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Hilbert Coefficients of Hilbert Filtrations
199, 40]61 Ž1998. JA977194
JOURNAL OF ALGEBRA ARTICLE NO.
Hilbert Coefficients of Hilbert Filtrations A. Guerrieri* Dip. di Matematica, Uni¨ ersita ` di L’Aquila, Via Vetoio Coppito, 67010, L’Aquila, Italy
and M. E. Rossi † Dip. di Matematica, Uni¨ ersita ` di Geno¨ a, Via Dodecanesco, 35-16146, Geno¨ a, Italy Communicated by D. A. Buchsbaum Received June 25, 1996
INTRODUCTION Let Ž A, m. be a d-dimensional local ring with maximal ideal m. Let ` F s Fn4nG 0 be a filtration of A and denote by grF Ž A. s [ns0 FnrFnq1 the associated graded ring of F. If F is a Hilbert filtration, then grF Ž A. is a finite ArF1-module and for all n G 0 we denote by HF Ž n. s lŽ A r F1 .Ž FnrFnq1 . the Hilbert function of the local ring A with respect to F , where lŽ M . denotes the length of the A-module M. The generated series PF Ž z . s
Ý HF Ž n . z n jG0
is called the Hilbert series of F. It is well known that PF Ž z . is rational and that there exists a unique polynomial f F Ž z . g Zw z x with f F Ž1. / 0 such that PF Ž z . s
fF Ž z .
Ž1 y z.
* E-mail address: [email protected]. † E-mail address: [email protected]. 40 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
d
.
COEFFICIENTS OF HILBERT FILTRATIONS
41
For every j G 0 we define ej Ž F . s
f FŽ j. Ž 1 . j!
,
where f FŽ j. Ž1. denotes the jth formal derivative of the polynomial f F Ž z . in z s 1. The integers e j Ž F . are called the Hilbert coefficients of F. In the standard literature on local rings Žsee w7, 8, 13, 20, 21, 28x., only the integers e j Ž F . with 0 F j F d are considered but, as for the m-adic case Žsee w4, 5x., new and simple results can be obtained by considering the integers e j Ž F . also with j G d. The aim of this work is to study the Hilbert function and the Hilbert coefficients for a Hilbert filtration. Let us recall that if I is m-primary, the I-adic filtration is a Hilbert filtration; there are, however, several examples of non-I-adic filtrations, such as the Ratliff]Rush filtration, the filtration of integral closures Žprovided A is analytically unramified., and the filtration of tight closures Žprovided A contains a field.. It is interesting to notice that, in the current literature, many results in the I-adic case are found by using non-I-adic filtrations Žsee w7, 11, 18, 21x.. In the first section of this paper we extend most of the techniques of w4, 5x to the case of Hilbert filtrations and by doing so, we are also able to offer alternative simple proofs of some results of w7x. In the second section, thanks to the setting just described, if A is a local Cohen]Macaulay ring and F a Hilbert filtration, we give a sharp lower bound on e1Ž F . in terms of the multiplicity e0 Ž F ., of the length lŽ ArF1 ., and of a new invariant that extends the concept of embedding codimension of A. If the minimum value is reached, then grF Ž A. is Cohen]Macaulay and the Hilbert function is characterized Žsee Theorem 2.3.. This result was known for the m-adic filtration w4x. Furthermore if J is an ideal generated by a maximal superficial sequence, we find that e0 Ž F . s lŽ F1rF2 . q Ž 1 y d . lŽ ArF1 . q l Ž F2rJF1 . which extends the well known formulas by Abhyankar and Valla for the multiplicity. By using this equality and the bound found on e1Ž F ., we get e1 Ž F . G e0 Ž F . y l Ž ArF1 . G 0, which is well known for the I-adic filtration with I m-primary, w15x. Besides we characterize the Hilbert function when the equality holds Žsee w8, 16x in the m-primary case..
42
GUERRIERI AND ROSSI
In the last section we pursue the point of view that the extremal behaviour of the Hilbert coefficients forces the Hilbert function and, at the same time, gives information on the depth of the associated graded ring. Employing the results of the previous sections in the case of the I-adic filtration with I an m-primary ideal of a local Cohen]Macaulay ring, we characterize the Hilbert function when e1Ž I . y e0 Ž I . q lŽ ArI . s 1. In this case, we prove that the associated graded ring, denoted by grI Ž A., is Cohen]Macaulay if, and only if, I 2 l J s IJ with J an ideal generated by a maximal superficial sequence Žsee Proposition 3.1.. Similar results were obtained by Itoh w11x for I integrally closed, by Elias and Valla w4x for the m-adic filtration, and, if d G 2 and e2 Ž I . / 0, by Sally w20x. When I is integrally closed and e1Ž I . y e0 Ž I . q lŽ ArI . s 2 we completely describe the behavior of the Hilbert function and we give full information on the depth of the associated graded ring Žsee Theorem 3.2.. The latter completes, if I is integrally closed, a result proved by Sally in w21x in the particular case e2 Ž I . s 2 and d G 2. This result was obtained by Elias, Rossi, and Valla w5x in the case of the m-adic filtration. & If I is m-primary and F s I n4nG 0 is the Ratliff]Rush filtration, it is well known that e i Ž F . s e i Ž I . for i F d. We give a formula that relates e dq 1Ž F . and e dq1Ž I .. As a consequence we find a lower bound for e3 Ž I . in the 2-dimensional case. An example given by Narita w14x shows that this bound is sharp. 1. PRELIMINARIES In this section we start by introducing the concept of Hilbert filtration. After having recalled several well known concepts related to this kind of filtration, we extend, to this setting, various past results and find some new ones. Let Ž A, m. be a local ring of dimension d and maximal ideal m. DEFINITION 1.1. A set of ideals F s Fi 4i G 0 of A is called a filtration if F0 s A, F1 / A and for all indexes i and j one has Fiq1 : Fi and Fi Fj : Fiqj . Given a filtration F one can construct the Rees algebra and the graded ring associated to it. Namely R F Ž A . s A [ F1 t [ F2 t 2 [ ??? grF Ž A . s ArF1 [ F1rF2 [ ??? .
