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%\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.
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\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}.