%August 3, 2000
\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 \Z{{\Bbb Z}}
 \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}}
%\NoBlackBox

 \rightheadtext{  Building Noetherian Integral Domains }
 \title
 Building Noetherian and Non-Noetherian \\
Integral Domains Using Power Series
\endtitle
 \author
 William Heinzer, Christel Rotthaus and Sylvia
 Wiegand  \endauthor
\date Dedicated to Jim Huckaba on the occasion of his retirement \enddate

 \thanks{The authors 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{ 
In this mainly expository article we 
describe a technique, dating back at least to
the 1930s, which  uses power series, homomorphic images and intersections
involving a Noetherian integral domain $R$ and
a homomorphic image $S$ of a power
series ring extension of $R$  to obtain a new 
integral domain $A$. Here  $A$ has 
the form $A:=L\cap S$, where $L$ is a 
field between the fraction field of $R$ and the total 
quotient ring of $S$. We give in certain circumstances 
necessary and sufficient conditions for $A$ to be  computable 
as a nested union of subrings of a specific form. We also prove that the 
Noetherian property for the associated  nested union is equivalent
to a flatness condition. 
We present several examples where this flatness condition holds,
and other examples where it fails to hold. In the first case
this produces a Noetherian integral domain and in the second 
case a non-Noetherian domain. }
\endabstract

 \endtopmatter

 \document

\baselineskip 18pt


\subheading{1. Introduction}  

Over the past sixty years, important examples of Noetherian integral 
domains have been constructed using power series, homomorphic
images  and intersections. The basic idea is to start with
a typical Noetherian integral domain $R$ such as 
a polynomial ring in several indeterminates over a field and to look for
unusual Noetherian and non-Noetherian extension rings
inside a homomorphic image $S$ of an ideal-adic completion $R^*$ of $R$. 
The constructed ring $A$ has the form $A := L \cap S$,
where $L$ is a field between the fraction field of $R$ 
and the total quotient ring of $S$. (The 
elements of $R^*$ are power series with coefficients in $R$.)

Several of our objectives are:
\roster
\item
To construct new examples of nontrivial Noetherian 
and non-Noetherian integral domains; 
this continues a tradition going back to Akizuki in the 1930s and 
Nagata in the 1950s,
\item
To study birational extensions (i.e., extensions inside the 
field of fractions) of a Noetherian local domain $R$; 
this is related to the work of Zariski going back to 
the 1930s and 1940s  on the problem of local
uniformization along a valuation domain birationally dominating $R$,
\item To consider the generic fiber
\footnote{The {\it generic fiber} is the fiber over 
the prime ideal $(0)$ of $R$. Thus if $U = R - (0)$,
then the generic fiber of the map $R \to R^*$ is 
the ring $U^{-1}R^*$.}  of the map $R \to  R^*$,
where $R^*$ is an ideal-adic  completion of the Noetherian
domain $R$, and investigate connections between this fiber  
and birational extensions of $R$. 
\endroster 
These objectives form a complete circle, since (3) is used to
accomplish (1).

The development of this technique  to
create new rings  from well-known ones
goes back to the work of  Akizuki  in \cite{A} and Nagata in \cite{N1}
and has been continued by  Ferrand-Raynaud\cite{FR}, Rotthaus
\cite{R1},\cite{R2},
Ogoma \cite{O1}, \cite{O2},
Brodmann-Rotthaus  \cite{BR1}, \cite{BR2},
Heitmann \cite{H1},
Weston \cite{W},  and the authors \cite{HRW1}, $\ldots$, \cite{HRW5}
to produce a wide variety of Noetherian rings.
In the  work of Akizuki,  Nagata
and Rotthaus (and indeed in most of the papers cited above)
the description of the
constructed ring $A$ as an intersection is not explicitly stated.
Instead $A$ is defined as a direct limit or nested union of subrings. 

The fact that in certain circumstances the intersection domain
$A$ is computable
as a nested  union is important to the development of this technique; 
if this holds one may view $A$ from two different perspectives.
In general there is a natural direct limit domain $B$
associated with $A$. We examine conditions for  $A$ to be equal to $B$. 
This motivates our
formulation of  limit-intersecting properties
in \cite{HRW2, (2.5)} and \cite{HRW3, (5.1)}(see Section~5.2).  

A primary goal of our study is to  determine for a given $R$, $S$ and $L$
whether $A := L \cap S$ is Noetherian.   An important observation 
related to this goal is that the 
Noetherian property for the associated  direct limit ring $B$ is equivalent
to a flatness condition  \cite{HRW2, Theorem~2.12}, \cite{HRW5, Theorem~3.2}
(see Section~4.5).

It took about a page 
for Nagata \cite{N2, page 210} to establish the Noetherian property
of Example~3.1, and the original proof of the Noetherian property
of the example of Rotthaus described in  Example~3.3  is about 7 pages
\cite{R1, pages 112-118}. 
As noted in \cite{HRW2, Remark~3.4}, the Noetherian results of 
\cite{HRW2} and \cite{HRW5} described in Section~4.5  
gives  the Noetherian property in these examples more quickly. 

\subheading{2. Elementary examples}

We begin  by illustrating the construction with several  examples.
In these examples 
$R$ is a  polynomial ring over the field $\Q$ of rational numbers.

In the one-dimensional case the situation is fairly well understood:

\subheading{Example 2.1}
Let $y$ be a variable over $\Q$, let $R := \Q[y]$, 
and let $L$ be a subfield of  the field of fractions 
$\sQ(\Q[[y]])$ of $\Q[[y]]$ such that $\Q(y) \subseteq L$. 
Then the intersection domain  $A=L\cap \Q[[y]]$ is a 
rank-one discrete valuation domain (DVR)
with $y$-adic completion $A^* = \Q[[y]]$. For example,  if we work with our
favorite transcendental function and put $L=\Q(y, e^y)$,  then $A$ is a DVR 
having residue field $\Q$ and field of fractions $L$ of 
transcendence degree $2$ over $\Q$. 

The integral domain $A$ of Example~2.1 is perhaps
the simplest example of a local Noetherian domain on an algebraic
function field $L/\Q$ of two variables that is not essentially finitely
generated over $\Q$, i.e., $A$ is not the localization of a finitely
generated $\Q$-algebra. However $A$ does have a nice description as
an infinite nested union of 
localized polynomial rings in $2$ variables over $\Q$. Thus in a 
certain sense there is a good description of the elements of 
the intersection domain $A$ in this case.