COEFFICIENTS OF HILBERT FILTRATIONS
43
F is said to be a Noetherian filtration if R F Ž A. is a Noetherian ring. If F is Noetherian, then dim grF Ž A. s d Žsee w7, Sect. 2x.. It is well known that if Ž A, m. is a local ring, then each of the above rings has exactly one maximal homogeneous ideal and the residue ring of this ideal is a field. If M is the unique homogeneous maximal ideal of grF Ž A., by depth grF Ž A. we mean the depth of the local ring grF Ž A.M . Let us remark that if F is the I-adic filtration for some ideal I we denote, as usual, R F Ž A. and grF Ž A. by R I Ž A. and grI Ž A.. DEFINITION 1.2. A filtration F s Fi 4i G 0 is said to be a Hilbert filtration if F1 is m-primary and R F Ž A. is a finite R F1Ž A.-module. As a reference on Hilbert filtrations, see Hoa, Trung, Viet, and Zarzuela w6, 24, 26x. It is also useful to recall that it is possible to prove Žsee Theorem III.3.1.1 and Corollary III.3.1.4 of w3x. that R F Ž A. is a finite R F1Ž A.-module if, and only if, there exists an integer k such that Fn : Ž F1 . ny k for all n G k. Clearly, if I is an m-primary ideal and F is the I-adic filtration, then F is a Hilbert filtration. There are however several examples of non-I-adic Hilbert filtrations Žsee w7, 6x.. In particular, if I is an ideal that contains a non-zero divisor in a local ring Ž A, m., Ratliff and Rush w17x introduced the ideal I˜s
D Ž I kq 1 : I k . s x g A : xI h : I hq1 for some h4 .
kG1
&
&
Since I ns D k G 1Ž I kqn & : I k ., then I ns I n for all n large enough. If I is m-primary, then F s I n4 is a Hilbert filtration and I˜ is the largest ideal with the same Hilbert polynomial as I. We call this filtration the Ž A. ) 0 Žsee w7, Lemma Ratliff]Rush filtration. In this case, depth gr &F 4.12x., while depth grI Ž R . ) 0 if, and only if, I ns I n for all n ) 0. Another example of Hilbert filtration is F s I n4nG 0 where I n denotes the integral closure of I n in an analytically unramified local ring A. If F is a Hilbert filtration, then grF Ž A. is a finite ArF1-module and, for every n G 0, we denote by HF Ž n. [ lŽ FnrFnq1 . the Hilbert function of F where lŽ M . denotes the length of the A-module M. The higher iterated Hilbert functions are defined as
HFi
¡H Ž n. , ¢Ý H
Ž n . s~
if i s 0
F
n
iy1 F
js0
Ž n. ,
if i ) 0
for every n G 0.
44
GUERRIERI AND ROSSI
The Hilbert series of F is PF Ž z . s
HF Ž n . z n ,
Ý nG0
and the iterated Hilbert series are PFi Ž z . s
HFi Ž n . z n .
Ý nG0
Since HFi Ž n. y HFi Ž n y 1. s HFiy1 Ž n. it is easy to see that PFi Ž z . s Ž1 y z . PFiq1 Ž z . for every i G 0. Since grF Ž A. is a finite ArF1-graded module it is well known Žsee w2, Corollary 4.1.8x. that there exists a unique polynomial f F Ž z . g Zw z x with f F Ž1. / 0 such that PF Ž z . s
fF Ž z .
Ž1 y z.
d
.
sŽ F . Clearly, if f F Ž z . s Ý is0 a i Ž F . z i with a i Ž F . g Z, then
a0 Ž F . s l Ž ArF1 .
and
a1 Ž F . s l Ž F1rF2 . y d lŽ ArF1 . .
As in the case of the m-adic filtration, we call Ž a0 Ž F ., . . . , a sŽ F .Ž F .. the h-¨ ector of F , f F Ž z . the h-polynomial of F , and we denote by sŽ F . the degree of the h-polynomial of F. For the higher iterated Hilbert series we have PFi Ž z . s
fF Ž z .
Ž1 y z.
dqi
.
For every i G 0, HFi Ž n. is a polynomial function for large n and we denote by h iF Ž x . g Qw x x its associated polynomial. This is a polynomial of degree d q i y 1 and it is called the i-Hilbert polynomial of A with respect to F. As in w2, Proposition 4.1.9x, it can be proved that dqiy1
h iF Ž x . s
Ý Ž y1. j e j Ž F .
js0
ž
xqdqiyjy1 , dqiyjy1
/
where, if we denote by f FŽ j. the jth formal derivative of the h-polynomial of F , ej Ž F . s
f FŽ j. Ž 1 . j!
45
COEFFICIENTS OF HILBERT FILTRATIONS
for j ) 0 and e0 Ž F . s f F Ž1.. In fact, by using the Taylor expansion in z s 1 one gets t
fF Ž z . s
Ý
f FŽ j. Ž 1 . j!
js0
t
j Ž z y 1. s
Ý e j Ž F . Ž z y 1. j js0
for every t G sŽ F . where e j Ž F . s 0 if j ) sŽ F .. Consequently PFi Ž z . s
Ýtjs0 e j Ž F . Ž z y 1 .
Ž1 y z.
j
dqi
dqiy1
s
Ý js0
j Ž y1. e j Ž F .
Ž1 y z.
dqiyj
q Qi Ž z . ,
where Q i Ž z . g Qw z x. Thus PFi Ž z . s
Ý
HFi Ž n . z n
nG0 dqiy1
Ý Ž y1. j e j Ž F . Ý
s
js0
nG0
dqiy1
s
Ý nG0
ž ž ž
žÝ
j Ž y1. e j Ž F .
js0
nqdqiyjy1 n z q Qi Ž z . dqiyjy1 nqdqiyjy1 dqiyjy1
/ / //
z n q Qi Ž z .
and hence for n 4 0 dqiy1
Ý Ž y1. j e j Ž F .
HFi Ž n . s
js0
ž
nqdqiyjy1 . dqiyjy1
/
If we define for every integer q G 0
ž
Ž x q q . Ž x q q y 1 . ??? Ž x q 1 . xqq s q q!
/
then we have dqiy1
h iF Ž x . s
Ý Ž y1. j e j Ž F .
js0
ž
xqdqiyjy1 . dqiyjy1
/
The integers ej Ž F . s
f FŽ j. Ž 1 . j!
sŽ F .
s
Ý ksj
k a F j kŽ .
