164

数理解析研究所講究録
第 713 巻 1990 年 164-180

MULTIPLICITY OF FILTERED RINGS*

筑波大学 数学系

昌孝

泊(MASATAKA TOMARI)\S 1 Introduction and results.
(1.1) Let (V, ) be a germ of a projective variety at a closed point
$p$

$p$

fundamental problem to study the ring theoretic properties of the local ring

. It is a$O_{V,p}$bymeans of resolution of singularities ; $\psi:(\tilde{V}, E)arrow(V,p)$ . In the case $\dim V=2$ , Artin’s

fundamental cycle for the resolution
singularities. Let$Z_{0}$

is important and gives many information of$\psi$

be Artin’s fundamental cycle foris independent of choice of the resolution$-(Z_{0})^{2}\leq the$$\psi$$\psi$. For example the degree$(Z_{0})^{2}$and we have the relation

multiplicity of(V, )$p$(Ph. Wagreich cf. [Wagreich]). Unfortunately no higher dimensional analogue of thisobject are studied. In this note we will study the multiplicity of singularities by filteredblowing-ups. We prove an inequality (1.6) which gives a lower bound of multiplicity by
the data of tangent cone of the filtration.
An application of our results to a purely eUiptic singularity of special type ([IW] [Y] [T1]
$)$

$wiU$

be given in another note for the talk of “COMMUTATIVE RING THEORY ;JAPAN NO.II”.
(1.2)

Throughout this note we $wiU$ fix the following situation. Our singularity(V, ) or local ring $(A, m)=(O_{(V,p)},m)$ is always assumed as the material coming from$p$some scheme over a field$k$. Further we $wiU$ assume $(A, m)$ is analytically unramified

after (1.6). In particular,$(A,m)$ :$d$-dimensional Noether local ring over a field ,$F=\{F^{h}\}_{h\geq 0}$$k$: a filtration of ideals as foUows ;$(F^{O}=A\supset F^{1}=m,$$F$

This is a preliminary version.
graded rings ’$*1$).
$(^{*})$

為$\supset F^{h+1},$$F^{h}F^{j}$$t^{k}1^{\backslash ^{J}}\prime_{g}0\rangle$$t_{1}\cdot\uparrow t_{\xi}^{:}eL_{\backslash ^{\wedge 7}}^{s_{;\llcorner}}tT$$\subset F^{h+j}$,$\dotplus_{\ }\prime_{\downarrow C}t_{-}^{\neg}Lt,r\grave{*}\backslash ,$$\{\circ$$\iota$$\bigwedge_{\backslash }*g\cdot\neq$$\mathfrak{l}_{\sim J}^{\sim\star_{\chi}}7_{\llcorner}*$$\llcorner\gamma_{\sim}\backslash 0$”Multiplicity of normal

165$R=\oplus_{l\geq 0}F^{l}.T^{l}\subset A[T]$is a finitely generated$A$-algebra, where$T$

is an indeter-

minate.
There is an integer $N>0$ with $(F^{N})^{m}=F^{Nm}$ for $m\geq 0$ .
$F^{N}$: $m$ -primary)

By these assumption, $G+=\oplus_{l\geq 1}F^{i}/F^{l+1}$ is the homogeneous maximal ideal of

Problem (1.2.1).

$G$.

Study the multiplicity $e(m, A)$ of $(A, m)$ from the associatedgraded ring $gr_{F}A=G=\oplus_{h\geq 0}F^{h}/F^{h+1}$ and compare the integers $e(m, A)$ and $e(G_{+}, G)$ .
(We hope that these are very near when $G$ is a “good ” ring. )First we shall prove the following.
Let the situation $be$ as above. Then

FACT (1.3).

$l(A/m^{l+1})\leq l(G/(G_{+})^{l+1})$

for$l\geq 0$.

In partic ular we obtain the relations $e(m, A)\leq e(G_{+}, G)$embdim

$G$

Proof

an

$d$

embdim.
The induced filtration on $A/m^{l+1}$ by $F=\{F^{h}\}$ is given as follows:$0arrow m^{l+1}arrow Aarrow A/m^{l+1}arrow 0$$\cup i[$$\cup|\{$$(J^{((}$$0arrow m^{l+1}\cap F^{h}arrow F^{h}arrow F^{h}(A/m^{l+1})arrow 0$

Hence we obtain $g^{r}r_{F}(A/m^{l+1})=gr_{F}(A)/gr_{F}(m^{1+1})$ . Here we see$g^{l}r_{F}(m^{l+1})=\oplus_{h\geq 0}m^{l+1}\cap F^{h}/m^{l+1}\cap F^{h+1}$$\cong\oplus_{h\geq 0}\frac{F^{h}\cap m^{l+1}+F^{h+1}}{F^{h+1}}$.

$(G_{+})^{l+1}= \oplus_{h\geq l+1^{\ovalbox{\tt\small REJECT}}}\sum m_{1}+\ldots+m_{l+1}=h,m_{F^{h+}}:\geq 1F_{1}^{m_{1}}\ldots F^{m_{l+1}}+F^{h+1}$

Clearly we have$\sum_{m_{1}+\ldots+m_{l+1}=h,m:\geq 1}F^{m_{1}}\ldots F^{m_{l+1}}\subset F^{h}\cap m^{l+1}$

2

.

$A$

$\leq$

Hence$(G_{+})^{l+1}\subset gr_{F}(m^{l+1})$.

Therefore.

