数理解析研究所講究録
第 713 巻 1990 年 206-212

NORMAL HILBERT POLYNOMIALSby Shiroh ITOH

1. Normal Hilbert Polynomials.

This note is a short summary of my recent work [5]. Throughtout this note$(A, m)$ will be a Cohen-Macaulay local ring of dimension $d\geq 2$ , andwill bea parameter ideal for , i.e.
is an m-primary ideal generated by elements.
Assume that is analytically unramified and $A/m$ is infinite. For an ideal in$A,$denotes the integral closure of , i.e., $\overline{J}=\{x\in A|x^{n}+a_{1}x^{n-1}+\cdots+a_{n}=$for some}.
It is well known that there exist uniquely determined integers
such that
$I$

$A$

$d$

$I$

$A$

$J$

$J$

$\overline{J}$$a_{i}\in J^{i}$$0$$\overline{e}_{0}(I),$$\cdots,\overline{e}_{d}(I)$$Iength_{A}(A/\overline{I^{n+1}})=\overline{e}_{0}(I)(\begin{array}{l}n+dd\end{array})-\overline{e}_{1}(I)(\begin{array}{l}n+d-1d-1\end{array})+\cdots+(-1)^{d}\overline{e}_{d}(I)$for all large$n$. We say that$P(I, n)=\overline{e}_{0}\{I)(\begin{array}{l}n+dd\end{array})-\overline{e}_{1}(I)(\begin{array}{ll}n+d -1d-1 \end{array})+$$+(-1)^{d}\overline{e}_{d}(I)$is a well-known number called theis the normal Hilbert polynomial of I.
multiplicity of i.e., $\overline{e}_{0}(I)=e(I)=length_{A}(A/I)$ . Our purpose of this note isand E3(Z). Our results are contained in
to report some properties ofthe following two theorems.
$\overline{e}_{0}(I)$$I$$\overline{e}_{1}(I),$THEOREM 1. (1)if and only if(2)for every $n\geq 0$ .

$\overline{e}_{2}(I)$$\overline{e}_{1}(I)-length_{A}(\overline{I}/I)\geq length_{A}(\overline{I^{2}}/I\overline{I})$$\overline{I^{n+2}}=I^{n}\overline{I^{2}}$for every$\overline{e}_{2}(I)\geq\overline{e}_{1}(I)-Iength_{A}(\overline{I}/I)$$n\geq 0$, and the equahty holds.

, and the equahty holds if and only if$\overline{I^{n+2}}=$$I^{n}\overline{I^{2}}$, andTHEOREM 2. (1)$n\geq0$.
every(2) Assume thatis Gorenstein andfor every $n\geq 0$ .
$\overline{e}_{3}(I)\geq 0$$if\overline{e}_{3}(I)=0$$A$$\overline{I^{n+2}}=I^{n}\overline{I^{2}}$1, then$\overline{I}=m$$\overline{I^{n+2}}$

is contained in$I^{n}$for. Then E3(Z) $=0$ if and only ifthe normal sectional genus of
Ooishi called the numberby.
and denoted it
In two dimensional case, Huneke remarked in his paper that$\overline{e}_{1}(I)-Iength_{A}(\overline{I}/I)$$I$$\overline{g}_{s}(I)$$Iength_{A}(A/\overline{I^{n+1}})=Iength_{A}(A/I)(\begin{array}{ll}n +2 2\end{array})$$-( \sum_{r\geq 0}Iength_{A}(\overline{I^{r+1}}/I\overline{I^{\prime r}}))(\begin{array}{ll}n +1 1\end{array})+ \sum_{r\geq 1}Iength_{A}(\overline{I^{\prime r+1}}/I\overline{I^{r}})r$

for all large$n$([3]). Therefore$\sum_{r\geq 1}1ength_{A}(\overline{I^{r+1}}/I\overline{I^{r}})r\geq\overline{g}_{s}(I)=\sum_{r\geq 1}length_{A}(\overline{I^{r+1}}/I\overline{I^{r}})$$\geq length_{A}(\overline{I^{2}}/I\overline{I})$;thus our Theorem 1 is a natural generalization of this fact to high dimensionalcase, and the main diMculty is how we reduce the problem to two dimensionalcase.

