%From ratliff@math.ucr.edu Fri Nov 11 10:19:39 1994
\magnification=\magstephalf
\input amstex
\documentstyle{amsppt}
\hoffset .25 true in
\NoBlackBoxes
\def\CE{corner-element\ }
\def\CES{corner-elements\ }
\def\R{\text{Rad}}
\topmatter
\title PARAMETRIC DECOMPOSITION OF MONOMIAL IDEALS (I)
\endtitle
\author
William Heinzer, L. J. Ratliff, Jr. and Kishor Shah
\endauthor
\thanks{The first author's research on this paper was supported in part by the National Science Foundation, Grant DMS-9101176.}
\endthanks
\address
\break
{Department of Mathematics, Purdue University, W. Lafayette, IN 47907}\break
{Department of Mathematics, University of California, Riverside, CA 92521}\break
{Department of Mathematics, Southwest Missouri State University, Springfield, MO 65804}
\endaddress
\subjclass AMS (MOS) Subject Classification Numbers: Primary: 13A17, 13C99.
Secondary: 13B99, 13H99\endsubjclass
\abstract
Emmy Noether showed that every ideal in a Noetherian ring admits a decomposition
into irreducible ideals. In this paper we explicitly calculate this
decomposition in a fundamental case. Specifically, let $R$ be a commutative
ring with identity, let $x_1,\dots,x_d$ $(d>1)$ be an $R$ -sequence,
let $X=(x_1,\dots,x_d)R$, and let $I$ be a monomial ideal
(that is, a proper ideal
generated by monomials $x_1^{e_1}\cdots x_d^{e_d})$ such
that $Rad(I)$ $=$ $Rad(X)$. Then the main result gives a canonical and
unique decomposition of $I$ as an irredundant finite intersection of ideals of
the form $(x_1^{n_1},\dots,x_d^{n_d})R$, where the
exponents $n_1,\dots,n_d$ are
positive integers. Specifically, if $z_1,\dots,z_m$ are the monomials
in $(I:X)-I$, and if $z_j=x_1^{a_{j,1}-1}\cdots x_d^{a_{j,d}-1}$,
then $I$ $=$ $\cap\{(x_1^{a_{j,1}},\dots,x_d^{a_{j,d}})R$; $j$ $=$
$1,\dots,m\}$. We also calculate the decomposition of
the ideals $I^{[k]}$ generated by the $k$ -th powers of the monomial
generators of $I$.
The methods we use are algebraic, but they were suggested by the geometry
of lattices.
\endabstract
\endtopmatter
\baselineskip 18pt
\noindent
{\smc{\bf 1. Introduction}}. {\it Throughout this paper}, $R$ is a
commutative ring with identity $1\not= 0$, $x_1,\dots, x_d$ $(d>1)$ is
an $R$ -sequence, $X=(x_1,\dots,x_d)R$, and $I$ is a monomial ideal (that
is, a proper ideal generated by monomials $x_1^{e_1}\cdots x_d^{e_d})$ such
that $Rad(I)$ $=$ $Rad(X)$.
It is known (for example, see [HRS2, (3.15)]) that in a regular local
ring $R$ of altitude two, irreducible ideals are parameter ideals.
Therefore in altitude two regular local rings, Emmy Noether's fundamental
decomposition theorem
[N, Satz IV] shows that each open ideal in $R$ is a finite intersection of
parameter ideals (but of course the $x$'s may vary). One consequence of
our main result, (4.1), is that a similar statement holds for open monomial
ideals in a Cohen-Macaulay local ring.
Monomial ideals are important in several areas of current research, and they
have been studied in their own right in several papers (for example, \cite{EH}
and \cite{T}), so many useful results are known about such ideals. In the
present paper we are interested in giving an explicit decomposition of $I$ as
an irredundant finite intersection of parameter ideals. We do this in
Section 2 for the special case when $I=X^n$ ($n$ a positive integer), and it
is shown that $X^n$ is the irredundant intersection of
the $\left(\smallmatrix n+d-2\\ d-1\endsmallmatrix\right)$ parameter
ideals $(x_1^{a_1},\dots, x_d^{a_d})R$, where $a_1,\dots a_d$ are
positive integers that sum to $n+d-1$.
To extend this result to an arbitrary monomial ideal $I$ (such
that $Rad(I)$ $=$ $Rad(X)$), in Section 3 we introduce and study the $J$
-corner-elements of a
monomial ideal $J$. We show that they are the monomials in $(J:X)-J$, that
there are only finitely many of them, and that if $(x_1,\dots,x_{d-1})R$
$\subseteq$ $Rad(J)$, then their $J$ -residue classes are a minimal
basis, in any order, of $(J:X)/J$. Also, if $Q$ is an open monomial
ideal in a regular local ring $(R,M)$ of altitude two,
then $v((Q:X)/Q)$ $=$ $v(Q)-1$ (where $v(J)$ denotes the
number of elements in a minimal basis of the ideal $J$), and if $t$ is an
integer such that $v(Q)-1$ $\le$ $t$ $\le$ $2v(Q)-1$, then $Q$ can be
chosen such that $v(Q:X)=t$. Finally, we give a geometric interpretation
of $I$ -corner-elements, an algebraic construction of them, and then close this
section with several examples of such elements.
In Section 4 we show that if $z_1,\dots z_m$ are the $I$ -corner-elements,
then $I$ is the irredundant intersection of the $m$ parameter
ideals $P(z_j)$ $=$ $(x_1^{a_{j,1}},\dots x_d^{a_{j,d}})R$,
where $z_j$ $=$ $x_1^{a_{j,1}-1}\cdots x_d^{a_{j,d}-1}$.