ž/
46
GUERRIERI AND ROSSI
are called Hilbert coefficients of A with respect to F. In particular sŽ F . e0 Ž F . s f F Ž1. s Ý ks0 a k Ž F . is called the multiplicity with respect to F. It is clear that knowing the h-vector of F one knows the Hilbert coefficients and vice versa. Whenever F is an I-adic filtration with I m-primary we use the notation e i Ž I ., PIi Ž z ., HIi Ž n., sŽ I ., a i Ž I ., etc. If L is an ideal of A, we denote by FrL the filtration Ž Fn q L.rL4nG 0 . If F is a Hilbert filtration, then FrL is a Hilbert filtration too. In the following F will always be a Hilbert filtration. If x g F1 , it is easy to see that the following extension of Singh’s equality holds for all n G 0 HF Ž n . s HF1 Ž n . y l Ž Ž Fnq1 : x . rFn . , where F s FrŽ x .. DEFINITION 1.3. element x g F1 is integer c such that
Let Ž A, m. be a local ring of positive dimension d. An called a superficial element for F if there exists an Ž Fn : x . l Fc s Fny1 for all n ) c.
It is easy to see that a superficial element x is not in F2 and that x is superficial for F if, and only if, the initial form x* in F1rF2 does not belong to the relevant associated primes of grF Ž A.. Moreover if grade F1 G 1 and x is superficial for F , then x is a regular element of A, Ž Fnq 1 : x . s Fn for all n 4 0, and we have h F s h F1 where F s FrŽ x .. 1 Consequently e j Ž F . s e j Ž F . for every j s 0, . . . , d y 1. Last, note that if the residue field Arm is infinite, then a superficial element exists. The result below, proved in w7, Proposition 3.5x, is a generalization of a very well known result of Valabrega and Valla w27, Corollary 2.7x. PROPOSITION 1.4. Let F be a Noetherian filtration and x s x 1 , . . . , x k elements of F1. Then xU1 , . . . , xUk is a regular sequence if, and only if, x is a regular sequence and Ž x . l Fn s Ž x . Fny1 for all n G 1. The next result extends Proposition 1.2 in w5x to the case of Hilbert filtrations. PROPOSITION 1.5. filtration F. Then
Let x be a regular element superficial for a Hilbert
d d Ž y1. e d Ž F . s Ž y1. e d Ž F . y
Ý l Ž Ž Fjq1 : x . rFj . , jG0
where F s FrŽ x .. Furthermore the following facts are equi¨ alent: Ž1. Ž2.
e d Ž F . s e d Ž F ., e j Ž F . s e j Ž F . for all j G 0,
COEFFICIENTS OF HILBERT FILTRATIONS
Ž3. Ž4.
47
x* is a regular element in grF Ž A., PF Ž z . s Ž1 y z . PF Ž z ..
Proof. Since HF Ž n. s HF1 Ž n. y lŽŽ Fnq1 : x .rFn . for all n G 0, it is easy to see that n
HFiq1 Ž n . s HFiq2 Ž n . y
Ý js0
ž
nyjqi l Ž Ž Fjq1 : x . rFj . i
/
for all positive integers n, i. Hence for n 4 0 dqi
Ý Ž y1. j e j Ž F .
js0
ž
nqdyjqi dyjqi
dqi
s
Ý Ž y1. j e j Ž F .
js0
y
Ý jG0
ž
ž
/
nqdyjqi dyjqi
/
nyjqi l Ž Ž Fjq1 : x . rFj . . i
/
Since e j Ž F . s e j Ž F . for j s 1, . . . , d y 1, we get dqi
Ý Ž y1. j e j Ž F .
jsd
ž
nqdyjqi dyjqi
dqi
s
Ý Ž y1. j e j Ž F .
jsd
y
Ý jG0
ž
ž
/
nqdyjqi dyjqi
/
nyjqi l Ž Ž Fjq1 : x . rFj . . i
/
If i s 0 one has d d Ž y1. e d Ž F . s Ž y1. e d Ž F . y
Ý l Ž Ž Fjq1 : x . rFj . . jG0
From the previous formulas one obtains the following equivalent facts. First of all notice that e d Ž F . s e d Ž F . if, and only if, Ž Fjq1 : x . s Fj for all j G 0, or equivalently if, and only if, x* is regular in grF Ž A.. From the above equalities it is simple to see that Ž Fjq1 : x . s Fj for all j G 0 is also equivalent to e j Ž F . s e j Ž F . for all j G 0. Thus Ž1., Ž2., and Ž3. are all equivalent. Moreover, by Singh’s formula, x* is regular in grF Ž A. if, and only if, HF Ž n. s HF1 Ž n. for all n G 0 or equivalently if, and only if, PF Ž z . s PF1 Ž z . s PF Ž z .rŽ1 y z .. Then Ž3. and Ž4. are equivalent.
48
GUERRIERI AND ROSSI
DEFINITION 1.6. A sequence x 1 , . . . , x k is called a superficial sequence for F if x 1 is superficial for F and x i is superficial for FrŽ x 1 , . . . , x iy1 . for 2 F i F k. As we proved for one superficial element in Proposition 1.5, it is easy to see that if x 1 , . . . , x k is a superficial sequence for F , then xU1 , . . . , xUk is a regular sequence in grF Ž A. if, and only if, PF Ž z . s Ž1 y z . k PF Ž z . where F s FrŽ x 1 , . . . , x k .. If grade F1 G k and x 1 , . . . , x k is a superficial sequence for F , then e i Ž F . s e i Ž F . for 0 F i F d y k where F s FrŽ x 1 , . . . , x k . Žsee w9, Lemma 4x.. Recall that if A is a d-dimensional Cohen]Macaulay ring and F a Hilbert filtration, then there always exists a superficial sequence x 1 , . . . , x d for F. In w19x is proved that if x 1 , . . . , x d is a superficial sequence for the I-adic filtration with I m-primary, then Ž x 1 , . . . , x d . is a minimal reduction of I, that is, I n s Ž x 1 , . . . , x d . I ny 1 for all n 4 0. In the following we denote by J the ideal generated by a maximal superficial sequence x 1 , . . . , x d for F. If d s 0, we let J s Ž0.. We now recall a property of superficial sequences, shown by Huckaba and Marley in w7, Lemma 2.2x, that will be fundamental in the following. PROPOSITION 1.7. Let F be a Noetherian filtration and x 1 , . . . , x k a superficial sequence for F. If depth grF rŽ x 1 , . . . , x k .Ž ArŽ x 1 , . . . , x k .. ) 0, then depth grF Ž A. G k q 1. In w23; 22, Theorem 2.3, Corollary 2.4x. Sally already introduced the above ‘‘machine’’ in the m-adic case for k s d y 1 and she used it as a tool to reduce to dimension 1 the problem of proving the Cohen] Macaulayness of grmŽ A.. LEMMA 1.8. If A is a 1-dimensional Cohen]Macaulay local ring, F a Hilbert filtration, e0 Ž F . the multiplicity with respect to F , then HF Ž n . s e0 Ž F . y lŽ Fnq1rxFn . , where x is a superficial element for F. Proof. For this it is enough to notice that for each n G 0 one has HF Ž n . s lŽ FnrFnq1 . s lŽ ArFnq1 . y lŽ ArFn . s lŽ ArxFn . y l Ž Fnq1rxFn . y l Ž ArFn . s lŽ ArxA . q lŽ xArxFn . y lŽ Fnq1rxFn . y lŽ ArFn . . In fact e0 Ž F . s lŽ ArxA. and since x is regular lŽ xArxFn . s lŽ ArFn ..
