%From swiegand@math.unl.edu  Mon Nov 18 08:12:58 2002


\input amstex
  \documentstyle{amsppt}
   \magnification=\magstephalf
  \def \m{{\text {\bf m}}}
   \def \n{{\text {\bf n}}}
   \def \q{{\text {\bf q}}}
   \def \a{{\text {\bf a}}}
   \def \q{{ \text{\bf q}}}
   \def \p{{ \text{\bf p}}}
   \def \m{{ \text{\bf m}}}
   \def \w{{ \text{\bf w}}}
   \def \u{{ \text{\bf u}}}
   \def \Q{{\Bbb Q}}
   \def \N{{\Bbb N}}
   \def \C{{\Bbb C}}
   \def \b{{ \text{\bf b}}}
   \def \d{{ \text{\bf d}}}
   \def \bn{{\Bbb N}}
   \def \sQ{{\Cal Q}}
   \def \I{{\Cal I}}
   \def \R{{\Cal R}}
   \def \Q{{\Bbb Q}}
  \long\def\alert#1{\smallskip\line{\hskip\parindent\vrule%

\vbox{\advance\hsize-2\parindent\hrule\smallskip\parindent.4\parindent%
  \narrower\noindent#1\smallskip\hrule}\vrule\hfill}\smallskip}
   \def \hgt{\operatorname{ ht}}
   \def \dim{\operatorname{ dim}}
   \def \ker{\operatorname{ ker}}
   \def \max{\operatorname{ max}}
   \def \Ass{\operatorname{ Ass}}
   \def \Spec{\operatorname{ Spec}}
   \def \spec{\operatorname{ Spec}}
   \def \rad{\operatorname{ rad}}
   \def \char{\operatorname{ char}}
   \def \deg{\operatorname{ deg}}
   \def \max{\operatorname{ max}}
   \def \Rad{\operatorname{ Rad}}
   \def \Ass{\operatorname{ Ass}}
  \NoBlackBoxes
  \rightheadtext{ Non-finitely generated primes}\topmatter

   \title
  Non-finitely generated prime ideals in \\
  subrings of   power series rings
  \endtitle

   \author
   William Heinzer, Christel Rotthaus and Sylvia
   Wiegand
  \endauthor

 %\date  January 3, 2003
 %\enddate

   \thanks{The authors would like to thank the National Science
   Foundation and the National Security Agency for
   support for this research. In addition they are grateful for
  the hospitality and cooperation of Michigan State, Nebraska
   and Purdue, where several work sessions on this research were
   conducted.}
   \endthanks

   \address{Department of Mathematics, Purdue
   University,
   West Lafayette, IN 47907-1395}
   \endaddress

   \address{Department of Mathematics, Michigan State University, East
  Lansing,
   MI 48824-1027}
   \endaddress

   \address{Department of Mathematics and Statistics,
   University of Nebraska,
   Lincoln, NE 68588-0323}
   \endaddress

\abstract{Given a power series ring $R^*$ over a Noetherian
integral domain $R$ and an intermediate field $L$ between $R$ and
the total quotient ring of $R^*$, the integral domain $A = L \cap
R^*$ often (but not always) inherits nice properties from $R^*$
such as being Noetherian. For certain fields $L$ it is possible to
approximate $A$ using a localization $B$ of a particular nested
union of polynomial rings over $R$ associated to $A$;  if $B$ is
Noetherian then $B = A$. If $B$ is not Noetherian, we can
sometimes identify   which prime ideals of $B$ are not
finitely generated.  We have obtained in this way,  for
each positive integer $n$, a three-dimensional quasilocal unique
factorization domain $B$ such that   the
maximal ideal of $B$ is two-generated, $B$ has precisely $n$ prime
ideals of height two, each prime ideal of $B$ of height two is not
finitely generated and all the other prime ideals of $B$ are finitely
generated. We examine the structure of the map $\Spec A \to \Spec
B$ for this example. We also present a generalization of this example
to higher dimensions.}
  \endabstract

   \endtopmatter

   \document

   \baselineskip 18pt

\subheading{1. Introduction } In this paper we analyze the prime
ideal structure of particular non-Noetherian integral domains
arising from a general construction developed in
our earlier papers \cite{3},\dots, \cite{9}. Briefly, with
this technique two types of integral domains are constructed: (1)
the intersection of a power series ring with a field yields an
integral domain $A$ as in the abstract, and (2) an approximation
of the domain $A$  by a nested union $B$ of localized polynomial
rings has the second form described in the abstract. Classical
examples such as those of Akizuki \cite{1} and Nagata \cite{12,
pages~209-211} were created using the second (nested union)
description of this construction. As we observe in \cite{8}, it
is also possible to realize
these classical examples as the intersection domains of the first
description.