$l(A/m^{l+1})\leq l(gr_{F}(A/m^{l+1}))=l(G/gr_{F}(m^{\iota+1}))\leq l(G/(G_{+})^{l+1})$

Example(1.4). We shall introduce a filtration$k[[x, y, z]]$$F$on the regular local ring $A=$

with $m=(x,y, z)A$ by means of the associated order function$\nu$

as in thefollowing (cf. [Rees]) :$\nu(x)=\nu(y)=\nu(z)=1$ ,$\nu(\alpha)\geq k\}\subset A$

and$\nu(x^{2}+y^{2}+z^{2})=3(>2)$ with $F^{h}=\{\alpha\in A$$|$.

We can easily check that $G=g^{l}r_{F}(A)\cong k[x,y, z, w]/x^{2}+y^{2}+z^{2},$ $e(m, A)=1$ and$e(G_{+}, G)=2$.

In fact we have$F^{0}=A$$F^{1}=m$$F^{2}=m^{2}$$F^{3}=m^{3}+(x^{2}+y^{2}+z^{2})A$$F^{4}=m^{4}+(x^{2}+y^{2}+z^{2})m$$F^{5}=m^{5}+(x^{2}+y^{2}+z^{2})m^{2}$$F^{6}=m^{6}+(x^{2}+y^{2}+z^{2})m^{3}+(x^{2}+y^{2}+z^{2})^{2}A$

hence $x^{2}+y^{2}+z^{2}\in F^{3}-F^{4}$ can not be represented by

ring

$G$

. To compute$G$$x,$ $y,$ $z\inF^{1}-F^{2}$

as in the $ass$ ertion, remark that if we regard$A\cong k[[x, y, z, w]/(w-x^{2}-y^{2}-z^{2})$, then$F$$A$

in thein the form

is the induced filtration from the filtration

on $k[[z, y, z, w]]$ by the degree of monomials as $F^{h}=\{x^{a}y^{b}z^{c}w^{d}\in k[x, y, z,w]$$|$$a+b+c+3d\geq k\}A$ .

Example(1.5). We introduce the filtration $F$ on the local ring
$A(=k[[b, c, y, z]]/b^{2}+(y^{3}+z^{7}+c^{21})c)\cong k[[a, b, c,y, z]]/(a+y^{3}+z^{7}+c^{21}, b^{2}-ac)$ with$m=(a, b, c, y, z)A$ by the order function

$\nu$

as:

3

.

$\nu(a)=\nu(y^{S}+z^{7}+c^{21})=23,$ $\nu(b)=12,$ $\nu(c)=1\nu(y)=7$ and $\nu(z)=3$

Now $G=gr_{F}(A)\cong k[a,b, c, y, z]/(y^{3}+z^{7}+z^{21},b^{2}-ac),$ $e(m, A)=2$ and $e(G_{+}, G)=6$ .

Further one can see thatis a normal domain.

$G$

These examples say that the integers $e(m, A)$ and $e(G_{+}, G)$ are different ,in general,
even if we assume that $G$ is a normal Gorenstein domain. The next is the main resultof this note which gives a lower bound of $e(m, A)$ from the data of $G$ .
Let the situation be as in (1.2). Further we assume that

THEOREM (1.6).

cally un ramified and that$k$is an infinite field. Let a system of elements$x_{1},$$A$$\ldots$

is analyti-

,

$x$

.

be a minimal homogeneo us generator system $ofG+witA$ degx $1\leq degx_{2}\leq$ ... $\leq degx$$s\geq d=\dim A=\dim$ G. Then we have the following

with

.

$\in c_{+}$(I)$( \prod_{:=1}^{d}degx_{*}\cdot)\lim_{\lambdaarrow 1}(1-\lambda)^{d}P(G, \lambda)$$\leq(i)e(m, A)$$\leq e(G_{+}, G)\leq(i:)(degx.)^{d}\lim_{\lambdaarrow 1}(1-\lambda)^{d}P(G, \lambda)$.

where $P(G, \lambda)=\sum_{h\geq 0}l(G_{h})\lambda^{h}\in Z[[\lambda]]$ .
(2) If the $eq$ uali $ty$ holds in (i), then $e(m, A)=e(G_{+}, G)$ and there is a parameter, ... ,of A whose $ini$ tial form gives a homogeneous parameter system, in $(y_{d})ofG$ such that $\deg in(y_{*}\cdot)=degx_{i}$ for $i=1,$
, .
in
$(S)$
and is normal with G.C. $D.$ ( $degX_{1},$, degx,) $=1$ ,If the equality holds in
is a homogeneous $ring$ . That is $degx_{i}=1$ holds forthen $e(m, A)=e(G_{+}, G)$ andsystem$(y_{1}),$$y_{1}$$y_{d}$$d$$\ldots$$\ldots$$(\ddot{n})$$G$$\ldots$$G$$i=1,$$s$.

In general we have the following.

Remark (1.7)(1) Let $R=R(E, D)$ be a normal d-dimensional graded ring with

Demazure’s description.
$D^{d-1}=\lambdaarrow 1hm(1-\lambda)^{d}P(R, \lambda)$

where$P(R, \lambda)=\sum_{h\geq 0}\dot{l}(R_{h})\lambda^{h}\in Z[[\lambda]]$(2), with $d=\dim R$ .

For a graded complete intersection$R=k[x_{1}, x_{d+}.]/(f_{1}, f.)$ ,4

where$f_{1},$$\ldots$

,

.

