# Symbolic Rees algebras and
set-theoretic complete intersections

Clare D’Cruz∗ Chennai Mathematical Institute, Plot H1 SIPCOT IT Park,
Siruseri, Kelambakkam 603103, Tamil Nadu, India<EMAIL_ADDRESS>,  Mousumi
Mandal† Department of Mathematics, Indian Institute of Technology Kharagpur,
721302, India<EMAIL_ADDRESS>and  J. K. Verma  Department of
Mathematics, Indian Institute of Technology, Mumbai, 400076, India
<EMAIL_ADDRESS>
###### Abstract.

In this paper we extend a result of Cowsik on set-theoretic complete
intersection and a result Huneke, Morales and Goto and Nishida about
Noetherian symbolic Rees algebras of ideals. As applications, we show that the
symbolic Rees algebras of the following ideals are Noetherian and the ideals
are set-theoretic complete intersections: (a) the edge ideal of a complete
graph, (b) the Fermat ideal and (c) the Jacobian ideal of a certain hyperplane
arrangement.

###### Key words and phrases:

Symbolic Rees algebra, set-theoretic complete intersections, Fermat ideal,
Jacobian ideal, edge ideal

###### 2010 Mathematics Subject Classification:

Primary: 13A30, 1305, 13H15, 13P10

∗ partially supported by a grant from Infosys Foundation

† Supported by SERB(DST) grant No.: $\mbox{MTR}/2020/000429$, India

##  1\. Introduction

Throughout this paper $(R,\mathfrak{m})$ is a Noetherian local ring of
positive dimension $d$ with infinite residue field and ${\bf a}\subset R$ is
an ideal. We say that ${\bf a}$ is defined set-theoretically by $s$ elements
if there exists $f_{1},\ldots,f_{s}\in\mathfrak{m}$ such that $\sqrt{{\bf
a}}=\sqrt{(f_{1},\ldots,f_{s})}$. If $s=h:=\operatorname{ht}({\bf a})$ and
$\sqrt{{\bf a}}=\sqrt{(f_{1},\ldots,f_{h})}$ then ${\bf a}$ is called set-
theoretic complete intersection.

One of the remarkable results on set-theoretic complete intersection was
proved by R. C. Cowsik and M. V. Nori. They showed that if the characteristic
of a field $\Bbbk$, $\operatorname{char}(\Bbbk)=p>0$, then the defining ideal
of any affine curve in $\mathbb{A}_{\Bbbk}^{n}$ is a set-theoretic complete
intersection ([7, Theorem 1]). J. Herzog proved that all monomial curves in
the affine space $\mathbb{A}^{3}_{\Bbbk}$ over any field $\Bbbk$ are set-
theoretic complete intersections. This was also proved by H. Bresinsky [3]. It
has been a long standing open problem to characterise ideals that are set-
theoretic complete intersections. We shall review a few of these results.

Symbolic powers of ideals have played an important role in connection with the
problem of set-theoretic complete intersections. The $n$-th symbolic power of
an ideal ${\bf a}$ is defined as ${\displaystyle{\bf
a}^{(n)}=\bigcap_{\mathfrak{p}\in\operatorname{Ass}({\bf a})}({\bf
a}^{n}R_{\mathfrak{p}}\cap R)}$. It has also been of importance because of its
connection with algebraic geometry which goes back to the work of O. Zariski
[31] and M. Nagata [22]. If $\Bbbk$ is an algebraically closed field, then the
$n$-th symbolic power of a given prime ideal consists of functions that vanish
up to order $n$ on the algebraic variety defined by the prime ideal [9].

Let ${\displaystyle\mathcal{R}_{s}({\bf a}):=\bigoplus_{n\geq 0}{\bf
a}^{(n)}t^{n}}$ denote the symbolic Rees algebra of an ideal ${\bf a}$. The
study of the symbolic Rees algebras of ideals in a Noetherian local domain was
of interest due to a result of D. Rees which gave a counter-example to a
problem of Zariski using symbolic Rees algebras. Rees showed that if
$\mathfrak{p}$ is a prime ideal with $\operatorname{ht}(\mathfrak{p})=1$ in a
Noetherian normal local domain of dimension two [25] then the symbolic Rees
algebra is Noetherian if and only if some symbolic power of $\mathfrak{p}$ is
principal. Rees’ work was generalized by A. Ooishi [24]. The above results
were studied under the condition $\operatorname{dim}(R/{\bf a})=1$. In 1990,
Goto. et. al studied the Noetherian property of any unmixed ideal in a
Noetherian local ring [11]. The symbolic Rees algebra of prime ideals has also
been studied in [16], [26], [18], [28], [29], [27] [13].

In $1981$ R. C. Cowsik proved the following remarkable result that rejuvenated
interest in symbolic powers of an ideal:

######  Theorem 1.1.

[5] Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d$ and
${\bf a}\not=\mathfrak{m}$ a radical ideal. If the symbolic Rees algebra
$\mathcal{R}_{s}({\bf a}):=\oplus_{n\geq 0}{\bf a}^{(n)}$ is Noetherian, then
${\bf a}$ is the radical of an ideal generated by $d-1$ elements. In
particular, if ${\bf a}$ has height $d-1$, then it is a set-theoretic complete
intersection.

The converse of Cowik’s result need not be true [12, Corollary 1.2]. One of
the main results in this paper is to generalize Cowsik’s result on set-
theoretic complete intersections. Cowsik’s work motivated several researchers
to investigate the Noetherian property of the symbolic Rees algebra. In 1987,
C. Huneke gave necessary and sufficient conditions for
$\mathcal{R}_{s}(\mathfrak{p})$ to be Noetherian when $R$ is a $3$-dimensional
regular local ring and $\mathfrak{p}$ is a height two prime ideal [17].
Huneke’s result was generalised for $\operatorname{dim}R\geq 3$ by M. Morales
in [21] and later by S. Goto [11]. We state the most general form of the
result. Recall that a local ring $R$ is called unmixed if all the associated
primes of the completion of $R$ have the same dimension.

######  Theorem 1.2.

[11, Theorem 1.1] Let $(R,\mathfrak{m})$ be a local ring of dimension $d$ and
$\mathfrak{p}$ a prime ideal of height $d-1$.

  1. (1)

Suppose $\mathcal{R}_{s}(\mathfrak{p})$ is Noetherian. Then there exists a
system of parameters $x,f_{1},\ldots,f_{d-1}$ such that elements
$f_{i}\in\mathfrak{p}^{(k_{i})}$, $i=1,\ldots,d-1$ with $k_{i}>0$, satisfying

$\displaystyle
e_{(x,f_{1},\ldots,f_{d-1})}(R)=e(R_{\mathfrak{p}}){\lambda}\left(\frac{R}{(\mathfrak{p},x)}\right)\prod_{i=1}^{d-1}k_{i}.$
(1.3)

  2. (2)

If $R$ is an unmixed local ring containing a system of parameters
$x,f_{1},\ldots,f_{d-1}$ satisfying (1.3), then
$\mathcal{R}_{s}(\mathfrak{p})$ is Noetherian.

These results provided a useful way to find Noetherian symbolic Rees algebras
and to find ideals that are set-theoretic complete intersections.

In this paper we extend the work of Huneke [17], Morales [21] and Goto and
Nishida [11] for any arbitrary ideal ${\bf a}$ such that
$1\leq\operatorname{ht}{\bf a}\leq d-1$ (Theorem 2.4, Theorem 2.8). We give
necessary and sufficient conditions for the Noetherian property of
$\mathcal{R}_{s}({\bf a})$. For a radical ideal ${\bf a}$, we prove in Theorem
2.3 that if $\mathcal{R}_{s}({\bf a})$ is Noetherian and $R/{\bf a}^{(n)}$ is
Cohen-Macaulay for large $n$, then ${\bf a}$ is set theoretic complete
intersection. We demonstrate our results with a few examples. In Section 3,
using our main theorem we prove that the symbolic Rees algebra of the edge
ideal of complete graph is Noetherian. In Section 4, we give a new proof for
the Noetherian property of the symbolic Rees algebra of the Fermat ideal. In
section 5, we prove that the symbolic Rees algebra of the Jacobian ideal the
defining polynomial of certain hyperplane arrangement is Noetherian.