49
COEFFICIENTS OF HILBERT FILTRATIONS
The previous lemma allows us to obtain, in a very simple way, some results due to Huckaba and Marley w7, Proposition 4.6, Lemma 4.12x. PROPOSITION 1.9. Let F be a Hilbert filtration of a Cohen]Macaulay local ring of positi¨ e dimension d. If depth grF Ž A. G d y 1, then ek Ž F . s
ž
Ý jGky1
j r ky1 j
;k G 1,
/
where r j s lŽ Fjq1rJFj .. Proof. Since depth grF Ž A. G d y 1, xU1 , . . . , xUdy1 is a regular sequence in grF Ž A., and e k Ž F . s e k Ž F . for all k G 0 where F s FrŽ x 1 , . . . , x dy1 . s Fn4nG 0 . Using Proposition 1.4 one obtains Fn Fnq 1rx d Fn ( Fnq1r Ž x d Fn q Fnq1 l Ž x 1 , . . . , x dy1 . . s Fnq1rJF and we can reduce ourselves to the 1-dimensional case. Thus we have PF Ž z . s
Ý sjs0 a j Ž F . z j
Ž1 y z.
and by Lemma 1.8 we deduce that for all n G 0 one has HF Ž n. s e0 Ž F . y rn s Ý nj G 0 a j Ž F . where a j Ž F . s 0 for j ) s. It follows that a j Ž F . s r jy1 y r j for every j G 1. Hence for every k G 1 ek Ž F . s
Ý jGk
j a ŽF. s k j
ž/
Ý jGk
j k
ž /Ž
r jy1 y r j . s
Ý jGky1
ž
j r. ky1 j
/
COROLLARY 1.10. Let Ž A, m. be a 2-dimensional Cohen]Macaulay local & ring and I an m-primary ideal of A. If F s I n4nG 0 is the Ratliff]Rush filtration, then e1 Ž I . s
&
&
Ý l ž I nq 1rJI n /
nG0
e2 Ž I . s
&
&
Ý n l ž I nq 1rJI n / .
nG1
Proof. Since, as previously noted, depth grF Ž A. ) 0, by Proposition 1.9 we have e1 Ž F . s
&
&
Ý l ž I nq 1rJI n /
nG0
e2 Ž F . s
&
&
Ý n l ž I nq 1rJI n / .
nG1
50
GUERRIERI AND ROSSI
&
However, I ns I n for n 4 0 implies that h1I s h1F and we conclude that e1 Ž I . s e1 Ž F . s
&
&
Ý l ž I nq 1rJI n /
nG0
e2 Ž I . s e2 Ž F . s
&
&
Ý n l ž I nq 1rJI n / .
nG1
Note that if x 1 , . . . , x k is a superficial sequence for the Ratliff]Rush filtration, then it is also a superficial sequence for the I-adic filtration. As a consequence of this setting we obtain the following result which is a generalization of Proposition 2.1 of w5x and of most of the results of Marley w12x concerning the properties of the Hilbert coefficients with respect to the I-adic filtration when I is m-primary and depth grI Ž A. G d y 1. Using Proposition 1.9 the proof is as in w5x. PROPOSITION 1.11. Let F be a Hilbert filtration of a Cohen]Macaulay local ring of positi¨ e dimension d and suppose depth grF Ž A. G d y 1. Then Ž1. e k Ž F . G 0 for e¨ ery k G 0. Ž2. If e k Ž F . s 0 for some k G 1, then e j Ž F . s 0 for e¨ ery j G k. Ž3. Žy1. k wŽÝ kjs 0 Žy1. j e j Ž F .. y lŽ ArF1 .x G 0, for e¨ ery k s 0, . . . , sŽ F .. Ž4. If Ý kjs0 Žy1. j e j Ž F . s lŽ ArF1 . for some k G 0, then sŽ F . F k. 2. BOUNDS ON e0 Ž F . AND e1Ž F . Let Ž A, m. be a d-dimensional Cohen]Macaulay local ring. In w4, Theorem 2.1x, Elias and Valla showed that if we denote by h [ dim k Ž mrm2 . y d the embedding codimension of A, then e1 Ž m . G 2 e 0 Ž m . y Ž h q 2 . . They proved that, if the equality holds, not only grmŽ A. is Cohen]Macaulay, but also the Hilbert series is characterized. Notice that, in this case, the embedding codimension h coincides with a1Ž m., the second term of the h-vector. It is also worth mentioning that the previous bound extends the well known inequality of Northcott, e1Ž m. G e0 Ž m. y 1 by using the well known inequality e0 Ž m. G h q 1, see w1x. Let us introduce the following extension of the concept of embedding codimension by defining the integer h Ž F . [ lŽ F1rF2 . y d lŽ ArF1 . q l Ž Ž F2 l J . rJF1 . , where J is the ideal introduced in the previous section.