$f$

is a homogeneous$re$gular sequence of $k[ae_{1}, x_{d+}.]$ , we have$P(R, \lambda)=\frac{(1-\lambda^{d\epsilon gf_{1}})...\cdot.\cdot..\cdot(1-\lambda^{degf}\cdot)}{(1-\lambda^{d\epsilon gx_{1}}).(1-\lambda^{degaea}+\cdot)}$Hence$\lim_{\lambdaarrow 1}(1-\lambda)^{d}P(R, \lambda)=\frac{(degf_{1})...\cdot.\cdot..\cdot(degf.)}{(degx_{1}).(degx_{d+}.)}$.

(1.8) By using (1.7), we will observe (1.4) and (1.5).
(1.8.1) For$G$of (1.4), we have $degx=degy=degz=1$ and $degw=3$ . Hence

1.1.1. $\lim_{\lambdaarrow 1}(1-\lambda)^{d}P(G, \lambda)=1.1.1.\frac{2}{1.1.1.3}=\frac{2}{3}(\leq 1=e(m, A))$ .
(1.8.2) For $G$ of (1.5), we have $dega=23$ , $degb=12$ , $degc=1$ , $degy=7$ and $degz=3$ .

Hence

1.3.7. $\lim_{\lambdaarrow 1}(1-\lambda)^{d}P(G, \lambda)=1.3.7.\frac{21.24}{1.3.7.12.23}=\frac{42}{23}(\leq 2=e(m, A))$ .
COROLLARY (1.9). Let the situation be as in (1.6).
(I) If the condition
th round up of the number $( \prod_{i=1}^{d}degx_{i})\lim_{\lambdaarrow 1}(1-\lambda)^{d}P(G, \lambda)=e(G_{+}, G)$$e$holds, then th $eequaIitye(m, A)=e(G_{+}, G)$ holds.
(2)$IfG$ is $a$ Aypersurface with th $e$ isolated singularity at $G_{+}$ , then $e(m, A)=e(G_{+}, G)$.

Proof. (1) is obvious from (1) of Theorem (1.6). (2) Let us represent as$=k[x_{1}$ , ... , $x_{d+1}]/f$ with $degf=h$ and $degX_{i}=q_{i}$ . Let us represent
by a linear
$G$

$f$

combination of monomials of the form $x^{M}= \prod:_{=1}x_{i}^{m:}$ with$f=$$\sum$$anr^{x^{M}}$$m_{i}\geq 0$with $a_{Af}\in k$$M\in(z_{\geq 0})^{d+1}$

We define the Newton support of

$f$

by

Support(f) $=\{M\in(z_{\geq 0})^{d+1}|a_{M}\neq 0\}$

5

as

$G$

169

The condition $\sum_{:}=1q_{i}m_{i}=h$ implies$q_{1}$.

$\frac{h}{q_{d+1}}\leq\sum_{i=1}m_{i}\leq\frac{h}{q_{1}}$$\ldots q_{d}\lim_{\lambdaarrow 1}(1-\lambda)^{d}P(G, \lambda)=\frac{h}{q_{d+1}}\leq multiplicity$.

$x^{M}\in Suppcrt(f)\leq\underline{h}=q_{2}$$\ldots.q_{d+1}D^{d-1}$

Hence we haveof $f=minimum$ of$\sum_{i=1}^{d+1}m_{i}$for. Since $\{f=0\}$ has only isolated singularity$q_{1}$, a monomial of form
with $j(i)\in\{1, \ldots d+1\}$ is contained in Support(f)for each (K. Saito [S1], V.I. Arnold, P. Orlik- Ph. Wagreich, and Fletcher[Fletcher]at$0$$x_{i}^{m_{i}}x_{j(*)}$

$i$

$)$

. In particular$[h/q_{d+1}]$$x_{d+1}$$x_{j(d+1)}\in$round up of the rational number
ExampleLet(1.10).

$A$

Support(f). Hence the multiplicity of$f$equals the$\frac{h}{q_{d+1}}$

a normal graded complete intersection ”) asbe (follows : $A=k[[x, y, z, w, u]]/(f_{1}, f_{2})$ with the filtration $F$ on$A$naturally induced as$degx=degy$ $=degz$ $=degw$ $=1$ , $degu=2$ and $degf_{1}=degf_{2}=3$ . We have
$G=k[x, y,z,w, u]/(f_{1}, f_{2})$ with $degx=degy$ $=degz$ $=degw$ $=1$

, $degu=2$ anddegfi $=degf_{2}=3$ . By (1.6) we obtain$\frac{9}{2}\leq e(m, A)\leq e(G_{+}, G)\leq 36$.

Since $(A, m)$ is not tangential complete intersection with respect to the maximal-idealadic filtration on , the lower bound is the best. But the upper bound of this implication
$a$

$A$

is very bad.
In the rest of this note we give a outline of proof of Theorem (1.6) and state some

generalities on the rational number
graded ring$R$$( \prod_{i=1}^{d}degx_{i})\lim_{\lambdaarrow 1}(1-\lambda)^{d}P(G, \lambda)$

in terms of Demazure’s description of$R$

for the normal.

\S 2 The openness of reduction property.
The purpose of this section is to prove (2.7) which we will use in \S 4.
(2.1)

loc

$a1$

Let

$I$

Let (V, ) be a singularity over a field$p$ring. We assume that$p$$I.O_{\overline{V}}$$o_{\gamma_{p}},\cdot$

is a locally principalLet$O_{\tilde{V}}$$\pi$$\tilde{V}$$\overline{V}$over the field$A$-module. We wiu represent $I.O_{V}$, as

.
6

$k$

.