##  2\. The Main Result

In this section we prove the main result of this paper. We generalise a result
of Cowsik, Huneke, Morales and Goto to any arbitrary ideal ${\bf a}$ of height
$h$, where $1\leq h\leq d-1$ and give necessary and sufficient condition for
the Noetherian property of $\mathcal{R}_{s}({\bf a})$. In order to prove the
main theorem first we prove few preliminary results. For an ideal ${\bf a}$,
the associated graded ring is defined as ${\displaystyle\mathcal{G}({\bf
a}):=\bigoplus_{n\geq 0}{\bf a}^{n}/{\bf a}^{n+1}}$ and for any element
$x\in{\bf a}^{n}\setminus{\bf a}^{n+1}$, let $x^{\star}$ denote the image of
$x$ in ${\bf a}^{n}/{\bf a}^{n+1}.$

######  Lemma 2.1.

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d\geq 1$ and
${\bf a}$ be an ideal with $\operatorname{dim}R/{\bf a}=s\geq 1$. Let $R/{\bf
a}^{n}$ be Cohen-Macaulay for all $n\geq 1$.

  1. (1)

Then there exist $x_{1},\ldots,x_{s}$ such that
$x_{1}^{\star},\ldots,x_{s}^{\star}$ is a $\mathcal{G}({\bf a})$-regular
sequence and $x_{1},\ldots,x_{s}$ is an $R$-regular sequence and

  2. (2)

$\operatorname{dim}R/{\bf a}=\operatorname{dim}R-\operatorname{ht}{\bf a}$.

###### Proof.

(1) We prove (a) by induction on $\operatorname{dim}(R/{\bf a})=s$. First
assume that $\operatorname{dim}(R/{\bf a})=1$, then we show that there exists
$x_{1}\in R$ such that $x_{1}^{\star}$ is a $\mathcal{G}({\bf a})$-regular
element and $x_{1}$ is an $R$\- regular element. As $R/{\bf a}^{n}$ is Cohen-
Macaulay for all $n\geq 1,$ $\operatorname{Ass}(R/{\bf
a}^{n})=\\{P_{1},\ldots,P_{t}\\}$ where $P_{1},P_{2},\ldots,P_{t}$ are the
minimal primes of ${\bf a}.$

By prime avoidance lemma choose
$x\in\mathfrak{m}\setminus\displaystyle{\cup_{i=1}^{t}P_{i}}$. Then $x$ is
$R/{\bf a}^{n}$-regular for all $n\geq 1$. To show $x$ is $\mathcal{G}({\bf
a})$-regular, it is enough to show that $x$ is regular on ${\bf a}^{n}/{\bf
a}^{n+1}$ for all $n\geq 1$. From the exact sequence

$0\longrightarrow{\bf a}^{n}/{\bf a}^{n+1}\longrightarrow R/{\bf
a}^{n+1}\longrightarrow R/{\bf a}^{n}\longrightarrow 0\hskip 14.45377pt(n\geq
1),$

$x$ is ${\bf a}^{n}/{\bf a}^{n+1}$-regular for all $n\geq 0$ and hence $x^{*}$
is $\mathcal{G}({\bf a})$-regular. We now show that $x$ is $R$-regular. There
exists $n\geq 0$ so that $x\in{\bf a}^{n}\setminus{\bf a}^{n+1}.$ If there is
a $y\in R$ such that $xy=0$ and $y\neq 0$ then $y\in{\bf a}^{m}\setminus{\bf
a}^{m+1}$ for some $m.$ As $(xy)^{*}=x^{*}y^{*}=0,$ $y^{*}=0$ which means
$y\in{\bf a}^{m+1}$ which is a contradiction. Hence $x$ is $R$-regular. Now
assume that $\operatorname{dim}R/{\bf a}>1$. Using the above argument choose
$x_{1}$ so that $x_{1}^{\star}$ is $\mathcal{G}({\bf a})$-regular and $x_{1}$
is $R$-regular. As $x_{1}$ is $R/{\bf a}^{n}$\- regular for all $n\geq 1$, we
have $(x_{1})\cap{\bf a}^{n}=x_{1}{\bf a}^{n}$ for all $n\geq 1$ which implies
that $\mathcal{G}({\bf a}\overline{R})\cong\mathcal{G}({\bf
a})/x_{1}^{*}\mathcal{G}({\bf a})$, where $\overline{R}=R/(x_{1})$. Since
$x_{1}$ is $R/{\bf a}^{n}$-regular for all $n\geq 1$, we have
$\overline{R}/\overline{{\bf a}^{n}}$ is Cohen-Macaulay of dimension $s-1$ for
all $n\geq 1$. By induction hypothesis, we can choose $x_{2},\ldots,x_{s}\in
R$ such that $\overline{x_{2}}^{*},\ldots,\overline{x_{s}}^{*}$ is
$\mathcal{G}({\bf a}\overline{R})\cong\mathcal{G}({\bf
a})/x_{1}^{*}\mathcal{G}({\bf a})$-regular sequence and
$\overline{x_{2}},\ldots,\overline{x_{s}}$ is $\overline{R}$-regular sequence.
Thus $x_{1}^{*},\ldots,x_{s}^{*}$ is a $\mathcal{G}({\bf a})$-regular sequence
and $x_{1},\ldots,x_{s}$ is a an $R$-regular sequence. This proves (1).

(2) By part (1), since there exists an $R/{\bf a}$-regular sequence
$\underline{x}=x_{1},\ldots,x_{s},$ $\operatorname{dim}(R/{\bf
a}+(\underline{x}))=0.$ Hence $\overline{{\bf a}}$ is an
$\mathfrak{m}/(\underline{x})$-primary ideal in
$\overline{R}=R/(\underline{x})$. Since $\underline{x}$ is an $R$\- regular
sequence,

$\operatorname{ht}({\bf a})\geq\operatorname{ht}(\overline{{\bf
a}})=\operatorname{dim}(R/(\underline{x}))=\operatorname{dim}(R)-\operatorname{dim}R/{\bf
a}=\operatorname{ht}({\bf a}).$

Therefore $s=\operatorname{dim}R/{\bf
a}=\operatorname{dim}R-\operatorname{ht}{\bf a}$. ∎

######  Example 2.2.

Lemma 2.1 need not hold true if we drop the assumption that $R/{\bf a}^{n}$ is
Cohen-Macaulay for all $n\geq 1$. Let
$R=k[[x,y,z]]/(xy,xz)=k[[\overline{x},\overline{y},\overline{z}]]$ and
$I=(\overline{x})$. Then $R/I$ is Cohen-Macaulay, but $R/I^{n}$ is not Cohen-
Macaulay for all $n\geq 2$. Since $\operatorname{dim}(R/I)=2$ and
$\operatorname{depth}(R)=1,$ we cannot find a regular sequence of length two
in $R$. ∎

In [10, Theorem 3.3], Goto et. al proved that if $R$ is an unmixed local ring
and ${\bf a}$ an ideal such that $\ell({\bf a}^{(k)})=\operatorname{ht}({\bf
a}^{(k)})$, for some $k\geq 1$, then $\mathcal{R}_{s}({\bf a})$ is Noetherian
where $\ell({\bf a})$ denote the analytic spread of ${\bf a}$. With the
additional assumption that $R/{\bf a}^{(n)}$ is Cohen-Macaulay for all $n\gg
0$, they proved the converse [10, Theorem 3.6]. In the next lemma we give an
alternate proof of the converse.

######  Theorem 2.3.

Let $(R,\mathfrak{m})$ be a local ring of dimension $d,$ and $R/\mathfrak{m}$
is infinite. Let ${\bf a}$ be an ideal of positive height $h$. If $R/{\bf
a}^{(n)}$ is Cohen-Macaulay for all $n\gg 0$ and $\mathcal{R}_{s}({\bf a})$ is
Noetherian, then $\ell({\bf a}^{(k)})=\operatorname{ht}({\bf a}^{(k)})$ for
some $k\geq 1$ and ${\bf a}$ is a set-theoretic complete intersection.

###### Proof.

Since $\mathcal{R}_{s}({\bf a})$ is Noetherian, by [10, Theorem 3.2] there
exists $m\geq 1$ such that ${\bf a}^{(m)r}={\bf a}^{(mr)}$ for all $r\geq 1$.
By our assumption, there exists an $n_{0}$ such that for all $n\geq n_{0}$,
$R/{\bf a}^{(n)}$ is Cohen-Macaulay. Choose $k\geq\\{m,n_{0}\\}$ such that
both the assumptions are satisfied. Put $I={\bf a}^{(k)}$. Then for all $n\geq
1$, $R/I^{n}$ is Cohen-Macaulay. Hence by Lemma 2.1(2),
$\operatorname{dim}R/I=\operatorname{dim}R-\operatorname{ht}I$. Since
$R/I^{n}$ is Cohen-Macaulay we have
$\operatorname{depth}(R/I^{n})=\operatorname{dim}R/I^{n}=d-\operatorname{ht}(I)$.
By Burch’s inequality [4], we get

$\ell(I)\leq d-\inf_{n}\operatorname{depth}(R/I^{n})=\operatorname{ht}(I).$

As $\operatorname{ht}(I)\leq\ell(I)$, equality holds which implies that
$\ell({\bf a}^{(k)})=\operatorname{ht}({\bf a}^{(k)})$. Since $R/\mathfrak{m}$
is infinite, there is a minimal reduction $J=(f_{1},\ldots,f_{h})$ of ${\bf
a}^{(k)}$. Hence $\sqrt{{\bf a}}=\sqrt{{\bf a}^{(k)}}=\sqrt{J}$. ∎

The rest of this section is dedicated for a generalization of results in [17]
and [21] and [11]. We first prove some preliminary results.