The two-dimensional (two variable) case is more interesting.
The  following theorem of Valabrega \cite{V} is useful in 
considering this case.
 
\proclaim{Theorem} Let $C$ be a DVR, let $y$ be an indeterminate 
over $C$, and let 
$L$ be a subfield of $\sQ(C[[y]])$ such that  $\sQ(C)(y) \subseteq L$.  Then the 
integral domain  $ D= L\cap C[[y]]$
is a two-dimensional regular local domain  
having  completion
\footnote{By the {\it completion } of a local ring,
we mean  the ideal-adic completion with respect to the 
powers of its maximal ideal.}  $\widehat D =\widehat C[[y]]$, where 
$\widehat C$ is the completion of $C$.
\endproclaim

Applying Valabrega's theorem,  we see that the intersection 
domain is a two-dimensional regular local domain 
with the ``right" completion in the following two examples:

\subheading{Example 2.2} Let $x$ and $y$ be 
indeterminates over $\Q$ and   let  $C  = \Q(x, e^x) \cap \Q[[x]]$.
Then  $A_1 := \Q(x,e^x,y)\cap  C[[y]]$ is a two-dimensional
regular local domain having  completion $\Q[[x,y]]$.

\subheading{Example 2.3} Let $x$ and $y$ be  
indeterminates over $\Q$ and   let  $C  = \Q(x, e^x) \cap \Q[[x]]$
as in Example~2.2. 
Then $A_2 :=  \Q(x,y,e^x,e^y)\cap  C[[y]]$ is a 
two dimensional regular local domain having completion $\Q[[x,y]]$.

There is a significant difference, however,  between the 
integral domains $A_1$ of Example 2.2 and $A_2$ of 
Example 2.3. As is shown in  \cite{HRW3, Section 2},
the 2-dimensional regular local domain $A_1$ of
Example 2.2 is,  in a natural way,   a nested  union of 
3-dimensional regular local domains. It is possible therefore to
describe $A_1$ rather explicitly. On the other hand,
the 2-dimensional regular local domain $A_2$ of Example~2.3
contains, for example, the element 
$(e^x-e^y)/(x-y)$. As discussed in  
\cite{HRW3, Section 2}, the associated nested union domain $B$
naturally associated with $A_2$ is 3-dimensional and 
non-Noetherian.

\subheading{Remark 2.4} 
It is shown in \cite{HRW1, Theorem 3.9} that  if we go 
outside the range of Valabrega's
theorem, that is, if we take more 
general subfields $L$ of $\sQ(\Q[[x,y]])$ such that $\Q(x,y) \subseteq L$,
then  the intersection domain $A = L\cap \Q[[x,y]]$ can
be, depending on $L$, a localized 
polynomial ring in $n \ge 3$ variables over $\Q$ or 
even a localized  polynomial ring in
infinitely many variables over $\Q$. In particular $A = L \cap \Q[[x,y]]$
need not be Noetherian.

\subheading{3. Historical examples}

A Noetherian local domain $R$ is said to be {\it analytically
irreducible}  if its completion $\widehat R$ is again
an integral domain. Related to singularities of algebraic
curves, there are classical examples of one-dimensional
local Notherian domains $R$ such that $\widehat R$ is
not an integral domain, i.e., $R$ is analytically reducible.
For example, let $X$ and $Y$ be variables over $\Q$ and let
$R = \Q[X,Y]_{(X,Y)}/(X^2 - Y^2 - Y^3)$. Then $R$ is a
one-dimensional Noetherian local domain since the polynomial
$X^2 - Y^2 - Y^3$ is irreducible in the polynomial ring
$\Q[X,Y]$. Let $x$ and $y$ denote the images in $R$ of $X$
and $Y$, respectively. Then $\widehat R$ is also the
$y$-adic completion of $R$ and is equal to
$R^* = \Q[X][[Y]]/(X^2 - Y^2((1 + Y))$. Since
$(1 + Y)^{1/2} \in \Q[[Y]]$, we see that
$X^2 - Y^2(1 + Y)$ factors in $\Q[X][[Y]]$
as $(X - Y(1+Y)^{1/2})\cdot(X + Y(1+Y)^{1/2})$. Thus
$R^*$ is not an integral domain.

In this example, the integral domain $R$  is not {\it normal}
or equivalently,  {\it integrally closed}.
That is, there are monic polynomials in the polynomial
ring $R[Z]$ that
have roots in the fraction field of $R$ that are not 
in $R$. For example, the polynomial $Z^2 - (1+y) \in R[Z]$
has $x/y$ as a root.

If $R$ is a normal one-dimensional Noetherian local domain,
then $R$ is a rank-one discrete valuation domain (DVR) and
it is well-known that the completion of $R$ is again a DVR.
Thus $R$ is analytically irreducible.

Zariski showed (cf. \cite{ZS, pages 313-320}) that the
normal Noetherian local domains that
occur in algebraic geometry are {\it analytically normal},
i.e., the completion of such a domain is again a normal domain.
In particular, the normal local domains occuring in algebraic
geometry are analytically irreducible.

This motivated the question of whether 
there exists a normal  Noetherian local domain for
which the completion is not a domain.
Nagata produced such examples. He also 
pinpointed sufficient conditions
for a normal Noetherian local domain to be
analytically irreducible \cite{N, (37.8)}. 

In Example~3.1, we present a special case of a
construction of Nagata  \cite{N2, Example 7, pages 209-211}
of a 2-dimensional normal Noetherian local domain
that is analytically reducible.



\subheading{Example 3.1 (Nagata)}
Let $x$ and $y$ be algebraically independent over 
$\Q$ and let $R$ be the localized polynomial ring  
$R=\Q[x,y]_{(x,y)}$. Then the 
completion of $R$ is $\widehat R=\Q[[x,y]]$.
Let $\alpha\in y\Q[[y]]$ be an element that is 
transcendental over $\Q(x,y)$,  e.g., $\alpha=e^y-1$. 
Let $\rho=x+\alpha.$
Now define
\footnote{The original definition by Nagata is in
terms of a nested union of subrings. He then proves
that this nested union is Noetherian with completion 
$\Q[[x,y]]$. It then follows that the nested union
is an intersection as defined here.}
 $A :=\Q(x,y,\rho^2)\cap \Q[[x,y]]$. Then $A$ is Noetherian
(in fact a 2-dimensional regular local domain
\footnote{ This example constructed by  Nagata (historically) is  the
first occurence  of a 2-dimensional regular local domain
containing a field of characteristic zero that fails to
be pseudo-geometric. As such, the example fails to satisfy one of 
the conditions in the definition of an excellent ring.
For the definition and information on excellent rings
see \cite{M1, Chapter 13}, \cite{M2, Section~32} and \cite{R4}.}).
Moreover  $\rho^2$ is a prime element of $A$, so  if $D:=
(A[z]/(z^2-\rho^2))_{(x,y,z)}$, then
$D$ is an integral domain. As is shown by 
Nagata, $D$ is,  in fact,  a normal
Noetherian local domain. The element   $z^2$, however, 
factors as a square in  $\widehat D$: $z^2 =(x+\alpha)^2$ in $\widehat D$.
Thus $D$ has completion $\widehat D =
\Q[[x,y,z]]/(z-(x+\alpha))(z+(x+\alpha))$ which is not an
integral domain.