COEFFICIENTS OF HILBERT FILTRATIONS
51
Clearly, hŽ F . coincides with the usual concept of embedding codimension in the case of the m-adic filtration. When F2 l J s JF1 Že.g., in the case of the I-adic filtration with I an integrally closed m-primary ideal., then hŽ F . s a1Ž F .. In general, as shown below ŽProposition 2.1., hŽ F . s a1Ž FrJ .. Using hŽ F . and the methods described in Section 1, we obtain a lower bound for e1Ž F . which generalizes the bound in w4x to the case of Hilbert filtrations and we characterize the minimal case. PROPOSITION 2.1. Let Ž A, m. be a Cohen]Macaulay local ring of dimension d and F s Fn4nG 0 a Hilbert filtration. Then e1 Ž F . G 2 e0 Ž F . y Ž h Ž F . q 2 l Ž ArF1 . . . Proof. Clearly, e1 Ž FrJ . s
Ý ja j Ž FrJ . jG1
s2
žÝ
a j Ž FrJ . y 2 a0 Ž FrJ . y a1 Ž FrJ .
jG0
q
/
Ý Ž j y 2. a j Ž FrJ . . jG3
Since ArJ is an Artinian ring, the a j Ž FrJ .’s are all non-negative. Thus e1 Ž FrJ . G 2 e0 Ž FrJ . y 2 l Ž ArF1 . y l Ž Ž F1 q J . rJr Ž F2 q J . rJ . s 2 e0 Ž F . y 2 lŽ ArF1 . y l Ž F1r Ž F2 q J . . . If d s 0 the result is proved. Let now d G 1 and notice that
l Ž F1r Ž F2 q J . . s l Ž F1rF2 . y l Ž Jr Ž F2 l J . . s lŽ F1rF2 . y l Ž JrJF1 . q l Ž Ž F2 l J . rJF1 . , where actually lŽ JrJF1 . s d lŽ ArF1 ... It is now clear that a1Ž FrJ . s lŽ F1rŽ F2 q J .. s hŽ F .. Since e1 Ž F . s e1 Ž Fr Ž x 1 , . . . , x dy1 . . G e1 Ž FrJ . one may eventually deduce that e1 Ž F . G 2 e0 Ž F . y 2 lŽ ArF1 . y l Ž F1r Ž F2 q J . . s 2 e0 Ž F . y Ž h Ž F . q 2 lŽ ArF1 . . .
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GUERRIERI AND ROSSI
Let us remark that when F is the m-adic filtration, then, from Proposition 2.1, one obtains the bound, already given in w4x, e1 Ž m . G 2 e Ž m . y Ž h q 2 . . THEOREM 2.2. Let Ž A, m. be a Cohen]Macaulay local ring of dimension d and F s Fn4n G 0 a Hilbert filtration. Then the following are equi¨ alent Ž1. Ž2.
e1Ž F . s 2 e0 Ž F . y Ž hŽ F . q 2 lŽ ArF1 .. F2 l J s JF1 and sŽ F . F 2.
If either of the abo¨ e conditions holds, then grF Ž A. is a Cohen]Macaulay ring. Proof. We have e1Ž F . s Ý j G 1 ja j Ž F . and, clearly, Ž2. implies Ž1. because hŽ F . s a1Ž F ., a i Ž F . s 0 for every i G 3 and a2 Ž F . s e0 Ž F . y hŽ F . y lŽ ArF1 .. To prove that Ž1. implies Ž2. one must observe that in proving Proposition 2.1 we obtained the following particular inequality e1 Ž F . G e1 Ž FrJ . s 2 e0 Ž F . y Ž h Ž F . q 2 l Ž ArF1 . . q
Ý Ž j y 2. a j Ž FrJ . . jG3
Thus when e1Ž F . attains its minimal value, then a j Ž FrJ . s 0 for j G 3. If d s 0, the result follows; if d G 1, then e1 Ž F . s e1 Ž Fr Ž x 1 , . . . , x dy1 . . s e1 Ž FrJ . and Proposition 1.5 guarantees that depth gr F r Ž x 1 , . . . , x d y 1 . Ž ArŽ x 1 , . . . , x dy1 .. ) 0. Now by Proposition 1.7 one has that grF Ž A. is a Cohen]Macaulay ring. In particular we get that PF Ž z .Ž1 y z . d s PF r J Ž z . and F2 l J s JF1. Now the result follows from the 0-dimensional case. Let us further remark that if e1Ž F . is minimal, then, being grF Ž A. Cohen]Macaulay, F2 l J s JF1. Therefore hŽ F . s a1Ž F . and e1 Ž F . s 2 e0 Ž F . y Ž a1 Ž F . q 2 l Ž ArF1 . . . Actually the above condition by itself is not sufficient to guarantee the Cohen]Macaulayness of grF Ž A.. Take, for example, A s k w x, y x, I s Ž x 6 , x 4 y, xy 5, y 6 . A and F the I-adic filtration. In this case J s Ž x 4 y, x 6 q y 6 . is a minimal reduction of I. Then, as shown in w10, Example 6.1x, I / I˜ and depth grI Ž A. s 0. However, using a computer algebra program as COCOA or MACAULAY, one can check that e0 Ž I . s 30, e1Ž I . s 10, e2 Ž I . s 3, and lŽ IrI 2 . s 50, so that e1Ž I . s 2 e0 Ž I . y lŽ IrI 2 . s 2 e0 Ž I . y Ž a1Ž I . q 2 lŽ ArI .. while lŽŽ I 2 l J .rIJ . s 3.
53
COEFFICIENTS OF HILBERT FILTRATIONS
The next result gives the extension to filtrations of Abhyankar’s and Northcott’s inequalities w1, 15x. PROPOSITION 2.3. Let Ž A, m. be a Cohen]Macaulay local ring of dimension d and F a Hilbert filtration. Then Ž1. Ž2.
e0 Ž F . G lŽ F1rF2 . q Ž1 y d . lŽ ArF1 . e1Ž F . G e0 Ž F . y lŽ ArF1 ..
Proof. To prove Ž1. we just extend to filtrations a well known formula of Valla w25, Lemma 1x. In fact considering the following short exact sequences 0 ª F1rJ ª ArJ ª ArF1 ª 0, 0 ª F2rJF1 ª F1rJF1 ª F1rF2 ª 0 0 ª JrJF1 ª F1rJF1 ª F1rJ ª 0 and recalling that lŽ JrJF1 . s d lŽ ArF1 ., one immediately obtains e0 Ž F . s l Ž F1rF2 . q Ž 1 y d . l Ž ArF1 . q lŽ F2rJF1 . .
Ž ).
Note that e0 Ž F . s a0 Ž F . q a1Ž F . q lŽ F2rJF1 .. In order to show Ž2. it is enough to use the inequality in Proposition 2.1 and Ž).. In fact we get e1 Ž F . y e0 Ž F . q l Ž ArF1 . G l Ž F2rJF1 . y l Ž Ž F2 l J . rJF1 . G 0.
Ž )).
