1.0in\hbox to .3in{\hss$Q_1$\hss}}%
   \rlap{\kern3.6in\lower1.0in\hbox to .3in{\hss$Q_2$\hss}}%
   \rlap{\kern4.5in\lower1in\hbox to .3in{\hss$P_0$\hss}}%
   \rlap{\kern2.3in\lower1.8in\hbox to .3in{\hss$(0)$\hss}}%
   \rlap{\special{pa 250 80} \special{pa 650 870} \special{fp}}%1
   \rlap{\special{pa 850 40} \special{pa 650 870} \special{fp}}%4
   \rlap{\special{pa 850 40} \special{pa 1650 870} \special{fp}}%4
   \rlap{\special{pa 850 40} \special{pa 4550 870} \special{fp}}%4
   \rlap{\special{pa 1650 80} \special{pa 1650 870} \special{fp}}%4
   \rlap{\special{pa 2150 40} \special{pa 650 870} \special{fp}}%4
   \rlap{\special{pa 2150 40} \special{pa 2650 870} \special{fp}}%4
   \rlap{\special{pa 2650 80} \special{pa 2650 870} \special{fp}}%4
   \rlap{\special{pa 3650 80} \special{pa 3650 870} \special{fp}}%4
   \rlap{\special{pa 4200 40} \special{pa 2650 870} \special{fp}}%4
   \rlap{\special{pa 4200 40} \special{pa 3650 870} \special{fp}}%4
   \rlap{\special{pa 4200 40} \special{pa 4550 870} \special{fp}}%4
   \rlap{\special{pa 4650 80} \special{pa 4550 870} \special{fp}}%4
   \rlap{\special{pa 50 1080} \special{pa 2350 1670} \special{fp}}%10
   \rlap{\special{pa 650 1080} \special{pa 2350 1670} \special{fp}}%11
   \rlap{\special{pa 1650 1080} \special{pa 2350 1670} \special{fp}}%11
   \rlap{\special{pa 2650 1080} \special{pa 2350 1670} \special{fp}}%11
   \rlap{\special{pa 3650 1080} \special{pa 2350 1670} \special{fp}}%11
   \rlap{\special{pa 4550 1080} \special{pa 2350 1670} \special{fp}}%14
}}

\box4

\vskip 18 pt

That is, for the examples that arise from this construction, if the 
transients $T_1$ and $T_2$ are contained in distinct maximals containing the 
survivor, then $T_1$ and $T_2$ are comaximal in the glued ring.


\Refs 
%\widestnumber\key{HLW2}

\ref \no 1 \by W. Heinzer, D. Lantz, and S. Wiegand
\paper Projective lines over one-dimensional semilocal domains
and spectra of birational extensions
\inbook Algebraic Geometry and Its Applications:
Collection of papers from Abhyankar's 60th birthday conference
\ed  C. Bajaj
\publ Springer Verlag \publaddr New York  \pages 309--325 \yr 1994
\endref

\ref\no 2 \by W. Heinzer, D. Lantz, and S. Wiegand
\paper Prime ideals in birational extensions of polynomial rings
\pages 73-93
\inbook Commutative Algebra: Syzygies, Multiplicities,
and Birational Algebra: Proceedings of a summer research conference
on commutative algebra held July 4-10, 1992 
\ed  W. Heinzer, C. Huneke, J.D. Sally
\bookinfo Contemporary Mathematics \vol 159
\publ Amer. Math. Soc. \publaddr Providence \yr 1994
\endref

\ref \no 3 \by W. Heinzer and S. Wiegand
\paper Prime ideals in two-dimensional polynomial rings
\pages 577-586\jour Proc. Amer. Math. Soc. \vol 107(3)\yr 1989
\endref

\ref \no 4 \by E. Kunz
\book Introduction to Commutative Algebra and Algebraic Geometry
\publ Birkha\" user \publaddr Boston \yr 1985
\endref

\ref \no 5 \by H. Matsumura
\book Commutative Algebra,  Second Edition
\publ Benjamin \yr 1980
\endref

\ref \no 6 \by H. Matsumura
\book Commutative Ring Theory
\bookinfo Cambridge Studies in Advanced Mathematics {\bf 8}
\publ Cambridge University Press \yr 1989
\endref

\ref \no 7 \by R. Wiegand
\paper Homeomorphisms of affine surfaces over a finite field
\pages 28-32\jour J. London Math. Soc.(2) \vol 18 \yr 1978
\endref

\ref \no 8 \by R. Wiegand
\paper The prime spectrum of a two-dimensional affine domain
\pages 209--214 \jour J. Pure Appl. Alg.  \vol 40 \yr 1986
\endref

\ref \no 9 \by S. Wiegand
\paper Intersections of prime ideals in Noetherian rings
\pages 1853-1876 \jour Communications in Algebra  \vol 11(16) \yr 1983
\endref

\endRefs

\enddocument