Three interesting corollaries are: $\cup\{Ass(R/I^n)$;
$n\ge$ $1\}$ $\subseteq$ $Ass(R/X)$; and, if $R$ is
a Gorenstein local ring with maximal ideal $M$, if $X$ is generated by
a system of parameters, and if $I$ is open, then $v((I:M)/I)$\break
$=$ $v((I:X)/I)$, and $I$ is
irreducible if and only if there exists exactly one $I$\break
-corner-element, and
then $I$ is generated by a system of parameters. Also, unique factorization
holds in the sense that if $I$ $=$ $\cap\{P(z_j)$; $j$
$=$ $1,\dots,m\}$ $=$ $\cap\{P(w_i)$; $i$ $=$ $1,\dots,n\}$, then $n$
$=$ $m$ and $\{z_1,\dots z_m\}$ $=$ $\{w_1,\dots,w_n\}$. Further,
if $k$ is a positive integer and $I^{[k]}$ is the ideal generated by
the $k$-th powers of the monomial generators of $I$, then $I^{[k]}$
$=$ $\cap\{(P(z_j))^{[k]}$; $j$ $=$ $1,\dots,m\}$ and
the $I^{[k]}$ -corner-elements are
the $m$ monomials $z_j^{(k)}$ $=$
$x_1^{ka_{j,1}+k-1}\cdots x_d^{ka_{j,d}+k-1}$.
In Section 5 a related decomposition of $I$ as an irredundant finite
intersection of irreducible ideals is proved. Specifically, with the notation
of the preceding paragraph, if $R$ is local with maximal ideal $M$ and
if $Q_j$ is maximal in $S_j$ $=$ $\{Q$; $Q$ is an ideal in $R$,
$P(z_j)$ $\subseteq$ $Q$, and $z_j$ $\notin$ $Q\}$ for $j$
$=$ $1,\dots,m$, then each $Q_j$ is
irreducible, $\cap\{Q_j$; $j$ $=$ $1,\dots,m\}$ is an irredundant
intersection, and $(\cap\{Q_j$; $j$ $=$ $1,\dots,m\})$ $\cap$
$(I:X)$ $=$ $I$ $+$ $M(I:X)$. It then follows that if $R$ is a regular
local ring and $X$ $=$ $M$, then $Q_j=P(z_j)$ for $j$ $=$ $1,\dots m$.
Finally, in Section 6 we show that: if $I$ is irreducible, then $I$ is a
parameter ideal; $I$ is a parameter ideal if and only if $I$ has exactly one
corner-element; and, if $R$ is a Gorenstein local ring of altitude $d$,
then $I$ is a parameter ideal if and only if $I$ is irreducible. Also, the
parameter ideals that are minimal with respect to containing $I$ are
the ideals $P(z)$, where $z$ is an $I$ -corner-element.
The authors have been fascinated by the historic and fundamental decomposition
theorems of Emmy Noether, and this fascination gave rise to the results in
[HRS1, HRS2, HRS3, HRS4] and the present paper. We are pursuing further topics
in this area (in particular, in [HMRS]), and we hope this theory turns out to be
fascinating and useful to others.
\bigskip
\noindent
{\smc{\bf 2. Parametric Decompositions of Powers of an $R$ -Sequence.}}
The main result in this section, (2.4), shows that if $X$ is an ideal
generated by an $R$ -sequence, then $X^n$ is the irredundant intersection
of $\left(\smallmatrix n+d-2\\ d-1\endsmallmatrix\right)$ parameter ideals.
To prove this, we need a few preliminary results, so we begin with these.
\medskip
\noindent
{\bf{(2.1) Definition.}}
Let $R$ be a ring, let $x_1,\dots,x_d$ $(d>1)$ be an $R$ -sequence,
and let $X=(x_1,\dots,x_d)R$. Then:
{\pmb{(2.1.1)}} A {\bf monomial} (in $x_1,\dots x_d$) is a power
product $x_1^{e_1}\cdots x_d^{e_d}$, where $e_1,\dots,e_d$ are nonnegative
integers (so a monomial is either a nonunit or the element 1), and
a {\bf monomial ideal} is a proper ideal generated by monomials.
{\pmb{(2.1.2)}} A {\bf parameter ideal} (in $x_1,\dots,x_d$) is an ideal of the
form $(x_1^{a_1},\dots, x_d^{a_d})R$, where $a_1,\dots, a_d$ are positive
integers (so the parameter ideal $(x_1^{a_1},\dots, x_d^{a_d})R$ is
a monomial ideal generated by the $R$ -sequence $x_1^{a_1},\dots, x_d^{a_d})$.
If $f=x_1^{e_1}\cdots x_d^{e_d}$ is a monomial, then we
let $\bold{P(f)}$ denote the parameter
ideal $(x_1^{e_1+1},\dots,x_d^{e_d+1})R$. (Note that
if $f=1$, then $P(f)=X$.) And if $a_1,\dots a_d$ are positive integers,
then we let ${\bold {P(a_1,\dots,a_d)}}$ denote the parameter
ideal $(x_1^{a_1},\dots, x_d^{a_d})R$ (so $P(a_1,\dots,a_d)$ $=$ $P(f)$,
where $f=x_1^{a_1-1}\cdots x_d^{a_d-1})$.
\medskip
\noindent
{\bf{(2.2) Remark.}}
Let $f$ and $g$ be monomials. Then:
{\pmb{(2.2.1)}} If $f_1,\dots,f_n$ are monomials
then $f\in (f_1,\dots,f_n)R$ if and only if $f\in f_iR$ for some
$i$ $=$ $1,\dots,n$.