In \cite{8} we observe that  in certain
applications of this technique  flatness of a map of associated
polynomial rings implies that the constructed domains are
Noetherian and that $A = B$.  In \cite{8} and \cite{9}, we
apply this observation to the construction of  examples of both
Noetherian and non-Noetherian integral domains.

This paper represents a continuation of this project. We present
several examples constructed using this technique of building
non-Noetherian integral domains inside a power series ring for
which the prime ideal structure  can be described explicitly.

  In Section 2 we give the notation for a simplified adaptation of
the construction  which we use  to produce these examples. This
adaptation produces  ``insider" examples $A^{\prime\prime}$ and
$B''$  having the form $A$ and $B$ described above and fitting
inside an easier and more straightforward Noetherian domain
$A'=B'$ (where the two forms of the construction are equal). We
have shown in our previous work that the question of  whether or
not the insider domains $A''$ and $B''$ that arise in this
adaptation are Noetherian or equal is related to  flatness of a
map of polynomial rings  corresponding to the extension
$B''\hookrightarrow B'$ and having the form $\varphi: S := R[\underline
f]\hookrightarrow  T := R[\underline x],$ where $\underline
x:=(x_1,\dots,x_n)$ is a tuple of indeterminates over $R$ and
$\underline f:=(f_1,\dots,f_m)$ consists of polynomials $f_i$ in
$R[\underline x]$ that are algebraically independent over the
field of fractions $\Cal Q(R)$ of $R$.

In Sections 3 and 4, we analyze and provide more details about an
``insider'' example $B$ constructed in \cite{9}. For each positive
integer $n$, this construction produces an example of a
3-dimensional quasilocal unique factorization domain that is a
generalized local ring in the sense of Cohen \cite{2} and has as
its completion a 2-dimensional regular local domain. Moreover, $B$
is not catenary and has precisely $n$ prime ideals of height two.
The associated intersection domain $A$ for this construction is a
2-dimensional regular local domain. In Section 4, we examine the
prime spectrum map $\Spec A \to \Spec B$ and describe three types
of height-one prime ideals of $B$ by how they relate to primes of
$A$.

In Section 5 we prove for each positive integer $n  \ge 2$ and
each positive integer $t$, the existence of a
  non-Noetherian integral domain  $B$ such that:
\roster
  \item"(a)" $\dim B  = n+1$.
  \item"(b)" The maximal ideal of $B$ is generated by $n$ elements.
  \item"(c)" $B$ has exactly $t$ prime ideals of height $n$ and each
of these primes is not finitely generated.
  \item"(d)" $B$ is a factorial domain.
  \item"(e)" The completion of $B$ is a regular local domain of
dimension $n$.
  \item"(f)" $B$ is a birational extension of the localized polynomial
ring   over a field in $n+1$ variables.
\endroster






   \subheading{2. Background and Notation}

  We begin this section by recalling some  details from Section 3 of
  \cite{9} concerning the insider adaptation of the construction
  utilized in this article \footnote{For more detail see
  \cite{8} and  \cite{9}}.

  We use the following setting throughout the paper.

 \subheading{2.1 General Setting}
Let $k$ be a field and let $x, y_1, \dots, y_s$ be indeterminates
over $k$; for convenience, we abbreviate the $y_i$ by $\underline
y$. (Often we assume  $k$  to have characteristic zero. This
ensures excellence of the constructed domains in the simplest
applications.) Let $R :=  k[x, y_1, \dots,
y_s]_{(x,y_1,\dots,y_s)}=k[x,\underline y]_{(x,\underline y)}$ and
let $R^*$ be the $(x)$-adic completion of $R$, that is,  $R^*:=
k[\underline y]_{(\underline y)}[[x]]$, a power series ring in
$x$. We write $\Cal Q(R), \Cal Q(R^*)$, etc, for the total rings
of quotients of $R,R^*$, etc, respectively.  Let $\tau_1,
\dots,\tau_n$ be elements of $xk[[x]]$ which are algebraically
independent over $k(x)$; we abbreviate them by $\underline\tau$.
Let $T := R[\underline\tau]$. We define the {\it intersection
domain} $A_{\underline\tau}$ corresponding to $\underline\tau$ by
$A_{\underline\tau}:= k(x, \underline y, \underline\tau) \cap
R^*=\Cal Q(T)\cap R^*$.

Next we select elements $f_1,\dots, f_m$ of $R[\underline\tau]$
which are algebraically independent over $\Cal Q(R)$; we
abbreviate them by $\underline f$.  The {\it intersection domain}
corresponding to $\underline f$ is $A_{\underline f}=\Cal
Q(R[\underline\tau])\cap R^*$.