######  Theorem 2.4.

Let $(R,\mathfrak{m})$ be a a local ring of dimension $d\geq 1$ with infinite
residue field $k.$ Let ${\bf a}$ be an ideal of height $h$ with $1\leq h\leq
d-1$ such that $R/{\bf a}^{(m)}$ is Cohen-Macaulay for large $m$ and
$\mathcal{R}_{s}({\bf a})$ is Noetherian. Then there exist
$x_{1},\ldots,x_{d-h},$ a system of parameters for $R/{\bf a},$ and
$f_{i}\in{{\bf a}}^{(k_{i})}$ for $i=1,2,\ldots,h$ such that

  1. (1)

$x_{1},\ldots,x_{d-h}$ is a regular sequence in $R$.

  2. (2)

$x_{1},\ldots,x_{d-h},f_{1},\ldots,f_{h}$ is a system of parameters for $R$.

  3. (3)

Let $\operatorname{MinAss}(R/{\bf a}):=\\{P_{1},\ldots,P_{s}\\}$. Then

$\displaystyle
e_{(x_{1},\ldots,x_{d-h},f_{1},\ldots,f_{h})}(R)=\left(\displaystyle{\prod_{j=1}^{h}k_{j}}\right)\displaystyle{\sum_{i=1}^{s}e_{{\bf
a}R_{P_{i}}}(R_{P_{i}})e_{(x_{1},\ldots,x_{d-h})}(R/P_{i})}.$

###### Proof.

(1) As $\mathcal{R}_{s}({\bf a})$ is Noetherian, by Theorem 2.3 there exists
$r\geq 1$ such that $\ell({\bf a}^{(r)})=h$. By our assumption there exists
$n_{0}$ such that $R/{\bf a}^{(n)}$ is Cohen-Macaulay for all $n\geq n_{0}$.
Choose $k=\max\\{r,n_{0}\\}$ and put $I={\bf a}^{(k)}$. Then by the proof of
Lemma 2.1, we can choose an $R$-regular sequence
$x_{1},\ldots,x_{d-h}\in\mathfrak{m}$.

(2) As $k$ is infinite and $\ell(I)=h,$ we can choose a minimal reduction
$J=(f_{1},\ldots,f_{h})R$ of $I$. Since $\operatorname{dim}R/I=d-h$, by Lemma
2.1, we can choose $x_{1},\ldots,x_{d-h}\in\mathfrak{m}$ whose images in $R/I$
form a system of parameters for $R/I$. As $J$ is a minimal reduction of $I$ we
have $\operatorname{MinAss}(R/I)=\operatorname{MinAss}(R/J)$, which implies
that $x_{1},\ldots,x_{d-h}$ is also a system of parameters for $R/J$. Thus
$x_{1},\ldots,x_{d-h},f_{1},\ldots,f_{h}$ is a system of parameters of $R$.

(3) Put $B=R/(\underline{x})$ where $\underline{x}=(x_{1},\ldots,x_{d-h})$.
Then $JB$ is a reduction of $IB$. Since $R/I^{n}$ Cohen-Macaulay and
$\underline{x}$ is a system of parameters in $R/I^{n}$ we have

$\displaystyle e_{({\underline{x},J)}}(R)=e_{J}(B)$ $\displaystyle=$
$\displaystyle e_{I}(B)\hskip 142.26378pt\mbox{ (as $J$ is a reduction of
$I$)}$ $\displaystyle=$
$\displaystyle{\underset{{n\to\infty}}{\lim}}\left[\frac{h!}{n^{h}}{\lambda}_{B}\left(\frac{B}{I^{n}B}\right)\right]$
$\displaystyle=$
$\displaystyle{\underset{{n\to\infty}}{\lim}}\left[\frac{h!}{n^{h}}{\lambda}_{R}\left(\frac{R}{(\underline{x})+I^{n}}\right)\right]$
$\displaystyle=$
$\displaystyle{\underset{{n\to\infty}}{\lim}}\frac{h!}{n^{h}}\left[e_{\underline{x}R}\left(\frac{R}{I^{n}}\right)\right]$
$\displaystyle=$
$\displaystyle{\underset{{n\to\infty}}{\lim}}\frac{h!}{n^{h}}\left[\displaystyle{\sum_{i=1}^{s}{\lambda}_{R_{P_{i}}}\left(\frac{R_{P_{i}}}{I^{n}R_{P_{i}}}\right)e_{\underline{x}}\left(\frac{R}{P_{i}}\right)}\right]\hskip
14.45377pt\mbox{(by \cite[cite]{[\@@bibref{}{mat}{}{}, Theorem 14.7]})}$
$\displaystyle=$
$\displaystyle{\underset{{n\to\infty}}{\lim}}\frac{k^{h}h!}{(kn)^{h}}\left[\sum_{i=1}^{s}{\lambda}_{R_{P_{i}}}\left(\frac{R_{P_{i}}}{{{\bf
a}}^{kn}}\right)e_{\underline{x}}\left(\frac{R}{P_{i}}\right)\right]\hskip
14.45377pt\mbox{[since ${{\bf a}}^{(kn)}R_{P_{j}}={{\bf a}}^{kn}R_{P_{j}}$]}$
$\displaystyle=$ $\displaystyle
k^{h}\left[\displaystyle{\sum_{i=1}^{s}e_{{{\bf
a}}R_{P_{i}}}(R_{P_{i}})e_{\underline{x}}(R/P_{i})}\right].$ (2.6)

∎

In order to prove the converse of the above theorem we need the following
result proved by E. Böger. An ideal $I$ is said to be equimultiple ideal if
$\operatorname{ht}I=\ell(I)$.

######  Theorem 2.7 ([2]).

Let $J$ be an equimultiple ideal of a quasi-unmixed local ring
$(R,\mathfrak{m})$. Let $I\supseteq J$ be an ideal. Then $J$ is a reduction of
$I$ if and only if $e(I_{p})=e(J_{p})$ for every minimal prime $p$ of $I$ and
$\sqrt{I}=\sqrt{J}.$

######  Theorem 2.8.

Let $(R,\mathfrak{m})$ be a quasi-unmixed local ring of dimension $d\geq 1$.
Let ${\bf a}$ be an ideal of height $h$ with $1\leq h\leq d-1$ and
$\operatorname{MinAss}(R/{\bf a})=\\{P_{1},\ldots,P_{s}\\}$ such that
$\operatorname{ht}P_{i}=h$ for all $i=1,\ldots,s$. If there exist
$x_{1},\ldots,x_{d-h}$ a system of parameters of $R/{\bf a}$ and $f_{i}\in{\bf
a}^{(k_{i})}$ such that conditions $(\ref{main theorem one})$, $(\ref{main
theorem two})$ and $(\ref{main theorem three})$ of Theorem 2.4 are satisfied,
then $\mathcal{R}_{s}({\bf a})$ is Noetherian.