{\pmb{(2.2.2)}} If $f$ $\in$ $gR$, then there exists a
monomial $k$ (possibly $k=1$) such that $f$ $=$ $gk$.
{\pmb{(2.2.3)}} If $h$ is a monomial such that $fh=gh$, then $f$ $=$ $g$.
{\pmb{(2.2.4)}} If $fx_j$ $=$ $gx_i$ for some $i$ $\not=$
$j$ in $\{1,\dots,d\}$, then $f$ $\in$ $x_iR$ and $g$ $\in$ $x_jR$.
\demo{Proof} It is shown in [T, Theorem 1] that if $r$ $\in$ $R$ and $rf$
$\in$ $(f_1,\dots,f_n)R$, then either $f$ $\in$ $f_iR$ for some $i$
$=$ $1,\dots,n$ or $r$ $\in$ $(x_1,\dots,x_d)R$. (2.2.1) readily follows
from this.
(2.2.2)--(2.2.4) readily follow by the ``independence'' of power products in
an $R$ -sequence (that is, $x_1^{e_1}\cdots x_d^{e_d}$
$=$ $x_1^{a_1}\cdots x_d^{a_d}$ if and only if $a_i$ $=$ $e_i$ for $i$
$=$ $1,\dots,d$), \qed
\enddemo
\medskip
\proclaim{(2.3) Lemma}
Let $f$ and $g$ be monomials. Then $g\in P(f)$ (see (2.1.2)) if and only if
$f\notin gR$.
\endproclaim
\demo{Proof}
Let $f=x_1^{e_1}\cdots x_d^{e_d}$. Then $f\notin x_i^{e_i+1}R$ for $i=1,\dots,d$, since $e_i < e_i+1$ for $i=1,\dots, d$, so (2.2.1) shows that
$f\notin P(f)$. Therefore if $g\in P(f)$, then $f\notin gR$.
For the converse assume that $g\notin P(f)$ and let $g=x_1^{a_1}\cdots x_d^{a_d}$. Then $a_i< e_i+1$ for $i=1,\dots, d$, so $e_i\ge a_i$ for
$i=1,\dots,d$, hence $f\in gR$, \qed
\enddemo
\medskip
(2.4), the main result in this section, extends [HRS2, (3.5)] (where it is
shown that in a regular local ring $(R,M=(x,y)R)$,
$M^n$ $=$ $\cap \{(x^{n+1-i},y^i)R$; $i=1,\dots,n\}$).
Concerning the ideals $P(a_1,\dots,a_d)$ in (2.4), see (2.1.2).
\medskip
\proclaim{(2.4) Theorem}
Let $X$ be an ideal that is generated by an $R$ -sequence $x_1,\dots,x_d$
$(d>1)$ and let $n$ be a positive integer. Then $X^n=\cap\{P(a_1,\dots,a_d);
a_1+\cdots+ a_d= n+d-1\}$ and this intersection is irredundant. Therefore
$X^n$ is the irredundant intersection of $\left(\smallmatrix n+d-2\\ d-1\endsmallmatrix\right)$ parameter ideals.
\endproclaim
\demo{Proof}
If $n=1$, then this is clear, so it will be assumed that $n>1$.
Let $J=\cap\{P(a_1,\dots,a_d); a_1+\cdots + a_d=n+d-1\}$. Then since $X^n$
is generated by the monomials $x_1^{e_1}\cdots x_d^{e_d}$, where $e_1,\dots,
e_d$ are nonnegative integers that sum to $n$, to show that $X^n\subseteq J$
it suffices to show that each such monomial is in $J$. For this, fix
$f=x_1^{e_1}\cdots x_d^{e_d}$ and consider any of the ideals $P(a_1,\dots,a_d)$.
Then $e_i\ge a_i$ for some $i=1,\dots,d$ (since otherwise $n=e_1+\cdots +e_d a_i$ for some $i=1,\dots,d$, so
$x_1^{b_1 -1}\cdots x_d^{b_d -1}\in P(a_1,\dots,a_d)$, hence it follows that
$x_1^{b_1 -1}\cdots x_d^{b_d -1}\in \cap\{P(a_1,\dots,a_d)$; $a_1+\cdots + a_d$
$=$ $n+d-1$
and $a_i\not= b_i$ for some $i\}$, and $x_1^{b_1 -1}\cdots x_d^{b_d -1}\notin
P(b_1,\dots,b_d)$, by (2.3), so $x_1^{b_1 -1}\cdots x_d^{b_d -1}\notin \cap
\{P(a_1,\dots,a_d); a_1+\cdots + a_d=n+d-1\}$.
For the final statement, each ideal $P(a_1,\dots,a_d)$ is a parameter ideal,
by (2.1.2). And the preceding paragraph shows that they are distinct for
distinct $d$-tuples $(a_1,\dots,a_d)$ of positive integers. To compute
the number of these ideals, since we are only interested in the number of ideals, it may be assumed that $X=(x_1,\dots,x_d)R$ is the maximal ideal $M$
in a regular local ring $(R,M)$. Then by [HRS2, (2.3.2) and (2.4)] the
number of ideals is $d(X^n)=\dim_{R/M}((X^n:X)/X^n)=\dim_{R/M}(X^{n-1}/X^n)=
v(X^{n-1})=\left(\smallmatrix n+d-2\\ d-1\endsmallmatrix\right)$, \qed
\enddemo
\medskip
\proclaim{(2.5) Corollary}
If $R$ is a Gorenstein local ring and altitude $(R)=d$, then $X^n=\cap\ \{P
(a_1,\dots, a_d)$; $a_1+\cdots + a_d = n+d-1\}$ is an irredundant intersection of
$\left(\smallmatrix n+d-2\\ d-1\endsmallmatrix\right)$ irreducible ideals.
