数理解析研究所講究録
第 1604 巻 2008 年 125-130

125

A note on Schmitt-Vogel lemma
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

be a polynomial ring over an infinite field , andmial ideal of . The arithmetical rank of is defined by

Let

$k$

$S$

$I$

a squarefree mono-$I$

$S$

ara

$I$

$:= \min\{$ $r$: there existFor ideals $J\subset I\subset S,$$s\in N$ such that$J$

$a_{1},$$\ldots,$, ) $=\sqrt{I}\}$ .
such that $\sqrt{(a_{1}}$$a_{r}\in I$$a_{r}$is said to be a reduction of$I^{s+1}=JI^{s}$$I$

if there exists some.

Note that when this is the case, $\sqrt{J}=\sqrt{I}$ holds. The analytic spread of isdefined by$l(I)$ $:= \min$ { $\mu(cI)$ :is a reduction of },where $\mu(J)$ denotes the minimal number of generators of . The existence ofthe minimal reduction shows ara $I\leq l(I)$ . On the other hand, it is known
by Lyubeznik [3] that $pd_{S}S/I\leq$ ar.a , where pd $s^{S}/I$ denotes the projective
dimension of $S/I$ . Therefore we have the following inequalities:
$I$

$I$

$J$

$J$

$I$

pd $s^{S}/I\leq$ ara $I\leq l(I)$ .

In the study of the arithmetical rank, Schmitt-Vogel lemma [5, Lemma, pp.
249] is an important and useful tool, because it gives a sufficient condition forideals $J\subset I$ to hold $\sqrt{J}=\sqrt{I}$ . In this report, we give a sufficient condition
with $J\subset I$ to be a reduction of by refining Schmitt-Vogelfor an ideallemma. As an application of our theorem, wc prove $l(I)=pd_{S}S/I$ for theideal$I=(x_{11}, \ldots, x_{1i_{1}})\cap\cdots\cap(x_{q1}, \ldots, x_{qi_{q}})$ ,are variables in pairwise distinct. Schmitt and Vogel [5]where$I=$$s^{S}/I$for this ideal using their lemma.
pdproved ara
$I$

$J$

$x_{11},$ $\ldots,$$x_{qi_{q}}$

$S$

$I$

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1. MAIN THEOREM
In this section, we consider an arbitrary commutative ring
Our main result of this report is the following:

Theorem 1.1. Let be a commutative ring with unitary. Let
be finite subsets, and we set
$R$

$R$

with unitary.

$P_{0},$$P_{1},$$\ldots,$$P_{r}\subset$$R$,$P= \bigcup_{=p0}P_{\ell}$$gp= \sum_{a\in P_{\ell}}a_{7}$$\ell=0,1,$ $\ldots,$ $r$.

Assume that(Cl) $\# P_{0}=1$ .
(C2) For all $P>0$ and $a”\in- P_{\ell}(a\neq a’’)$ , there exist some $\ell’(0\leq\ell’<P)$ ,, and $b\in(P)$ such that $aa”=a’b$ .
Then we haveis a reduction of $(P)$ .
$a,$$a’\in P_{\ell’}$$(g_{0}, g_{1}, \ldots, g_{r})$On the other hand, Schmitt-Vogel lemma is the following:
Proposition 1.2 (Schmitt-Vogel [5, Lemma, pp. 249]). Let $R$ be a commutative ring with unitary. Let
be finite subsets, and we set$P_{0},$$P_{1},$$\ldots,$$P= \bigcup_{\ell=0}^{r}P_{\ell}$$gp= \sum_{a\in P_{\ell}}a$$\mathcal{A}ssume$$P_{r}\subset R$,,$P=0,1,$$\ldots,$

$r$

.

that

(Cl) $\# P_{0}=1$ .
(C2)’ For all $\ell>0$ and $a,$ $a”\in P_{\ell}(a\neq a’’)_{Z}$ there exist some $\ell’(0\leq\ell/<\ell)$andsuch that $aa^{;/}\in(a’)$ .
Then we have $\sqrt{(g_{0},g_{1}}$, ) $=\sqrt{(P)}$ .
$a’\in P_{\ell’}$$g_{r}$Second condition of Theorem 1.1- is stronger than that of Schmitt-Vogellemma, but Theorem 1.1 has a stronger conclusion than Schmitt-Vogel lemma.
Remark 1.3. Schmitt-Vogel lemma allows us to add some exponent $e(a)$ foreachin the sum , i.e., we may put$a\in P_{\ell}$$g_{\ell}$$g_{\ell}= \sum_{a\in P_{\ell}}a^{e(a)}$.

