数理解析研究所講究録
第 1562 巻 2007 年 104-113

104

Arithmetical rank of squarefree monomial ideals
名古屋大学・大学院多元数理科学研究科

木村杏子 (Kyouko KIMURA)

Graduate School of Mathematics
Nagoya University
佐賀大学・文化教育学部
寺井直樹 (Naoki TERAI)

Department of Mathematics
Faculty of Culture and Education
Saga University
名古屋大学・大学院多元数理科学研究科
吉田健一 (Ken-ichi YOSHIDA)

Graduate School of Mathematics
Nagoya University

INTRODUCTION
This report complements our paper [4]. Throughout this report, is an
infinite field and is a polynomial ring over . We denote variables of by
and $y_{i}(i=1,2, \ldots)$ . Let denote a squarefree monomial ideal (i.e., the ideal
generated by monomials in which the exponent of each variable is at most 1).
For example, $I=(x_{1}x_{2}, x_{2}x_{3}x_{4}, x_{1}x_{4}x_{5})$ is a squarefree monomial ideal.
The arithmetical rank of , denoted by ara , is defined by
$k$

$R$

$R$

$k$

$x_{i}$

$I$

$I$

$I$

ara $I= \min\{r$ : there exists$a_{1},$$\ldots$,$a_{r}\in I$, )such that $\sqrt{(a_{1}}$$a_{r}$$=\sqrt{I}\}$.

That is, ara is the minimal number of elements in which generate up
to radical. The arithmetical rank has the following geometric interpretation.
Assume is an algebraic closed field, and put $R=k[x_{1}, \ldots, x_{n}]$ . Then thealgebraic variety associated to is defined by
$I$

$I$

$I$

$k$

$I$

$V(I)=$ {$(z_{1},$$\ldots$,$z_{n})\in k^{n}$: $f(z_{1},$$\ldots,$$z_{n})=0$, )As $V(\sqrt{I})=V(I)$ , if $r=araI$ and $\sqrt{(a_{1}}$$a_{r}$(0.1)

for all $f\in I$ }.

$=\sqrt{I}$, then$V(I)=V((a_{1}, \ldots, a_{r}))=V((a_{1}))\cap\cdots\cap V((a_{r}))$.

So, $V(I)$ can be written as an intersection ofjust hypersurfaces set-theoretically.
Moreover, (0.1) shows an importance to know explicitly elements generate
up to radical.
In general, it is difficult to determine the arithmetical rank. If we findelements which generate up to radical, then such gives an upper bound, the minimal number of generators of , isfor ara , and in particular,a trivial upper bound. On the other hand, the following fact is known (seeLyubeznik [5]).
$r$

$I$

$r$

$r$

$I$

$I$

$r$

$\mu(I)$$I$105Fact 0.1. If I is a squarefree monomial ideal, then(0.2)

$pd_{R}R/I\leq araI$ ,where $pd_{R}R/I$ is the projective dimension of $R/I$ .
The projective dimension is easy to compute. So, the importance of thisinequality is to give a lower bound for the arithmetical rank. Here, we consider
the following problem.

Problem 0.2. Does ara $I=pd_{R}R/I$ hold?
If $\mu(I)$ -height $I=0$ , then the problem is trivially true. Moreover, it is ako
known that ara $I=pd_{R}R/I$ holds in the case $\mu(I)$ –height $I=1,2$ ; see [4].
But, in general, there is a counter-example for this problem.
Example 0.3 ([11]). Let be the Stanley-Reisner ideal of Reisner’s triangu(see Figure 1). That is, is the squarefree monomial ideal inlation of$I$$\mathbb{P}^{2}(\mathbb{R})$$I$

FIGURE 1. Reisner’s triangulation of$k[x_{1}, \ldots, x_{6}]$$\mathbb{P}^{2}(\mathbb{R})$generated by following 10 monomials:$x_{1}x_{2}x_{3},$ $x_{1}x_{2}x_{4},x_{1}x_{3}x_{5},$ $x_{1}x_{4}x_{6},x_{1}x_{5}x_{6},$ $x_{2}x_{3}x_{6},$ $x_{2}x_{4}x_{5},$ $x_{2}x_{5}x_{6},$ $x_{3}x_{4}x_{5},x_{3}x_{4}x_{6}$.