\subheading{Remark 3.2}
The two-dimensional regular  local domain $A$ of Example~3.1 has 
a  principal prime ideal $\rho^2A$ that factors  in
$\widehat A = \Q[[x,y]]$ as the square of the prime
element  $\rho$ of $\widehat A$. Therefore the map  
$A \to \widehat A = \Q[[x,y]]$ is not a regular morphism.
\footnote{ A homomorphism $\phi : S \to T$ of Noetherian rings is said
to be  {\it regular } if it is flat with geometrically
regular fibers \cite{M2, page~255}.}

The existence of examples such as the normal Noetherian
local domain $D$ of Example~3.1 naturally motivated
the question: Is  a {\it pseudo-geometric domain } (in 
the terminology of Grothendieck,  a {\it universally Japanese domain}; in 
that of Matsumura,  a {\it Nagata domain})
necessarily excellent? It was shown by Rotthaus in \cite{R1} that
pseudo-geometric domains need not be excellent.  

In Example~3.3, we present a special case of the 
construction of Rotthaus   \cite{R1}
of a 3-dimensional regular local domain $A$ such that
the formal fibers of $A$ are geometrically reduced, but are 
not geometrically regular. The integral domain $A$ is pseudo-geometric
but is not excellent. 

\subheading{Example 3.3 (Rotthaus)} 
Let $x, y , z$  be algebraically independent over 
$\Q$ and let $R$ be the localized polynomial ring  
$R=\Q[x,y,z]_{(x,y,z)}$. 
Let $\tau_1 = \sum_{i=1}^\infty a_iy^i \in \Q[[y]]$
and 
$\tau_2 = \sum_{i=1}^\infty b_iy^i \in \Q[[y]]$ be power
series such that $y, \tau_1, \tau_2$ are algebraically
independent over $\Q$, for example, $\tau_1 = e^y - 1$
and $\tau_2 = e^{y^2} - 1$. 
Let  $u := x + \tau_1$ and $v := z + \tau_2$. 
Define  
\footnote{In \cite{R1} the example is constructed  as a direct limit. The
fact that it is Noetherian implies it is also this intersection.}
$ A  := \Q(x,y,z,uv) \cap  (\Q[x,z]_{(x,z)]}[[y]])$.
It is shown in \cite{R1} that $A$ is Noetherian and that  the 
completion of $A$ is $\widehat A = \Q[[x,y,z]]$, so $A$ is a 
3-dimensional regular local domain.  Since $u, v$ are part of a 
regular system of parameters of $\widehat A$, it is clear that 
$(u,v)\widehat A$ is a prime ideal of height two. It is shown
in \cite{R1}, that $(u,v)\widehat A \cap A = uvA$. Thus 
$uvA$ is  a prime ideal and 
$\widehat A_{(u,v)\widehat A}/uv\widehat A_{(u,v)\widehat A}$ is
a non-regular formal fiber of $A$. Therefore $A$ is not excellent.


\subheading {4.  Constructions, pictures and Noetherian results}

Let $R$ be a Noetherian integral domain and let $a \in R$ be a 
nonzero nonunit. Then the $a$-adic completion of $R$ is the ring 
$R^* :=  R[[y]]/(y-a)$ \cite{N2, (17.5)}. Thus the
elements of $R^*$ are power series in $a$ with 
coefficients in $R$, but without the uniqueness of
expression as power series that occurs in the formal power
series ring $R[[y]]$.   


We usually reserve the notation $\widehat R$
for the situation where $R$ is a local ring with maximal ideal $\m$
and $\widehat R$ is the $\m$-adic completion of $R$. If 
$\m$ is generated by elements $a_1, \dots,a_n$, then 
$\widehat R$ is realizable by taking the $a_1$-adic
completion $R^*_1$ of $R$, then the $a_2$-adic completion
$R^*_2$ of $R^*_1$, $\ldots$, and then the $a_n$-adic
completion of $R^*_{n-1}$.

Given a Noetherian domain
$R$ and a  nonzero nonunit $a\in R$,
there are two forms of the construction 
associated with the $a$-adic completion $R^*$ of $R$. Thus
there are two methods for the construction. 

\subheading{Method 4.1} 
Suppose $\tau_1,\dots,\tau_s\in aR^*$ are
algebraically independent over the fraction field 
$\Cal Q(R)$ of $R$.  Let
$L=\Cal Q(R)(\tau_1,\dots,\tau_s)$, and 
define $A := L\cap   R^*$. 

\subheading{Method 4.2} Suppose   $I $ is an ideal of $ R^*$  
having the property that  
$P\cap R=(0)$ for each $P \in \Spec R^*$ that is associated to $I$.
Define $A := \Cal Q(R)\cap  ( R^*/I)$. 

The condition in (4.2), that $P\cap R=(0)$ for every prime ideal $P$
of $R^*$ that is associated to $I$ implies that
the fraction field $\Cal Q(R)$
of $R$ embeds in  the  total quotient  ring $ \Cal Q(R^*/I)$
of $R^*/I$.
For then  $R\to R^*/I$ is
an injection and regular
elements of $R$ remain regular as elements of  $R^*/I$.

\subheading {Pictures} Diagrams for these constructions are drawn below.