In particular, we can takeas homogeneous if $R$ is graded. But a similarstatement does not hold for our theorem.
$g_{\ell}$Instead of proving Theorem 1.1, we will give a detailed explanation of anexample in Section 3, which illustrates the outline of the proof of the theorem.
See also [2].

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2. AN APPLICATIONIn this section, we apply Theorem 1.1 to some ideals and calculate theanalytic spread of them.
Consider the ideal$I=(x_{11}, \ldots, x_{1i_{1}})\cap\cdots\cap(x_{q1}, \ldots, x_{qi_{q}})$ ,
(2.1)
where

$x_{11},$ $\ldots,$$x_{qi_{q}}$are variables inpairwise distinct.

$S$Lemma 2.1. For the above ideal ,$I$$pd_{S}S/I=\sum_{s=1}^{q}i_{s}-q+1$

Proof. For an integer$q\geq 1$.

, we set$I_{q}=(x_{11}, \ldots,x_{1i_{1}})\cap\cdots\cap(x_{q1}, \ldots, x_{qi_{9}})\wedge$$r_{q}= \sum_{s=1}^{q}i_{s}-q+1$’.

We prove the lemma by induction on . The case $q=1$ is clear. Supposethat $q\geq 2$ . If we put $P=(x_{q1}, \ldots, x_{qi_{q}})$ , then $I_{q}=I_{q-1}\cap P$ and $r_{q}=$height $P-1=r_{q-1}+$ pd $s^{S}/P-1$ . Consider Mayer-Vietoris sequence$q$$\gamma_{q-1}+$$0arrow S/I_{q}arrow S/I_{q-1}\oplus S/Parrow S/(I_{q-1}+P)arrow 0$ .

Since $pd_{S}S/T=\max\{i:To1_{i}^{S}(k, S/I)\neq 0\}$ , the long exact sequence. . . $arrow 0=Tor_{r_{q}+1}^{S}(k, S/I_{q-1})\oplus Tor_{r_{q}+1}^{S}(k, S/P)arrow Tor_{r_{q}+1}^{S}(k, S/(I_{q-1}+P))$$arrow Tor_{r_{q}}^{S}(k, S/I_{q})arrow T_{o1_{r_{q}}^{S}}\cdot(k, S/I_{q-1})\oplus Tor_{r_{q}}^{S}(k, S/P)=0arrow\cdots$

implies $r_{q}=pd_{S}S/I_{q}$ .

口Schmitt-Vogel [5] proved ara $I=pd_{S}S/I$ (see also Schenzel-Vogel [4]).
They proved it by applying Schmitt-Vogel lemma to$P_{p}=\{X_{1}\ell_{1}X_{2}\ell_{2}\ldots X_{q}\ell_{q}:\ell_{1}+\cdots+l_{q}=l+q\}$,$P=0,1,$$\ldots,$$r$,

where $r= \sum_{s=1}^{q}i_{s}-q$ . Thesealso satisfy the assumption ofTheorem 1.1. Thus $J=(g_{0}, g_{1,\ldots,9r})$ is a reduction of . Since$P_{0},$$P_{1},$$P_{r}$$\ldots,$$I$$r+1=pd_{S}S/I=$ ara $I\leq l(I)\leq r+1$ ,we have $1(I)=pd_{S}S/I$ . Therefore we have the following corollary:
Corollary 2.2. Let $I=(x_{11}, \ldots, x_{1i_{1}})\cap\cdots\cap(x_{q1}, \ldots, x_{qi_{q}})$ . Then we have$l(I)=pd_{s}S/I$ .

In particular,$(g_{0}, g_{1}, \ldots, g_{r})$is a minimal reduction of .

Note that we have a minimal reduction of

$I$

$I$

explicitly.

Remark 2.3. In general, $l(I)\neq pd_{S}S/I$ for a squarefree monomial ideal . For
example, if $\mu(I)$ -height $(I)=1$ and $S/I$ is Cohen-Macaulay, then height $(I)=$$pd_{S}S/I=$ ara $I<l(I)=\mu(I)$ ; see [1].
$I$

128
3. AN EXAMPLE
In this sectToii, we give one example to illustrate the outline of the proof ofTheorem 1.1.
Let us consider the ideal$I=\backslash (x_{1}, x_{2}, x_{3})\cap(y_{1}, y_{2}, y_{3})$.

This is a special form of the ideal (2.1). The minimal graded resolution of $S/I$
is$0arrow S(-6)arrow S(-5)^{6}arrow S(-4)^{1_{J}^{r}}’arrow S(-3)^{18}arrow S(-2)^{9}arrow Sarrow S/Iarrow 0$

and $pd_{S}S/I(=3+3-2+1)=5$ . Then
$P_{0}=\{x_{1}y_{1}\}$,$P_{1}=\{x_{1}y_{2}, x_{2}y_{1}\}$,$P_{2}=\cdot\{x_{1}y_{3}, x_{2}y_{2}, x_{3}y_{1}\}$$P_{3}=\{x_{2}y_{3}, x_{3}y_{2}\}$$P_{4}=\{X_{3}l/3\}$,,.