Then, $\mu(I)=10$ and height $I=3$ , so, the difference is rather big. If thecharacter of is not 2, then $R/I$ is Cohen-Macaulay and $pd_{R}R/I=3$ . But Z.
Yan [11] showed ara $I=4$ using the \’etale cohomology. Therefore $pd_{R}R/I<$
$k$

ara .
$I$

Now let us explain the organization of this report. In Section 1, we recall
the notion of the Alexander duality, and explain the following inequality:
indeg $I\leq regI\leq arith\deg I$ .
In Section 2, we prove the main theorem of this report, which asserts that
Problem 0.2 is true in the case arithdeg $I=reg$ by giving ara generators(up to radical). See also [4, Theorem 4.1]. In Section 3, we construct anotherara generators in special cases, which is different from ones constructed in
Section 2. These generators do not contain no redundant elements in somesense. Finally, as an appendix, we consider the analytic spread in the casearithdeg $I=in\deg I$ . Note that contents in Section 3 and Appendix A are notincluded in [4].
$I$

$I$

$I$

106
1. ALEXANDER DUALITY
In this section, we recall the notion of the Alexander duality and introduce
an inequality corresponding to (0.2).
be a simpli cial
Set $R=k[x_{1}, \ldots, x_{n}]$ and $[n]=\{1, \ldots, n\}$ . Let$[n]$$i\in[n];(b)F\in\Delta$,, that is, (a)for allcomplex with the vertex set$G\subset F$ implies $G\in\Delta$ . The Alexander dual complex of, is, denoted bydefined by$\Delta^{*}=\{F\subset[n] :[n]\backslash p\not\in\Delta\}$ ,associated to is defined byand the Stanley-Reisner ideal$\Delta\subset 2^{[n]}$$\{i\}\in\Delta$$\Delta^{*}$$\Delta$$-$$\Delta$$I_{\Delta}\subset R$).
: $1\leq i_{1}<\cdots<i_{r}\leq n$ such that
It is clear that is a squarefree monomial ideal. Conversely, for any squarefree monomial ideal $I\subset R$ , there exists the unique simplicial complex on$[n]$ such thatwhen indeg $I\geq 2$ . Then if height $I\geq 2$ , we can define, the Alexander dual ideal of . It is known that $I^{**}=I$ .
We shall see the correspondence between Alexander dual ideals and originalones. If admits the prime decomposition($I_{\Delta}=$$\{i_{1},$$x_{i_{1}}\cdots x_{i_{r}}$$\ldots,$$i_{r}\}\not\in\Delta$$I_{\Delta}$$\Delta$$I=I_{\Delta}$$I$$I^{*}=I_{\Delta^{*}}$$I$$I= \bigcap_{l=1}^{q}(x_{t_{\ell 1}},x_{t\ell 2},$ $\ldots,x_{t_{\ell j\ell}})$then $I^{*}=$$(m_{1}, \ldots , m_{q})$, where$m \ell=\prod_{i=1}^{j\ell}x_{t_{\ell i}}$,

. It is easy to see that$\mu(I^{*})=$and height $I=in\deg I$ “, where $Ass_{R}R/I$ is the set of the associatedprime ideal of $R/I$ and indeg , the initial degree of , is the minimal degree ofminimal generators of . Since is a squarefree monomial ideal, the anithmetic
degree of , denoted by arithdeg , is equal to $\#Ass_{R}R/I$ .
$\#Ass_{R}R/I$

$I$

$I$

$I$

$I$

$I$

$I$

Example 1.1. Consider$I=(x_{1}, x_{2})\cap(x_{2}, x_{3}, x_{4})\cap(x_{1}, x_{4}, x_{5})=(x_{1}x_{2},x_{1}x_{3}, x_{1}x_{4},x_{2}x_{4}, x_{2}x_{5})$,then$I^{*}=(x_{1}x_{2}, x_{2}x_{3}x_{4}, x_{1}x_{4}x_{5})=(x_{1}, x_{2})\cap(x_{1},x_{3})\cap(x_{1},x_{4})\cap(x_{2},x_{4})\cap(x_{2}, x_{5})$.