2. Key Theorem.

Detailed studies in$\overline{e}_{i}(I)s$are based on the following theorem and lemma:

THEOREM 3. There exists a system of generatorsof I such that,
if we put $C=A(T)/( \sum_{i}x_{i}T_{i})$ and $J=IC_{J}$ where $A(T)=A[T]_{m[T]}$ with $T=$$(T_{1}, \cdots T_{d})d$ indeterminates, then(1)for every $n\geq 0$ ;
(2)
;
(3)
for all large ;(4)
is normal if is analytically normal and $dimA\geq 3$ .
$x_{1},$$\cdots x_{d}$$\overline{J^{n}}\cap A=\overline{I^{n}}$$\overline{J}=\overline{I}C$$\overline{J^{n}}=\overline{I^{n}}C\cong\overline{I^{n}}A(T)/(\sum_{i}x_{i}T_{i})\overline{I^{n-1}}A(T)$$C$$n$$A$,LEMMA 4. Choose a system of minimal generatorsof and put$B=A[x_{1}/x_{2}],$ $R=R(A, I)=A[It,t^{-1}]$ (the Rees ring of), $D=R_{P}$ , where$P=(t^{-1},m)R$ , and $nD=$ the maximal ideal of D. Let: $Barrow D$ be acanonical $hom$ omorphism which maps
. Thento$x_{1},$$\cdots$

$I$

$h$

$x_{1}/x_{2}$$x_{1}t/x_{2}t$(0) $nD\cap B=m[x_{1}/x_{2}]$ .
We here put $C=Bm[x_{1}/x_{2}]$ and $J=IC$ . Then(1)every $n\geq 0$ ,(2)
and$\overline{J^{n}}\cap A=\overline{I^{n}}fo\tau$$\overline{J}=\overline{I}C$2$x_{d}$$I_{J}$

208(3)

for every$\overline{I^{n+r}}\subseteq\overline{I^{n}}$$n\geq 0$

if and only

$if\overline{J^{n+r}}\subseteq\overline{J^{n}}$for every$n\geq 0$.

Applying the theorem (and the lemma), we have the following results.
PROPOSITION 5. With the same notation as in Theorem 3, we have the followingassertions.
(1)
(2)
(3)

$\overline{e}_{i}(I)=\overline{e}_{i}(J)$$\overline{g}_{s}(I)=\overline{g}_{s}(J)$;$\overline{I^{2}}A(T)/I\overline{I}A(T)$$\geq Ierigth_{C}(\overline{J^{2}}/I\overline{J})$.

is a submodule ofis contained infor every $n\geq 0$ .

(4)$\overline{J^{n}}$for every $i\leq d-1$ ;$\overline{I^{n+r}}$$I^{n}$

for every$\overline{J^{2}}/J\overline{J}$$n\geq 0$, in articular$Iength_{A}(\overline{I^{2}}/I\overline{I})$$p$

if and only

$if\overline{J^{n+r}}$

is contained inPROOF. We put $z= \sum_{i}x_{i}T_{i}$ for simplicity. (1) and (2) follow from Theorem.
3. (3): It is enough to show that
$A(T)=I\overline{I}A(T)+\overline{I^{2}A(T)}\cap
zA(T)=I\overline{I}A(T)+\overline{IA(T)}z=I\overline{I}A(T)$. (4)(II, )
follows from Lemma 4(3).
$\overline{I^{2}}A(T)\cap(I\overline{I}, z)A(T)=I\overline{I}A(T)$$\overline{I^{2}}A(T)\cap$$z$

It is known that$d=dimA$ , we have
COROLLARY 6.

$\overline{e}_{2}(I)\geq\overline{g}_{s}(I)$$\overline{e}_{2}(I)\geq\overline{g}_{s}(I)$if $dimA=2$ ; therefore by the induction on.

As proved in [4, Proposition 10],$(*)length_{A}(A/\overline{I^{n+1}})$$\geq Iength_{A}(A/I^{n+1}\overline{I^{2}})$$=Iength_{A}(A/I)(\begin{array}{l}n+dd\end{array})-(length_{A}(\overline{I}/I)+length_{A}(\overline{I^{2}}/I\overline{I}))(\begin{array}{ll}n+d -1d -1\end{array})$$+Iength_{A}(\overline{I^{2}}/I\overline{I})(\begin{array}{ll}n+d -2d -2\end{array})$

for all$n$. Therefore$\overline{e}_{1}(I)\geq Iength_{A}(\overline{I}/I)+Iength_{A}(\overline{P}/I\overline{I})i.e.$,$\overline{g}_{s}(I)=\overline{e}_{1}(I)-length_{A}(\overline{I}/I)\geq length_{A}(\overline{I^{2}}/\Pi^{-})$

We give here the proof of (1) in Theorem 1.
PROPOSITION 7. (1)$\overline{g}_{s}(I)\geq Iength_{A}(\overline{I^{2}}/I\overline{I})$

3.

.