In order to obtain $A_{\underline\tau}$ as a nested union of
polynomial rings over $k$ in $s+m+1$ variables, we recall some of
the details of the construction.\footnote{Again more details may
be found in \cite{9}.}


\subheading{2.2 Approximation technique} With $k, x, \underline y,
s, R$ and $R^*$ as in (2.1), let $\rho_1, \dots,\rho_m$ be
elements of $R^*$ which are algebraically independent over $\Cal
Q(R)$; we abbreviate them by $\underline \rho$. (Thus the {\it
intersection domain} $A_{\underline \rho}$ corresponding to
$\underline \rho$ is $A_{\underline \rho} := k(x, \underline y,
\underline \rho) \cap R^*=\Cal Q(R[\underline \rho])\cap R^*$.)
Write each $\rho_i: =\sum^{\infty}_{j=1} b_{ij}x^j$, with the
$b_{ij}\in R$. There are natural sequences
$\{\rho_{ir}\}_{r=0}^\infty$ of elements in $R^*$, called the
$r^{\text{th}}$ {\it endpiece}s for the $\rho_i$, which
``approximate" the $\rho_i$, defined by:
$$\text{For each }i\in\{1,\dots,n\}\text{ and } r\ge 0,\quad
\rho_{ir}:=\sum_{j=r+1}^{\infty} (b_{ij}x^j)/x^r.\tag{2.2.1}$$

Now,  for each $r \ge 0$, define $U_r:=R[\rho_{1r},\dots,\rho_{n r}]$
and set $B_r$ to be $U_r$ localized at the multiplicative system $
1 + xU_r$. Then set $U := \cup_{r=0}^\infty U_r$ and
$B_{\underline \rho}= \cup_{r=0}^\infty B_r$. Thus $U$ is a nested
union of  polynomial rings over $R$ and $B_{\underline \rho}$ is a
nested  union of localized polynomial rings over $R$; clearly
$U\subseteq B_{\underline\rho}\subseteq A_{\underline\tau}$. We
refer to $B_{\underline\rho}$ as the {\it nested union domain}
 corresponding to $\underline\rho$. The
definition of the $U_r$ (and hence also of $B_r$, $U$ and $B$) is
independent of the representation of the $\rho_i$ as power series
with coefficients in $R$ \cite{4, Proposition~2.3}.

For the two sequences of power series in (2.1), namely
$\underline\tau$ and $\underline f$, we have the approximation
sequences $\underline\tau_i$ and $\underline f_i $ and the nested
union domains $B_{\underline\tau}$ and $B_{\underline f}$, which
are localizations of $\cup _{r=0}^\infty R[\tau_{1r},\dots,\tau_{n
r}]$ and of $\cup _{r=0}^\infty R[f_{1r},\dots,f_{m r}]$
respectively. Clearly $B_{\underline f} \subseteq
B_{\underline\tau}$ are quasilocal domains with $B_{\underline \tau}$
dominating $B_{\underline f}$.


 \proclaim{2.3 Proposition}
\cite{5, Proposition~4.1} Assume the set-up as in (2.1). Then  $T
\to R^*[1/x]$ is flat, $A_{\underline\tau}=B_{\underline\tau}$
and $A_{\underline\tau}$ is a Noetherian regular local ring;
moreover, if $\char k = 0$, then $A_{\underline\tau}$ is
excellent.
\endproclaim

\subheading{2.4 Insider construction details} Let
$\underline\tau = \tau_1,\dots,\tau_n$ and $\underline f =
f_1,\dots, f_m$ be as in (2.1) and (2.2). We assume
the constant terms in $R = k[x, \underline y]$ of the $f_i$ are
zero. Let $S := R[\underline f]$. The inclusion map
$S \hookrightarrow T$ is an injective $R$-algebra homomorphism, and
$m \le n$.

Let $A$ be the intersection domain for ${\underline f}$, i.e.
$A:=A_{\underline f}=   \Cal Q(S) \cap R^*$. Let
$B:=B_{\underline f}$ be the nested union domain
associated to the $f_1, \dots,f_m$,
as in (2.2). We consider the inclusion maps
$\phi :S \hookrightarrow T$ and $\psi :T \hookrightarrow R^*[1/x]$
as shown in the following diagram.