###### Proof.

By passing to high powers of the $f_{i}$’s we may assume that $k=k_{i}$ for
all $1\leq i\leq h$. Fix $k$ and put $I={{\bf a}}^{(k)}$ and
$J=(f_{1},\ldots,f_{h})$. We will show that $J$ is a reduction of $I$. Let
$B=R/(x_{1},\ldots,x_{d-h})$. Then as $\underline{x}=(x_{1},\ldots,x_{d-h})$
is $R$-regular, by [19, Theorem 14.11] and condition (3) we have

$\displaystyle e_{J}(B)=e_{(\underline{x})+J}(R)=k^{h}\sum_{i=1}^{s}e_{{\bf
a}R_{P_{i}}}(R_{P_{i}})e_{\underline{x}}\left(\frac{R}{P_{i}}\right).$ (2.9)

Let $F=\operatorname{MinAss}_{R}(R/J)$. Then for all $n\geq 1$,
$F=\operatorname{MinAss}_{R}(R/J^{n})$. Hence we get

$\displaystyle{e_{J}(B)}$ $\displaystyle=$
$\displaystyle\left[\underset{{n\to\infty}}{\lim}\frac{h!}{n^{h}}{\lambda}_{B}(B/J^{n})\right]$
(2.10) $\displaystyle=$
$\displaystyle\underset{{n\to\infty}}{\lim}\left[\frac{h!}{n^{h}}{\lambda}_{R}(R/(\underline{x})+J^{n})\right]$
$\displaystyle\geq$
$\displaystyle\underset{{n\to\infty}}{\lim}\frac{h!}{n^{h}}\left[e_{\underline{x}R}\left(\frac{R}{J^{n}}\right)\right]$
$\displaystyle=$
$\displaystyle\underset{{n\to\infty}}{\lim}\frac{h!}{n^{h}}\left[\displaystyle{\sum_{P\in
F}{\lambda}_{R_{P}}(R_{P}/J^{n}R_{P})e_{\underline{x}}(R/P)}\right]\mbox{by
\cite[cite]{[\@@bibref{}{mat}{}{}, Theorem~{}14.7]}}$ $\displaystyle=$
$\displaystyle{\displaystyle{\sum_{P\in
F}e_{JR_{P}}(R_{P})e_{\underline{x}}(R/P)}}$

As $\operatorname{ht}I=\operatorname{ht}J=h$ and $J\subseteq I$ for all
$i=1,\ldots,s$, $P_{i}\in F$,

$\displaystyle\sum_{P\in
F}e_{JR_{P}}(R_{P})e_{\underline{x}}(R/P)\geq\sum_{i=1}^{s}e_{JR_{P_{i}}}(R_{P_{i}})e_{\underline{x}}(R/P_{i}).$
(2.11)

Now from (2.10) and (2.11) we get

$\displaystyle{e_{J}(B)}\geq\sum_{i=1}^{s}e_{JR_{P_{i}}}(R_{P_{i}})e_{\underline{x}}(R/P_{i}).$
(2.12)

Since $J\subseteq I$, for $1\leq i\leq s$,

$\displaystyle e_{JR_{P_{i}}(R_{P_{i}})}$ $\displaystyle\geq$ $\displaystyle
e_{IR_{P_{i}}}(R_{P_{i}})$ (2.13)

Since $I^{n}R_{P_{i}}=({{\bf a}}^{(k)}R_{P_{i}})^{n}={{\bf a}}^{kn}R_{P_{i}}$,
we have

$\displaystyle
e_{IR_{P_{i}}}(R_{P_{i}})=\underset{{n\to\infty}}{\lim}\frac{h!}{n^{h}}{\lambda}(R_{P_{i}}/I^{n}R_{P_{i}})=\underset{{n\to\infty}}{\lim}\frac{h!}{n^{h}}{\lambda}(R_{P_{i}}/{{\bf
a}}^{kn}R_{P_{i}})=k^{h}e_{{{\bf a}}R_{P_{i}}}(R_{P_{i}}).$ (2.14)

Thus from (2.12), (2.13), (2.14) and (2.9) we have

$\displaystyle e_{J}(B)$ $\displaystyle\geq$
$\displaystyle\displaystyle{\sum_{i=1}^{s}e_{JR_{P_{i}}}(R_{P_{i}})e_{\underline{x}}(R/P_{i})}$
$\displaystyle\geq$
$\displaystyle\displaystyle{\sum_{i=1}^{s}e_{IR_{P_{i}}}(R_{P_{i}})e_{\underline{x}}(R/P_{i})}$
$\displaystyle=$ $\displaystyle\displaystyle{\sum_{i=1}^{s}e_{{{\bf
a}}R_{P_{i}}}(R_{P_{i}})k^{h}e_{\underline{x}}(R/P_{i})}$ $\displaystyle=$
$\displaystyle e_{J}(B)$

Therefore we get $F=\\{P_{1},\ldots,P_{s}\\}$ and
$e_{JR_{P_{i}}}(R_{P_{i}})=e_{IR_{P_{i}}}(R_{P_{i}})$ for all $1\leq i\leq s.$
Now since $R$ is quasi-unmixed and $J$ is an equimultiple ideal, by Theorem
2.7, $J$ is a reduction of $I$, which implies that $\ell(I)=h$. Thus by
Theorem [10, Theorem 3.6], $\mathcal{R}_{s}({\bf a})$ is Noetherian. ∎

######  Example 2.15.

Theorem 2.8 may not be true if $R$ is quasi-unmixed but
$\operatorname{ht}({\bf a})=0$. Let
$R=k[[x,y]]/(xy)=k[[\overline{x},\overline{y}]]$ and let ${\bf
a}=(\overline{x})$. Then $J=(\overline{x})$ is a minimal reduction of $I$ and
$f=\overline{x+y}$ is a parameter for $R$ as $\operatorname{dim}(R)=1$. Since
${\bf a}^{m}R_{{\bf a}}\cap R={\bf a}=(\overline{x})$ for all $m\geq 1$,
$R/{\bf a}^{(m)}$ is Cohen-Macaulay for all $m\geq 1$, but
$\mathcal{R}_{s}({\bf a})$ is not Noetherian. ∎

######  Example 2.16.

Theorem 2.8 does not hold true if $R$ is not quasi-unmixed. Let
$R=k[[x,y,z]]/(xy,xz)=k[[\overline{x},\overline{y},\overline{z}]]$ and let
${\bf a}=(\overline{x},\overline{y})$. Put $x_{1}=(\overline{x+y})$. Then
$(x_{1}){\bf a}={\bf a}^{2}.$ Hence $(x_{1})$ is a minimal reduction of ${\bf
a}$. Moreover, if $f_{1}=\overline{x+y+z}$, then $x_{1},f_{1}$ is a system of
parameters for $R.$ Since ${\lambda}(R/(x_{1},f_{1})^{n})=\binom{n+1}{2}+n,$
$e_{(x_{1},f_{1})}(R)=1.$ As ${\bf a}$ is a prime ideal, $h=1$, $k_{1}=1$,
$s=1$, we get

$\displaystyle\left(\displaystyle{\prod_{j=1}^{h}k_{j}}\right)\displaystyle{\sum_{i=1}^{s}e_{{\bf
a}R_{P_{i}}}(R_{P_{i}})e_{(x_{1},\ldots,x_{d-h})}(R/P_{i})}=e_{(\overline{x},\overline{y})}R_{(x,y)}\cdot
e_{(\overline{x+y+z)}}(R/(\overline{x},\overline{y}))=e_{(x_{1},f_{1})}(R)=1.$

As ${\bf a}^{n}R_{{\bf a}}\cap R=(\overline{x},\overline{y^{n}})$ for all
$n\geq 1,$ $\mathcal{R}_{s}({\bf a})$ is not Noetherian. ∎

##  3\. The symbolic Rees algebra of the edge ideal of complete graph

Let $\Bbbk$ be a field and $S=\Bbbk[x_{1},\ldots,x_{n}]$, a polynomial ring in
$n$ indeterminates. Let $G$ be a complete graph on $n$ vertices and
$I=I(G)=(x_{i}x_{j}\mid\\{i,j\\}\text{ is an edge of }G)\subset S$ be the
corresponding edge ideal of $G$. By a result of Herzog, Hibi and Trung [15,
Corollary 1.3], $\mathcal{R}_{s}(I)$ is Noetherian. We give another proof of
this result.

Let ${\bf x}:=x_{1},\ldots,x_{n}$ and $\sigma_{1}({\bf
x}),\ldots,\sigma_{n}({\bf x})$ be the elementary symmetric functions of $\bf
x.$

######  Proposition 3.1.

The functions $\sigma_{j}({\bf x})$ for $j=1,2,\ldots,n$ form a homogeneous
system of parameters of $S$ and

${\lambda}(S/(\sigma_{1},\sigma_{2},\ldots,\sigma_{n}))=n!.$

###### Proof.

Note that each $x_{j}$, for $j=1,2,\ldots,n$, is a root of the monic
polynomial

$x^{n}-\sigma_{1}x^{n-1}+\sigma_{2}x^{n-2}-\cdots+(-1)\sigma_{n}.$

Hence $S$ is an integral extension of the polynomial ring
$R=\Bbbk[\sigma_{1},\sigma_{2},\ldots,\sigma_{n}].$ Therefore
$J=(\sigma_{1},\ldots,\sigma_{n})S$ is
$\mathfrak{m}=(x_{1},x_{2},\ldots,x_{n})S$-primary. Thus
$\sigma_{1},\sigma_{2},\ldots,\sigma_{n}$ is a homogeneous system of
parameters of $S.$

(2) Since $\sigma_{1},\sigma_{2},\ldots,\sigma_{n}$ is an $S$-regular
sequence, the Hilbert series of $S/J$ is given by

$H(S/J,u)=\sum_{n=0}^{\infty}{\lambda}((S/J)_{n})u^{n}=\frac{(1-u)(1-u^{2})\ldots(1-u^{n})}{(1-u)^{n}}=\prod_{j=1}^{n-1}(1+u+u^{2}+\cdots+u^{j}).$

Therefore ${\lambda}(S/J)=H(S/J,1)=n!.$ ∎

######  Example 3.2.

Let $S=\Bbbk[x_{1},\ldots,x_{n}]$. Put $R=S_{\mathfrak{m}}$ and
$\mathfrak{a}=I(G)R$ where $\mathfrak{m}=(x_{1},\ldots,x_{n})$.