\endproclaim
\demo{Proof}
If $R$ is Gorenstein, then each open parameter ideal is irreducible, so the
conclusion follows from (2.4), \qed
\enddemo
\medskip
\noindent
{\bf{(2.6) Remark.}}
It follows from (2.4) that the cardinality of $\{x_1^{e_1}\cdots x_d^{e_d}$;
$e_1,\dots, e_d$ are positive integers that sum to $n+d-1\}$ is
$\left(\smallmatrix n+d-2\\ d-1\endsmallmatrix\right)$.
\bigskip
\noindent
{\smc{\bf 3. J-Corner-Elements}}. We now want to extend (2.4) to an arbitrary
monomial ideal $I$ such that $Rad(I)$ $=$ $Rad(X)$. (It should be noted that
$Rad(I)$ $=$ $Rad(X)$ is a necessary condition to extend (2.4), since the radical of
each parameter ideal is the radical of $X$, and in (4.1) we show that this
condition is also sufficient.) To accomplish this extension, we have found it
useful to use ``corner-elements''. So in this section we introduce such elements and derive some of their basic properties, and then use some of these
properties in the proof of (4.1) to give the desired extension of (2.4).
We think ``corner-elements will be of interest and use in other problems, so
in this section we prove several results
concerning them. Specifically, we show in (3.2) and (3.7) that if $J$ is a
monomial ideal, then there exist only finitely many $J$ -corner-elements,
that they are the monomials in $(J:X)-J$, and that if $(x_1,\dots, x_{d-1})R$
$\subseteq$ $Rad(J)$, then the $J$ -residue classes of these corner-elements are a
minimal basis, in any order, of $(J:X)/J$. We then apply these results to the case
when $J$ is an open monomial ideal in a regular local ring $(R,M)$ of
altitude two, give a geometric interpretation of $I$ -corner-elements and an
algebraic construction of them, and then close this section with several examples of such elements.
We begin with the definition.
\medskip
\noindent
{\bf{(3.1) Definition.}}
Let $J$ be a monomial ideal. Then a $J$ -{\bf corner-element} is a monomial
$z$ such that $z\notin J$ and $zx_i\in J$ for $i=1,\dots,d$.
\medskip
(The name ``corner-element'' is suggested by the geometric interpretation
in (3.13), where a corner-element is an element $z=x^ay^b$ with coordinates
$(a,b)$ such that $(a,b+1),(a+1,b)$, and $(a+1,b+1)$ are the coordinates of
points in $I$ and $z\notin I$.)
Concerning (3.1), note that $1$ is the unique $X$ -corner-element (since each
nonunit monomial is in $X$). Also, if $J$ is a monomial ideal and $1$ is a
$J$ -corner-element, then $1x_i\in J$ for $i=1,\dots, d$, so $J=X$.
In (3.2) we characterize the $J$ -corner-elements and show that there are only
finitely many of them. (It follows from (3.2) that $J$ uniquely determines its
corner-elements. The converse of this is proved in (4.2) when $Rad(J)$
$=$ $Rad(X)$.)
\medskip
\proclaim{(3.2) Proposition}
If $J$ is a monomial ideal, then the $J$ -corner-elements are the monomials in
$(J:X)-J$. Also, if $z,z'$ are distinct $J$ -corner-elements, then
$zR\not\subseteq z'R$ and $z'R\not\subseteq zR$, so there exist only finitely
many $J$ -corner-elements.
\endproclaim
\demo{Proof}
Let $\bold C$ be the set of $J$ -corner-elements (so each element in $\bold C$ is a
monomial). Then it is clear from (3.1) that $\bold C\subseteq (J:X)-J$. And if
$z$ is a monomial in $(J:X)-J$, then $z\notin J$ and $zx_i\in J$
for $i=1,\dots,d$, so $z$ is a $J$ -corner-element, hence
$z\in \bold C$. Therefore $\bold C$ is the set of monomials in $(J:X)-J$.
Now let $z$ and $z'$ be distinct $J$ -corner-elements and suppose that
$zR\subseteq z'R$. Then (2.2.2) shows that $z=z'f$ for some monomial $f$
(and $f\not= 1$, since $z\not=z'$). But this implies that $z=z'f\in J$
(since $z'$ is a $J$ -corner-element), and this is a contradiction. Therefore
$zR\not\subseteq z'R$ and, similarly, $z'R\not\subseteq zR$.
Finally, the ideal generated by the $J$ -corner-elements (viewed as elements
in $Z_k[x_1,\dots,x_d]$, where $k$ is the characteristic of $R$ and where
$Z_k$ is the ring generated by the identity of $R$) is finitely generated, so
since there are no inclusion relations among the ideals they generate, (2.2.1)
shows that there are only finitely many of them, \qed
\enddemo
\medskip
\proclaim{(3.3) Corollary}
Let $J$ be a monomial ideal and let $z_1,\dots,z_m$ be the $J$ -corner-elements.
Then for $j=1,\dots,m$ it holds that $(z_1,\dots,z_{j-1},z_{j+1},\dots,z_m)R
\subseteq P(z_j)$ and $z_j\notin P(z_j)$. Therefore $\cap\{P(z_j); j=1,\dots,
m\}$ is an irredundant intersection of parameter ideals.