  \vskip 10 pt
  \setbox4=\vbox{\hbox{%
     \rlap{\kern3.2in\lower0in\hbox to .3in{\hss$R^*[1/x] $\hss}}%
     \rlap{\kern1.5in\lower.6in\hbox to .3in{\hss$R$\hss}}%
     \rlap{\kern1.73in\lower.6in\hbox to .3in{\hss$\subseteq$\hss}}%
     \rlap{\kern2.2in\lower.6in\hbox to .3in{\hss$S:=R[\underline
f]$\hss}}%
     \rlap{\kern3.4in\lower.6in\hbox to .3in{\hss$T:=R[\underline\tau]
  $\hss}}%
     \rlap{\kern0.0in\lower.40in\hbox to .3in{\hss%
  $(2.4.1) $\hss}}%
  \rlap{\kern3.1in\lower.45in\hbox to .3in{\hss%
  $\overset{\psi}\to{\phantom{a}} $\hss}}%
     \rlap{\kern2.5in\lower.3in\hbox to .3in{\hss%
  $\overset\alpha:=\psi\varphi\to{\phantom{a}} $\hss}}%
     \rlap{\kern2.8in\lower.6in\hbox to .3in{\hss%
  $\overset\varphi\to{\phantom{a}}$\hss}}%
     \rlap{\special{pa 2700 550} \special{pa 3100 550} \special{fp}}%1
     \rlap{\special{pa 3250 80} \special{pa 3250 430} \special{fp}}%3
     \rlap{\special{pa 2400 430} \special{pa 3100 20} \special{fp}}%3
     \rlap{\special{pa 3079 529} \special{pa 3100 550} \special{fp}}%1
     \rlap{\special{pa 3079 571} \special{pa 3100 550} \special{fp}}%1
     \rlap{\special{pa 3250 80} \special{pa 3229 101} \special{fp}}%3
     \rlap{\special{pa 3250 80} \special{pa 3271 101} \special{fp}}%3
     \rlap{\special{pa 3100 20} \special{pa 3066 08} \special{fp}}%3
     \rlap{\special{pa 3100 20} \special{pa 3092 55} \special{fp}}%3
     }}

  \box4 \vskip 10 pt

\proclaim{2.5 Theorem}  \cite{3, Theorem 2.2}, \cite{9, Theorem 3.2}
\roster
  \item $B$ is
  Noetherian and $B=A$ if and only if the map $\alpha : S \to
  R^*[1/x]$ in (2.4.1)  is flat.
\item
  For $Q^* \in \Spec(R^*[1/x])$, the localization $\alpha_{Q^*}: S \to
  (R^*[1/x])_{Q^*}$ of the map $\alpha$ in (2.4.1) is flat if and only
if
the localization
  $\varphi_{Q^*\cap
  T}$  of the map $\varphi$ in (2.4.1) is flat.
  \item
  The following are equivalent:

  $\phantom{x}$ (i) $B$ is Noetherian and $A = B$.

  $\phantom{x}$ (ii) $B$ is Noetherian.

  $\phantom{x}$ (iii) The localized map $\varphi_{Q^*\cap T}$ from
(2.4.1) is
  flat for every maximal ideal
   $Q^* \in \Spec(R^*[1/x])$.
  \endroster
  \endproclaim

Proposition 2.6 is helpful for testing whether  the map  $\phi$
of (2.4.1) is flat.

\proclaim{2.6  Proposition} \cite{9, Proposition  2.4} Let $R$ be
a  Noetherian ring and let $x_1,\dots,x_n$ be indeterminates over
$R$. Assume that
   $f_1,\dots,f_m\in R[x_1,\dots,x_n]$ are algebraically
  independent over $R$.  Then \roster
  \item
  $\varphi : S:=R[f_1,\dots,f_m]\hookrightarrow T:=R[x_1,\dots,x_n]
  $ is flat if and only if, for each prime ideal $ P$ of $T$, we
  have  $\text{ht}(P)\ge \text{ht}(P\cap S)$.
  \item
  For $Q \in \Spec T$, $\varphi_Q : S \to   T_Q $ is flat if and
  only if  for each prime ideal $ P \subseteq Q$ of $T$, we have
  $\text{ht}(P)\ge \text{ht}(P\cap S)$.
  \item If $\varphi_x: S\to T[1/x]$ is flat, then $B$ is Noetherian
  and $B=A$.
  \endroster
  \endproclaim

 \subheading{3. Non-Noetherian  Examples}
  We review the
  construction of a series of examples of non-Noetherian integral
  domains inside power series rings given in \cite{9, Examples 5.1}.

\subheading{3.1 Specific construction details for the examples of
\cite{9}} This construction is a localized version of (2.4), with
$s = 1$. Thus $k$ is a field, $R = k[x, y]_{(x,y)}$ is a
two-dimensional regular local ring and $R^* = k[y]_{(y)}[[x]]$ is
the $(x)$-adic completion of $R$. Let $\tau = \sum_{j=1}^\infty
c_jx^j \in xk[[x]]$ be algebraically independent over $k(x)$.
(It is easy to see that (2.3)
still holds; that is, the modified $A_{\tau}:=\Cal Q(R[\tau])\cap
R^*=\bigcup_{r=0}^\infty R[\tau_r]_{(-)}=B_{\tau}$.)

For the insider domains, let
$p_i \in R \setminus xR$ be such that $p_1R^*, \ldots, p_nR^*$ are $n$ distinct prime
ideals. For example, we could take $p_i = y - x^i$. Let $q = p_1
\cdots p_n$. We set  $f := q\tau$ (i.e., $m=1$).

Let $B:=B_f$ be the nested union domain associated to $f$ as in (2.4).