: (V, ) $arrow(V,p)$ be a projective morphism$I.O_{\tilde{V}}=O_{\tilde{V}}(-D(I, \pi))$by a Cartier divisor on

and $(O_{V,p},m)$ be the associatedis a closed point of a projective variety

be an $m$ -primary ideal ofsuch that$k$$1?0$

Let th situation below as above. Asumme $d=\dim O_{\tilde{V}}$ . Then

THEOREM (2.2).

$e$$e(I, O_{V,p})=(-1)^{d+1}D(I, \pi)^{d}$.

Proof. Let a projective variety be a compactification of andwith center . There is a natural morphism :the blowing-up ofandsatisfies the relation. We have$V$$\overline{V}$$I$$\overline{V}$for arbitrary large integer$k$:$\overline{V}_{1}arrow\overline{V}$$\overline{\tilde{V}}arrow\overline{V}_{1}$$\tau$$I^{h}=\psi_{*}(I^{h}O_{V}’)$$\pi=\psi.\tau$$)$$\psi$bewhich$R^{i}\psi_{*}(I^{h}O_{\tilde{V}})=0(i\geq 1$(EGA III). We have $l(I^{h}/I^{h+1})=\chi(\overline{V}, I^{h}/I^{k+1})$ . By

Leray’s spectral sequence$E_{2}^{p,q}=H^{p}(\overline{V}, R^{q}\psi_{*}(I^{h}O_{\overline{V}_{1}}))\Rightarrow_{p}H^{n}(\overline{V}_{1}, I^{h}O_{\overline{V}_{1}})$,

we have.

$\sum_{q\geq 0}(-1)^{q}\chi(\overline{V}, R^{q}\psi_{*}(I^{h}O_{\overline{V}_{1}}))=\chi(\overline{V}_{1}, I^{h}O_{\overline{Y}_{1}})$Hence for $k>>0$ , we obtain$l(I^{h}/I^{h+1})=\chi(\overline{V}, I^{h}O_{\overline{V}})-\chi(\overline{V}, I^{h+1}O_{\overline{V}})$$=\chi(\overline{V}_{1}, I^{h}O_{\overline{V}_{1}})-\chi(\overline{V}_{1}, I^{h+1}O_{\overline{V}_{1}})$

Let$P\in \mathbb{Q}[t]$be the Hilbert-Samuel polynomial defined as.

$P(k)=\chi(\overline{V}_{1}, I^{h}O_{\overline{V}_{1}}/I^{h+1}O_{\overline{V}_{1}})$

for $k>>0[K1]$ . We have degree $P=d-1$ . Let us set of polynomials$1\leq m\leq d$$\Delta^{(m)}P$

as ; $\Delta^{(1)}P(k)=P(k)-P(k-1),$ . $..,$ $\Delta^{(m)}P=\Delta(\Delta^{(m-1)}P)$ , inductively.

Here $\Delta^{(d-1)}P$ is the constant function $e(I, O_{V,p})$ . Further we have$\Delta^{(1)}P(k)$$=\chi(\overline{V}_{1}, I^{h}O_{\overline{V}_{1}})-\chi(\overline{V}_{1}, I^{h+1}O_{\overline{V}_{1}})-\chi(\overline{V}_{1}, I^{h-1}O_{\overline{V}_{1}})+\chi(\overline{V}_{1}, I^{h}O_{\overline{V}_{1}})$.

$=-\chi(\overline{V}_{1}, I^{h+1}O_{\overline{V}_{1}})+2.\chi(\overline{V}_{1}, I^{h}O_{\overline{V}_{1}})-\chi(\overline{V}_{1}, I^{h-1}O_{\overline{V}_{1}})$

An by similar calculations we obtain$\Delta^{(d-1)}P(k)=-\sum_{=:0}^{d}()(-1)^{i}\chi(\overline{V}_{1}, I^{k+1-:}O_{\overline{V}_{1}})di$$=-\deg_{o_{V}(-D(I,\psi))}(O_{ba\tau V_{1}})$$=-(O_{V_{1}}(-D(I,\psi))^{d}.O_{V_{1}})_{V_{1}}$$=(-1)^{d+1}D(I,\psi)^{d}$Since$\tau$

for( the intersection symbol [K1]).

is birational, $(-1)^{d+1}D(I,\psi)^{d}=(-1)^{d+1}D(I, \pi)^{d}$.

Hence $e(I, O_{V,p})=(-1)^{d+1}D(I, \pi)^{d}$ . Q.E.D.

7

171

We will apply (2.2) to the following.
Let

integerbe $m$ -primary ideals of$J\subset I$$\prime r>0$

such that$\Gamma J=\Gamma^{+1}$$O_{V,p}$. Recall is a reduction of$J$

if there is an$I$(Northcott-Rees [NR]).

THEOREM (2.3) (D. REES $[HIO][REES],$ SEE ALSO J. LIPMAN [LIPMAN]). Assume

that $(A, m)$ is analyticaUy unramified. Then$J$

is a reduction of I if and only if th$e$equali $tye(J, O_{V,p})=e(I, O_{V,p})$ holds.

COROLLARY (2.4). For m-primary $ide$ alsequi valent each other.

$J\subset I$, the following three conditions are(1) The $equ$ allty $e(J, O_{V,p})=e(I,O_{V,p})$ holds.

is a reduction of .
(3) There exists a birationaJ morphism
(2)

$I$

$J$

:

$\psi$$\tilde{V}arrow Vsu$

ch that the relation$J.O_{\gamma}’=I.O_{\tilde{V}}$holds.

Proof.