  1. (1)

$\mathfrak{a}$ is unmixed and $R/\mathfrak{a}^{(k)}$ is Cohen-Macaulay for all
$k\geq 1$.

  2. (2)

$\mathcal{R}_{s}(\mathfrak{a})$ is Noetherian.

  3. (3)

$\mathfrak{a}$ is a set-theoretic complete intersection.

###### Proof.

(1) Put $\mathfrak{p}_{i}=(x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{n})$, $1\leq
i\leq n$. By [30, Corollary 3.35] it follows that for all $n\geq 1$,

$\displaystyle\mathfrak{a}^{(k)}=\bigcap_{i=1}^{n}\mathfrak{p}_{i}^{k}.$

As $\operatorname{ht}\mathfrak{p}_{i}=n-1$ for all
$\mathfrak{p}_{i}\in\operatorname{Ass}(S/{\bf a})$, ${\bf a}$ is an unmixed
ideal. Since $x=\sigma_{1}\not\in\mathfrak{p}_{i}$ for all $i=1,\ldots,n$ it
is a nonzerodivisor on $R/\mathfrak{a}^{(k)}$ for all $k\geq 1$. Hence $R/{\bf
a}^{(k)}$ is Cohen-Macaulay for all $k\geq 1$.

(2) Let $x=\sigma_{1}.$ Then for all $j=2,\ldots,n$,

$\displaystyle\sigma_{j}({\bf x})$ $\displaystyle\in$ $\displaystyle
I^{(j-1)}\hskip 36.135pt\mbox{\cite[cite]{[\@@bibref{}{bocci}{}{}, Lemma
2.6]}}$ (3.3) $\displaystyle e_{(x)}(R/\mathfrak{p}_{i})$ $\displaystyle=$
$\displaystyle{\lambda}\left(\frac{R}{(x_{1},\ldots,x_{i-1},x,x_{i+1},\ldots,x_{n})}\right)=1$
(3.4) $\displaystyle e_{\mathfrak{a}}(R_{\mathfrak{p}_{i}})$ $\displaystyle=$
$\displaystyle{\lambda}\left(\frac{R_{\mathfrak{p}_{i}}}{\mathfrak{a}R_{\mathfrak{p}_{i}}}\right)={\lambda}\left(\frac{R_{\mathfrak{p}_{i}}}{\mathfrak{p}_{i}R_{\mathfrak{p}_{i}}}\right)=1$
$\displaystyle e_{(x,\sigma_{1},\ldots,\sigma_{n})}(R)$ $\displaystyle=$
$\displaystyle\lambda\left(\frac{R}{(\sigma_{1},\ldots,\sigma_{n})R}\right)=n!.$
(3.5)

By the equation (3.3), $k_{j}:=j-1$ for all $j=2,\ldots,n$. Hence by
Proposition 3.1

$\displaystyle\prod_{j=2}^{n}k_{j}\sum_{i=1}^{n}e_{\mathfrak{a}}(R_{\mathfrak{p}_{i}})e(x,R/\mathfrak{p}_{i})=(n-1)!n=n!=e_{(x,\sigma_{2},\ldots,\sigma_{n})}(R).$
(3.6)

By Proposition 3.1, equation (3.6) and Theorem 2.4 we conclude that
$\mathcal{R}_{s}(\mathfrak{a})$ is Noetherian.

(3) By (1), (2) and Theorem 2.3, ${\bf a}$ is a set-theoretic complete
intersection. ∎

##  4\. The symbolic Rees algebra of the Fermat ideal

Let $S=\mathbb{C}[x,y,z]$ be the polynomial ring and $\mathfrak{m}=(x,y,z)$.
Let $n\geq 3$ and

$\displaystyle r_{n}:=y^{n}-z^{n},\hskip 14.45377pts_{n}=z^{n}-x^{n},\hskip
14.45377ptt_{n}=x^{n}-y^{n},\text{ and }J_{n}=(xr_{n},ys_{n},zt_{n}).$

The ideal $J_{n}$ is called the Fermat ideal. The ideal $J_{n}$ defines a set
of $n^{2}+3$ points in $\mathbb{P}^{2}$. For $n=3$, this ideal was studied in
[8] in relation with the containment problem. In [23] the authors study the
the symbolic Rees algebra of $J_{n}$.

######  Lemma 4.1.

Let $R=S_{\mathfrak{m}}$ and $I_{n}=J_{n}R$. Then

  1. (1)

$J_{n}$ is a radical ideal.

  2. (2)

$e(R/I_{n})=e(S/J_{n})=n^{2}+3$.

###### Proof.

Let $\eta$ be a primitive $n^{\rm th}$ root of unity. Put
$P_{ij}=(y-\eta^{i}z,z-\eta^{j}x)$ ($i,j=0,\ldots,n-1$), $P_{1}=(y,z)$,
$P_{2}=(x,z)$, $P_{3}=(x,y)$ and

$\displaystyle K_{n}=\bigcap_{i,j=0}^{n-1}P_{ij}R\bigcap P_{1}R\bigcap
P_{2}R\bigcap P_{3}R.$ (4.2)

Then clearly, $I_{n}\subseteq K_{n}$. To prove the lemma it is enough to show
that equality holds. A minimal free resolution of $J_{n}$ is

$\displaystyle 0\rightarrow S[-n-3]\oplus
S[-2n]\xrightarrow{\displaystyle\begin{pmatrix}yz&x^{n-1}\\\ xy&z^{n-1}\\\
-xz&-y^{n-1}\end{pmatrix}}S[-n-1)]^{3}\xrightarrow{\displaystyle\begin{pmatrix}xr^{n}&ys^{n}&zt^{n}\\\
\end{pmatrix}}S\rightarrow\frac{S}{J_{n}}\rightarrow 0.$ (4.3)

Since $\mathfrak{m}$ is a maximal ideal in $S$,
$\operatorname{depth}(S/J_{n})_{\mathfrak{n}}\leq\operatorname{dim}(S/J_{n})_{\mathfrak{m}}=1$.
From (4.3) the projective dimension of $(S/J_{n})_{\mathfrak{m}}=2$. By the
Auslander-Buchsbaum formula, $(S/J_{n})_{\mathfrak{m}}$ is Cohen-Macaulay.
Using the exact sequence (4.3), we obtain the Hilbert series of $S/J_{n}$. Let
$u$ be an indeterminate. Then

$H(S/J_{n},u)=\frac{1-3u^{n+1}+u^{n+3}+u^{2n}}{(1-u)^{3}}$

Since $\operatorname{dim}S/J_{n}=1,$ we can write
$H(S/J_{n},u)=\frac{f(u)}{1-u}$ for some $f(u)\in\mathbb{Z}[u].$ Let
$g(u)=1-3u^{n+1}+u^{n+3}+u^{2n}.$ Then $g(u)=(1-u)^{2}f(u).$ Therefore

$e(S/J_{n})=f(1)=\lim_{u\to 1}\frac{g(u)}{(1-u)^{3}}=n^{2}+3.$

Every minimal prime of $K_{n}$ is also that of $J_{n}.$ Let $a=n^{2}+3$ and

$J_{n}=\bigcap_{i,j=0}^{n-1}Q_{ij}R\bigcap Q_{1}R\bigcap Q_{2}R\bigcap
Q_{3}R\bigcap_{j=a+1}^{b}Q_{j}$

be a reduced primary decomposition of $J_{n}$ and let $Q_{j}$ be
$\mathfrak{p}_{j}$-primary for all $j=1,2,\ldots,b.$ By the associativity
formula for multiplicities, we have

$e(\mathfrak{m},R/I_{n})=\sum_{j=1}^{b}e(\mathfrak{m},R/\mathfrak{p}_{j}){\lambda}((R/I_{n})_{\mathfrak{p}_{j}})=n^{2}+3.$

This shows that $a=b=n^{2}+3$ and $Q_{j}=\mathfrak{p}_{j}$ for all
$j=1,2,\ldots,a.$ Therefore $I_{n}=K_{n}$ and $I_{n}$ is a radical ideal. ∎

######  Example 4.4.

Let $R=S_{\mathfrak{m}}$ and $I_{n}=J_{n}R$. Then

  1. (1)

For all $n\geq 3,$ the analytic spread of $I_{n}$ is $3$.