If $\tau_r = \sum_{j= r+1}^\infty \frac{c_jx^j}{x^r}$ is the
$r^{th}$ endpiece of $\tau$, then $f_r := q\tau_r$ is the
$r^{th}$ endpiece of $f$. For each $r \ge 0$, let $B_r =
R[f_r]_{(x, y, f_r)}$. Then each $B_r$ is a 3-dimensional
regular local ring and $B = \bigcup_{r = 0}^\infty B_r$.

In this example, the intersection domain $A_f$ associated to $f$
is the same as that associated to $\tau$; that is,
$A=A_\tau=\Cal Q(R[\tau])\cap R^*=\Cal Q(R[f])\cap R^*=A_f$, since
$\Cal Q(R[\tau])=\Cal Q(R[f])$.

\proclaim{3.2 Proposition} \cite{9}  For each positive integer
$n$, the nested union domain $B$ constructed in (3.1) is a
three-dimensional quasilocal unique
factorization domain  such that
\roster
  \item $B$ is not catenary.
  \item The maximal ideal of $B$ is two generated.
  \item $B$ has precisely $n$ prime ideals of height two.
  \item Each prime ideal of $B$ of height two is not
  finitely generated.
  \item Each height-one prime ideal of $B$ is principal and
  each nonzero prime ideal of $B$ is the union of the prime ideals
  of height one that it contains, so $B$ has infinitely many prime
  ideals of height one.
  \item For every non-maximal prime $P$ of $B$, the ring $B_P$ is
Noetherian.
  \endroster
\endproclaim

\subheading{3.3 Notes}
(1) The nested union domain $B$ is not Noetherian by (2.5.1)
and (2.6.2); each $p_iR^*[1/x]$ is a height-one prime of $R^*[1/x]$,
but
  $p_iR^*[1/x]\cap S=(p_i,f)S$ has height two, thus the
  map $\alpha:S\to R^*[1/x] $ is not flat.

 (2) The maximal ideal of $B$ is $(x,y)B$, because $B/xB=R/xR$.

(3) As noted above, $B_\tau= A_\tau=A =R^*\cap
  \Cal Q(R(\tau))=R^*
  \cap  \Cal Q(R(f))=A_f$ in the notation of (2.1) and (2.2), so that
$A_f$ is a
nested union of three-dimensional regular local domains  (although
we will see that it is not equal to the nested union domain
$B_f$).

  We consider the inclusion map $B \hookrightarrow A$
   and the map $\Spec A \to \Spec B$.  The following
  Proposition is proved in \cite{9}.

\proclaim{3.4 Proposition }  With the notation of (3.1) and
$A = R^* \cap \Cal Q(R(f))$, we have
\roster
  \item $A$ is a two-dimensional regular local domain
  with maximal ideal $\m_A = (x, y)A$.
  \item
  $\m_A$  is the unique prime of $A$ lying over $\m_B  = (x,y)B$,
  the maximal ideal of $B$.
  \item If  $P \in \Spec B$ is nonmaximal, then
  $\hgt(PR^*) \le 1$ and $\hgt(PA) \le 1$. Thus every nonmaximal
  prime of $B$ is contained in a nonmaximal prime of $A$.
  \item
  If $P \in \Spec B$ and $xq \not\in P$, then $\hgt P \le 1$.
  \item If $P \in \Spec B$, $\hgt P =  1$ and $P \cap R \ne 0$,
  then $P = (P \cap R)B$.
  \item If $pA$ is a height-one
  prime of $A$ with $pA \not\in \{p_1A, \ldots p_nA \}$, then
  $A_{pA}  = B_{pA \cap B}$ and $\hgt(pA \cap B) = 1$. However,
   $p_iA \cap B$ has  height two and is not finitely
  generated.\item Each $p_iB$ is prime in $B$.
  \item
  $p_iB$ and $Q_i := (p_i,f_1, f_2, \ldots )B = p_iA \cap B$
  are the only primes of $B$ lying over $p_iR$ in $R$.
  \item
   $Q_i$ has height two and is not finitely generated.
  \endroster\endproclaim

\subheading{3.5 Notes} (1) With regard to the
  birational inclusion $B \hookrightarrow A$ and the map $\Spec A
  \to \Spec B$, we  remark that the following hold: Each $Q_i$
  contains infinitely many height-one primes of $B$ that are the
  contraction of primes of $A$ and infinitely many that are not.
  Among the primes that are not contracted from $A$ are the $p_iB$.
  In the terminology of \cite{14, page 325}, $P$  is {\it not lost}
  in $A$ if $PA \cap B = P$.  Since $p_iA \cap B = Q_i$ properly
  contains $p_iB$,  $p_iB$ is lost in $A$. Since $(x, y)B$ is the
  maximal ideal of $B$ and $(x, y)A$ is the maximal ideal of $A$ and
  $B$ is integrally closed, a version of Zariski's Main Theorem
  \cite{13} implies that $A$ is not essentially finitely
  generated as a $B$-algebra.