Now we are interested in characterizing the Hilbert functions in the minimal case in both previous inequalities. In the case of the maximal ideal see w23, Theorem 1; 4, Corollary 2.2 Ž2.x; in the more general case of the m-primary ideal see w8, Theorem 2.1; 16x. Eventually, in the case of the Ratliff]Rush filtration, the following result is described in w11, Proposition 8x with d G 2. COROLLARY 2.4. Let Ž A, m. be a Cohen]Macaulay local ring of dimension d and F a Hilbert filtration. The following are equi¨ alent Ž1. Ž2.
e1Ž F . y e0 Ž F . q lŽ ArF1 . s 0, sŽ F . F 1.
If either of the abo¨ e conditions holds then grF Ž A. is Cohen]Macaulay. Proof. Clearly, Ž2. implies Ž1.. If Ž1. holds we have that e1Ž F . is minimal also with respect to the bound in Proposition 2.1. Thus Theorem
54
GUERRIERI AND ROSSI
2.2 implies that grF Ž A. is Cohen]Macaulay, F2 l J s JF1 and sŽ F . F 2. Hence Ž1. and Ž)). force F2 s JF1 and we get e0 Ž F . s a0 Ž F . q a1Ž F ., thus a2 Ž F . s 0. Next we recover a well known result for the I-adic filtration with I m-primary. COROLLARY 2.5. Let Ž A, m. be a Cohen]Macaulay local ring of dimension d and I an m-primary ideal. The following are equi¨ alent Ž1. Ž2. Ž3.
e1Ž I . y e0 Ž I . q lŽ ArI . s 0, e0 Ž I . s lŽ IrI 2 . q Ž1 y d . lŽ ArI ., sŽ I . F 1.
If either of the abo¨ e conditions holds then grI Ž A. is Cohen]Macaulay. Proof. In fact it is enough to notice that by Corollary 2.4, Ž1. is equivalent to Ž3. and that if Ž2. holds, then e0 Ž I . s a0 Ž I . q a1Ž I . and I 2 s JI. Consequently grI Ž A. is Cohen]Macaulay and Ž3. follows easily. It is now natural to ask if the result above described holds for Hilbert filtrations. The problem is to characterize the Hilbert functions when e0 Ž F . is minimal with respect to the bound given by Ž1. in Proposition 2.3. In particular we do not know if the condition F2 s JF1 Žequivalent to e0 Ž F . s lŽ F1rF2 . q Ž1 y d . lŽ ArF1 .. forces the associated graded ring to be Cohen]Macaulay. This problem does not exist for the I-adic filtration thanks to its multiplicative structure. Actually we are able to give a positive answer for those Hilbert filtrations that are definitely F1-adic, i.e., Fn s F1n
for all n 4 0.
We call these filtrations the F1-Hilbert filtrations. Clearly, for such filtrations h1F s h1F1. Examples of this type are given by the Ratliff]Rush filtration or F s Ž I nq 1 : Ž x 1 , . . . , x k ..4nG 0 where x 1 , . . . , x k is a superficial sequence with respect to the I-adic filtration. PROPOSITION 2.6. Let Ž A, m. be a Cohen]Macaulay local ring of dimension d and F a F1-Hilbert filtration. The following are equi¨ alent Ž1. Ž2. Ž3.
e1Ž F . y e0 Ž F . q lŽ ArF1 . s 0, e0 Ž F . s lŽ F1rF2 . q Ž1 y d . lŽ ArF1 ., sŽ F . F 1.
If either of the abo¨ e conditions holds then grF Ž A. is Cohen]Macaulay. Proof. Part Ž1. is equivalent to Ž3. by Corollary 2.4. Thus to conclude it is enough to show that Ž2. implies Ž3. since the converse is trivial. If Ž2.
COEFFICIENTS OF HILBERT FILTRATIONS
55
holds then F2 s JF1 and F12 : F2 s JF1 : F12 . We get F12 s F2 s JF1 and, in particular, e0 Ž F1 . s lŽ F1rF12 . q Ž1 y d . lŽ ArF1 .. By Corollary 2.5 we get sŽ F1 . F 1. However, a0 Ž F1 . s a0 Ž F . and a1Ž F1 . s a1Ž F . because F12 s F2 . Since e0 Ž F . s e0 Ž F1 ., if d s 0 we conclude. If d G 1 we have e1Ž F . s e1Ž F1 . since h1F s h1F1. Thus e1Ž F . y e0 Ž F . q lŽ ArF1 . s e1Ž F1 . y e0 Ž F1 . q lŽ ArF1 . s 0 by Corollary 2.5. Now the result follows by Corollary 2.4.
3. THE I-ADIC FILTRATION In this section we want to use the previously developed techniques in order to gain information on the I-adic case. We characterize the Hilbert function when e1Ž I . attains values close to the minimal one with respect to Nothcott’s inequality. As in w4x the point of view is that the extremal behaviour of the Hilbert coefficients forces the Hilbert function and gives information on the depth of the associated graded ring. According with the previous notation J s Ž x 1 , . . . , x d . denotes the ideal generated by a superficial sequence of I; if d s 0, we let J s Ž0.. PROPOSITION 3.1. Let Ž A, m. be a Cohen]Macaulay local ring of dimension d and I an m-primary ideal such that I 2 l J s IJ. The following are equi¨ alent: Ž1. e1Ž I . y e0 Ž I . q lŽ ArI . s 1, Ž2. e 0 Ž I . s lŽ IrI 2 . q Ž1 y d . lŽ ArI . q 1 and gr I Ž A . is a Cohen]Macaulay ring, Ž3. e1Ž I . s lŽ IrI 2 . q Ž1 y d . lŽ ArI . q 2, e2 Ž I . s 1, e i Ž I . s 0 ; i G 3, Ž4. PI Ž z . s Ž lŽ ArI . q Ž lŽ IrI 2 . y d lŽ ArI .. z q z 2 .rŽ1 y z . d. Proof. Note that Ž4. implies Ž1. trivially. In the same way it is clear that Ž3. and Ž4. are equivalent. Thus we just need to show that Ž1. implies Ž2. which implies Ž4.. If Ž1. is true, then e1 Ž I . s e0 Ž I . y l Ž ArI . q 1 G 2 e0 Ž I . y Ž l Ž IrI 2 . q Ž 2 y d . Ž ArI . . . It follows that e0 Ž I . F lŽ IrI 2 . q Ž1 y d . lŽ ArI . q 1. However, by Ž). in Proposition 2.3 applied to the case of the I-adic filtration, we have e0 Ž I . s l Ž IrI 2 . q Ž 1 y d . lŽ ArI . q l Ž I 2rIJ . .
