Iv, a system of parameters in q such that w(i) is a superficial element of q( ) . Then n + d fl. i d - 1 i2. Ii. 142(Q/ci n+. 1 i f ). I i(2kV.(d + 1))( d. ⢠⢠â¢. E (Ai + Bd ,). i d - 1 ...
J . M ath. K yoto Univ. (JMKYAZ) 30-3 (1990) 475-479
A note on the coefficients of Hilbert polynomial By Ravinder KUMAR *
Let (Q, m) be a local ring of dimension d and q an m-primary ideal. It is w e ll known that for sufficiently large values of n (notation : n » 0) 1,2 (Qlqn + 1 ), the length of the Q-module Q / q n +1 is a uniquely determined polynomial of degree d, called the Hilbert polynom ial and is given by lQ (Qlqn + 1 ) = e 0 (
n+ d —
(n + d — e (\ — 1 )+ l
( -1)ded
where the integers e„, e , ed depend on q and are known as normalized Hilbert coefficients. ei is som etim es written a s ei (q) to emphasize its dependence o n q. It is easily seen that e , is p o sitiv e . In ca se Q is a Cohen-Macaulay ring, N orthcott [2] showed that e is non-negative and N arita [1] show ed that in this case e2 is non-negative a s w e ll. N arita gave an example of a Cohen-Macaulay ring and an m -prim ary ideal q such that e 3 (q) is negative. The purpose of this note is to show that for any integer d 3, it is possible to construct an example of a Cohen-Macaulay ring (Q, m) of dimension d and an mprimary ideal q such that e ( q ) is negative. W e u se th e a rg u e m e n ts given i n [ 1 ] to o b ta in e x p lic it values of the normalized Hilbert coefficients and use them subsequently to test our examples for th e claim m ade a b o v e . T o g iv e a general treatm ent w e m ust introduce some auxilliary notations which a re explained at th e appropriate p la c e . Throughout this note (Q, m) denotes a Cohen-M acaulay ring with infinite residue field. l
l
§ 1: T o start w ith w e quote th e following result
th a t (i) (ii) (iii)
1.1 (Northcott [2]). L et dim Q = d and w a superfical elem en t of q. S u p p o se w is n o t a z ero divisor. L et Q =- Q lQ w and = q lQ w . T h en w e have lo -(Q 1 4 ') = l Q (q" -": w lq " ." ) f o r a ll n > 0 /6 (Q/4 1 ) = 1,2 (421e 1 ) — 1.2 (Q lq") f o r a ll n>> 0 ei (q) = e J O , 0 i d — 1.
Received, Nov. 4, 1988 * This work was done during author's visit to the Dept. Math., Faculty of Science, Kyoto University, Kyoto, JAPAN.
Ravinder Kumar
476
Let w 1 , w 2 , , w
Let dim Q = d and q an m-primary ideal.
d
b e a system of
W e denote by Q( 0 the quotient ring of Q modulo
in q. parameters in
Qwi . q
(i)
i=
denotes the image of q in Q( 0 a n d w( denotes the im age of wi in Q . F o r t h e sake of uniformity Q( , ) , q ( l ) a n d w( 1 ) w ill denote Q , q a n d w , respectively. )
1). 1.2. T heorem . L et (Q, m ) b e a Cohen-M acaulay ring of dim ension d Iv, a sy stem of param eters in q such Suppose q is an m-primary ideal and w 1 , w 2 , th at w( i ) i s a superf icial elem ent of q( ) . Then
142 (Q/ci n+
n+ 1 if) i(2kV.(d + 1))( d
I
fl
d
Il
E
id - 2
=
i2
id - 2
1
=0
E
it
=0
...
—
0
id - 2
?I
Bd
-
1,i
—
E B1,1 i=0
ii = 0 1 = 0
0 id -3 = 0
d
w here A i =1,2 (Q1 E Qw,) — 142 (121e
+
1
E
k =1
k= 1
(A i + B d ,)
i= 0
il
1 — E E
d
1o
Q w k ),
and B i ,1 =1 Q ( i ) (q u t) : w ( J ) /q ) ), j = 1, 2, ... , d; i i
Ii
i2 • • •
id -
_
id - 1
0
1
W e shall denote the terms A i and corresponding letters. Bi ,1 f o r Q (2 b y bars over the Suppose d = 1. Write w, as w . Since w is a superficial element which is not a zero divisor, we have P ro o f . The proof is by induction on d. )
1(2(Q1Qw + q 1 ')
1 Q (Qlq
1
1
1
= 1 Q (Qlq = 1Q (Qlq
i + 1
) — 1,2 (Qw +
1
Iqi + 1 )
) — 1,2 ( Q / q ': w ) ) — 1Q (Q10 + 1,2 (q' : wig')
Thus 1
0 2 / q
i +1)
4 2 ( Q
i)
i c i
v Q
/
Q
w
q
i +1)
B
1
, 1
n a n d summing up the respective sides, we get
Putting i = 0, 1, ,
i,(2/ q "+1)= E 1 (Q 1=0
(2 )
Q
Iq V ) — i E 0 B 1 ,1
In + 1 )
= lQ(Q(20(
(1) n
7J
j
:
Ø
A 1—
E
i=o
B 1 ,1
Assume the result holds when dim Q = d — 1. Since dim(Q ( 2 ) ) given hypothesis ensures that 1 Q (2 )(Q (2 )/ q( 2) j +
1
) —
1Q(2)((Q(2))(d)
(n + d — 1 d—1
1d-
2
(A E3 = 0• • • ii EO E i=0
id - 2 = 0 id -
i+
—
d — 1, the
Coefficients of Hilbert polynomial 11
i2
i d -3
id_3=0 E
477
il
EE
E gi,i
ii =0 1 = 0 1
id - 4 = 0
=0
Using (1) we get n
k
k=0
i d -2 = 0
+d —1
n
E
1Q (Qlgn + 1 ) = 10 2 ) (02( 4 , 0 (
d—
i=0 12
E E
it
fi
E E
k = 0 id -3 = 0
1
) )
1
j
.;
(A i+ Bd — Ii
n
= 0 1= 0
k
E E
d — 2,i — ***
ii = 0 1 = 0
N o w observe th a t B _ = B produces th e desired result.