  (2) The quasilocal domains $B$ constructed in Example
  3.1 are {\it generalized local rings} in the sense of Cohen \cite{2,
  page 56}, that is, the maximal ideal $\m$ of $B$ is finitely generated
  and the intersection of the powers of the maximal ideal is zero.
Cohen proves in \cite{2} that the completion of a generalized
local ring is Noetherian. In our situation, the $\m_B$-adic
completion of $B$ is equal to the $\m_A$-adic completion of $A$
and is a 2-dimensional regular local domain.

  \subheading{4. Further analysis of $B$ for $n=1$}

In this section we consider the ring $B = \bigcup_{r=0}^\infty B_r$ of
the previous   section in the case where $n = 1$ and $q= p_1 = y$.
Thus
  $f=y\tau$, $R=k[x,y]_{(x,y)}$, $B_r = R[y\tau_r]_{(x, y, y\tau_r)}$
  and $B=\bigcup_{r=0}^\infty R[y\tau_r]_{(x,y, y\tau_r)}$.
  This ring $B$ has exactly one prime ideal $Q = (y,\{y\tau_r \}_{r
  =0}^\infty)B$   of height 2. Moreover, $Q$ is not finitely
  generated and is the only prime ideal of $B$ that is not finitely
  generated. We also have    $Q =yA\cap B$, and $Q\cap
  B_r=(y,y\tau_r)B_r$ for each $r \ge 0$.

  If $q$ is a height-one prime of $B$, then $B/q$ is Noetherian if
  and only if $q$ is not contained in $Q$. This is clear since  $Q$
  is the unique prime of $B$ that is not finitely generated and a
  ring is Noetherian if each prime ideal of the ring is finitely
  generated.

  The height-one primes $q$ of  $B$ may be  separated into three
  types as follows:

  \subheading{Type I} The primes $q \not \subseteq Q$, such as $xB$.
  As mentioned above, $B/q$ is Noetherian. These primes are
  contracted from $A$. To see this, consider  $q=gB$ where $g
  \not\in Q$. Then $gA$ is contained in a height one prime $P$ of
  $A$. Then  $g \in (P \cap B) \setminus Q$ so $P \cap B \ne Q$.
  Since $\m_B A = \m_A$, we have $P \cap B \ne \m_B$. Therefore $P
  \cap B$ is a height-one prime containing $q$, so $q = P \cap B$
  and $B_q = A_P$.

  There are infinitely many primes $q$ of type I, because  every
  element of $\m_B \setminus Q$ is contained in a prime $q$ of type
  I. Thus $\m_B \subseteq Q \cup \bigcup\{ q \text{ of type I} \}$.
  Since $\m_B$ is not the union of finitely many strictly smaller
  prime ideals, there are infinitely many primes $q$ of type I.

  If $q$ is a height-one prime of $B$ not of type I, then $\overline
  B= B/q$ has precisely three prime ideals. These prime ideals form
  a chain: $(\overline 0)\subset \overline Q\subset \overline
  {(x,y)B} = \overline{\m_B}$.

   \subheading{Type II} The
  primes  $q \subset Q$,  where $q$ has height  one and is
  contracted from a prime  $p$ of $A=k(x,y)\cap R^*$, for example,
  the prime   $y(y+\tau)B$. For $q$ of this type, $B/q$ is dominated
  by the one-dimensional Noetherian local domain $A/p$. Thus $B/q$
  is a non-Noetherian generalized local ring in the sense of Cohen.

  For $q$ of Type II,   the maximal ideal of $B/q$ is not principal.
  This follows because a quasilocal generalized local domain having
  a principal maximal ideal is a DVR \cite{12, (31.5)}.

  There are infinitely many height-one primes of type II, for
  example, $y(y + x^n\tau)B$ for each $n \in \N$. For $q$ of type II,
  the DVR $B_q$ is birationally dominated by $A_p$. Hence $B_q =
  A_p$ and the ideal $\sqrt{qA} = p \cap yA$.

  \subheading{Type III} The primes  $q \subset Q$, where $q$ has
  height one and is  not contracted from $A$, for example, the prime
  $yB$ and the prime $(y+x^ny\tau)B$ for $n \in \N$. Since the
  elements $y$ and $y + x^ny\tau$ are in $\m_B$ and are not in
  $\m_B^2$ and since $B$ is a UFD, these elements are necessarily
  prime. Also the prime ideals $(y + x^ny\tau)B$ and $(y +
  x^my\tau)B$ are distinct for $n$ and $m$ distinct positive
  integers, for if $W$ is a prime ideal of $B$ that contains them
  both and if $m > n$, then $x^{m-n}y\tau \in W$. But $x \not\in Q$
  and $Q$ and $\m_B$ are the only primes of $B$ containing $(y,
  y\tau)B$. For $q$ of type III, we have $\sqrt{qA} = yA$.