So, $\mu(I^{*})=arith\deg I=3$ and height $I=in\deg I^{*}=2$ .

We now recall the following inequalities:
height $I\leq pd_{R}R/I\leq\mu(I)$ .

(1.1)

Then the notion which corresponds to the projective dimension is the regularity
reg of :reg $I= \max\{j-i$ : $(Tor_{i}^{R}(I, k))_{j}\neq 0\}$ .
$I$

$I$

Theorem 1.2 (N. Terai [9, Corollary 0.3]). Let I be a squarefree monomial
ideal with height $I\geq 2$ . Then we have

reg $I^{*}=pd_{R}R/I$ .
From (1.1), we obtain the following corollary.

107-Terai[2, TheoCorollary 1.3 (Hoa-Trung [3, Theorem 1.1],
rem 3.8]). Let I be a squarefree monomial ideal. Then we have
$R\text{\"{u}} hbis- Kr\ddot{u}ger$

indeg $I\leq regI\leq arith\deg I$ .

(1.2)

2. MAIN THEOREM
We consider Problem 0.2 in the case arithdeg $I=regI$ .
Theorem 2.1 ([4, Theorem 4.1]). Let I be a squarefree monomial ideal Uttharithdeg $I=reg$ I. Then we haveara $I=pd_{R}R/I$ .
Remark 2.2. This theorem has been already proved by Terai [10, Theorem 3.3],
but our proof gives ara generators of up to radical.
$I$

$I$

Remark 2.3. The case arithdeg $I=$ indeg , which is contained in this casebecause of (1.2), is solved by Schenzel-Vogel [7] and Schmitt-Vogel [8]. In thiscase, ara generators have been already known; see Section 3.
$I$

$I$

From now on, we prove Theorem 2.1. We use the following lemma.
Lemma 2.4 (Hoa-Trung [3, Theorem 2.6]). Let I be as in Theorem 2.1. Then
I can be rewri tten in the following form by changing the notation of variables
in :
$R$

$I=(y_{1}, x_{t_{11}}, \ldots, x_{t_{1j_{1}}})\cap(y_{2}, x_{t_{21}}, \ldots, x_{t_{2j_{2}}})\cap\cdots\cap(y_{q}, x_{t_{q1}}, \ldots, x_{t_{qjq}})$where$y_{\ell}$and$x_{t}$

are variables of , and$R$$y_{\ell}$is different from other$y_{\ell’}$andFrom this lemma, we also have
$pd_{R}R/I=\#\{x_{t_{11}},$ $\ldots,$ $x_{t_{1j_{1}}},x_{t_{21}},$ $\ldots,$ $x_{t_{2j_{2}}},$ $\ldots,$ $x_{t_{q1}},$ $\ldots,$ $x_{t_{qjq}}\}+1$

Now to prove the theorem, it is enough to find $pd_{R}R/I$ generators.

Proof of Theorem 2.1. By Lemma 2.4, we can write$I=Q_{1}\cap\cdots\cap Q_{q}$,We denote the number of variables$Q_{\ell}=(y_{l}, x_{t_{\ell 1}}, \ldots, x_{t_{\ell j\ell}})$$x_{t}$

appearing in$I$.

by , that is,$s$$s=\#\{x_{t_{11}},$ $\ldots,$ $x_{t_{1j_{1}}},$ $\ldots,$ $x_{t_{q1}},$ $\ldots,x_{t_{qjq}}\}$.

Then $pd_{R}R/I=s+1$ . Set$P_{\epsilon-\ell}= \{x_{11}\cdots x_{i\ell}\prod_{j}y_{j}$

where

$j$

runs through: $1\leq i_{1}<..$ .

$x_{1_{1}}\cdots x_{i_{\ell}}\not\in Q_{j}$$g_{\ell}= \sum_{a\in P_{\ell}}a$,$<i_{\ell}\leq s\}$, and set
$P= \bigcup_{\ell=0}^{\epsilon}P_{\ell}$

.

,

,

$0\leq\ell\leq s$,.

$x_{t}$.