56
GUERRIERI AND ROSSI
Thus by comparing the two expressions we get lŽ I 2rIJ . F 1. If it were lŽ I 2rIJ . s 0 we would have e0 Ž I . s lŽ IrI 2 . q Ž1 y d . lŽ ArI . and by Corollary 2.5 this would be equivalent to e1Ž I . y e0 Ž I . q lŽ ArI . s 0 which is not our case. We are forced to conclude that Ž1. implies lŽ I 2rIJ . s 1, e0 Ž I . s lŽ IrI 2 . q Ž1 y d . lŽ ArI . q 1, and e1Ž I . s 2 e0 Ž I . y Ž lŽ IrI 2 . q Ž2 y d . lŽ ArI ... Since by hypothesis I 2 l J s IJ, we can apply Theorem 2.2 to the case of the I-adic filtration to conclude that grI Ž A. is Cohen]Macaulay, so Ž1. implies Ž2.. Let now grI Ž A. be a Cohen]Macaulay ring and e0 Ž I . s lŽ IrI 2 . q Ž1 y . d lŽ ArI . q 1. By Ž)., we get lŽ I 2 rIJ . s 1. Since gr I Ž A . is Cohen]Macaulay all the terms a j Ž I ., 0 F j F sŽ I ., in the h-vector are positive. As known a0 Ž I . s lŽ ArI . and a1Ž I . s lŽ IrI 2 . y d lŽ ArI ., thus, since
l Ž IrI 2 . q Ž 1 y d . lŽ ArI . q 1 s e0 Ž I . s a0 Ž I . q ??? qa sŽ I . Ž I . we deduce that a2 Ž I . q ??? qa sŽ I .Ž I . s 1. Then a2 Ž I . s 1 and a j Ž I . s 0 for all j G 3. In w20, Theorem 1.4x, Sally showed that if d G 2, e2 Ž I . / 0, and e1Ž I . y Ž e0 I . q lŽ ArI . s 1 then depth grI Ž A. G d y 1. Itoh recently obtained the result described in Proposition 3.1 in the case of integrally closed ideals Žsee w11, Corollary 14x.. We note that in order to get the Cohen]Macaulayness of the associated graded ring is enough to have I 2 l J s IJ. This condition is automatically satisfied when I is the maximal ideal of A and in fact, precisely in this case, the previous result had already been obtained by Elias and Valla in w4, Corollary 2.2 Ž2.x. Let us remark that the assumption I 2 l J s IJ is essential and indeed, as illustrated in the following example due to Sally w20, Example 3.3x, the condition e1Ž I . s e0 Ž I . y lŽ ArI . q 1 alone does not guarantee the Cohen]Macaulayness of grI Ž A.. Let k be a field, A s w < t 4 , t 5, t 6 , t 7
57
COEFFICIENTS OF HILBERT FILTRATIONS
Elias, Rossi, and Valla w5, Proposition 2.4x, found that, in the case of the maximal ideal, the possible values for e2 Ž I . are either 2 or 3. Moreover they determined the Hilbert function in both cases. In the result below we extend the results in w5x to the case of an integrally closed m-primary ideal. THEOREM 3.2. Let Ž A, m. be a Cohen]Macaulay local ring of dimension d and I an integrally closed m-primary ideal. Suppose that e1Ž I . s e0 Ž I . y lŽ ArI . q 2. Then either Ž1. PI Ž z . s Ž lŽ ArI . q Ž lŽ IrI 2 . y d lŽ ArI .. z q 2 z 2 .rŽ1 y z . d and grI Ž A. is Cohen]Macaulay, or Ž2. PI Ž z . s Ž lŽ ArI . q Ž lŽ IrI 2 . y d lŽ ArI .. z q z 3 .rŽ1 y z . d and depth grI Ž A. s d y 1 holds. Proof. Note that by our assumption we have I 2 l J s IJ. Since e1 Ž I . s e0 Ž I . y l Ž ArI . q 2 G 2 e0 Ž I . y Ž l Ž IrI 2 . q Ž 2 yd . l Ž ArI . . , it follows that e0 Ž I . F l Ž IrI 2 . q Ž 1 y d . lŽ ArI . q 2. If the equality holds, then e1Ž I . s 2 e0 Ž I . y Ž lŽ IrI 2 . q Ž2 y d . lŽ ArI .. and we get Ž1. because of Theorem 2.3. Let us now assume e0 Ž I . - lŽ IrI 2 . q Ž1 y d . lŽ ArI . q 2. This means that e0 Ž I . s l Ž IrI 2 . q Ž 1 y d . lŽ ArI . q l Ž I 2rIJ . F l Ž IrI 2 . q Ž 1 y d . l Ž ArI . q 1 which implies lŽ I 2rIJ . F 1. However, lŽ I 2rIJ . / 0 since otherwise e0 Ž I . s lŽ IrI 2 . q Ž1 y d . lŽ ArI . which, by Corollary 2.5, would imply e1Ž I . s e0 Ž I . y lŽ ArI .. Thus we have lŽ I 2rIJ . s 1 and e0 Ž I . s lŽ IrI 2 . q Ž1 y d . lŽ ArI . q 1. Then, by Proposition 3.1, we deduce that grI Ž A. is not Cohen]Macaulay and, in particular, d G 1. If d s 1, then H F Ž n . s e 0 Ž I . y lŽ I nq 1 rxI n . and e 1Ž I . s Ý nG 0 lŽ I nq 1rxI n . where x is a superficial element in A. It follows that Ý nG 1 lŽ I nq 1rxI n . s 2. Hence lŽ I 2rxI . s lŽ I 3rxI 2 . s 1, and PI Ž z . s
lŽ ArI . q Ž l Ž IrI 2 . y lŽ ArI . . z q z 3
Ž1 y z.
.