\vskip 18 pt
\setbox4=\vbox{\hbox{%
   \rlap{\kern2.0in\lower0in\hbox to .3in{\hss$\Cal Q(R^*)$\hss}}%
   \rlap{\kern3.2in\lower0in\hbox to .3in{\hss$R^*$\hss}}%
   \rlap{\kern4.8in\lower0in\hbox to .3in{\hss$\Cal Q(R^*/I)$\hss}}%
   \rlap{\kern0.8in\lower0.6in\hbox to .3in{\hss$R^*$\hss}}%
  \rlap{\kern1.9in\lower0.6in\hbox to .3in{\hss{$L=\Cal
Q(R)(\{\tau_i\})$}\hss}}%
   \rlap{\kern3.6in\lower0.6in\hbox to .3in{\hss$R^*/I$\hss}}%
   \rlap{\kern4.4in\lower0.6in\hbox to .3in{\hss$\Cal Q(R)$\hss}}%
   \rlap{\kern1.0in\lower1.2in\hbox to .3in{\hss\boxed{A=R^*\cap L}\hss}}%
  \rlap{\kern4.1in\lower1.2in\hbox to .3in{\hss\boxed{A=(R^*/I)\cap \Cal
Q(R)}\hss}}%
   \rlap{\kern0.5in\lower1.8in\hbox to .3in{\hss$R$\hss}}%
   \rlap{\kern2.6in\lower1.8in\hbox to .3in{\hss$R$\hss}}%
   \rlap{\special{pa 880 470} \special{pa 2150 80} \special{fp}}%1
   \rlap{\special{pa 1800 470} \special{pa 2150 80} \special{fp}}%1
   \rlap{\special{pa 3250 80} \special{pa 3550 470} \special{fp}}%3
  \rlap{\special{pa 3550 420} \special{pa 3550 470} \special{fp}}%3
  \rlap{\special{pa 3500 470} \special{pa 3550 470} \special{fp}}%3
  \rlap{\special{pa 3530 400} \special{pa 3530 450} \special{fp}}%3
  \rlap{\special{pa 3480 450} \special{pa 3530 450} \special{fp}}%3
   \rlap{\special{pa 4750 80} \special{pa 3750 470} \special{fp}}%3
   \rlap{\special{pa 4750 80} \special{pa 4550 470} \special{fp}}%4
   \rlap{\special{pa 850 650} \special{pa 1150 1040} \special{fp}}%4
   \rlap{\special{pa 1650 650} \special{pa 1250 1040} \special{fp}}%4
   \rlap{\special{pa 3650 650} \special{pa 3950 1040} \special{fp}}%4
   \rlap{\special{pa 3550 650} \special{pa 2680 1670} \special{fp}}%4
   \rlap{\special{pa 4450 650} \special{pa 4050 1040} \special{fp}}%9
   \rlap{\special{pa 1050 1280} \special{pa 580 1670} \special{fp}}%10
   \rlap{\special{pa 3850 1280} \special{pa 2680 1670} \special{fp}}%13
   \rlap{\special{pa 3250 80} \special{pa 2680 1670} \special{fp}}%13
}}

\box4
\vskip 12 pt
\centerline{(4.1) $A := L\cap   R^*\phantom{Actually the form in (2)}$(4.2)
$A := \Cal Q(R)\cap  ( R^*/I)$}
\vskip 18 pt

\noindent
{\bf Remark 4.3.} 
The papers \cite{HRW2} and \cite{HRW3} feature  the construction described
in (4.1) which realizes the intersection
domain $A:=\sQ(R)(\tau_1,\dots,\tau_s )\cap R^*$. 
The construction given in  (4.2) includes
that given in  (4.1) as a special case.
To see this, let $R, a $ and $R^*$ be as in (4.1).
Let $t_1,\dots,t_s$ be indeterminates over $R$, 
define $S := R[t_1, \dots, t_s]$, let 
$S^*$ be the $a$-adic completion of $S$ and 
let $I$ denote the ideal $(t_1-\tau_1,\dots,t_s-\tau_s )R^*$.
Consider the following diagram where $\lambda$ is the 
$R$-algebra isomomorphism that maps $t_i \to \tau_i$ for $i = 1,\dots,s$. 


$$
\CD {}  @. S:= R[t_1,\dots,t_s ] @>>> D:=\sQ(R)(t_1,\dots,t_s)\cap (S^*/I)
@>>>S^*/I \\
@.  @V{\lambda}VV   @V{\lambda}VV   @V{\lambda}VV \\
R @>>> R[\tau_1,\dots,\tau_s] @>>> A:=\sQ(R)(\tau_1,\dots,\tau_s)\cap R^*@>>>
R^*.     \endCD  \tag{4.4}
$$
Since $\lambda$ maps $D$ isomorphically onto $A$, we see that the
construction given in (4.2) includes as a special case that of (4.1).

\subheading{4.5. Noetherian results }
Suppose  $R$ is a Noetherian local 
domain and let the notation be as in (4.1). 
We prove in \cite{HRW2, Theorem 2.12}:
 
\proclaim{Theorem 4.5.1} The canonical map   
$R[\tau_1,\dots, \tau_s] \to R^*[1/a]$
is flat if and only if $A$ is Noetherian and is a 
nested union of localized polynomial rings 
in $s$ variables over $R$ as defined in Section~5.2 .
\endproclaim

Later, in \cite{HRW5}, we prove an analogous result
in the more general setting of (4.2). With notation as in (4.2),
we prove in \cite{HRW5, Theorem 3.2}:

\proclaim{Theorem 4.5.2} The canonical map  $R \to (R^*/I)[1/a]$ is
flat if and only if $A$ is simultaneously Noetherian and a 
localization of a subring of $R[1/a]$.
\endproclaim

The proof of the more general result in the setting of (4.2) is
actually shorter and more direct than the earlier proof in \cite{HRW2}.
In \cite{HRW5} and \cite{HRW6} examples are constructed of the
form (4.2) that cannot be realized by means of (4.1).


\subheading{5. Flatness, approximations  and universality}

\subheading{5.1. Flatness}
The concept of flatness was  introduced by Serre in
the 1950's in an appendix to
his paper \cite{S}.
Mumford writes in \cite{Mu, page~424}: ``The concept of flatness is a
riddle that comes out of algebra, but which technically is the
answer to many prayers.'' An $R$-module $M$ is {\it flat} over
$R$ if tensoring with $M$ preserves exactness of 
every exact sequence of $R$-modules. Equivalently,
$M$ is flat over $R$ if for every $m_1, \dots,m_n \in M$
and $a_1,\dots,a_n \in R$ such that $\sum a_im_i = 0$,
there exist for some integer $k$ elements $b_{ij} \in R$
and elements $m_1', \dots, m_k' \in M$ such that $m_i = \sum_{j=1}^kb_{ij}m_j'$
for each $i$  
and $\sum_{i=1}^na_ib_{ij} = 0$ for each $j$.
A finitely generated module over a local ring is flat
if and only if it is free \cite{M1, Proposition 3.G}.
If $S$ is obtained as a localizaton of $R$, then $S$ is flat
as an $R$-module \cite{M1, (3.D)}.
If $I$ is an ideal of a Noetherian ring $R$, then the $I$-adic
completion of $R$ is flat over $R$ \cite{M1, Corollary 1, page 170}. 
Thus flatness  is a property that holds for several standard 
constructions for extensions of a Noetherian ring. 