108
Then Schmitt-Vogel lemma (Lemma 2.5) means $\sqrt{(g_{0},g_{1}}$, ) $=\sqrt{(P)}$ .
$P$$=\sqrt{I}$$\sqrt{(g_{0},g_{1}}$generates , we have
Since
, )
. Therefore ara $I\leq$$pd_{R}R/I$ . This complete the proof.
$g_{s}$$I$$g_{s}$$\square$Lemma 2.5 (Schmitt-Vogel [8, Lemma, pp.249]). Let be a ring and isa finite subset of R. Suppose subsets
of satisfy the following
conditions:$R$$P_{0},$ $P_{1},$

$P$

$P$

$P_{s}$$\ldots,$(SV-1);$\#
P_{0}=1$(SV-2);(SV-3) For all $p(0<\ell\leq s)$ and for all $a”\in Pp,$ $a\neq a’’$ , there vist$(0\leq\ell’<\ell)$ andsuch that $a\cdot a’’\in(a’)$ .
Then setting $g_{\ell}= \sum_{a\in P_{\ell}}a^{e(a)}(\ell=0,1, \ldots, s)$ , where $e(a)$ is an arbitrary, we haveelement in$P= \bigcup_{\ell=0}^{s}P_{p}$$\ell’$$a,$$a’\in P_{\ell’}$$\mathbb{Z}_{>0}$$\sqrt{(g_{0},g_{1},,g_{s})}=\sqrt{(P)}$.

Example 2.6. Consider$\#\{x_{1},x_{2}, x_{3}\}+1=4$ . In this case,$I=(y_{1}, x_{1}, x_{2})\cap(y_{2}, x_{1}, x_{3})\cap(y_{3}, x_{3})$$P_{0}=\{x_{1}x_{2}x_{3}\}$,$P_{1}=\{x_{1}x_{2}y_{3}, x_{1}x_{3},x_{2}x_{3}\}$$P_{2}=\{x_{1}y_{3},x_{2}y_{2}y_{3}, x_{3}y_{1}\}$$P_{3}=\{y_{1}y_{2}y_{3}\}$. Then $pd_{R}R/I=$

,

,

.

Let check conditions of Schmitt-Vogel lemma. From our setting, (SV-1) and(SV-2) are clear. We shall see (SV-3). For example, we take,then their product is, and.
$x_{1}x_{2}y_{3},$ $x_{1}x_{3}\in P_{1}$$x_{1}x_{2}y_{3}\cdot x_{1}x_{3}=x_{1}^{2}x_{2}x_{3}y_{3}\in(x_{1}x_{2}x_{3})$Take$x_{1}y_{3},$$x_{2}y_{2}y_{3}\in P_{2}$$x_{1}x_{2}x_{3}\in P_{0}$, then their product is.
andincrease the variety of variables ,, then each elements $a,a”\in P_{\ell}$ also$x_{1}y_{3}\cdot x_{2}y_{2}y_{3}=x_{1}x_{2}y_{2}y_{3}^{2}\in(x_{1}x_{2}y_{3})$Thus, the product of 2 elements $a,$and if the element
divisible bydivisible by the same variable .
Moreover, if we set,$x_{1}x_{2}y_{3}\in P_{1}$$a”\in P_{\ell}$$a’\in P_{\ell’}$$y_{j}$$x_{i}$$y_{j}$,$e(x_{1}x_{2}y_{3})=e(x_{2}y_{2}y_{3})=2$ ,$e(x_{1}x_{2}x_{3})=e(y_{1}y_{2}y_{3})=1$$e(x_{1}x_{3})=e(x_{2}x_{3})=e(x_{1}y_{3})=e(x_{3}y_{1})=3$,

then we have homogeneous generators.

3. IRREDUNDANT GENERATORS IN THE CASE
arithdeg $I=regI=in\deg I+1$
In the previous section we constructed ara generators in the case arithdeg $I=$
reg . However, we needed many “redundant” elements in there in somesense; see Example 3.3. In this section, we will give another generators
$I$

$I$

$I$

109
which consists of irredundant elements of in the case arithdeg $I=regI=$
indeg $I+1$ .
Before stating our result, we now consider the case arithdeg $I=$ indeg .
Notice that this condition implies that arithdeg $I=regI$ . Thus our methodin the previous section (see the proof of Theorem 2.1) gives at least one ara
generators of (up to radical). On the other hand, if arithdeg $I=in\deg I$ ,then it is known that can be written by the following form:
$I$