The equivalence of (1) and (2) are due to $(2.3).Assume$ the condition (3)holds. Ther exists a birational morphism$e$$\tau$:$V’arrow\tilde{V}$such that $I.O_{V’}=J.O_{V^{r}}$ is local

principal. By (2.2) we have $e(J, O_{V,p})=e(I, O_{V,p})$ . Next we assume there is an integer$r>0$ such that $\Gamma J=\Gamma^{+1}$ . Let

is local principal. Then we have$\varphi$: $V‘arrow V$ be a birational morphism such that $I.O_{V’}$$J.\Gamma.O_{V’}=\Gamma^{+1}.O_{V^{\ell}}$and have $J.O_{V’}=I.O_{V’}$ .

By this we can see the reduction property of ideals are open condition as in thefollowing sense.
DEFINITION (2.5).

over a scheme$\rho^{-1}(0)=J$

$Y$

Let

$J$

be an ideal ofwith a reference point$O_{V,p}.$$0$

A deformation ofideal

is an ideal$\tilde{J}$of$O_{V\cross Y}$$J\rho$:$\tilde{J}arrow Y\ni c$

at $p\cross Y$ such that.
Let J C I be $m$ -primary ideals of

PROPOSITION (2.6).

deformation of ideal of J. Suppose
neighborhood

$U$

of

$c$

in

$Y$

where

Proof. There is an integer$J$$O_{V,p}$and$\rho$:$\tilde{J}arrow Y\ni o$be a

is a reduction of I. Then there is a Zariski open$\tilde{J}_{y}=\rho^{-1}(y)$$\prime r>0$

is a red uction of I for any pointsuch that 1‘ $J=\Gamma^{+1}$ and$\varphi$

$y$

of .
$U$

: $V’arrow V$ be abirational morphism such that $I.O_{V’}$ is local principal and $V’$ is normal. Then we have

J.Ovt $=I.Ov’$ . Consider the morphism$I.O_{V^{\ell}\cross Y}$

is defined as an invertible$\tilde{\varphi}$$O_{V’\cross Y}$: $V’\cross Yarrow V\cross Y$ with I.OV’

in a trivial extension of I.

8$O_{V’}$$\cross Y\supset\tilde{J}$. Here. Now we have

172

the relation J $O_{V’\cross Y}+\varphi^{-1}(m_{o})I.O_{V’\cross Y}=I.O_{V’\cross Y}$ . Henceequal at each generic points of $V’\cross Y$ which contains a point ofparticular the reflexive hull
non-reflexive locus of$(\tilde{J}.O_{V^{\ell}\cross Y})^{**}$$\tilde{J}.O_{V’\cross Y}$. Then$S$are$\tilde{\varphi}^{-1}(0)=V’\cross 0$

. In

and

equals $I.O_{VxY}$ . Let S C $V’\cross Y$ be thedoes not intersects $V’\cross c$ andcontain the point . By Corollary (2.4) at any point $y\in Y-\varphi(S)$$0$I.

$\tilde{J}.O_{V’\cross Y}$$I.O_{V’\cross Y}$$\tilde{J}_{y}$$\varphi(S)$does notis a reduction of

Q.E.D.

CORO LLARY (2.7).

Let I be an $m$ -primary ideal of

Suppose that $O_{V,p}c$ ontains a field$J=$ $(y_{1}, \ldots , y_{m})O_{V,p}$$k$gener ated as $I=(f_{1} , f_{*})$ .

and th at th $ere$ is a reduction $JofI$ written as

where

,

$wi$

th

Then there is a Zariski open neighborhood$U$$y_{i}= \sum_{j=1}^{\cdot}a:i^{t_{j}}$$a_{i,j}\in k1\leq i\leq m,$of$(a_{i,j})$$1\leq j\leq s$.

in $k^{\ell m}sucA$ th at $J_{b}=$is a reduction $ofI$ for $z_{i}= \sum_{j=1}b_{i,j}x_{j}$ , with $(b_{i,j})\in U$$(z_{1}, z_{m})O_{V,p}$

Proof.

$O_{V,p}$Define the deformation ofassertion follows from Proposition 3.

$J$

by

]

$\tilde{J}=$$j_{b\in k^{m}}.J_{b}$over$k^{m}$. The theQ.E.D.

We state the following which is a higher dimensional analogue of a theorem of
Laufer (cf. [L1]):
THEO $REM(2.8)$ .
$(x_{1}, \ldots , z_{d})$Let $(W, w)$ be a normal d-dim ension

a parameter system $ofO_{W,w}$ . Letwith $normaJX$ an $dE=\psi^{-1}(w)$ . We write$\psi$$e$assume that th $edi$ visor
If

$tr$

$div_{X}(x_{*}\cdot O_{X})$by$i=1,$$d$$\ldots,$,

ansform of $\{x_{i}=0\}$ and $D(x_{i}O_{W}, \psi)$ is the part $ofE$ . We

$W_{x:},\psi$$W_{g_{1}},\psi\cap\ldots\cap W_{x_{d}},\psi$

singularity and: $Xarrow W$ be a projective modification$div_{X}(x:O_{X})=D(x_{i}O_{W},\psi)+W_{x:},\psi$

where $Wae:,\psi$ is th strict

$aI$

is

$Q$

-Cartier for $i=l,$$\ldots d$.

is empty, we $have$ the relation$e((x_{1}, \ldots, x_{d}), O_{W,w})=(-1)^{d+1}D(x_{1}O_{W}, \psi)\ldots D(x_{d}O_{W},\psi)$

We omit the proof.