An important fact for the Noetherian results 
described in Section~4.5  is that if 
$\phi : C \to D$ is a flat homomorphism of rings, i.e., $D$ is a
flat $C$-module,  
then  $\phi$ satisfies the going-down theorem \cite{M1, (5.D)}.
This implies that
for each  $P \in \Spec D$
the height of $P$ in $D$ is greater than or equal to the height of
$\phi^{-1}(P)$ in $C$.


\subheading{5.2. Approximations}
Let $R$ be a Noetherian integral domain with field of
fractions $K$, let $a\in R$ be a nonzero nonunit,
and let
$R^*$ denote  the $a$-adic
completion of $R$. 
Associated with the constructions  described in (4.1) and (4.2)
there are subrings of the intersection domain 
$A$ which approximate $A$. 

In (4.1), the elements $\tau_1, \dots \tau_s \in aR^*$ 
are algebraically independent over $\Cal Q(R)$. 
Hence $U_0 := R[\tau_1, \dots , \tau_s] \subseteq A$ is a 
polynomial ring in $s$ variables over $R$.
Each  $\tau_i \in aR^*$ has a representation
$\tau_i = \sum_{j=1}^\infty r_{ij}a^j$, where the  
$r_{ij} \in R$.  For each
positive integer $n$, we associate 
with this representation of $\tau_i$ 
the $n$-th {\it endpiece}, $\tau_{in} = \sum_{j=n+1}^\infty r_{ij}a^{j-n}$.
Then $U_n := R[\tau_{1n}, \dots, \tau_{sn}]$ is a polynomial ring in 
$s$ variables over $R$, and for each $n$ we have a birational 
inclusion of polynomial rings $U_n \subseteq U_{n+1}$. 
Since $a$ is in the Jacobson radical of $R^*$ \cite{M1, (24,B)}, 
the localization 
$B_n := (1 + aU_n)^{-1}U_n$ of $U_n$ is also a subring of $A$. 
We define
$U := \cup_{n=1}^\infty U_n$ and $B = \cup_{n=1}^\infty B_n$, 
and we say the construction (4.1) is {\it limit-intersecting } 
if $B = A$.

The limit-intersecting property depends on the choice of
the elements $\tau_1, \dots, \tau_s \in aR^*$. For example,
if $R$ is the localized polynomial ring $k[a,y]_{(a,y)}$, 
$s = 1$ and $U_0 = R[\tau_1]$, and if we define
$U_0' := R[y\tau_1,]$, then $\Cal Q(U_0) = 
\Cal Q(U_0')$, so the intersecton domain 
$A = \Cal Q(U_0)  \cap R^* = \Cal Q(U_0') \cap R^*$. 
However the approximation domain $B'$ associated to $U_0'$
is properly contained in the approximation domain $B$
associated to $U_0$. Therefore $B' \subsetneq B \subseteq A$
and the limit-intersecting property fails for the element 
$y\tau_1$.

 
For the construction (4.2), one no longer has
an approximation of $A$ by a nested union of 
polynomial rings over $R$. Indeed, in (4.2) the
extension $R \subseteq A$ is birational. 
However, there is an analogous approximation. 
We are given 
an ideal $I$ of $R^*$ with the property that
$P \cap R = (0)$ for each
$P \in\Ass(R^*/I)$.
Let $I:=(\sigma_1,\dots,\sigma_t)R^*$, where each
$\sigma_i:
=\sum^{\infty}_{j=0} a_{ij}a^j$, and the $a_{ij}\in R$.
We define $\sigma_{in}$, the $n^{\text{th}}$
{\it frontpiece} for $\sigma_{i}$,  to be
$$\sigma_{in}:=\sum_{j=0}^n (a_{ij}a^j)/a^n.$$

As an element of
the total quotient ring of $R^*/I$,  it is observed in \cite{HRW5}
that the frontpiece
$\sigma_{in}$  is the negative of
the $n^{\text{th}}$ {\it endpiece} of $\sigma_i$ as defined
in \cite{HRW2, (2.1)}; that is,
$$ -\sigma_{in}=\sum_{j=n+1}^\infty (a_{ij}a^j)/a^n=
\sum_{j=n+1}^\infty a_{ij}a^{j-n}\qquad\text{(mod }I).$$  It
follows that
$\sigma_{in}\in K \cap (R^*/I)$.

We define $$U_n:= R[ \sigma_{1n},\dots,
\sigma_{tn}],\qquad
\text{and } B_n:= (1+aU_n)^{-1}R[ \sigma_{1n},\dots, \sigma_{tn}]_{(
  \sigma_{1n},\dots, \sigma_{tn})},$$
 where these rings  are considered to be subrings of  $R^*/I$.

Now $\sigma_{in}= - aa_{i,n+1}+ a\sigma_{i,n+1}$, and so
 $R \subseteq U_0 \subseteq \cdots U_n\subseteq U_{n+1}$ and $B_n\subseteq B_{n+1}.$
Set
$$U:= \cup_{n=1}^\infty  U_n,\qquad
B:= \cup_{n=1}^\infty  B_n = (1+aU)^{-1}U,
\qquad {\text{and }}A:=K \cap (R^*/I). $$
Again  the fact that $a$ is in the Jacobson radical of $R^*$ 
implies that $B \subseteq A$. We say the construction (4.2) 
is {\it limit-intersecting } if $B = A$.