$I$

$I$

$I$

$I$

$I=(x_{11}, \ldots, x_{1j_{1}})\cap(x_{21}, \ldots, x_{2j_{2}})\cap\cdots\cap(x_{q1}, \ldots, x_{qj_{q}})$,,are distinct variables of .
where
Schenzel-Vogel [7] and Schmitt-Vogel [8] showed that any squarefree monomial
ideal with arithdeg $I=$ indeg satisfies ara $I=pd_{R}R/I$ using this fact.
Indeed, Schenzel-Vogel [7, Lemma 2] showed that such an ideal satisfies$pd_{R}R/I=s+1$ , where $s= \sum_{i=1}^{q}j_{1}-q$ , and Schmitt-Vogel [8, Proposition,
pp.248] showed that if we set$x_{11},$$x_{21},$$\ldots,$ $x_{1j_{1}},$$\ldots,$ $x_{2j_{2}},$ $\ldots,$ $x_{q1},$$R$$x_{qj_{q}}$$\ldots$

$I$

$I$

$I$

$P_{\ell}=\{x_{1\ell_{1}}x_{2\ell_{2}}\cdots x_{q^{p_{q}}} :\ell_{1}+\ell_{2}+\cdots+\ell_{q}=q+\ell\}$,$g_{\ell}= \sum_{a\in P_{\ell}}a$,, , then $\sqrt{(g_{0},g_{1}}$, ).
for $P=0,1,$
Each polynomial appearing in the ara generators given by SChmitt-Vogelis described as a sum of several elements in the minimal set of monomial
generators of . Therefore they consist of irredundant terms in some sense.
We now consider the case arithdeg $I=regI=in\deg I+1$ . In this case, we
can classify the following two cases; see [4, Lemma 5.2].
$\ldots$$s$$g_{\epsilon}$$=\sqrt{I}$
$I$

$I$

case l:$I_{1}$$=$$(x_{11}, x_{12}, \ldots , x_{1j_{1}})\cap(x_{21}, x_{22}, \ldots, x_{2j_{2}})\cap\cdots\cap(x_{q1}, x_{q2}, \ldots, x_{qj_{q}})$$\cap(x_{q+11},$ $x_{q+12},$$\ldots,$$x_{q+1j_{q+1}}$,$x_{11},$ $x_{12},$ $\ldots,$ $x_{1i_{1}},$ $x_{21},$ $x_{22},$ $\ldots,$ $x_{2i_{2}},$ $\ldots,$ $x_{p1},$ $x_{p2},$ $\ldots,$

where $1\leq p\leq q,$ $1\leq ip<jp(\ell=1,2, \ldots,p),$case 2:$I_{2}$$=$$j_{p+1},$$\ldots$$x_{\dot{\mu}_{p}}$),, $j_{q},j_{q+1}\geq 1$ .

$(x_{11}, x_{12}, \ldots, x_{1j_{1}}, y_{1}, y_{2}, \ldots , y_{p})$$\cap(x_{21}, x_{22}, \ldots, x_{2j_{2}}, y_{1}, y_{2}, \ldots, y_{p})$$\cap(x_{31}, x_{32}, \ldots, x_{3j_{3}})\cap\cdots\cap(x_{q1}, x_{q2}, \ldots, x_{qj_{q}})$,$\cap(x_{q+11}, x_{q+12}, \ldots,x_{q+1j_{q+1}}, x_{11}, x_{12}, \ldots, x_{1:_{1}},x_{21}, x_{22}, \ldots,x_{2i_{2}})$where $q\geq 2,$ $p\geq 1,1\leq i_{\ell}<j_{\ell}(\ell=1,2),$
Set$s_{1}= \sum_{i=1}^{q+1}j_{i}-(q+1)$

then $pd_{R}R/I_{i}=s_{i}+1$ for $i=1,2$ .

,$j_{3},$$\ldots$, $j_{q},j_{q+1}\geq 1$ .