Let us see conditions of Theorem 1.1. Since $\# P_{0}=1$ , (Cl) is satisfied. For theassumption (C2), we have the following equations:
(3.1)
:
,
$x_{1}y_{2}\cdot x_{2}y_{1}=x_{1}y_{1}\cdot x_{2}y_{2}\in(P_{0})(P_{2})$$P_{1}$

(3.2)
(3.3)

$P_{2}:\{\begin{array}{l}x_{1’}y_{3}\cdot x_{2}y_{2}=x_{1}y_{2}\cdot x_{2}y_{3}\in(P_{1})(P_{3}),x_{1}y_{3}\cdot x_{3}y_{1}=x_{1}y_{1}\cdot x_{3}y_{3}\in(P_{0})(P_{4}),x_{2}y_{2}\cdot x_{3}y_{1}=x_{2}y_{1}\cdot x_{3’}y_{2}\in(P_{1})(P_{3}),\end{array}$$P_{3}$:$x_{2}y_{3}\cdot x_{3}y_{2}=x_{2}y_{2}\cdot x_{3}y_{3}\in(P_{2})(P_{4})$.

Thus (C2) is also satisfied.
Now we shall see $J=(g_{0}, g_{1}, g_{2}, g_{3}, g_{4})$ is a reduction of , where$I$,$g_{0}=x_{1}y_{1}$$gl=x_{1}y_{2}+x_{2}y_{1}$,$g_{2}=x_{1}y_{3}+x_{2}y_{2}+x_{3}y_{1}$$g_{3}=x_{2}y_{3^{-}}+x_{3}y_{2}$$g_{4}=x_{3}y_{3}$,,.

We put$P=0,1,2,3,4$ .

,$I_{\ell}=( \bigcup_{j=0}^{\ell}P_{j})$Note that $I_{4}=I$ . It is enough to show
$I_{\ell}^{2^{\ell}}\subset JI^{2^{i}-1}$

in order to see thatfact, we show

$J$,$\ell=0,1,2,3,4$

is a reduction of . We show this by induction on$I$$I_{\ell}^{2^{f}}\subset I_{\ell-1}^{2^{\ell-1}}I^{2^{t}-2^{\ell-1}}+JI^{2^{\ell}-1}$,$P=0,1,2,3,4$ .

$p$.

In

129
Step 1: The case $P=0$ . In this case, $I_{0}=(P_{0})=(x_{1}y_{1})=(g_{0})\subset J$ .
Step 2; $Tht$ case $p=1$ . We want to show $I_{1}^{2}\subset I_{0}I+JI$ . To see this, it
is enough to show that $a_{1}a_{2}\in I_{0}I+JI$ for all(we do not assume$(3.1)$ shows. When. When $a_{1}=a_{2}=a$ , we use. For example,$a_{1},$$a_{1}\neq a_{2})$$a_{1}\neq a_{2},$$a_{2}\in P_{1}$$a_{1}a_{2}\in I_{0}I$$g_{1}$.

$(x_{1}y_{2})^{2}=(g_{1}-x_{2}y_{1})x_{1}y_{2}=g_{1}x_{1}y_{2}-x_{2}y_{1}\cdot x_{1}y_{2}=g_{1}x_{1}y_{2}-x_{1}y_{1}\cdot x_{2}y_{2}\in JI+I_{0}I$

Therefore $I_{1}^{2}\subset I_{0}I+JI$ holds.

Step 3: The case $\ell=2$ . We want to show
. To see this, weonly checkfor all. There aretwo cases:(i);(ii) for some.
In case (i), there are two cases dividing large. The first one is that
andby renumbering. In this case, it is easy to checkbecause of (3.2). For example,$I_{2}^{4}\subset I_{1}^{2}I^{2}+JI^{3}$$a_{1}a_{2}a_{3}a_{4}\in I_{1}^{2}I^{2}+JI^{3}$$a_{1},$$a_{2},$$a_{3},$$a_{1)}a_{2},$$a_{3},$$a_{4}\in P_{0}\cup P_{1}\cup P_{2}$$a_{4}\in P_{2}$$i,$$a_{i}\in P_{0}\cup P_{1}$$a_{1}\neq a_{2}$$a_{3}\neq a_{4}$$a_{1},$ $a_{2},$$a_{3},$ $a_{4}$$a_{1}a_{2}a_{3}a_{4}\in I_{1}^{2}I^{2}$$(x_{1}y_{3})^{2}x_{2}y_{2}\cdot x_{3}y_{1}=(x_{1}y_{3}\cdot x_{2}y_{2})(x_{1}y_{3}\cdot x_{3}y_{1})=(x_{1}y_{2}\cdot x_{1}y_{1})(x_{2}y_{3}\cdot x_{3}y_{3})\in I_{1}^{2}I^{2}$

The second one is that there are no such a renumbering on
this case, we use
as in the case $\ell=1$ . For example,$a_{1},$$a_{2},$$a_{3},$$a_{4}$.