  2. (2)

For all $k\geq 1$, $R/I_{n}^{(k)}$ is Cohen-Macaulay.

  3. (3)

Put ${\bf a}=I_{n}$. Then

    1. (a)

$\mathcal{R}_{s}({\bf a})$ is Noetherian.

    2. (b)

${\bf a}$ is a set-theoretic complete intersection.

###### Proof.

(1) Suppose that $\ell(I_{n})=2.$ Since $I_{n}$ is a radical ideal and
$R_{\mathfrak{p}}$ is regular for all minimal primes of $I_{n},$ by [6,
Theorem], $I_{n}$ is generated by two elements. This is a contradiction. Hence
$\ell(I_{n})=3.$
(2) Since $I_{n}R$ is a height two ideal of a $3$-dimensional local ring,
$I_{n}^{(k)}$ is an unmixed ideal for all $k\geq 1.$ Hence $R/I_{n}^{(k)}$ is
a Cohen-Macaulay local ring for all $k\geq 1.$

(3a) Put $r=r_{n}$, $s=s_{n}$, $t=t_{n}$, $x_{1}=x+y+z$,
$f_{1}=rs^{n-2}t-2r^{n-2}st$ and
$f_{2}=rs^{n-2}t+2(s^{n-2}r^{2}x^{n}+t^{n-2}s^{2}y^{n}+r^{n-2}t^{2}z^{n}).$ We
claim that $x_{1},f_{1},f_{2}$ is a system of parameters in $R$. We can write

$\displaystyle rs^{n-2}t-2r^{n-2}st=rst(s^{n-3}-2r^{n-3})$

Let $\mathfrak{p}$ is a prime ideal and
$\mathfrak{p}\supseteq(x_{1},f_{1},f_{2})$. If $r\in\mathfrak{p}$, then as
$r+s+t=0$,

$\displaystyle\mathfrak{p}\supseteq(x_{1},r,f_{2})=(x_{1},r,t^{n-2}s^{2}y^{n})=(x_{1},r,(-r-s)^{n-2}s^{2}y^{n})=(x_{1},r,s^{n}y^{n}).$

Let $\eta$ be a primitive $n^{\rm th}$ root of unity. Then we can write

$\displaystyle r$ $\displaystyle=$ $\displaystyle
y^{n}-z^{n}=\prod_{i=1}^{n}(y-\eta^{i}z)$ $\displaystyle s$ $\displaystyle=$
$\displaystyle z^{n}-x^{n}=\prod_{j=1}^{n}(z-\eta^{j}x).$

Hence, for some $i,j=1,\ldots,n$,

$\displaystyle\mathfrak{p}$ $\displaystyle\supseteq$
$\displaystyle(x_{1},y-\eta^{i}z,(z-\eta^{j}x)y)$ (4.5) $\displaystyle=$
$\displaystyle(x+y+z,y-\eta^{i}z,(z-\eta^{j}(x+y+z-y-z))y)$ $\displaystyle=$
$\displaystyle(x+y+z,y-\eta^{i}z,(z-\eta^{j}(-y-z))y)$ $\displaystyle=$
$\displaystyle(x+y+z,y-\eta^{i}z,(z+\eta^{j}(y+z))y)$ $\displaystyle=$
$\displaystyle(x+y+z,y-\eta^{i}z,(z+\eta^{j}(\eta^{i}z+z))(\eta^{i}z))$
$\displaystyle=$
$\displaystyle(x+y+z,y-\eta^{i}z,\eta^{i}z^{2}((1+\eta^{i+j}+\eta^{j})))$
$\displaystyle=$ $\displaystyle(x+y+z,y-\eta^{i}z,z^{2}).$

Hence $\mathfrak{p}=\mathfrak{m}$. Similarly, if $s\in\mathfrak{p}$ or
$t\in\mathfrak{p}$, then using the similar argument as in (4.5) we get
$\mathfrak{p}=\mathfrak{m}$.

We now compute $e_{(x_{1})}(R/(f_{1},f_{2}))$. Since $x,f_{1},f_{2}$ is a
system of parameters,

$\displaystyle e_{(x_{1})}(R/(f_{1},f_{2}))$ (4.6) $\displaystyle=$
$\displaystyle e_{(x_{1},f_{1},f_{2})}(R)$ $\displaystyle=$
$\displaystyle\begin{cases}e_{(x_{1},r,f_{2})}(R)+e_{(x_{1},s,f_{2})}(R)+e_{(x_{1},t,f_{2})}(R)&n=3\\\
e_{(x_{1},r,f_{2})}(R)+e_{(x_{1},s,f_{2})}(R)+e_{(x_{1},t,f_{2})}(R)+e_{(x_{1},s^{n-3}-2r^{n-3},f_{2})}(R)&n>3\\\
\end{cases}.$

Let $n>3$ and $s^{n-3}-2r^{n-3}\in\mathfrak{p}$. Then

$\displaystyle\mathfrak{p}$ $\displaystyle\supseteq$
$\displaystyle(x_{1},s^{n-3}-2r^{n-3},f_{2})$

Let $\zeta$ be the primitive $(n-3)^{\rm th}$ root of unity. Then

$\displaystyle s^{n-3}-2r^{n-3}=\prod_{i=0}^{n-3}(s-\sqrt[n-3]{2}\zeta^{i}r)$

Fix $i$ ($0\leq i<n-3$). Put $\xi=\zeta^{i}$ and $v=s-ar$ where
$a=\sqrt[n-3]{2}\xi r$. Since $r+s+t=0$,

$\displaystyle t$ $\displaystyle=$ $\displaystyle-(r+s)=-(1+a)r-v$
$\displaystyle f_{2}$ $\displaystyle=$ $\displaystyle
rs^{n-2}t+2(s^{n-2}r^{2}x^{n}+t^{n-2}s^{2}y^{n}+r^{n-2}t^{2}z^{n})$ (4.7)
$\displaystyle=$ $\displaystyle
r(v+ar)^{n-2}(-r(1+a)-v)+2(v+ar)^{n-2}r^{2}x^{n}$
$\displaystyle+2(-(1+a)r+v)^{n-2}(ar)^{2}y^{n}+2r^{n-2}(-(1+a)r-v)^{2}z^{n}$
$\displaystyle=$
$\displaystyle-a^{n-2}(1+a)r^{n}+2a^{n-2}x^{n}r^{n}+(-1)^{n-2}2a^{2}(1+a)^{n-2}y^{n}r^{n}+2(1+a)^{2}z^{n}r^{n}+g(v)$
$\displaystyle=$
$\displaystyle[a^{n-2}(1+a)+2a^{n-2}x^{n}+(-1)^{n-2}2a^{2}(1+a)^{n-2}y^{n}+2(1+a)^{2}z^{n}]r^{n}+g(v),$

where $g(v)$ denotes the terms in $f_{2}$ which involve positive powers of
$v$.

Since
$a^{n-2}(1+a)+2a^{n-2}x^{n}+(-1)^{n-2}2a^{2}(1+a)^{n-2}y^{n}+2(1+a)^{2}z^{n}$
is a unit in $R$, if $v\in\mathfrak{p}$, then
$\mathfrak{p}\supseteq(x_{1},v,f_{2})=(x_{1},v,r^{n})=(x+y+z,s-ar,r^{n})$.
Hence, $\mathfrak{p}\supseteq(x+y+z,s,r)$ which implies that for some
$j,k=0,\ldots,n-1$, $\mathfrak{p}\supseteq(x+y+z,z-\eta^{j}x,y-\eta^{k}z)$.
Hence $\mathfrak{p}=\mathfrak{m}$.