$s_{2}= \sum_{i=1}^{q+1}j_{i}+p-(q+1)$,

110
Proposition 3.1. Consider the ideal$P_{\ell}=\{$ $x_{1\ell_{1}X_{2}\ell_{2}\cdots X_{q}p_{q}}$:$g_{p}= \sum_{a\in P\ell}a$. For $\ell=0,1,$$\ldots$,$s_{1}$, we set$\ell_{1}+\cdots+l_{q}=\ell+q$$p_{t}\leq i_{t}$for some $t=1,2,$:$\cup\{X_{1}\ell_{1}X_{2}\ell_{2}\cdots X_{q}\ell_{q}X_{q+1l_{q+1}}$

and$I_{1}$$\ldots,p\}$$2,.,p\}$ ,$l_{1}+\cdots+\ell_{q}+\ell_{q+1}=l+q.+.1i_{t}<\ell_{t}\leq j_{t}forallt=1,$. Then we have$\sqrt{(g_{0},g_{1},,g_{s_{1}})}=\sqrt{I_{1}}$.

generates . Hence it is enough to check conditions of Schmitt-Vogel lemma. (SV-1) is nothing. Since $P_{0}=\{x_{11}x_{21}\cdots x_{q1}\}$ ,and the latteras(SV-2) is clear. For (SV-3), we set the former set of. For any $\ell>0$ , we take $a”\in P_{\ell}(a\neq a’’)$ . If both and $a”$ lie in
one as, then we can write

Proof. It is clear that$I$

$P= \bigcup_{\ell=0}^{s_{1}}P_{\ell}$$P_{\ell}^{(1)}$$P_{l}$$P_{p}^{(2)}$

$a$

$a,$

$P_{\ell}^{(1)}$$a=x_{1\ell_{1}}x_{2\ell_{2}}\cdots x_{q\ell_{q}}$,”$a=x_{1}p_{1}\prime\prime x_{2\ell_{2}’’}\cdots x_{q\ell_{q}’’}$

, and
Since
. Thenthere exists $u\in\{1,2, \ldots, q\}$ such thatlie insatisfies the condition. The case both and, then we can write
andsimilarly. If$(\ell_{1}, p_{2}\ldots,\ell_{q})\neq(\ell_{1’ 2}^{\prime\prime p\prime\prime}, \ldots,\ell_{q}’’)$$\ell_{1}+\ell_{2}+\cdots+\ell_{q}=p_{1}’’+\ell_{2}’’+\cdots+\ell_{q}’’$,$a’=x_{1}x_{u}\prime\prime\cdots x_{q\ell_{q}}$$\ell_{u}>\ell_{u}’’$

$a”$

$a$

$a\in P_{\ell}^{(2)}$.

$P_{p}^{(2)}$can be checked$a”\in P_{\ell}^{(1)}$$a=x_{1\ell_{1}}x_{2\ell_{2}\cdots X_{q}p_{q}X_{q+1\ell_{q+1}}}$,,$a”=x_{1}\prime x\prime\prime\cdot$

where $\ell_{1}+\ell_{2}+\cdots+\ell_{q}+\ell_{q+1}=\ell+q+1,$. Thenexists $t\in\{1,2, \ldots,p\}$ such that$\ell_{1}’’+\ell_{2}’’+\cdots+\ell_{q}’’=\ell+q$$\ell_{t}’’\leq i_{t}$$p_{t}’’<p_{t}$, and there

and therefore.

$\ell_{1}+\cdots+p_{t}\prime\prime+\cdots+\ell_{q}<\ell+q+1-\ell_{q+1}\leq\ell+q$

So,$a’=X_{1}p_{1}\cdots X_{t}p_{t}\prime\prime\cdots X_{q\ell_{q}}$satisfies the condition.

$\square$Proposition 3.2. Consider the ideal . For $i=1,2,$ $\ldots,p$ , we set $y_{i}=$, , we set$x_{1j_{1}+i}=x_{2j_{2}+i}$ . For $\ell=0,1,$$I_{2}$$\ldots$$s_{2}$$P_{\ell}=\{\begin{array}{lllllll} \ell_{1}+ \cdots +\ell_{q}=\ell+qX_{l}p_{1}X_{2}p_{2} \cdots x_{q\ell_{q}} p_{l} \leq i_{2} or\ell_{2}\end{array}\}$:$U\{$ $x_{1\ell_{1}}x_{2p_{2}}\cdots x_{q\ell_{q}}x_{q+1l_{q+1}}$$\cup\{$ $y_{i}x_{3\ell s}\cdots x_{q^{p_{q}}}x_{q+1l_{q+1}}$and$g_{p}= \sum_{a\in P_{\ell}}a$$\ell_{1}+\cdots+\ell_{q}+\ell_{q+1}=\ell+q+1\}$$i_{t}<\ell_{t}\leq j_{t}$for all $t=1,2$: $j_{1}+j_{2}+i+\ell_{3}+\cdots+l_{q}+\ell_{q+1}=\ell+q+1\}$$1\leq i\leq p$. Then we have$\sqrt{(g_{0},g_{1},,g_{s_{2}})}=\sqrt{I_{2}}$.