\endproclaim
\demo{Proof}
(It follows from (3.2) that there are only finitely many $J$ -corner-elements.
Also, if $m=1$ and $z_1=1$, then $P(z_1)=X$, $(0)$ (the ideal generated by
the empty set) is contained in $X$, and $1\notin P(1)=X$, so the conclusion
holds in this case.)
Fix $j$ $\in$ $\{1,\dots,m\}$. Then it follows from (3.2) that
if $i$ $\in$ $\{1,\dots,j-1,\break
j+1,\dots,m\}$, then $z_j$ $\notin$ $z_iR$, so (2.3) shows
that $z_i$ $\in$ $P(z_j)$ (hence $(z_1,\dots,z_{j-1},\break
z_{j+1},\dots,z_m)R$ $\subseteq$ $P(z_j)$) and $z_j$
$\notin$ $P(z_j)$. This shows that $\cap\{P(z_j);j=1,\dots,m\}$ is an
irredundant intersection, and (2.1.2) shows that the ideals $P(z_j)$ are parameter ideals, \qed
\enddemo
\medskip
In (3.4) we specify the $X^n$ -corner-elements.
\medskip
\proclaim{(3.4) Corollary} If $n>1$ is a positive integer, then the $X^n$ -corner-elements are the $\left(\smallmatrix n+d-2\\ d-1\endsmallmatrix
\right)$ generators $x_1^{e_1}\cdots x_d^{e_d}$ of $X^{n-1}$ (so
$e_1,\dots,e_d$ are nonnegative integers such that $e_1+\cdots+ e_d=n-1$).
\endproclaim
\demo{Proof}
By (3.2) the $X^n$ -corner-elements are the monomials in $X^{n-1}-X^n$ (since
$X^n:X=X^{n-1}$), and since $X$ is generated by an $R$ -sequence of length $d$
it follows that there are $\left(\smallmatrix n+d-2\\ d-1\endsmallmatrix\right)$ distinct such elements, \qed
\enddemo
\medskip
It follows from (3.5) that if $z$ is a $J$ -corner-element, then
the $d$ elements $zx_1,\dots, zx_d$ are members of distinct
principal ideals generated by
monomials in $J$.
\medskip
\proclaim{(3.5) Proposition}
Let $f$ and $g$ be monomials and let $i\not= j\in\{1,\dots,d\}$. If
$fx_i\in gR$ and $fx_j\in gR$, then $f\in gR$.
\endproclaim
\demo{Proof}
If $fx_i\in gR$ and $fx_j\in gR$, then (2.2.2) shows that there exist
monomials $h_i$, $h_j$ such that $fx_i=gh_i$ and $fx_j=gh_j$. Then
$fx_ix_j=gh_ix_j=gh_jx_i$, so $h_ix_j=h_jx_i$ by (2.2.3). (2.2.4) then shows
that $h_i\in x_iR$, so $h_i=kx_i$ for some monomial $k$ by (2.2.2). Therefore
$fx_ix_j=gh_ix_j=g(kx_i)x_j$, so $f=gk\in gR$ by (2.2.3), \qed
\enddemo
\medskip
\proclaim{(3.6) Corollary}
If $J$ is a monomial ideal that has a corner-element, and if $f_1,\dots,f_n$
are monomials that generate $J$, then $n\ge d$.
\endproclaim
\demo{Proof}
This follows immediately from (3.5), \qed
\enddemo
\medskip
In (3.7) it is shown that if $(x_1,\dots,x_{d-1})R$ $\subseteq$ $Rad(J)$,
then the
$J$ -residue classes of the $J$ -corner-elements are a minimal basis, in any
order, of $(J:X)/J$. (In this regard, note that if $J:X=J$, then there are no
$J$ -corner-elements, and the empty set does generate the ideal $(J:X)/J=J/J$.
On the other hand, if $Rad(J)$ $=$ $Rad(X)$ then $J:X\not=J$.)
\medskip
\proclaim{(3.7) Theorem}
Let $J\not=X$ be a monomial ideal such that $(x_1,\dots,x_{d-1})R$
$\subseteq$ $Rad(J)$. Then the $J$ -residue classes of the
$J$ -corner-elements are a minimal basis, in any order, of $(J:X)/J$.
\endproclaim
\demo{Proof}
(By ``minimal basis'', we mean a basis such that no proper subset is a
generating set of the ideal.) Since $Rad((x_1,\dots,x_{d-1})R)$
$\subseteq$ $Rad(J)$,
[T, Theorem 6] shows that $J:X$ is a monomial ideal, so it follows from (3.2)
that the $J$ -residue classes of the $J$ -corner-elements generate $(J:X)/J$.
Let $\bold C=\{z_1,\dots, z_m\}$ be the set of $J$ -corner-elements ($\bold C$ is a finite
set by (3.2)) and suppose that there exists a permutation $\pi 1,\dots,\pi m$
of $1,\dots,m$ such that $z_{\pi1}\in (z_{\pi2},\dots,z_{\pi m})R$. Then
$z_{\pi 1}\in z_{\pi k}R$ for some $k=2,\dots,m$ by (2.2.1), so (2.2.2) shows
that $z_{\pi 1}=z_{\pi k}f$ for some monomial $f$ ($f\not= 1$, since
$z_1,\dots,z_m$ are distinct). However, this implies that
$z_{\pi 1}=z_{\pi k}f\in J$ (since $z_{\pi k}$ is a $J$\break
-corner-element), and
this is a contradiction. Therefore $z_{\pi 1}\notin (z_{\pi2},\dots,z_{\pi m})R$
for all permutations $\pi 1,\dots,\pi m$ of $1,\dots,m$.