Since $r+s+t=0$, $(x_{1},r,f_{2})=(x_{1},r,t^{n-2}s^{2}y^{n})$,
$x_{1},r,f_{2}$ is a system of parameters. Hence

$\displaystyle e_{(x_{1},r,f_{2})}(R)$ $\displaystyle=$ $\displaystyle
e_{(x_{1},r,t^{n-2}s^{2}y^{n})}(R)$ (4.8) $\displaystyle=$ $\displaystyle
e_{(x_{1},r,y^{n})}(R)+e_{(x_{1},r,t^{n-2})}(R)+e_{(x_{1},r,s^{2})}(R)$
$\displaystyle=$ $\displaystyle n^{2}+n^{2}(n-2)+2n^{2}=n^{3}+n^{2}.$

Similarly,

$\displaystyle e(x_{1},s,f_{2})=e(x_{1},t,f_{2})=n^{3}+n^{2}.$ (4.9)

We now compute $e(x_{1},s^{n-3}-2r^{n-3},f_{2})$. Since
$x_{1},s^{n-3}-2r^{n-3},f_{2}$ is a system of parameters,

$\displaystyle
e(x_{1},s^{n-3}-2r^{n-3},f_{2})=ne(x_{1},s^{n-3}-2r^{n-3},r)=ne(x_{1},s^{n-3},r)=n^{3}(n-3).$
(4.10)

Combining (4.6), (4.8), (4.9) and (4.10) we get that for all $n\geq 3$

$e_{(x_{1})}(R/(f_{1},f_{2}))=3(n^{3}+n^{2})+n^{4}-3n^{3}=n^{4}+3n^{2}=n^{2}(n^{2}+3).$
(4.11)

As $f_{1},f_{2}\in I^{(n)}$, $k_{1}=k_{2}=n$ and we have

$\left(\displaystyle{\prod_{j=1}^{2}k_{j}}\right)\displaystyle{\sum_{i=1}^{n^{2}+3}e_{{\bf
a}R_{P_{i}}}(R_{P_{i}})e_{(x_{1})}(R/P_{i})}=n^{2}(n^{2}+3)$ (4.12)

Then by (4.11) and Theorem 2.4 we have $\mathcal{R}_{s}({\bf a})$ is
Noetherian.

(3b) follows from (2), (3a) and Theorem 2.3. ∎

##  5\. The Jacobian ideal of hyperplane arragements

This example was motivated by the paper by J. Migliore, U. Nagel, and H.
Schenck [20].

######  Example 5.1.

Let $S=k[x,y,z,w]$ and $f=w(x+y)(x+y+z+w)$ and $(J(f))S$, the Jacobian ideal
of $f$. Put $R=S_{\mathfrak{m}}$ where $\mathfrak{m}=(x,y,z,w)$ and ${{\bf
a}}=(J(f))R$. Then

  1. (1)

${{\bf a}}$ is a height two unmixed ideal and $R/{{\bf a}}^{(n)}$ is Cohen-
Macaulay for all $n\geq 1$.

  2. (2)

The symbolic Rees algebra $\mathcal{R}_{s}({{\bf a}})$ is Noetherian.

  3. (3)

${{\bf a}}$ is a set-theoretic complete intersection.

###### Proof.

(1) One can verify that $f_{x}=f_{y}$ and

$\displaystyle{{\bf a}}$ $\displaystyle=$
$\displaystyle(f_{x}=2xw+2yw+zw+w^{2},f_{z}=xw+yw,f_{w}=x^{2}+2xy+y^{2}+xz+yz+2xw+2yw)$
$\displaystyle=$ $\displaystyle(zw+w^{2},x^{2}+2xy+y^{2}+xz+yz,xw+yw)$
$\displaystyle=$ $\displaystyle(w(z+w),(x+y)(x+y+z),w(x+y))$ $\displaystyle=$
$\displaystyle(z+w,x+y)\cap(w,x+y)\cap(w,x+y+z).$

Put $\mathfrak{p}_{1}=(z+w,x+y)$, $\mathfrak{p}_{2}=(w,x+y)$ and
$\mathfrak{p}_{3}=(w,x+y+z)$. Then

$\displaystyle{{\bf
a}}^{(n)}=\mathfrak{p}_{1}^{n}\cap\mathfrak{p}_{2}^{n}\cap\mathfrak{p}_{3}^{n}.$

Consider the exact sequences:

$\displaystyle
0{\longrightarrow}\frac{R}{\mathfrak{p}_{1}^{n}\cap\mathfrak{p}_{2}^{n}}{\longrightarrow}\frac{R}{\mathfrak{p}_{1}^{n}}\oplus\frac{R}{\mathfrak{p}_{2}^{n}}{\longrightarrow}\frac{R}{\mathfrak{p}_{1}^{n}+\mathfrak{p}_{2}^{n}}{\longrightarrow}0,$
(5.2) $\displaystyle
0{\longrightarrow}\frac{R}{\mathfrak{p}_{1}^{n}\cap\mathfrak{p}_{2}^{n}\cap\mathfrak{p}_{3}^{n}}{\longrightarrow}\frac{R}{\mathfrak{p}_{1}^{n}\cap\mathfrak{p}_{2}^{n}}\oplus\frac{R}{\mathfrak{p}_{3}^{n}}{\longrightarrow}\frac{R}{(\mathfrak{p}_{1}^{n}\cap\mathfrak{p}_{2}^{n})+\mathfrak{p}_{3}^{n}}{\longrightarrow}0.$
(5.3)

Note that $R/\mathfrak{p}_{1}^{n}$, $R/\mathfrak{p}_{2}^{n}$ and
$R/\mathfrak{p}_{3}^{n}$ are Cohen-Macaulay. Since

$\displaystyle\mathfrak{p}_{1}^{n}+\mathfrak{p}_{2}^{n}$ $\displaystyle=$
$\displaystyle((x+y)^{n-i}(w^{i},(z+w)^{i}):i=0,\ldots,n)$

$\operatorname{ht}(\mathfrak{p}_{1}^{n}+\mathfrak{p}_{2}^{n})=3$ and
$\sqrt{\mathfrak{p}_{1}^{n}+\mathfrak{p}_{2}^{n}}=(x+y,z,w)$. We claim that
$x$ is a nonzerodivisor on $R/\mathfrak{p}_{1}^{n}+\mathfrak{p}_{2}^{n}$.
After a change of variables let $y^{\prime}=x+y$, $z^{\prime}=w+z$. Then we
have

$\displaystyle(\mathfrak{p}_{1}^{n}+\mathfrak{p}_{2}^{n})S$ $\displaystyle=$
$\displaystyle((y^{\prime})^{n-i}(w^{i},(z^{\prime})^{i}):i=0,\ldots,n)\subseteq
k[x,y^{\prime},z^{\prime},w],$
$\displaystyle((\mathfrak{p}_{1}^{n}\cap\mathfrak{p}_{2}^{n})+\mathfrak{p}_{3}^{n})S$
$\displaystyle=$
$\displaystyle((z^{\prime},y^{\prime})^{n}\cap(w,y^{\prime})^{n})+(w,y^{\prime}+z^{\prime})^{n}\subseteq
k[x,y^{\prime},z^{\prime},w].$

We can write $\mathfrak{p}_{1}^{n}+\mathfrak{p}_{2}^{n}=\cap Q_{j}$ (finite
intersection) where each $Q_{j}$ is generated by pure powers of the variables
(see [14, Theorem 1.3.1]. Then as
$(y^{\prime})^{n},(z^{\prime})^{n},w^{n}\in\mathfrak{p}_{1}^{n}+\mathfrak{p}_{2}^{n}$
and $x$ does not occur in any of generators of
$\mathfrak{p}_{1}^{n}+\mathfrak{p}_{2}^{n}$, each $Q_{j}$ is of the form
$((y^{\prime})^{n_{1,j}},(z^{\prime})^{n_{2,j}},w^{n_{3,j}})$ for some
${n_{1,j}},{n_{2,j}}{n_{3,j}}\in\mathbb{N}$. Since
$\sqrt{\mathfrak{p}_{1}^{n}+\mathfrak{p}_{2}^{n}}S=\cap\sqrt{Q_{j}}S$ and
$\sqrt{Q_{j}}S=(y^{\prime},z^{\prime},w)$ for all $j$, localizing we conclude
that $(y^{\prime},z^{\prime},w)R$ is the only associated prime of
$(\mathfrak{p}_{1}^{n}+\mathfrak{p}_{2}^{n})R$ and hence $x$ is a
nonzerodivisor on $R/(\mathfrak{p}_{1}^{n}+\mathfrak{p}_{2}^{n})$. By depth
lemma applied to the exact sequence 5.2, we see that
$R/(\mathfrak{p}_{1}^{n}\cap\mathfrak{p}_{2}^{n})$ is Cohen-Macaulay.
Similarly, we conclude that $x$ is a nonzerodivisor on
$R/(\mathfrak{p}_{1}^{n}\cap\mathfrak{p}_{2}^{n})+\mathfrak{p}_{3}^{n}$. By
the depth lemma applied to the exact sequence (5.3), it follows that $R/{\bf
a}^{(n)}=R/(\mathfrak{p}_{1}^{n}\cap\mathfrak{p}_{2}^{n}\cap\mathfrak{p}_{3}^{n})$
is Cohen-Macaulay.