\subheading{Remark 5.3} The following results about the 
nested union approximation of an integral domain $A$  constructed 
as in (4.2) are given in \cite{HRW5} 
\roster
\item The definitions of $B$ and $U$ are  independent of
the choice of  generators  for $I$, and the
representation of the generators $\sigma_i$ of $I$ as power series in $a$.
\item  $a(R^*/I)\cap A=aA,\qquad a(R^*/I)\cap U=aU,\qquad a(R^*/I)\cap
B=aB.$
\item $U/a^nU = B/a^nB = A/a^nA = R^*/((a^n) + I)$.
All the rings  $A$, $U$,  $B$ have the same $a$-adic
completion, that is,
$A^*=U^*=B^*=R^*/I$.
\item $R_a=U_a$, $U = R_a \cap B = R_a \cap A$  and the fraction
fields of $R$, $U$, $B$ and $A$ are all equal to $K$.
\item The rings $U = R_a \cap (R^*/I)$ and $B = (1+aU)^{-1}U$ are
uniquely determined by $a$ and the ideal $I$ of $R^*$.
\item If $b \in B$ is a unit of
$A$, then $b$ is already a unit of $B$.
 \item We have the following diagram displaying the relationships among
the rings. Recall that $B = (1 + aU)^{-1}U$.
\endroster

$$
 \CD \Cal Q(R) @= \Cal Q(U)  @= \Cal Q(B)  @= \Cal Q(A) @>{\subseteq}>> \Cal
Q(R^*/I)\\
@AAA   @AAA   @AAA @AAA @AAA\\
R[1/a]  @= U[1/a]  @>{\subseteq}>> B[1/a] @>{\subseteq}>>
A[1/a] @>{\subseteq}>> (R^*/I)[1/a] \\
@AAA   @AAA   @AAA @AAA @AAA\\
R @>{\subseteq}>> U=\cup U_n @>{\subseteq}>> B @>{\subseteq}>>
A
@>{\subseteq}>> R^*/I.    \endCD  
$$


In connection with the flatness property, 
if  $U_0:=R[\tau_1,\dots, \tau_s]\hookrightarrow R^*[1/a]$
is flat,  then for each  $P \in \Spec R^*[1/a]$ one has that 
$\hgt P \ge \hgt(P \cap U_0)$. It is shown in 
\cite{HRW6, Theorem~2.2} that conversely this  height inequality in
certain contexts implies  flatness.



\subheading{5.4.  Universality}
Let $k$ be a field and let $L/k$ be a finitely generated
field extension.  A general question one may
ask with regard to $L/k$ is how to describe those
Noetherian local integral  domains $(A, \n)$
such that $k \subseteq A \subseteq L$, $A$ has fraction field $L$  and 
$k$ is a coefficient field for $A$, i.e., the canonical
map of $A \to A/\n$ maps $k$ isomorphically onto $A/\n$ ?
Given such a Noetherian local domain $(A,\n)$, it is easy to
find a Noetherian local domain $(R,\m)$ such that
\roster
\item $R$ contains $k$ and has fraction field $L$,
\item $R$ is contained in $A$  and $\m A = \n$,
\item $R$ is essentially finitely generated over $k$.
\endroster

There is then a relationship between $R$ and $A$ that
is realized by passing to completions;
the inclusion map $R \hookrightarrow A$
extends to a surjective homomorphism  $\widehat \phi: \widehat R \to
\widehat A$ of  the $\m$-adic completion
$\widehat R$ of $R$ onto the $\n$-adic completion $\widehat A$
of $A$ \cite{M2, Theorem~8.4, page~58}.
If  $I = \ker(\widehat \phi)$, then $L$
embeds in $\Cal Q(\widehat R/I)$, the total quotient ring of $\widehat R/I$
and $A = L \cap (\widehat R/I)$. The following commutative diagram,
where the vertical maps are injections,
displays the relationships among  these rings:

$$\CD
{} @.\widehat R @>{\hat\phi}>> \widehat R/I\cong \widehat A @>>> \Cal Q(
\widehat R/I)\\
@. @AAA @AAA @AAA \\
k @>{\subseteq}>> R @>>> A:=
 L\cap (\widehat R/I) @>>> \phantom{X}  L \phantom{X}.\endCD \tag{5.4.1}$$

\subheading{Summary 5.5}
\roster
\item 
Every Noetherian local domain $A$ whose  fraction field
$L$  is finitely generated over a coefficient field $k$ has the form 
$L \cap (\widehat R/I)$  for a domain $R$ which is essentially finitely
generated over $k$. That is, every such $A$ is realizable as
the intersection of $\Cal Q(R)$ with a homomorphic image of
$\widehat R$.

\item 
More generally,  if we drop the assumption that $R$ and $A$
have the same fraction field, then the above argument yields:
Every Noetherian local domain $A$ having a
coefficient field $k$ is realizable as  $L \cap (\widehat R/I)$,
where $R$ is a local domain essentially finitely
generated over $k$ and $I$ is an ideal of its
completion $\widehat R$  having   the property that
$P \cap R = (0)$ for each $P \in \Ass(\widehat R/I)$.
\endroster

\subheading{Remark 5.6}
A drawback with (5.5)  is that it is not true for 
each $R, L, I$ as in (5.5) that $L \cap (\widehat R/I)$ is
Noetherian (e.g, see part(4) of Remark~6.4  below).
To make the  classification
more satisfying an important goal is to  identify the
ideals $I$ of $\widehat R$ and fields $L$ 
such that $L \cap (\widehat R/I)$ is Noetherian.


In relation to Theorem~4.5.2, we display,  in the 
following diagram,  the inclusions that are always flat:
$$
 \CD \Cal Q(R) @= \Cal Q(U)  @= \Cal Q(B)  @= \Cal Q(A) @>{\subseteq}>> \Cal
Q(R^*/I)\\
@A{\text{flat}}AA   @A{\text{flat}}AA
@A{\text{flat}}AA @A{\text{flat}}AA @A{\text{flat}}AA\\
R[1/a]  @= U[1/a]  @>{\text{flat}}>>
B[1/a] @>{\subseteq}>> A[1/a] @>{\subseteq}>> (R^*/I)[1/a] \\
@A{\text{flat}}AA   @A{\text{flat}}AA
@A{\text{flat}}AA @A{\text{flat}}AA @A{\text{flat}}AA\\
R @>{\subseteq}>> U=\cup U_n @>{{\text{flat}}}>> B
@>{\subseteq}>> A
@>{\subseteq}>> R^*/I.    \endCD  $$
On the other hand, (i) if $B \subsetneq A$, then $B \to A$ is not flat, 
and (ii) the inclusion $A \subseteq R^*/I$ is flat if and only if $A$ 
is Noetherian.