And no $z_j$ in in $J$, so it follows from (2.2.1) that
the $J$ -residue classes of $z_1,\dots,z_m$ are a minimal basis,
in any order, of $(J:X)/J$, \qed
\enddemo
\medskip
In (3.8) we consider the case when $J=Q$ is open in a regular local ring $R$ of
altitude two. ((3.8) was noted in [HRS2, (3.3)] for the case $M=(x,y)R$,
and therein a homological proof using [HS, (2.1)] was sketched for an arbitrary
open ideal (in an altitude two regular local ring). (3.8) gives a non-homological proof for an arbitrary $R$ -sequence of length two, but only for
the case of an open monomial ideal.)
\medskip
\proclaim{(3.8) Corollary}
Let $(R,M)$ be a regular local ring of altitude two, let $x,y$ be an
$R$ -sequence, and let $Q\not=(x,y)R$ be an open monomial ideal in $x$ and $y$,
say $v(Q)=n$. Then $v((Q:X)/Q)=n-1$.
\endproclaim
\demo{Proof}
Let $Q=(f_1,\dots,f_n)R$ and lexicographically order the $f_i$ by saying that
$f_i0$, then $f_i\in z_{i-1}R$ for $i=s+1,\dots,n$ and $f_i\notin (z_1,\dots,
z_{n-1})R$ for $i=1,\dots,s$, so by (2.2.1) (and (3.8)) it readily follows that
$f_1,\dots,f_s, z_1,\dots z_{n-1}$ is a minimal basis of $Q:X$ so
$v(Q:X)=s+n-1=t$, \qed
\enddemo
\medskip
\proclaim{(3.12) Corollary}
Let $(R,M=(x,y)R)$ be a regular local ring of altitude two, let $Q\not= M$
be an open monomial ideal in $x$ and $y$, and let $n=v(Q)$. Then
$v((Q:M)/Q)=n-1$, $n-1\le v(Q:M)\le 2n-1$, and for each intermediate integer
$t$ there exists an ideal $Q$ such that $v(Q)=n$ and $v(Q:M)=t$.
\endproclaim
\demo{Proof}
This follows immediately from (3.8) and (3.11), since $M=X$, \qed
\enddemo
\medskip
In (3.13) we give a geometric interpretation of the $I$ -\CES for a monomial
ideal $I$ in an $R$ -sequence $x,y$ of length two such that $Rad(I)$
$=$ $Rad((x,y)R$).
\medskip
\noindent
{\smc{\bf (3.13) Geometric Interpretation}}.
Assume that $d=2$, let $x=x_1$ and $y=x_2$, let $f_1,\dots,f_n$ be a minimal
basis of $I$ (where the $f_l$ are monomials in $x$ and $y$, say $f_l=x^{i_l}
y^{j_l}$), and assume that $Rad(I)$ $=$ $Rad((x,y)R)$. Lexicographically order the
$f_l$ (as in the proof of (3.8)) and assume that $f_1<\cdots < f_n$. Plot the
$n$ points $(i_l,j_l)$ (corresponding to the $f_l$) in the first quadrant of
the $xy$-plane. Then for each of these $n$ points draw the horizontal line
segment connecting $(i_l,j_l)$, $(i_l+1,j_l)$, $(i_l+2,j_l),\dots,$ and draw
the vertical line segment connecting $(i_l,j_l)$, $(i_l,j_l+1)$, $(i_l,j_l+2),\dots$. (Then it is clear that there is a one-to-one correspondence from the set $\bold D =\{(a,b); a\ge i_l$ and $b\ge j_l \text{ for some } l=1,\dots,n\}$ to a subset $\bold M$ of the set of monomials in $Q$, and it
follows from (2.2.1) that, in fact, every monomial in $Q$ is in $\bold M$.) Since
$(i_l,j_l)< (i_{l+1},j_{l+1}),(i_{l+1},j_l)$ are the coordinates of the
intersection of the rightward extending horizontal line segment thru $(i_l,j_l)$ with the ascending vertical line segment thru $(i_{l+1},j_{l+1})$. Then
$z_l=x^{i_{l+1}-1}y^{j_l-1}\notin Q$, $z_ly$ has coordinates on the
rightward extending horizontal line segment thru $(i_l,j_l)$ (so $z_ly\in Q$),
and $z_lx$ has coordinates on the ascending vertical line segment thru
$(i_{l+1},j_{l+1})$ (so $z_lx\in Q$), hence $z_l$ is a $Q$ -corner-element.
And since a $Q$ -\CE must correspond to some $(a,b)$ with $0\le a < i_n$ and $0\le b1$, $b>1$, $c>1$, $d>1$ are integers.
Then the $Q$ -\CES are $x^{b-1}y^{c-1}z^{d-1}$, $w^{a-1} y^{c-1} z^{d-1}$,
$w^{a-1}x^{b-1}z^{d-1}$, and $w^{a-1}x^{b-1}y^{c-1}$. (This can be checked
by using (3.14).)
\medskip
\noindent
{\bf{(3.21) Example.}}
Let $(R,M=(x_1,\dots,x_d)R)$ be a regular local ring of altitude $d$ and let
$Q=(x_1^{a_1},\dots, x_d^{a_d})R$, where the $a_i$ are positive integers. Then
$Q$ is irreducible, so by (4.3) there is only one $Q$ -corner-element,
namely $z=x_1^{a_1-1}
\cdots x_d^{a_d-1}$.