(2) Put $x_{1}=x$, $x_{2}=z$, $g_{1}=w(z+w)$, $g_{2}=w(x+y)$,
$g_{3}=(x+y)(x+y+z)$, $f_{1}=g_{1}+g_{3}$ and $f_{2}=f=w(x+y)(x+y+z+w)$. Then
$f_{1}\in{{\bf a}}$. As

$\displaystyle zf_{2}$ $\displaystyle=$ $\displaystyle
zw(x+y)(x+y+z+w)=g_{1}g_{2}-g_{2}g_{3},$

$zf_{2}\in{\bf a}^{2}\subseteq{{\bf a}}^{(2)}\subset\sqrt{{{\bf
a}}^{(2)}}=\mathfrak{p}_{1}\cap\mathfrak{p}_{2}\cap\mathfrak{p}_{3}$. Since
$z\not\in\mathfrak{p}_{i}$, for $i=1,2,3$, $f_{2}\in{{\bf a}}^{(2)}$. Now,

$\displaystyle e_{(x_{1},x_{2},f_{1},f_{2})}(R)$ $\displaystyle=$
$\displaystyle e_{(x,z,g_{1}+g_{3},f)}(R)$ (5.4) $\displaystyle=$
$\displaystyle e_{(x,z,w(z+w)+(x+y)(x+y+z),w(x+y)(x+y+z+w))}(R)$
$\displaystyle=$ $\displaystyle e_{(x,z,w^{2}+y^{2},wy(y+w))}(R)$
$\displaystyle=$ $\displaystyle
e_{(x,z,w^{2}+y^{2},y)}(R)+e_{(x,z,w^{2}+y^{2},w)}(R)+e_{(x,z,w^{2}+y^{2},y+w)}(R)$
$\displaystyle=$ $\displaystyle 6.$

Since $k_{1}=1$ and $k_{2}=2$, we have

$\displaystyle\left(\displaystyle{\prod_{j=1}^{2}k_{j}}\right)\displaystyle{\sum_{i=1}^{s}e_{{\bf
a}R_{P_{i}}}(R_{P_{i}})e_{(x_{1},x_{2})}(R/P_{i})}$ $\displaystyle=$
$\displaystyle 1\cdot 2\cdot\sum_{i=1}^{3}e_{{\bf
a}R_{P_{i}}}(R_{P_{i}})e_{(x_{1},x_{2})}(R/\mathfrak{p}_{i})$ (5.5)
$\displaystyle=$ $\displaystyle
2\cdot\sum_{i=1}^{3}e_{{\mathfrak{p}_{i}}{R_{\mathfrak{p}_{i}}}}(R_{\mathfrak{p}_{i}})e_{(x,z)}(R/\mathfrak{p}_{i})$
$\displaystyle=$ $\displaystyle 2\cdot 3$ $\displaystyle=$ $\displaystyle 6.$

By (5.4), (5.5) and Theorem 2.8, $R_{s}({\bf a})$ is Noetherian.

(3) Since $R/{\bf a}^{(n)}$ is Cohen-Macaulay for $n\geq 1$ and $R_{s}({\bf
a})$ is Noetherian by Theorem 2.3, ${\bf a}$ is set theoretic complete
intersection. ∎

Acknoledgement: We thank Prof. S. Goto for many useful discussions during his
visit at IIT Bombay.

## References

  * [1] C. Bocci, S. Cooper, E. Guardo, B. Harbourne, M. Janssen, U. Nagel, A. Seceleanu, A. Van Tuyl and T. Vu, The Waldschmidt constant for squarefree monomial ideals, J. Algebraic Combin. 44 (2016), no. 4, 875-904.
  * [2] E. Böger, Minimalitätsbedingungen in der Theorie der Reduktionen von Idealen, Schr. Math. Inst. Univ. Münster, 40 (1968).
  * [3] H. Bresinsky, H. Monomial space curves in $\mathbb{A}^{3}$ as set-theoretic complete intersections, Proc. Amer. Math. Soc. 75 (1979), no. 1, 23-24.
  * [4] L. Burch. Codimension and analytic spread, Proc. Cambridge Philos. Sot. 72 (1972), 369-373.
  * [5] R. C. Cowsik, Symbolic powers and number of defining equations, Algebra and its applications (New Delhi, 1981), Lecture Notes in Pure and Appl. Math., 91, Dekker, New York, 1984.
  * [6] R. C. Cowsik and M. V. Nori, On the fibres of blowing up, Journal of the Indian Math. Soc. 40 (1976),217-222.
  * [7] R. C. Cowsik and M. V. Nori, Affine curves in characteristic $\mathtt{p}$ are set-theoretic complete intersections. Invent. Math. 45 (1978), 111-114.
  * [8] M. Dumnicki, T. Szemberg, Tomasz; H. Tutaj-Gasińska, Counterexamples to the $I^{(3)}\subseteq I^{2}$ containment. J. Algebra 393 (2013), 24-29.
  * [9] D. Eisenbud, Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995.
  * [10] S. Goto, M. Herrmann, K. Nishida and O. Villamayor, On the structure of Noetherian symbolic Rees algebras, Manuscripta Math. 67 (1990), 197-225.
  * [11] S. Goto and K. Nishida, The Cohen-Macaulay and Gorenstein Rees algebras associated to filtrations, Mem. Amer. Math. Soc. 110 (1994), 1–134.
  * [12] S. Goto, Shiro; K. Nishida, K. Watanabe, Non-Cohen-Macaulay symbolic blow-ups for space monomial curves and counterexamples to Cowsik’s question. Proc. Amer. Math. Soc. 120 (1994), 383-392.
  * [13] Eloísa Grifo and Alexandra Seceleanu, Symbolic Rees algebras, Commutative Algebra, Ed. Irena Peeva, Springer, 2021.
  * [14] J. Herzog and T. Hibi, Monomial Ideals, Grad. Texts in Math. 260, Springer, London, 2010.
  * [15] J. Herzog, T, Hibi and N. V. Trung, Symbolic powers of monomial ideals and vertex cover algebras. Adv. Math., 210 (2007), 304-322
  * [16] C. Huneke, On the finite generation of symbolic blow-ups, Math. Z. 179 (1982), 465-472.
  * [17] C. Huneke, Hilbert functions and symbolic powers, Michigan Math. J. 34 (1987), 293-318
  * [18] D. Katz and L.J. Ratliff Jr., On the symbolic Rees ring of a primary ideal. Comm. in Alg. 14 (1986), 959-970.
  * [19] Matsumura,H., Commutative Ring Theory, Cambridge University Press, 1989.
  * [20] J. Migliore, U. Nagel, and H. Schenck: Schemes supported on the singular locus of a hyperplane arrangement in $\mathbb{P}^{n}$. Int. Math. Res. Not. IMRN (2022), 140-170.
  * [21] M. Morales, Noetherian symbolic blow-ups, J. Algebra 140 (1991), 12-25.
  * [22] M. Nagata. Local rings Interscience Tracts in Pure and Applied Mathematics, No. 13 Interscience Publishers a division of John Wiley & Sons. New York-London 1962.
  * [23] U. Nagel and A. Seceleanu, Ordinary and symbolic Rees algebras for ideals of Fermat point configurations,  Journal of Algebra 468 (2016), 80-102.
  * [24] A. Ooishi, Noetherian property of symbolic Rees algebras, Hiroshima Math. J. 15, (1985), 581-584.
  * [25] D. Rees, On a problem of Zariski, Illinois J. Math. 2 (1958), 145-149.
  * [26] P. Roberts, A prime ideal in a polynomial ring whose symbolic blow-up is not Noetherian, Proc. Amer. Math. Soc. 94 (1985), no. 4, 589-592.
  * [27] A. Sannai and H. Tanaka, Infinitely generated symbolic Rees algebras over finite fields, Algebra Number Theory 13 (2019), 1879-1891.
  * [28] P. Schenzel, Examples of Noetherian symbolic blow-up rings, Rev. Roumaine Math. Pures Appl. 33 (1988), 375-383.
  * [29] P. Schenzel, Filtrations and Noetherian symbolic blow-up rings, Proc. Amer. Math. Soc. 102 (1988), 817-822.
  * [30] A.Van Tuyl, A beginners guide to edge and cover ideals. Monomial ideals, computations and applications,, 63-94, Lecture Notes in Math., 2083, Springer, Heidelberg, 2013.
  * [31] Oscar Zariski. A fundamental lemma from the theory of holomorphic functions on an algebraic variety,  Ann. Mat. Pura Appl. (4) 29 (1949), 187-198.