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(2)

$\overline{g}_{s}(I)=length_{A}(\overline{I^{2}}/I\overline{I})$if and only if$\overline{I^{n+2}}=I^{n}\overline{I^{2}}$for every$n\geq 0$.

PROOF. (2) We may assume that $A/m$ is infinite. We use the induction on $d=$$dimA$ . If $d=2$ , the assertion clearly holds. So assume that $d>2$ . if part follows
from. only if part: Choose a system of generators, of satisfying theconditions of Theorem 3, and put $z= \sum_{i}x_{i}T_{i},$ $C=A(T)/zA(T)$ and $J=IC$ .
Since
by Proposition 5, wehave. Thus
for every
and$(^{*})$$x_{1},$$\cdots$$x_{d}$$I$$Iength_{A}(\overline{I^{2}}/I\overline{I})\leq length_{C}(\overline{J^{2}}/J\overline{J})\leq\overline{g}_{s}(J)=\overline{g}_{s}(I)$$Iength_{C}(\overline{J^{2}}/J\overline{J})=\overline{g}_{s}(J)$$n\geq 0$. Then$\overline{J^{2}}=\overline{I^{2}}C$$\overline{I^{n+2}}A(T)=\overline{I^{n+2}A(T)}$$\overline{J^{n+2}}=J^{n}\overline{J^{2}}$is contained in $I^{n}\overline{I^{2}}A(T)+zA(T)$ , andhence $\overline{I^{n+2}}A(T)=\overline{I^{n+2}}A(T)\cap(I^{n}\overline{I^{2}}A(T)+zA(T))=I^{n}\overline{I^{2}}A(T)+z\overline{I^{n+1}}A(T)$ .
By the induction on $n,$, and therefore.
$\overline{I^{n+2}}=I^{n}\overline{I^{2}}$$\overline{I^{n+2}}A(T)=I^{n}\overline{P}A(T)$As we remarked in [4, Proposition 3],$(^{**})$

if$\overline{I^{n+2}}=I^{n}\overline{I^{2}}$for every $n\geq 0$ , then $G=R’/t^{-1}R’$ is Cohen-Macaulay.

Since$length_{A}(A/\overline{I^{n+1}})=Iength_{A}(A/I^{n+1})$$- \sum_{0\leq r\leq n}length_{A}(I^{n-r}\overline{I^{r+1}}/I^{n-r+1}\overline{I^{r}})$and$Iength_{A}(I^{n-r}\overline{I^{r+1}}/I^{n-r+1}\overline{I^{r}})$$\leq Iength_{A}((\overline{I^{r+1}}/I\overline{I^{r}})\otimes(I^{n-r}/I^{n-r+1}))$$=Iength_{A}(\overline{I^{r+1}}/I\overline{I^{r}})(\begin{array}{ll}n-r+d -1d-1 \end{array})$$=Iength_{A}(\overline{I^{r+1}}/I\overline{I^{r}})(\begin{array}{l}n+d-1d-1\end{array})-r(\begin{array}{ll}n-r+d -2d-2 \end{array})+Iower$

degree terms,

we have$\overline{e}_{1}(I)\leq\sum_{r\geq 0}Iength_{A}(\overline{I^{r+1}}/I\overline{I^{r}})$

i.e.,

.

$\overline{g}_{s}(I)=\overline{e}_{1}(I)-Iength_{A}(\overline{I}/I)\leq\sum_{r\geq 1}Iength_{A}(\overline{I^{r+1}}/I\overline{I^{r}})$PROPOSITION 8. (1)$\overline{g}_{s}(I)\leq\sum_{r\geq 1}Iength_{A}(\overline{I^{r+1}}/I\overline{I^{r}})$

4.

210

(2) If $depth_{M}R’\geq d_{J}$ then)where $M=(t^{-1}, It)R$ .
$\overline{g}_{s}\{I$$= \sum_{r\geq 1}Iength_{A}(\overline{I^{r+1}}/I\overline{I^{r}})$and$\overline{e}_{2}(I)=$$\sum_{r\geq 1}Iength_{A}(\overline{I^{r+1}}/I\overline{I^{r}})r_{f}$3.

$\overline{e}_{2}(I)$.

In this section, we shall give the proof of (2) in Theorem 1. By Corollary 6,the assertion remained to be proved is the following

PROPOSITION 9.

$\overline{e}_{2}(I)=\overline{g}_{s}(I)$

if and only if$\overline{I^{n+2}}=I^{n}\overline{I^{2}}$If part of the above proposition clearly holds by

from the following proposition.

$(^{*})$

for every$n\geq 0$.