9

\S 3 On Demazure’s description of normal graded rings.
(3.1) The purpose of this section is to collect the generalities of Demazure’s de-scription $R(E, D)$ of the normal graded ringin the connection with the number$R$$(_{*} I_{=}^{d}I_{1}^{degx}:)\lim_{\lambdaarrow 1}(1-\lambda)^{d}P(G, \lambda)$

As there are many good references on this subjects $[D],$ [$W$

.

a ],
$1$

$[W\otimes 2]$, we will review

a computation method for Demazure’s divisor $D$ by a tentative way as follows (cf.

[T1]): Let $R=\oplus_{k\geq 0}R_{h}$ be a normal d-dimensional graded ring with $R_{0}=k,$ $R+$

the homogeneous maximal ideal with a generator consisting in homogeneous elements$x_{1}$

,

... ,

$x$

.

as $R+=$$(x_{1}$, ... , $x.)R$ with

$x_{i}\in R_{q_{i}}$condition G.C.D. $(q_{1}, q.)=1$ . There are integers
We ch ose a homogeneous element
$0$

$T$

... , . We assume thefor $i=1$ ,$u_{1},$$\ldots$,$u$of the quotient field of.

$s$such that $\sum_{i=1}u_{i}q_{i}=1$ .
as$R$$T=\Pi_{i=1}^{f}(x_{i})^{\psi}$. We

represent$x:R= \bigcap_{Q\in HP(R)}Q^{\langle a_{iQ})}$

,

1,

$i=$

$\ldots,$$S$.

Here $HP(R)$ is the set of homogeneous prime ideals of height 1. By Demazure’s fundamental works we can represent $D$ as follows:
THEOREM (3.2)(DEMAZURE [D]). In the above situation, we define the divisor $D$ associated to $T$ as$D= \sum_{:=1}^{S}(\sum_{Q\in HP(R)}\frac{u:a_{iQ}}{N(Q)}.V(Q))\in Div(E)\otimes \mathbb{Q}$where $V(Q)$ is the integral Weil divisor on $E=Proj(R)$ defined by
the integer defin $ed$ as ;$N(Q)=G.C.D.$ { $n\in Z|n>0$ and $(R/Q)$. }$\neq 0$$Q$

and $N(Q)$ is the(cf. $(5.9.I)$ of $[TWl]$ ).

Then we obtain th $eequ$ ali $ty$
$R=\oplus_{k\geq 0H^{0}(E,O_{E}(kD)).T^{h}}$Example (3.3). Let$R$

Brieskorn type as follows :in $k(E)[T]$ .

is a normal -dimensional$d$$(d\geq 2)$ graded ring of the$R=C[x_{1}, \ldots, x_{d+1}]/\{(x_{1})^{a_{1}}+\ldots+(x_{d+1})^{a_{d+1}}\}$

10

where

$a_{1},$$\ldots,$$a_{d+1}$

are integers$\geq 2$. Introduce the weight of each monomials of

the degree of $x_{i}=L.C.M.(a_{1}, \ldots, a_{d+1})/a_{i}$ . We simply denote it as

Then the Demazure divisor $D$ associated to$T$$q_{i}$$C[z_{1}, \ldots, x_{d+1}]$, for $i=1,$

as

$d+1$

.

is written as.

$D= \sum^{d+1}\frac{u_{*}}{\wedge}$ $D_{i}\in Div(E)\otimes Q$$i=1G.C.D.(q_{1}, \ldots, i, \ldots,q_{d+1})$

where$D_{i}$

is the integral Weil divisor on $E=Proj(R)$ defined by the canonical morphism$D_{i}=Proj(R/x_{i}R)arrow Proj(R)=E$ for $i=1,$

LEMMA (3.4).

$\ldots,$$d+1$ (see (1.9) of [T1] for a proof).

Let $R=R(E, D)$ be a ormal d-dimensional graded ring with De$n$

mazure’s description. Let us consider th singularity of $\dot{S}pec(R)$ at $V(R_{+})$ . Let$e$$\psi$: $C=C(E, D)=Spec_{E}(\oplus_{h\geq 0O_{E}(kD))}arrow Spec(R)$be the partiaI resolu tion by the ffitered blowing-up of Spec $(R)$ with respect to the fil-tration induced by grading of Spec $(R)$ . Let$R_{R_{+}}$. Suppose$x_{1},$$\ldots$,$x_{\tau}$with$\prime r\leq d$, $x_{d}\in R$ be a param eter system at
be $hom$ ogeneous elemen . Then we Irave$x_{1},$$\ldots$$ts$$\dim W_{x_{1}},\psi\cap\ldots\cap W_{x_{d}},\psi\leq d-r-1$

Hence in th case $r=d,$$e$$W_{x_{1}},\psi\cap\ldots\cap W_{x_{d}},\psi$in$C(E, D)$.

is empty. In this as $ee((x_{1}, x_{d}),$$c$$R_{R+}$)

is computed by Theorem (2.8).
By (2.8) and (3.4) we obtain the following.

COROLLARY (3.5).

Let $R=R(E, D)$ be a $normaI$ d-dimensional graded ring with

Demazure’s description and$x_{1}$,$x_{d}\in R$

be a $hom$ ogeneous param eter system of$e((x_{1}, ..., x_{d}), R)=(-1)^{d+1}( \prod_{i=1}^{d}degx_{i})$Here$E^{d}$.

$E^{d}$$R$.

.

is the intersection multiplicity in $C=C(E, D)$ .

LEMMA (3.6).

Let $R=R(E, D)$ be a normal d-dimensional graded ring wi th De-

maznre’s description and$\psi$: $C=C(E, D)=Spec_{E}(\oplus_{k\geq 0}O_{E}(kD))arrow Spec(R)$

11

175

be the filtered blowing-up of Spec $(R)$ with respec to th fil tration indu $ced$ by grading
$t$

$e$

of Spec $(R)$ .
Then we $have$ th rela tion$e$.