%here is something I thought we may want to consider for another paper. I
%don't think it contains exciting news, but I feel it may be a good idea to
%compare all the methods we know of for constructing local Noetherian rings
%of finite transcendence degree over a field.
%If you prefer we can (also) include this in our expository paper.
%I don't know if I'm going to type my transparencies this time. If I do I'll
%send you a copy. I thought I would also include some of the stuff from below
%in my talk.  I hope everything is fine and that Mary Ann is doing ok.
%Best wishes, Christel


\subheading{ 6.  Explicit constructions}

Using the flatness results described in Section~4.5, we formulate
two methods for the construction of explicit examples. The first of these
methods uses the construction technique of (4.1) while the second
uses (4.2).

\subheading{Method 6.1} 
Let $k$ be a field, let $a,y_1,\hdots,y_n$ be variables over $k$, 
let $R$ be the localized polynomial ring 
$ R := k[a,y_1,\hdots,y_n]_{(a,y_1, \dots,y_n)}$ and let 
$\m$ denote the maximal ideal of $R$. 
Let $\tau_1, \dots, \tau_s \in ak[[a]]$ be formal power series
that are algebraically independent over $k(a)$ and let
$D_0 := R[\tau_1,\dots,\tau_s]_{(\m, \tau_1, \dots, \tau_s)}$ be
the associated localized polynomial ring over the field $k$ in $n+ s +1$
variables. Observe that   $D_0$ is contained in the $a$-adic 
completion $R^*$ of $R$. It is readily seen that 
the map $D_0 \to R^*[1/a]$ is flat and that 
$D := \Cal Q(D_0) \cap R^*$ is the localized 
polynomial ring $V[y_1, \dots, y_n]_{(a,y_1, \dots,y_n)}$,
where $V := k(a,\tau_1, \dots,\tau_s) \cap k[[a]]$ is a DVR. 
Thus the construction method of (4.1) gives a Noetherian
limit-intersecting domain $D$.

We now investigate the construction of examples inside $D$.
Let $f_1, \dots,f_r$ be elements of the maximal ideal of
$D_0$ that are algebraically independent over $\Cal Q(R)$, and
let $B_0 := R[f_1, \dots,f_r]_{(\m, f_1, \dots,f_r)}$ be the
associated localized polynomial ring. The
inclusion map $B_0 \hookrightarrow D_0$ is an injective local
$R$-algebra homomorphism. 

Let $A := \Cal Q(B_0) \cap R^*$ and let $B$ be the associated 
nested union domain. Then $B$ is Noetherian and $B=A$ if and only
if the map $B_0 \to R^*[1/a]$ is flat. This map factors as
$B_0 \to D_0 \to D_0[1/a] \to R^*[1/a]$ and the map $D_0 \to R^*[1/a]$ is flat.
Therefore $B$ is Noetherian (and so also $B = A$) 
if $B_0 \to D_0[1/a]$ is flat.

Since $B_0$ and $D_0$ are localized 
polynomial rings over a field, the nonflat locus of 
the inclusion map $B_0 \to D_0$ is closed \cite{M2, Theorem~24.3}
and is defined by the
ideal $J  := \cap\{P \in \Spec D_0 : B_0 \to (D_0)_P \text{ is not flat } \}$.  
Thus we have established the following theorem.


\proclaim{Theorem 6.2} With the notation above, we have
\roster
\item
If $JD_0[1/a] = D_0[1/a]$, then $B$ is Noetherian.
\item
$B$ is Noetherian if and only if $JD[1/a] = D[1/a]$ 
if and only if $JR^*[1/a] = R^*[1/a]$.
\endroster
\endproclaim

Using again the factorization of $B_0 \to R^*[1/a]$ through $D_0[1/a]$,
Theorem~5.5 of \cite{HRW3} implies the following result.

\proclaim{Theorem 6.3} With the notation above,  if 
$\hgt(JD_0[1/a])>1$, then $B = A$.
\endproclaim



\subheading{Remark 6.4} With the notation of this section, 
\roster
\item If $B$ is Noetherian, then $B$ is a regular local ring.

\item Example~3.1 of Nagata may 
be described by taking $n = s = r = 1, y_1 = y, \tau_1 = \tau$,
and $f_1 = f$. Then  $R = k[a, y]_{(a,y)}$,
$D_0 = k[a,y, \tau]_{(a, y, \tau)}$, $f = (y + \tau)^2$,
and $B_0 = k[a, y, f]_{(a, y, f)}$. The Noetherian property of
$B$ is implied by the flatness property of the map $B_0 \to D_0[1/a]$. 
In this case, $D_0$ is actually a free $B_0$-module with $<1, y+\tau>$ as
a free basis.


\item Example~3.3 of Rotthaus may be described by taking
$n = s = 2$, and $r =1$. Then 
$R = k[a, y_1, y_2]_{(a, y_1, y_2)}$, 
$D_0 = R[\tau_1, \tau_2]_{(\m, \tau_1, \tau_2)}$,
$f_1 = (y_1 + \tau_1)(y_2 + \tau_2)$ and $B_0 = R[f_1]_{(\m, f_1)}$.
Again the Noetherian property of $B$ is implied by 
the flatness property of the map $B_0 \to D_0[1/a]$.
\item The following example is given in \cite{HRW6, Section~4}. 
Let $n = s = r = 2$, let $f_1 = (y_1 + \tau_1)^2$ and 
$f_2 = (y_1 + \tau_1)(y_2 + \tau_2)$. It is shown in
\cite{HRW6} for this example that $B \subsetneq A$ and 
that both $A$ and $B$ are non-Noetherian.
\endroster



\subheading{Method 6.5}
Let $k$ be a field, let $a,y_1,\hdots,y_n$ be variables over $k$, 
let $R$ be the localized polynomial ring 
$ R := k[a,y_1,\hdots,y_n]_{(a,y_1, \dots,y_n)}$ and let 
$\m$ denote the maximal ideal of $R$. 
Let $\tau_1, \dots, \tau_s \in ak[[a]]$ be formal power series
that are algebraically independent over $k(a)$ and let
$D_0 := R[\tau_1,\dots,\tau_s]_{(\m, \tau_1, \dots, \tau_s)}$ be
the associated localized polynomial ring over the field $k$.

Assume that $I$ is an ideal of $D_0$ such that $P \cap R = (0)$ 
for each $P \in \Ass(D_0/I)$.
Let $A := \Cal Q(R) \cap (R^*/IR^*)$; 
then the  $a$-adic completion of $A$ is $R^*/IR^*$. 
Using the frontpiece approximations of a generating set for $I$,
it is shown in \cite{HRW5, Section~2} that there exists a
quasilocal integral domain $B = \cup_{n=1}^\infty B_n \subseteq A$ 
birationally dominating  $R$ 
such that the $a$-adic completion of $B$ is $R^*/IR^*$. 
By \cite{HRW5, Theorem~3.2}, 
$R \to (R^*/IR^*)[1/a]$ is flat if and only if $B$ is Noetherian.