. Only if part follows

PROPOSITION 10. Assume that $d\geq 3_{y}$ and choose a system of generators
satisfying the conditions of Theorem 1, and put $z= \sum_{i}x_{i}T_{i},$ $C=$$A(T)/zA(T)$ and $J=IC$ . If
thenfor every
every $n\geq 0$ .
$x_{1},$$\cdots$$x_{d}$$\overline{J^{n+2}}=J^{n}\overline{J^{2}}$$n\geq 0_{f}$$\overline{I^{n+2}}=I^{n}\overline{I^{2}}$$fo\tau$

PROPOSITION 11. Let
equivalent.

$r$

be either 1 or 2. Then the following assertions are(1)
for every $n\geq 0$ .
(2) $[H_{N}^{i}(R’)]_{j}=0$ for $i+j\geq r+1$ .
(3) $[H_{N}^{i}(R’)]_{j}=0$ for $i+j=r+1$ .
$\overline{I^{n+r}}=I^{n}\overline{I^{r}}$Proof of Proposition 10: Assume that, and choose a system of generators,
satisfying the conditions of Theorem 3, and put $z= \sum_{i}x_{i}T_{i},$ $C=$$A(T)/zA(T),$ $J=IC$ , as in Theorem 3. $S=A(T)\otimes R(=R(A(T), IA(T)))$ ,$x_{1},$$\cdots$$d\geq 3$$x_{d}$$S’=A(T)\otimes R’(=R’(A(T), IA(T))),$ $N’=ItS,$ $M’=(t^{-1},It)S’F=S/ztS(=$and $F’= \sum\overline{I^{n}C}t$ “ $(=R’(C, J))$ . Suppose that $[H_{N}^{i},(F’)]_{j}=0$ for$i+j=3$ . We shall prove $[H_{N}^{i}(R’)]_{j}=0$ for $i+j=3$ . Since $Aarrow A(T)$ isfaithfully flat, $H_{N}^{i}(R’)=H_{N}^{i},(S’)$ ; thus it is sufficient to prove that $[H_{N}^{i},(S’)]_{j}=$for $i+j=3$ . We first prove that $[H_{N}^{i-1}(S’/ztS’)]_{j+1}=0$ for $3\leq i\leq d$ and$j\geq 3-i$ . If this is proved, then $([H_{N}^{i}(S’)]_{j}=)[H_{N}^{i},(S’(-1))]_{j+1}arrow^{zt}[H_{N’}^{i}(S’)]_{j+1}$
is annihilated by some power of $zt$ ,is injective; since every element ofmust be . Since $dimF’/(S’/ztS’)=0$ , we have $H_{N}^{1},(S’/ztS’)=$$F’/(S’/ztS’)$ and $H_{N}^{i},(S’/ztS’)=H_{N}^{i},(F’)$ for $i\geq 2$ . Therefore, for $3\leq i\leq d$ ,$H_{N}^{i-1}(S’/ztS’)_{j+1}=H_{N’}^{i-1}(F’)_{j+1}=0$ , since $F’$ is a Cohen-Macaulay ring.
We next prove that $H_{N}^{2},(S’)_{1}=0$ . It is known that $H_{N}^{2},(S’)_{0}=0$ ; therefore$R(C, J))$$0$$H_{N}^{i},(S’)$$[H_{N}^{*}(S’)]_{j}$$0$5$0=H_{N}^{2},(S’)_{0}arrow^{zt}H_{N}^{2},(S’)_{1}arrow H_{N}^{2},(S’/ztS’)_{1}=0$

is also known that $H_{M}^{0},(S’)=H_{M}^{1},(S$Therefore $H_{N}^{i},(S’)=0$ for $i=0,1$ .

4.

$\overline{e}_{3}(I)$‘ $)$; hence $H_{N}^{2},(S’)1=0$ . It$=0$ and $H_{M}^{i},(S’)=H_{N}^{i},(S’)i=0,1$.

.

In general, it is known that$(-1)^{d} \overline{e}_{d}(I)=F(0)=\sum_{i}(-1)^{i}Iength_{A}([H_{N}^{i}(R’)]_{0})$.

If $d=3$ , then E3(I) $=Iength_{A}([H_{N}^{3}(R’)]_{0})=length_{A}(H^{2}(X, O_{X}))$ , because$[H_{N}^{2}(R’)]_{0}=[H_{M}^{2}(R’)]_{0}=0$ . Therefore by Proposition 5,$(^{***})\overline{e}_{3}(I)\geq 0$.