$D^{d-1}=(-1)^{d+1}E^{d}$

By using (1.7) we obtain the following.

COROLLARY (3.7).

in th situ a tion (3.6), assume$e$$x_{1},$$\ldots$, $x_{d}\in R$ be a homogeneous

parameter system.
$e((x_{1}, x_{d}),$ $R$ ) $=( \prod_{i=1}^{d}degx_{i}).D^{d-1}$$=(_{i}I_{=}^{d}I_{1}^{degx_{i}).\lim_{\lambdaarrow 1}(1-\lambda)^{d}P(R,\lambda)}\cdot$\S 4 Proof of Theorem (1.6).
(4.1) The inequality (i) of (1). Let a system of elements

ideal $m$ of$A$$G_{+}$$x$.

of the maximal

as follows ; $x_{i}\in F^{q_{i}}-F^{q:+1}$ and the initial forms

satisfies the relation $sG+=$$(\overline{x}_{1}, \ldots , \overline{x}.)G$

and$q_{1}\leq$...

see the relations $m=F^{n}+(x_{1} , x_{*})A$ for any positive integer$n$,$\ldots$

whose initial forms with respect to the filtration $F$ give the minimal

homogeneous generator of$=\overline{x}_{i}\in G_{q:}$$x_{1},$$\leq q.$$,$$n$$in_{F}(.z_{i})$. We can easily. There is an integer

such that $F^{n}Cm^{2}$ . Hence $m=(x_{1}, x.)$ by NAK.

There is a system of parameter

given as linear combination of$x_{1},$$\ldots$.

$1\leq j\leq s$

which is a minimal reduction of $m$ and$\ldots$,

as follows :$x$$y:= \sum_{j=1}^{l}a_{i,j}x_{j}$with $1\leq i\leq d$ ,,$y_{1},$$y_{d}$, where$a_{*,j}\in k$.

. By the openness of reduction property (Corollary (2.7)),

we may assume $A=(a_{i,j})_{1\leq i,j\leq d}$ is regular. So we can chooseform:$y_{i}=x_{i}+ \sum_{j=d+1}^{\cdot}a_{i,j}x_{j}$for $1\leq i\leq d$ from the beginning.

12, where$a_{i,j}\in k$$y_{i}$

in the following

Let

$L$

be a positive integer divided by L.C. $M.(degz_{1} , degz_{d})$ . By Leck’s lemma$e((y^{\frac{L}{1^{q_{1}}}}, ..., y^{\frac{L}{d^{*d}}}),A)=e((y_{1}\underline{L}.
$q_{1}$

Since$y_{i^{\frac{L}{g}}}\in F^{L}$\ldots.\underline{L}.

y_{d}), A)$.

$q_{d}$

for $1\leq i\leq d$, we have$e(F^{L},A) \leq\frac{L}{q_{1}}$There is an integer$L$.

$\ldots.\frac{L}{q_{d}}.e((y_{1}, y_{d}),$ $A$).

as above and satisfies the relation $F^{mL}=(F^{L})^{m}$ for any positive

integer $m$ , that is $(\oplus_{h\geq 0}F^{k}.T^{h})^{(L)}=A[F^{L}.T^{L}]$ . To finish the proof, it is sufficient toshow the following:
LEMMA (4.2).

Let

$L$

be a positive integer such th at the relation $F^{mL}=(F^{L})^{m}$ holds

for any positive integer $m$ . Then$e(F^{L}, A)=L^{d}.

Proof. By the assumption,\lim_{\lambdaarrow 1}(1-\lambda)^{d}P(G, \lambda)$$\oplus_{h\geq 0}F^{hL}/F^{(h+1)L}$.

is generated by $F^{L}/F^{2L}$Hence

we obtain the equality (see \S 13 and \S 14 of [Matsumura]):$e(F^{L}, A)= \lim_{\lambdaarrow 1}(1-\lambda)^{d}P(\oplus_{h\geq 0}F^{kL}/F^{(k+1)L}, \lambda)$.

Let us introduce the notation as $G^{(L)}=\oplus_{h\geq 0}F^{kL}/F^{kL+1}$ and, $L-1$ . Since there is an integer $M$ such that
$F^{L}.F^{b}=F^{L+b}$ holds for any $b\geq M,$, $L-1$ .
is a finite-module for $l=0,$
As graded-modules, we calculate the Poincare series ; $P(G^{(L,l)},\mu)\in Z[[\mu]]$ for$G^{(L,l)}=\oplus_{h\geq 0}F^{kL+l}/F^{kL+l+1}$

for $l=0,$$\ldots$$G^{(L,l)}$$G^{(L)}$$\ldots$$G^{(L)}$$l=0,$$L-1$ . For each$i,$$\lim_{\muarrow 1}(1-\mu)^{d}P(G^{(L,l)}, \mu)$is a finite number. We have therelations$(1- \mu)^{d}P(G, \mu)=\sum_{l=0}^{L-1}(1-\mu)^{d}P(G^{(L,l)}, \mu^{L}).\mu^{l}$$= \sum_{l=0}^{L-1}(1-\mu^{L})^{d}P(G^{(L,l)}, \mu^{L}).\frac{(1-\mu)^{d}}{(1-\mu^{L})^{d}}$.