\bigskip
\noindent
{\smc{\bf 4. Parametric Decompositions of Monomial Ideals.}}
(2.4) (together with (3.4)) shows that $X^n$ is the irredundant finite
intersection of the parameter ideals $P(z)$,
where $z$ is an $X^n$ -corner-element. The
main result in this section, (4.1), generalizes this to an arbitrary monomial
ideal $I$ such that $Rad(I)$ $=$ $Rad(X)$. And in (4.10) we show that such a
decomposition is unique.
\medskip
\proclaim{(4.1) Theorem}
Let $I$ be a monomial ideal such that $Rad(I)$ $=$ $Rad(X)$ and let
$z_1,\dots,z_m$ be the $I$ -corner-elements.
Then $I=\cap\{P(z_j); j=1,\dots,m\}$ is a
decomposition of $I$ as an irredundant intersection of parameter ideals.
\endproclaim
\demo{Proof}
Let $J=\cap \{P(z_j); j=1,\dots,m\}$. Then (3.3) shows that $J$ is the
irredundant intersection of the $m$ parameter ideals $P(z_j)$.
Now let $f$ be a monomial in $I$ and suppose that $f\notin P(z_j)$ for
some $j=1,\dots,m$. Then $z_j\in fR\subseteq I$, by (2.3), and this contradicts
the fact that $z_j\notin I$ (since $z_j$ is an $I$ -corner-element).
Therefore $I\subseteq
J$.
Finally, [T, Lemma 6] shows that $J$ is a monomial ideal (since $Rad(P(z_j))$
$=$ $Rad(X)$ for $j=1,\dots,m)$, so it suffices to show that each
monomial that is not in $I$ is not in $J$. For this, let $f$ be a monomial
that is not in $I$. Then (3.15) shows that there exists a monomial $g$
(possibly $g=1$) such that $fg$ is an $I$ -corner-element,
so $fg=z_j$ for some
$j=1,\dots,m$ (since (3.2) shows that the $I$ -\CES are finite in number and
uniquely determined by $I$). Then $f\notin P(z_j)$, by (2.3), so it follows
that $I\supseteq J$, hence $I=J$ by the preceding paragraph, \qed
\enddemo
\medskip
In (4.2) it is shown that the \CES of a monomial ideal $I$ determine $I$ when
$Rad(I)$ $=$ $Rad(X)$.
\medskip
\proclaim{(4.2) Corollary}
If $I$ and $J$ are monomial ideals such that $Rad(I)$ $=$ $Rad(X)$
$=$ $Rad(J)$ and if
$(I:X)-I=(J:X)-J$, then $I=J$.
\endproclaim
\demo{Proof}
If $(I:X)-I=(J:X)-J$, then $I$ and $J$ have the same corner-elements,
by (3.2), so this
follows immediately from (4.1), \qed
\enddemo
\medskip
\proclaim{(4.3) Corollary}
If $Q$ is an open monomial ideal in a Gorenstein local ring $R$ of altitude $d>1$,
then $Q$ is irreducible if and only if there exists exactly
one $Q$ -corner-element, and
then $Q$ is generated by a system of parameters.
\endproclaim
\demo{Proof}
Let $m$ be the number of $Q$ -corner-elements.
Then $Q$ is the irredundant intersection of
$m$ (open) parameter ideals, by (4.1). Since $R$ is Gorenstein, an open
parameter ideal is irreducible, so $Q$ is the irredundant intersection of $m$
(open) irreducible ideals. Since each such
decomposition of $Q$ has the same number of factors, $m=1$ if and only if
$Q$ is irreducible.
For the final statement, if $Q$ is irreducible, then $Q=P(z)$ is generated
by a system of parameters, where $z$ is the $Q$ -corner-element, \qed
\enddemo
\medskip
The next corollary is closely related to (2.5) and (4.3).
\medskip
\proclaim{(4.4) Corollary}
Let $I$ and $z_1,\dots,z_m$ be as in (4.1) and assume that $R$ is a
Gorenstein local ring of altitude $d$. Then $I=\cap\{P(z_j); j=1,\dots,m\}$
is an irredundant intersection of $m$ irreducible ideals.
\endproclaim
\demo{Proof}
If $R$ is Gorenstein, then each open parameter ideal is irreducible, so the
conclusion follows from (4.1), \qed
\enddemo
\medskip
In [T, Theorem 8] it is shown (among other things) that if $Rad(I)$
$=$ $Rad(X)$,
then $\cup\{P; P \in Ass(R/I)\}=\cup\{Q;Q\in Ass(R/X)\}$.
(4.1) yields a simple proof of the following closely related result.
\medskip
\proclaim{(4.5) Corollary}
If $I$ is as in (4.1), then $\cup\{Ass(R/I^n); n\ge 1\}\subseteq
Ass(R/X)$.
\endproclaim
\demo{Proof}
It is well known that if $Y$ and $Z$ are ideals that are generated by $R$ -sequences such that $Rad(Y)$ $=$ $Rad(Z)$, then $Ass(R/Y)=Ass(R/Z)$.
It therefore follows that if $z_1,\dots,z_m$ are the $I$ -corner-elements, then
$Ass(R/X)=Ass(R/P(z_j))$ for $j=1,\dots,m$ (since each
$P(z_j)$ is generated by powers of $x_1,\dots,x_d$). Therefore, since
$I=\cap\{P(z_j);j=1,\dots,m\}$, it follows that $Ass(R/I)=
Ass(R/(\cap\{P(z_j); j=1,\dots,m\}))\subseteq \cup\{Ass(R/P(z_j)); j=1,\dots,m\}=Ass(R/X)$. Finally, $I^n$ is generated by monomials for
all $n\ge 1$, so it follows from what was just shown that
$Ass(R/I^n)\subseteq Ass(R/X)$, \qed
\enddemo
\medskip
For the proof of the next corollary we need the following definition.