  If $q = yB$ or $q = (y + x^ny\tau)B$, then the image
  $\overline{\m_B}$ of $\m_B$ in $B/q$ is principal. It follows that
  the intersection of the powers of $\overline{\m_B}$ is $Q/q$ and
  $B/q$ is not a generalized local ring. For if $P$ is a principal
  prime ideal of a ring and $P'$ is a prime ideal properly contained
  in $P$, then $P'$ is contained in the intersection of the powers
  of $P$. \cite{11, page 7, ex. 5}.

\subheading{Another  way to examine the height-one primes of $B$}

Observe that $B[1/x]$ is a localization of the polynomial ring
$k[x,y,f]$ while $A[1/x]$ is
  a localization of $k[x,y,\tau]$. Thus  the embedding
  $B[1/x]\hookrightarrow A[1/x]$ is a localization of the map:
  $$ \phi: k[x,y,f]\longrightarrow k[x,y,\tau]$$
  which is defined by $\phi(f) = y\tau$. Obviously
the nonflat locus of
  $\phi$ is defined by the ideal $(y)$ of $k[x,y,\tau]$.

  If $q\in B$ is a height one prime ideal which is not contained in $Q$,
then   the extension $qA$ is not contained in $yA$ and for every
minimal prime   $p\subseteq A$ over $qA$ the induced map $ B_q
\hookrightarrow A_p$   is flat. In particular, $q$ is not lost in $A$.
If $q$ is a height-one prime of $B$ which is contained in $Q$ then $q$
is of   type II or III according to the minimal prime divisors of
$qA$:
If there is a minimal prime $p \in \Spec A$ of $qA$ which is different
from   $(y)$, then the map:   $ B_q\hookrightarrow A_p$
is flat, and $q$ is not lost in $A$. In this case, $q$ is of type II.
Note   that $yA$ is also a minimal prime divisor of $qA$.
If $yA$ is the only prime divisor  of $qA$, then $q$ is lost in $A$.
In this   case the  map
  $ B_Q\hookrightarrow A_{(y)}$ is a localization.

  We remark that $B/yB$ is a rank $2$ valuation ring. This can be seen
directly or else one may apply \cite{10, Prop. 3.5(iv)}. If we do
a similar construction with a prime contained in $\m^2$ (instead
of $y$), for example,   $f = (x^2+y^2)\tau$ (over $\Q$), then
$B/(x^2 + y^2)$ has a two-generated   maximal ideal and cannot be
a valuation ring.


  \subheading{5. A More General Construction}

In Theorem 5.1, we obtain a generalization of the construction
that gives the examples considered in Sections 3 and 4.

\proclaim{5.1 Theorem}
  Let $n  \ge 2$ be a positive integer. For each
positive integer $t$, there exists  a
  non-Noetherian integral domain  $B$ such that:
\roster
  \item"(a)" $\dim B = n+1$.
  \item"(b)" The maximal ideal of $B$ is generated by $n$ elements.
  \item"(c)" $B$ has exactly $t$ prime ideals of height $n$ and each
of these primes is not finitely generated.
  \item"(d)" $B$ is a factorial domain.
  \item"(e)" The completion of $B$ is a regular local domain of
dimension $n$.
  \item"(f)" $B$ is a birational extension of the localized polynomial
ring   over a field in $n+1$ variables.
  \endroster
\endproclaim

\demo{Proof}
  Let $k$ be a field and let $x,y_1,\hdots,y_{n-1}$ be variables over
  $k$. Define:

  $$R = k[x,y_1,\hdots,y_{n-1}]_{(x,y_1,\hdots,y_{n-1})}$$
  and let $\tau_1,\hdots,\tau_{n-1}\in xk[[x]]$ be algebraically
  independent over $\Cal Q(R)$. For $1 \le i \le t$, let
  $ p_i = y_1 -x^i.$
  Set $q = p_1p_2\hdots p_t, $ and  consider the element
$$ f = q\tau_1 + y_2\tau_2 + \hdots y_{n-1}\tau_{n-1}.$$

  The map:

  $$ R[f] \longrightarrow R[\tau_1,\hdots, \tau_{n-1}]$$
  has as its  Jacobian ideal: $J= (q,y_2,\hdots, y_{n-1})$
  which is an ideal of  height $n-1$ that is the intersection of $t$
  prime ideals of height $n-1$:
$ J = Q_1\cap \hdots \cap Q_t$,  where
  $Q_i = (p_i,y_2,\hdots,y_{n-1}).$

  If we construct $B$ as usual as the union of localized polynomial
  rings of dimension $n+1$, then by \cite{9, Theorem 3.9}, $B$ is not
  Noetherian ( or use (2.5.1) and an argument similar to (3.3.1) ).
If $n > 2$ then $J$ has height $>1$ and it follows from \cite{6,
Theorem 5.5} or \cite{8, Theorem 6.3} that $ B = A = \Cal
Q(R[f])\cap R^*.$  It follows from \cite{6, Theorem 4.5} that $B$
is factorial.