i2
E
E E
B ii— 1 B
k= 0 1 = 01
=0
1 ,i
e tc . C o n se q u e n tly , th e la s t expression readily
1.3. Corollary. W ith the sam e assum ptions as in Theorem 1.2, w e have the following: eso
= V Q(d+ 00
e = l
(A + B d ,i ) i
i=o 00
e = 2
00
E
00
z
(A i + B d ,i ) —B
= 0 i = it +1 1
co
e3
—
1 =0
co
ao
E
E
(A i +
E
B d i) ,
E
E
Bd _ 1 1 .
ii = 0 i = i 1 + 1 1
12 = 0 i 1 = i 2 + 1 i = ii+ 1
Bd _ 2 1 ,
=0
• •• • •• 00
ed =
00
E i d -1
=
0 id -
2
.••
id -
o
00
00
000
(A i + B d ,i ) —
E i 1 = 12 + 1 i = i 1 + 1
1+1
d -i
02
0
E •••
I = 12 +
i d -2 = 0
1
i= ii
B
1= 0
P ro o f . A ssum e t h a t d = 1 . C le a rly , a ll b u t finitely m any A i a n d B 1 ,1 vanish. T aking n sufficiently large, we get 00
e0
= V Q ( 2 ) ) a n d el =
E (A
i=
i
+ B 1 ,1)
The corollary now follows o n using the inductive process and the following identity : If atm ost finitely many real numbers a i a r e nonzero then k
j
E
.J=0 1=01
00
ai =
E =0
00
a i (k +1 )—
E
a)
ai
j= 0 i=j+1
1.4. R em arks. 1. T heorem 1.2 includes a s a special case Proposition 6 of Narita [1].
Ravinder Kumar
478 3
2. W hen d = 3, e0 = 1 Q (QIE Qw ,) is also mentioned in N a rita [1]. i= §2: 2.1. E xam ple. Let x 1 , x , , x d , x , I b e d + 1 indeterminates over a n infinite fie ld F . L e t Q = F[[x 1 , x 2 , . . . , x d + 1 ]]/(x d + 1 ). T h e n Q is a l o c a l Co hen 2, M acaulay rin g o f d im e n s io n d. A ssum e d 3. L e t A = { x 1 , x 2 , , 4:1, 4 1} and B = {x _ x , x x }. P ut C = A u B . Further, suppose that for two sets U and V of monomials in the above indeterminates U V denotes the set of all products of monomials in U with all monomials in V. A simple calculation shows that d
2
d
-
d
d + 1
1
-
d + 1
d
C. C. ••• C = C. C. •-• C. A u T •••••■ ■ , . . „ , . . ■ ■ • • • "
d
d - 1 tim es
tim es
where T is a se t o f monomials in which x d + , occurs in degree d. d + 1 denote respectively the images of x,, x , ... Let modulo 4 + 1 . Let
Xd, Xd +
2
(
1
=
1,
e d
(1 ,
1)•
-1 -
'b d
A s argued in th e previous paragraph, it is immediate that q d = qd - 1
d -1 d -1
—,
d -1 )
W e claim that ed is negative. It is clear that (qto t) : 1
0 and j = 1, 2, ... , d — 2.
= .4 i ) f o r a ll t
Again it is easily seen that d +1
,d - 2 n q ( d - 1 ) I V (d - 1 )
(„ d -1 . 1) • '.(d - 1)1
-
-
and
1
w here e , 71 denotes the im age of e , in Q 1 ;, Q (d). Further, using (*) it is seen that ‘,61+ 1 i l l - 1 d +
- 1 ( 4
q t( ,+ 11 1 ) :c
(1 ,1- 1
1 ) ) = q t(d - 1)
( d -
1
and
(q
. (d )
1 )
Cid)
1
.
,d - 2
1\ —
(d )
1
1 (d )
n
Z d - 1
V (d ) d
+
1
a n d c , denotes the image of i - 1 d +
)
qt(d ) f o r t
d — 1.
Thus we find that B 1,
= B 2 , = • • • = B d - 2 ,i =
0 for all
0 and
Bd _ i d_ ,
2 — Bd ,d _ 2 0
O.
Now substituting these values in the expression for ed as deduced in Corollary 1.3, we find that e(q) =
B d - 1 ,d - 2