\medskip
\noindent
{\bf{(4.6) Definition.}}
If $P$ is a prime divisor of an ideal $I$ in a Noetherian ring, then
$\bold{D_P(I)}$ denotes the number of $P$ -primary ideals in a decomposition of
$I$ as an irredundant intersection of irreducible ideals.
\medskip
Concerning (4.6), a classical result of E. Noether [N, Satz VII] says that
$D_P(I)$ is well defined (that is, $D_P (I)$ is independent of the particular irredundant irreducible decomposition
of $I$).
\medskip
\proclaim{(4.7) Corollary}
Let $(R,M)$ be a Gorenstein local ring, let $X$ be an ideal generated by a
system of parameters $x_1,\dots,x_d$, and let $Q$ be an open monomial ideal.
Then $v((Q:M)/Q)=v((Q:X)/Q)$.
\endproclaim
\demo{Proof}
Let $m$ be the number of $Q$ -corner-elements. Then (3.7) shows that\linebreak
$v((Q:X)/Q)$ $=$ $m$, and (4.1) shows that $Q$ is the
irredundant intersection of
$m$ parameter ideals. Since $R$ is Gorenstein, each of these parameter ideals
is irreducible, so $Q$ is the irredundant intersection of $m$ irreducible
ideals, so $D_M(Q)=m$ (see (4.6)). However, [HRS2, (2.4)] shows that
$D_M(Q)=v((Q:M)/Q)$. Therefore $v((Q:X)/Q)=m=v((Q:M)/Q)$, \qed
\enddemo
\medskip
(4.1) shows that the $I$ -\CES determine a decomposition of $I$ as an
irredundant intersection of parameter ideals. (4.8) shows that the converse
also holds.
\medskip
\proclaim{(4.8) Proposition}
For $j=1,\dots,m$ let $\bold a_j=(a_{j,1},\dots,a_{j,d})$ be a $d$-tuple
of positive integers and let $I=\cap\{P(\bold a_j); j=1,\dots,m\}$ be a
decomposition of $I$ as an irredundant intersection of parameter ideals.
Then the $I$ -\CES are the $m$ elements $x_1^{a_{j,1}-1}\cdots x_d^{a_{j,d}-1}$.
\endproclaim
\demo{Proof}
(Note: $Rad(I)$ $=$ $Rad(X)$, since $Rad(P(\bold a_j))$
$=$ $Rad(X)$ for $j$ $=$ $1,\dots,m$.)
It will first be shown that each of the $m$ elements $z_j=x_1^{a_{j,1}-1}\cdots
x_d^{a_{j,d}-1}$ is an $I$ -corner-element.
For this, note first that $P(z_j)=P(\bold a_j)$ for $j=1,\dots,m$. Therefore
$z_i\notin z_jR$ for all $i\not=j\in\{1,\dots,m\}$ (for if
$z_i\in z_jR$, then $P(\bold a_i)=P(z_i)\subseteq P(z_j)=P(\bold a_j)$, and
this is a contradiction). Therefore (2.3) shows that $z_j\notin P(z_j)=P(\bold a_j)$ (so $z_j\notin I$) and that $z_j\in P(z_k)=P(\bold a_k)$ for
$k\in \{1,\dots,j-1,j+1,\dots,m\}$. Also, $z_jx_i\in x_i^{(a_{j.1}-1)+1}R
\subseteq P(\bold a_j)$ for $i=1,\dots,d$, so $z_jx_i\in\cap\{P(\bold a_h); h=1,\dots,m\}=I$. Therefore $z_j$ is an $I$ -corner-element, so it follows that
$z_1,\dots,z_m$ are among the $I$ -corner-elements.
Now let $w$ be an $I$ -corner-element. Then $w\notin I$, so $w\notin P(z_j)=P(\bold a_j)$ for
some $j=1,\dots,m$. Therefore $z_j\in wR$, by (2.3), so $z_j=wg$ for
some monomial $g$ by (2.2.2). If $g\not=1$, then $wg\in I$, since $w$ is an
$I$ -corner-element. But this implies that $z_j\in I$, and this contradicts the fact that
$z_j$ is an $I$ -corner-element. Therefore $g=1$, so $w=z_j$, so $z_1,\dots,z_m$ are all
the $I$ -corner-elements, \qed
\enddemo
\medskip
\proclaim{(4.9) Corollary}
Let $z_1,\dots,z_m$ be monomials such that $z_i\notin z_jR$ for $i\not=
j\in\{1,\dots, m\}$, let $J=(z_1,\dots,z_m)R$, and let $I=\cap\{P(z_j); j=1,\dots,m\}$. Then $z_1,\dots,z_m$ are the $I$ -\CES and $I:X=I+J$.
\endproclaim
\demo{Proof}
If $\cap \{P(z_j); j=1,\dots,m\}$ is an irredundant intersection, then it
follows from (4.8) that the $I$ -\CES are the elements $z_1,\dots,z_m$, so
$I:X=I+J$ by (3.2). Therefore it remains to show that this intersection is
irredundant.
For this, suppose, on the contrary, that it is redundant. Then by resubscripting the $z_j$, if necessary, it may be assumed that $I=\cap\{P(z_j); j=1,\dots,k\}$ for some $k