  Moreover, if we consider our usual set up:

  $$R[f] \longrightarrow R[\tau_1,\hdots,\tau_{n-1}][1/x]
\longrightarrow R^*[1/x]
$$
  the preimage in $R[f]$ of $JR^*[1/x]$ is the ideal $L=(J,
f)$ which
  has height $n$. The ideal $L$ corresponds to the intersection of $t$
prime ideals in $B$ of height $n$, these are the only primes in
$B$ of height   $n$ and they are not finitely generated.
\enddemo

\subheading{5.2 Remark} It would be interesting to know  whether
for $B$ as in Theorem 5.1, the prime ideals of $B$ of height $n$
are the only prime ideals of  $B$ that are not finitely generated.


\subheading{5.3 Remark}
  With the insider construction  given in (2.1)-(2.4), if
  the dimension of $B_f$ is greater than that of $R^*$, then $B_f$ is
not catenary.
  One way to see this is that in  $\Spec(B_f$) there is always
a saturated chain of prime ideals that includes $(x)$ and this
chain
  has length equal to  $\dim R^*$, while if $\dim B_f > \dim R^*$,
then there exists a saturated chain of prime ideals in $B_f$ of
length greater than $\dim R^*$.



  \Refs \widestnumber\key{HRW3}

\ref \key{1} \by Y. Akizuki
 \paper Einige Bemerkungen \"uber prim\"are Integritatsbereiche mit
 Teilerkettensatz \jour Proc. Phys-Math. Soc. Japan   \vol 17 \yr
 1935 \pages 327-336
 \endref

\ref\key{2} \by I. S. Cohen  \paper  On the structure and ideal
theory of complete local rings \jour Trans. Amer. Math. Soc. \vol
59 \yr 1946 \pages 54-106
\endref


\ref\key{3} \manyby W. Heinzer, C. Rotthaus and S. Wiegand \paper
Building Noetherian domains inside an ideal-adic
 completion  \inbook Abelian Groups, Module Theory and Topology
 Proceedings in honor of Adalberto Orsatti's 60th Birthday \publ Marcel
 Dekker, Inc. \ed Dikranjan and Salce \yr 1998 \publaddr New York
\pages 279-287
\endref

\ref\key{4} \bysame \paper Noetherian rings between a semilocal
domain and its  completion  \jour J. of Algebra \vol 198 \yr 1997
\pages 627-655
\endref

\ref\key{5} \bysame \paper Noetherian domains inside a homomorphic
image of a completion  \jour J. of Algebra \vol 215 \yr 1999
\pages 666-681
\endref

\ref\key{6} \bysame \paper Intermediate rings between a local
domain and its completion    \jour Illinois J. Math. \vol 43 \yr
1999 \pages 19-46  \endref

\ref\key{7} \bysame \paper Intermediate rings between a local
domain and its completion II   \jour Illinois J. Math. \vol 45 \yr
2001 \pages 965-979
\endref

\ref\key{8} \bysame \paper Building Noetherian and non-Noetherian
integral domains using power series \inbook Ideal theoretic
methods in commutative algebra in honor of James A. Huckaba's
retirement \publ Marcel Dekker, Inc. \ed Anderson and Papick \yr
2001 \publaddr New York \pages 251-264
\endref

\ref\key{9} \bysame \paper Examples of integral domains inside
power series rings \inbook Commutative ring theory and
applications \publ Marcel Dekker, Inc. \ed Fontana, Kabbaj and
Wiegand \yr 2002 \publaddr New York \pages 233-254
\endref

\ref\key{10} \by W. Heinzer and J. Sally \paper Extensions of
valuations to the completion of a local domain \jour J. Pure
Applied Algebra \vol 71 \yr 1991 \pages 175-185
\endref

\ref\key{11}\by I. Kaplansky  \book Commutative Rings  \publ Allyn
Bacon\publaddr Boston\yr 1970
  \endref

\ref\key{12} \by M. Nagata \book Local Rings \publ John Wiley
  \publaddr New York \yr 1962
  \endref

  \ref\key{13} \by C. Peskine \paper Une g\'en\'eralsation du ``main
  theorem'' de Zariski \jour Bull. Sci. Math. \vol 90 \yr 1966
  \pages 377-408
  \endref


  \ref\key{14} \by O. Zariski and P. Samuel  \book Commutative
  Algebra volume II \publ D. Van Nostrand \publaddr New York \yr
  1960
  \endref

  \endRefs

  \enddocument