It is then natural to ask when
. It follows from [4, Appendix 2] that
there exists a canonical graded homomorphism : $H_{N}^{d}(R’)arrow H_{m}^{d}(A)[t, t^{-1}]$ .
We denote by
the graded part of degree of . Then we have$\overline{e}_{3}=0$$\alpha$$j$$\alpha_{j}$LEMMA 12.
for every $n\geq 0$ .

$\alpha_{j}=0$$\alpha$(i.e., $[H_{M}^{d}(R’)]_{j}=[H_{N}^{d}(R’)]_{j}$ ) if and only if$\overline{I^{n+d-1+j}}\subseteq I^{n}$

Proof of Theorem 2. (1): By Proposition 5, we may assume that $d=dimA=3$ .

Then the assertion follows fromfollowing proposition.

$(^{***})$PROPOSITION 13.
Assume that
andare contained inandfor every $n\geq 0$ .

$\overline{I^{n+2}}$$m\overline{I^{n+1}}$$A$$I^{n}$and Lemma 12. (2): This follows from theis a Gorenstein local ring andfor every $n\geq 0$ , then$\overline{I^{2}}=I\overline{I}$. If$Iength_{A}(\overline{I^{2}}/\Pi^{-})=1$$\overline{I^{n+2}}=I_{l}^{n}\overline{I^{2}}$,, we have
PROOF. Since$Iength_{A}(\overline{I^{n+1}}/I^{n}\overline{I})\leq
1ength_{A}((I^{n}:m)/I^{n})=(^{n-1+d-1}d-1)$, becauseand hence
is Gorenstein. Therefore$m\overline{I^{n+1}}\supseteq I^{n}$$(I^{n} :m)/I^{n}\supseteq(\overline{I^{n+1}}+I^{n})/I^{n}=\overline{I^{n+1}}/I^{n}\overline{I}$$A$$length_{A}(A/\overline{I^{n+1}})$$=Iength_{A}(A/I^{n+1})-Iength_{A}((I^{n}\overline{I}/I^{n+1})-Iength_{A}(\overline{I^{n+1}}/I^{n}\overline{I})$$\geq Iength_{A}(A/I)(\begin{array}{l}n+dd\end{array})-length_{A}(\overline{I}/I)(\begin{array}{l}n+d\cdot-1d-1\end{array})-(\begin{array}{lll}n d-1+ -1 d-1 \end{array})$$=Iength_{A}(A/I)(\begin{array}{l}n+dd\end{array})-(Iength_{A}(\overline{I}/I)+1)(\begin{array}{l}n+d-1d-1\end{array})+(\begin{array}{ll}n+d -2d -2\end{array})$6.

(We have already proved in [4, Proposition 10] that $length_{A}((I^{n}\overline{I}/I^{n+1})=$$length_{A}(\overline{I}/I)(^{n+d-1}d-1).)$ Thus by
and $Iength_{A}(A/\overline{I^{n+1}})$$=length_{A}(A/I)(nd+d)-(Iength_{A}(\overline{I}/I)+1)1_{d-1}^{n+d-1})+(^{n+d-2}d-2)$ , and in particular,$(^{*}),$$\overline{I^{n+1}}=I^{n-1}\overline{I^{2}}$$length_{A}(\overline{I^{2}}/I\overline{I})=1$.

It is natural to ask whether the assertion (2) in Theorem 2 is true for any
parameter ideals.

CONJECTURE: Assume that is Gorenstein and $d=dimA\geq 3$ . Then E3(I) $=0$
if and only if
for every $n\geq 0$ .
$A$$\overline{I^{n+2}}=I^{n}\overline{I^{2}}$Assume that $d=3$ and$[H_{N}^{2}(R’)]_{1})=0$ , then: By Proposition 11, if $[H_{M}^{2}(R’)]_{1}(=$for every $n\geq 0$ .

$\overline{e}_{3}(I)=0$$\overline{I^{n+2}}=I^{n}\overline{I^{2}}$REFERENCES
1. H. Flenner, Die Stz von Bertini fr locale Ringe, Math. Ann. 229 (1977),
97-111.
2. S. Goto and K. Watanabe, On graded rings I, J. Math. Soc. Japan 30(1978),179-213.

3. C. Huneke, Hilbert functions and symbolic powers, Michigan Math. J. 24(1987) $,293- 318$ .
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$117(1988),390- 401$ .

5. S. Itoh, Coefficients of normal Hilbert polynomials, preprint.
6. J. Lipman and B. Teissier, Pseudo-rational local rings and a theorem ofBrianon-Skoda about integral closures of ideals,Michigan Math. J. 28(1981) $,97- 116$ .
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Math. J. $12(1982),624- 644$ .
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