Since the map $D_0 \to R^*[1/a]$ is flat, the map 
$D_0/I \to (R^*/IR^*)[1/a]$ is flat. Also the map
$R \to (R^*/IR^*)[1/a]$ factors as $R \to (D_0/I)[1/a] \to (R^*/I)[1/a]$.
Thus $R \to (R^*/IR^*)[1/a]$ is flat if $R \to (D_0/I)[1/a]$ is
flat. 
Since $R$ and $D_0/I := T$ are essentially finitely generated  
over a field, the nonflat locus of 
the inclusion map $R \to T$ is closed \cite{M2, Theorem~24.3} 
and is defined by the
ideal $J  := \cap\{P \in \Spec T : R \to T_P \text{ is not flat } \}$.  
This yields the following theorem.

\proclaim{Theorem 6.6} With the notation above,
if  $J(D_0/I)[1/a] = (D_0/I)[1/a]$, 
then  $B$ is Noetherian 
\endproclaim


\subheading{Remark 6.7} 
Let the  notation be as  in part(4) of Remark~6.4,
so $f_1 = (y_1 + \tau_1)^2$ and $f_2 = (y_1 + \tau_1)(y_2 + \tau_2)$,
and let $I := (f_1,f_2)R^*$.  
In \cite{HRW6, Section~4}
it is shown that $C := \Cal Q(R) \cap (R^*/I)$ is Noetherian
and limit-intersecting. Indeed, it is shown in 
\cite{HRW6, Proposition~4.5} that $C$ is a two-dimensional
Noetherian local domain for which the generic formal fiber
is not Cohen-Macaulay.








\Refs
\widestnumber\key{HRW2}

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

\ref \key{BR1} \manyby M. Brodmann and C. Rotthaus
\paper Local rings with bad sets of formal prime divisors
\jour J. Algebra \vol 75 \yr 1982 \pages 386-94
\endref

\ref \key{BR2} \bysame 
\paper A peculiar unmixed domain
\jour Proc. Amer. Math. Soc.  \vol 87 \yr 1983 \pages 596-600
\endref

\ref\key{FR} \by D. Ferrand and M. Raynaud
\paper Fibres formelles d'un anneau local Noetherian
\jour Ann. Sci. Ecole Norm. Sup.  \vol 3 \yr 1970 \pages 295-311
\endref

\ref\key{Gr}\by A.Grothendieck
\paper \'Elements de G\'eometrie algebrique $IV_4$
\jour Publ. Math. Inst. Hautes Etud. Sci.\vol 32\yr 1967
\endref

\ref\key{HRW1} \manyby W. Heinzer, C. Rotthaus and S. Wiegand
\paper Idealwise algebraic independence for elements of the
completion of a local domain \jour Illinois J. Math.  
\vol 41 \yr 1997 \pages 272-308
\endref


\ref\key{HRW2} \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{HRW3} \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{HRW4} \bysame  
\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 Dekker Inc \ed Dikranjan
and Salce
\endref

\ref\key{HRW5} \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{HRW6} \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{H1} \by R. Heitmann
\paper Examples of noncatenary rings
\jour Trans. Amer. Mat. Soc. \vol 247 \yr 1979 \pages 125-136
\endref


\ref\key{M1}\manyby H. Matsumura
\book Commutative Algebra 2nd Edition
\publ Benjamin/Cummings\publaddr Reading\yr 1980
\endref

\ref\key{M2}\bysame
\book Commutative Ring Theory
\publ Cambridge University Press\publaddr Cambridge\yr 1986
\endref

\ref\key{Mu}\by D. Mumford
\book Introduction to Algebraic Geometry
\publ Preliminary version of first 3 Chapters \yr 1968
\endref




\ref\key{N1} \manyby M. Nagata
\paper An example of a normal local ring which is analytically reducible
\jour Mem. Coll. Sci., Univ. Kyoto \vol 31 \yr 1958 \pages 83--85
\endref

\ref\key{N2}\bysame
\book Local Rings
\publ John Wiley\yr 1962
\endref

\ref\key{O1} \manyby T. Ogoma
\paper Non-catenary pseudo-geometric normal rings
\jour Japan J. Math. \vol 6 \yr 1980 \pages 147-163
\endref

\ref\key{O2}\bysame
\paper Cohen-Macaulay factorial domain is not necessarily
Gorenstein
\jour Mem. Fac. Sci. Kochi Univ. \vol 3 \yr 1982 \pages 65-74
\endref


\ref\key{R1} \manyby C. Rotthaus
\paper Nicht ausgezeichnete, universell japanische Ringe
\jour Math. Z. \vol 152 \yr 1977\pages 107-125
\endref

\ref\key{R2} \bysame
\paper Universell japanische Ringe mit nicht offenen
regul\"arem Ort
\jour Nagoya Math. J. \vol 74 \yr 1979 \pages 123-135
\endref

\ref\key{R3} \bysame
\paper Komplettierung semilokaler quasiausgezeichneter Ringe
\jour Nagoya Math. J. \vol 76 \yr 1980 \pages 173-180
\endref

\ref\key{R4} \bysame
\paper Excellent rings, Henselian rings, and the approximation property
\jour Rocky Mountain J. Math. \vol 27 \yr 1997 \pages 317-334
\endref

\ref\key{S} \by J.-P. Serre
\paper G\'eom\'etrie alg\'ebrique et g\'eom\'etrie analytique
\jour Ann. inst. Fourier  \vol 6 \yr 1955-56 \pages 1-42
\endref

\ref\key{V} \by P. Valabrega
\paper On two-dimensional regular local rings and a 
lifting problem
\jour Ann. Scuola Norm. Sup. Pisa  \vol 27 \yr 1973 \pages 1-21
\endref


\ref\key{W} \by D. Weston
\paper On descent in dimension two and non-split Gorenstein
modules
\jour J. Algebra \vol 118 \yr 1988 \pages 263-275
\endref

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




\endRefs



\enddocument

\subheading{Question} In 
analogy with Theorem~6.3, do we have in the homomorphic image 
context that with notation as above, then
$B = A$ if and only if $\hgt(J(D_0/I)[1/a] > 1$?
I'm not aware of
a result in the homomorphic image context similar to 
Proposition~5.2 of \cite{HRW3}.