$+ \sum_{l=0}^{L-1}(1-\mu^{L})^{d}P(G^{(L,l)},\mu^{L}).\frac{(1-\mu)^{d}}{(1-\mu^{L})^{d}}(\mu^{l}-1)$

13

Hence$\muarrow 1in_{1}(1-\mu)^{d}P(G, \mu)=\lim_{\muarrow 1}(1-\mu^{L})^{d}\sum_{l=0}^{L-1}P(G^{(L,l)}, \mu^{L}).\frac{1}{L^{d}}$$= \lim_{\muarrow 1}(1-\mu)^{d}P(\sum_{l=0}^{L-1}G^{(L,l)}, \mu).\frac{1}{L^{d}}$$= \lim_{\muarrow 1}(1-\mu)^{d}P(\oplus F^{kL}/F^{(h+1)L}, \mu).\frac{1}{L^{d}}$$*\underline{>}-O$$=e(F^{L}, A) \frac{1}{L^{d}}$(4.3) Proof of (2). Let$y_{1},$$\ldots$,.

be a parameter system of$y_{d}$$A$

as in the argumentsof (4.1). By the assumption we have the equality$e((y^{\frac{L}{1^{q_{1}}}} , ...

Hence$(y^{\frac{L}{1^{I1}}}, \ldots , y^{\frac{L}{d^{q_{d}}}})$y^{\frac{L}{d^{q_{d}}}}),$

is a reduction of$F^{L}$$A$) $=e(F^{L}, A)$ .

by a Theorem of Rees. There is an integer$r>0$ such that$(F^{L})^{\tau+1}=(F^{L})^{r}(y^{\frac{L}{1^{g_{1}}}} , y^{\frac{L}{d^{q_{d}}}})$

Let$\psi$in$A$.

: $X=Proj(\oplus_{k\geq 0}F^{k}.T^{h})arrow Spec(A)$ be the filtered blowing-up of Spec $(A)$ by $F$ .

We hav$e$$(F^{L})^{r+1}O_{X}=(F^{L})^{\tau}(y^{\frac{L}{1^{q_{1}}}}, \ldots , y^{\frac{L}{d^{q_{d}}}})O_{X}$Here $R^{L}O_{X}=O_{X}(L)$ is an invertible$O_{X}$

in$0_{X}$-module sheaf, we obtain the relation$F^{L}O_{X}=O_{X}(L)=(y^{\frac{L}{1^{q_{1}}}}$, ... ,$y^{\frac{L}{d^{q_{d}}}})O_{X}$.

We represent the strict transform of the scheme Spec $(A/y_{i})$ by$1,$, . Since$d$$\ldots$$(y^{\frac{L}{1^{q_{1}}}}, \ldots , y^{\frac{L}{d^{q_{d}}}})O_{X}$.

$\psi$

as$W_{y:},\psi$

for $i=$is locally free , $W_{y_{1}},\psi\cap\ldots\cap W_{y_{d}},\psi\cap E$ is empty.

Here$W_{y_{j}},\psi\cap E=Proj(G/In(y_{j})G)$

where In$(y_{j})$is the initial homogeneous element of

a parameter system of$G$$y_{j}$,. Therefore In, ... In$(y_{d})$is.

(4.4) Proof of the inequality (ii) of (i). There is an integer$G|_{mL}=(G|_{L})^{m}$$(y_{1})$$L$

satisfies the relation, that is $(G^{t})^{(L)}=G[G|_{L}]$ . Now we have $e(G|_{L}, G)=L^{d}. \lim_{\lambdaarrow 1}(1-$$14$

$\lambda)^{d}P(G, \lambda)$by (4.2). We an easily see $G|_{q.L}\subset(G_{+})^{L}$ . Hence$c$$L^{d}e(G_{+}, G)=e((G_{+})^{L}, G)\leq e(G|_{q.L}, G)=e((G|_{L})^{q}\cdot)=q_{l}^{d}e(G|_{L}, G)$$=q_{l}^{d}.L^{d}.

Therefore\lim_{\lambdaarrow 1}(1-\lambda)^{d}P(G, \lambda)$$e(G_{+}, G) \leq q_{l}^{d}.\lim_{\lambdaarrow 1}(1-\lambda)^{d}P(G, \lambda)$(4.5) $Proof$ of (3) By assumption$(G|_{L})^{q}$.

is a reduction ofarguments of (1) we consider the filtered blowing up$G$$\psi$$G_{+}^{L}$. As same as in theof Spec $(G)$ . By the assumption

is described by Demazure’s method as $G=R(E, D)$ . As in \S 3, we will represent$\psi$

as:$\psi$: $C=C(E, D)=s_{pec_{E}}(\oplus_{h\geq 0O_{E}(kD))}arrow Spec(R)$.

We obtain the relation$R_{+}^{L}O_{C}=(R|_{L})^{q}O_{C}=O_{C}$Since$x_{1}^{L}$$q_{1}=q.$

is not contained in$R|_{Lq_{1}+1}$( $-q.$ LE) on $C$ .

, we have the relation $R_{+}^{L}O_{C}=O_{C}(-Lq_{1}E)$ . Hence.

Acknowledgements.
$\Delta:!9\check{z}_{-\varpi n,)^{rightarrow}}t_{\vec{A}}tz\ovalbox{\tt\small REJECT}^{f_{\overline{A}}}B1_{D}^{-}\equiv$)}$\pm\Er3^{-}@02\mathfrak{l}^{arrow.aef_{\vee}^{\sim}- oT_{\backslash }}g\{x-\not\supset ffl-\#*$$arrowarrow a$vewaop$\ovalbox{\tt\small REJECT} brarrow\ovalbox{\tt\small REJECT}\langle\ovalbox{\tt\small REJECT}arrow-- l’\iota^{\backslash }\backslash \vee$

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17