Since the proof of this proposition is similar to Proposition 3.1, we omit
here.

111
Example 3.3. Let us compare the ara
these proposition.
Consider$I$

generators in previous section and$I=(x_{1}, x_{2}, x_{3})\cap(x_{4}, x_{5}, x_{6})\cap(x_{7}, x_{1}, x_{2}, x_{4})$.

To use the method of previous section, we can set $y_{1}=x_{3},$ $y_{2}=x_{6},$ $y_{3}=x_{7}$ .
, and . Thus $pd_{R}R/I=4+1=5$ .
Then other variables areare followings:$x_{1},$$g_{0},$ $g_{1},$$\ldots,$$x_{2},$$x_{4}$$x_{5}$$g_{4}$$\{\begin{array}{l}g_{0}=x_{1}x_{2}x_{4}x_{5}g_{1}=x_{1}x_{2}x_{4}+x_{1}x_{2}x_{5}+x_{1}x_{4}x_{5}+x_{2}x_{4}x_{5}g_{2}=x_{1}x_{2}\cdot x_{6}+x_{1}x_{4}+x_{1}x_{5}+x_{2}x_{4}+x_{2}x_{5}+x_{4}x_{5}\cdot x_{3}g_{3}=x_{1}\cdot x_{6}+x_{2}\cdot x_{6}+x_{6}\cdot x_{3}+x_{5}\cdot x_{3}x_{7}g_{4}=x_{3}x_{6}x_{7}\end{array}$

There are 16 elements of in the summand of
While Proposition 3.1 shows$I$$g_{0},$ $g_{1},$$\ldots$,$g_{4}$.

$\{\begin{array}{l}g_{0}=x_{1}x_{4}g_{1}=x_{1}x_{5}+x_{2}x_{4}g_{2}=x_{2}x_{5}+x_{1}x_{6}+x_{\theta}x_{4}g_{3}=x_{3}x_{5}x_{7}+x_{2}x_{6}g_{4}=x_{3}x_{6^{X}7}\end{array}$. These are
So, there are only 9 elements of in the summand ofgenerators.
ofminimal
We consider another example corresponding to Proposition 3.2. Set$I$$g_{0},$ $g_{1},$$\ldots,$$g_{4}$$I$$I=(x_{1}, x_{2}, x_{3}, x_{4})\cap(x_{5}, x_{6}, x_{4})\cap(x_{7}, x_{1}, x_{5})$.

For the method of previous section, we can set $y_{1}=x_{3},$ $y_{2}=x_{6},$ $y_{3}=x_{7}$ , then, and . So $pd_{R}R/I=5$ .
other variables are$x_{1},$$x_{2},$$x_{4}$$x_{5}$$\{\begin{array}{l}g_{0}=x_{1}x_{2}x_{4}x_{5}g_{1}=x_{1}x_{2}x_{4}+x_{1}x_{2}x_{5}+x_{1}x_{4}x_{5}+x_{2}x_{4}x_{5}g_{2}=x_{1}x_{2}\cdot x_{6}+x_{1}x_{4}+x_{1}x_{5}+x_{2}x_{4}\cdot x_{7}+x_{2}x_{5}+x_{4}x_{5}g_{3}=x_{1}\cdot x_{6}+x_{2}\cdot x_{6}x_{7}+x_{4}\cdot x_{7}+x_{5}\cdot x_{3}g_{4}=.x_{3}x_{6}x_{7}\end{array}$

There are 16 elements of in the summand of
While Proposition 3.2 shows$I$$g_{0},$ $g_{1},$$\{\begin{array}{l}g_{0}=x_{1}x_{5}g_{1}=x_{1}x_{6}+x_{2}x_{5}g_{2}=x_{1}x_{4}+x_{3}x_{5}+x_{2}x_{6}x_{7}g_{3}=x_{4}x_{5}+x_{3}x_{6}x_{7}g_{4}=x_{4}x_{7}\end{array}$$\ldots,$$g_{4}$.