. In$g_{2}$$(x_{1}y_{3})^{3}x_{2}y_{2}=(x_{1}y_{3})^{2}(g_{2}-x_{2}y_{2}-x_{3}y_{1})x_{2}y_{2}$$=g_{2}(x_{1}y_{3})^{2}x_{2}y_{2}-(x_{1}y_{3})^{2}(x_{2}y_{2})^{2}-(x_{1}y_{3})^{2}x_{3}y_{1}\cdot x_{2}y_{2}$$=g_{2}(x_{1}y_{3})^{2}x_{2}y_{2}-(x_{1}y_{3}\cdot x_{2}y_{2})^{2}-(x_{1}y_{3}\cdot x_{3}y_{1})(x_{1}y_{3}\cdot x_{2}y_{2})$$\in JI^{3}+I_{1}^{2}I^{2}$.

In case (ii), if there are two indices (say,) such that, thenandhold. Next, we consider the case that there
is only one such that $a_{i}\in P_{0}\cup P_{1}$ . We may assume $a_{1}\in P_{0}\cup P_{1}$ and. Then we need to make only one pair of distinct elements from. It is weaker requirement than that of case (i). In fact, to make one
pair of distinct elements, we only need two of. For example,$i$$a_{i_{l}}a_{i_{2}}\in I_{1}^{2}$$i_{1},$$a_{i}\in P_{0}\cup P_{1}$$i_{2}$$a_{1}a_{2}a_{3}a_{4}\in I_{1}^{2}I^{2}$$i$$a_{2},$ $a_{3},$$a_{2},$$a_{4}\in P_{2}$$a_{3},$ $a_{4}$$a_{2},$ $a_{3},$$a_{4}$$(x_{1}y_{3})^{2}=x_{1}y_{3}(g_{2}-x_{2}y_{2}-x_{3}y_{1})$$=g_{2}x_{1}y_{3}-x_{1}y_{3}\cdot x_{2}y_{2}-x_{1}y_{3}\cdot x_{3}y_{1}$$\in JI+I_{1}I$ .

Step 4: The case $P=3$ . We want to showsame argument as in Step 3 is also usable. We omit here.

$I_{3}^{8}\subset I_{2}^{4}I^{4}+JI^{7}$

Step 5; The case $\ell=4$ . It is clear that
Therefore we obtain that is a reduction of. In this case, the$I_{4}^{16}\subset I_{3}^{8}I^{8}+JI^{15}$

$J$

$I$.

Remark 3.1. The reduction number $r_{J}(I)$ is defined by$r_{J}(I):= \min\{s:I^{s+1}=JI^{s}\}$ .

since $\# P_{4}=1$ .

130Above argument gives an upper bound of $r_{J}(I)$ . But this is very big in general.
In fact, in the above argument, we only see $I^{2^{4}}=JI^{2^{4}-1}$ , that is, $r_{J}(I)\leq$$2^{4}-1=15$ . But $r_{J}(I)=3$ , i.e., $I^{4}=JI^{3}$ holds.
REFERENCES
1 K. Kimura, N. Terai, and K. Yoshida, Arithmetical rank of squarefree monomial idealsof small $a\tau\dot{v}thmetic$ degree, to appear in J. Algebraic Combin.
[2] K. Kimura, N. Terai, and K. Yoshida, Schmitt-Vogel’s lemma for reductions, in preparation.
3 G. Lyubeznik, On the local cohomology modules
for ideals a generated by monomials in an R-sequence, in Complete Intersections, Acireale, 1983 (S. Greco and R.
Strano eds., Lecture Notes in Mathematics No. 1092, Springer-Verlag, 1984 pp. 214$[$$]$$[$$]$$H_{\alpha}^{i}(R)$220.

4 P. Schenzel and W. Vogel, On set-theoretic intersections, J. Algebra 48 (1977), 401-408.
[5] T. Schmitt and W. Vogel, Note on set-theoretic intersections of subvarieties of projectivespace, Math. Ann. 245 (1979), 247-253.
$[$$]$