112
There are only 9 elements ofminimal generators of .

$I$

in the summand of$g_{0},g_{1},$ $\ldots,g_{4}$, and these are$I$

APPENDIX A. ANALYTIC SPREAD
In this section, we state the result that we get after the talk. We haveconsidered the inequality
height $I\leq pd_{R}R/I\leq araI\leq\mu(I)$ .

But there is an invariant which lies between ara
analytic spread $l(I)$ of .

$I$

and $\mu(I)$ , and that is the$I$

DefinItion A.1. Let be a homogeneous ideal of . Let $R[It]$ be a Rees ringof , that is, a subring of a polynomial ring $R[t]$ . Then $\ell(I)=\dim R[It]/\mathfrak{m}R[It]$is called the analytic spread of .
$R$

$I$

$I$

$I$

An ideal is called a reduction of if
and $I^{n+1}=JI^{n}$ for some $n\geq 1$ .
Moreover, is called a minimal reduction of if is a reduction of , anditself does not have any proper reductions. It is known that the cardinality ofthe minimal set of generators of minimal reductions of is constant, and thisnumber is equal to the analytic spread of .
As stated in the beginning of this section, the following inequality is known:(A.1)
height $I\leq pd_{R}R/I\leq araI\leq l(I)\leq\mu(I)$ .
We prove $l(I)=pd_{R}R/I$ for the squarefree monomial ideal with arithdeg $I=$
indeg .
$J$$I$

$J\subset I$

$J$

$I$

$J$

$I$

$J$

$I$

$I$

$I$

$I$

Theorem A.2. Let I be a squareffee monomial ideal with arithdeg $I=in\deg I$ .
Then we have$l(I)=pd_{R}R/I$ .

We prove this theorem by showing that ara generators as in previous section generate minimal reduction of . This result is stronger than ara $I=$
$I$

$I$

$pd_{R}R/I$.

REFERENCES[1] M. Barile, On the number of equations defining certain varieties, manuscripta math.
91 (1996), 483-494.

[2] A. Fr\"uhbis-Kr\"uger and N. Terai, Bounds for the regularity of monomial ideats, Mathematiche (Catania) 53 (1998), 83-97.
[3] L. T. Hoa and N. V. Trung, On the Castelnuovo-Mumford regularity and the arithme tic
degree of monomial ideals, Math. Z. 229 (1998), 519-537.
[4] K. Kimura, N. Terai and K. Yoshida, Arithme tical rank of squarefree monomial idealsof smdl arithmetic degree, preprint.
[5] G. Lyubeznik, On the local cohomology modules
for ideals a generated by monomials in an R-sequence, in Complete Intersections, Acireale, 1983 (S. Greco and R.
Strano eds., Lecture Notes in Mathematics No. 1092, Springer-Verlag, 1984 pp. 214$H\dot{i}(R)$220.

[6] P. Schenzel, Applications of Koszul homology to numbers of generators and syzygies, J.
Pure Appl. Algebra 114 (1997), 287-303.
[7] P. Schenzel and W. Vogel, On set-theoretic intersections, J. Algebra 48 (1977), 401-408.

113[8] T. Schmitt and W. Vogel, Note on set-theoretic intersections of subvarieties of projective
space, Math. Ann. 245 (1979), 247-253.
[9] N. Terai, Stanley-Reisner rengs of Alexander dual complexes, In: Proceedings of the
19th symposium on Commutative Algebra in Japan, 19 (1997), 53-66.
[10] N. Terai, Arithmetical rank of squarefree monomial ideals of small arithmetic degree,
In: Proceedings of the 26th symposium on Commutative Algebra in Japan, 26 (2004),

11-18.
[11] Z. Yan, An \’etale analog of the Goresky-Macpherson formula for subspace arrangements,
J. Pure Appl. Algebra 146 (2000), 305-318.