Notice also that, in this case, e2 Ž I . s 3. Assume now d ) 1. To prove Ž2. it is enough to show that depth grI Ž A. G d y 1. By Proposition 1.7 we just need to reduce ourselves to the
58
GUERRIERI AND ROSSI
2-dimensional case and to prove that, under the same assumptions, depth grI Ž A. ) 0. By Proposition 1.5, e2 Ž I . F 3 s e2 Ž IrŽ x 1 .. where x 1 is a superficial element in A. The result follows if, and only if, e2 Ž I . s 3. We now prove that e2 Ž I . - 3 does not occur. By Corollary 1.10 e1 Ž I . s
&
&
Ý l ž I nq 1rJI n /
e2 Ž I . s
and
nG0
&
&
Ý n l ž I nq1rJI n / .
nG1
Since I is integrally closed, then I˜s I and &
&
Ý l ž I nq 1rJI n / s e1Ž I . y lŽ IrJ . s 2.
nG1
If e2 Ž I . F 2, then e2 Ž I . s 2 and depth grI Ž A. s 0. This however is impossible by w21, Theorem 3.1x. Let us remark that, as already noted in the case of the maximal ideal, both the Hilbert functions given in Theorem 3.2 are realizable. Next we want to compare the Hilbert coefficients coming from the & I-adic filtration with those coming from the filtration F s I n4nG 0 . LEMMA 3.3. Let Ž A, m. be&a Noetherian local ring of dimension d ) 0, I an m-primary ideal, and F s I n4nG 0 . Then e dq 1 Ž I . s e dq1 Ž F . q Ž y1 .
dq 1
&
Ý l ž I irI i / .
iG1
&
Proof. Let N be the least integer for which I ns I n for all n ) N. In particular take n ) N big enough so that one also has HI2 Ž n. s h 2I Ž n. and HF2 Ž n. s h 2F Ž n.. Clearly, we have dq1
h2I Ž n . y h 2F Ž n . s
Ý Ž y1. i Ž ei Ž I . y ei Ž F . .
is0
Ý is0
nqdq1yi dq1yi
/
&
Ny1
s
ž
l I iq1rI iq1 .
ž
/
&
Being I ns I n for n 4 0 it is clear that h1I Ž n. s h1F Ž n. for all n and one has e i Ž I . s e i Ž F . for all 0 F i F d. Thus we immediately get
Ž y1.
dq 1
Ny1
&
Ž e dq 1Ž I . y e dq1Ž F . . s Ý l ž I iq1rI iq1 / . is0
In the following J denotes, as before, the ideal generated by a maximal superficial sequence.
59
COEFFICIENTS OF HILBERT FILTRATIONS
PROPOSITION 3.4. Let Ž A, m. be a 2-dimensional Cohen]Macaulay local, I m-primary. Then
e3 Ž I . s e2 Ž I . y e1 Ž I . q e0 Ž I . y l Ž ArI . q
Ý jG3
y
& & jy2 l I jrJI jy1 2
ž /
ž
/
&
Ý l ž I jrI j / .
jG2
&
Proof. Since when F s I n4nG 0 one has depth grF Ž A. G 1, then, by Proposition 1.9, one has
ek Ž F . s
Ý jGky1
ž
& j l I jq1rJI˜j ky1
/
ž
/
for all k&G 1. As by Lemma 3.4 we have that e3 Ž I . y e3 Ž F . s yÝ i G 1 lŽI irI i ., we may conclude that e3 Ž I . s e3 Ž F . y
&
Ý l ž I irI i / s Ý
iG1
jG3
& & jy1 l I jrJI jy1 2
ž /
ž
/
y Ý l Ž I˜irI i . . iG1
Moreover, using Corollary 1.10, we have that e2 Ž I . y e1 Ž I . q e0 Ž I . y l Ž ArI . s e2 Ž F . y e1 Ž F . q e0 Ž F . y l Ž ArI . s
&
&
Ý j l ž I jq1rJI˜j / y Ý l ž I jq1rJI˜j / q lŽ ArJ . y lŽ ArI .
jG1
s
jG0
&
˜ . q lŽ IrJ . Ý Ž j y 1. l ž I jq1rJI˜j / y l Ž IrJ
jG1
s
&
˜ .. Ý Ž j y 2. l ž I˜jrJI jy1 / y l Ž IrI
jG3
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GUERRIERI AND ROSSI
Now we can conclude by noticing the identities e3 Ž I . s
jG3
s
& jy1 l I˜jrJI jy1 y 2
ž /ž / Ý žž / ž // Ýž / ž˜ / Ý
jy2 jy2 q 2 1
jG3
s
jG3
&
l I˜jrJI jy1 y
ž
& jy2 l I jrJI jy1 q 2
˜ .y y l Ž IrI
Ý l Ž I˜jrI j . jG1
/
Ý l Ž I˜jrI j . jG1
&
Ý Ž j y 2. l ž I˜jrJI jy1 /
jG3
Ý l Ž I˜jrI j . jG2
s e2 Ž I . y e1 Ž I . q e0 Ž I . y l Ž ArI . q
Ý jG3
y
Ý l Ž I˜jrI j . .
& jy2 l I˜jrJI jy1 2
ž /
ž
/
B
jG2
PROPOSITION 3.5. Let Ž A, m. be a 2-dimensional Cohen]Macaulay local ring and I an m-primary ideal of A. Then e3 Ž I . G y
&
Ý l ž I jrI j / .
jG1
Proof. This comes simply from the fact that by Lemma 3.4 and by Corollary 1.10 e3 Ž I . s e3 Ž F . y
&
Ý l ž I irI i / s Ý
iG1
jG3
& & jy1 l I jrJI jy1 2
ž /
ž
/
&
y Ý l I irI i . iG1
ž
/
The following example due to Narita w14x shows that the bound is sharp. Let A s k w < x, y, z
COEFFICIENTS OF HILBERT FILTRATIONS
61
REFERENCES 1. S. Abhyankar, Local rings of high embedding dimension, Amer. J. Math. 89 Ž1967., 1073]1077. 2. W. Bruns and J. Herzog, ‘‘Cohen Macaulay Rings,’’ Cambridge Univ. Press, Cambridge, UK, 1993. 3. N. Bourbaki, ‘‘Commutative Algebra,’’ Addison]Wesley, Reading, MA, 1972. 4. J. Elias and G. Valla, Rigid Hilbert functions, J. Pure Appl. Algebra 71 Ž1991., 19]41. 5. J. Elias, M. E. Rossi, and G. Valla, On the coefficients of the Hilbert polynomial, J. Pure Appl. Algebra 108 Ž1996., 35]60. 6. L. T. Hoa and S. Zarzuela, Reduction numbers and a-invariant of good filtration, Comm. Algebra, in press. 7. S. Huckaba and T. Marley, Hilbert coefficients and the depths of associated graded rings, preprint, 1994. 8. C. Huneke, Hilbert functions and symbolic powers, Michigan Math. J. 34 Ž1987., 293]318. 9. K. Kubota, On the Hilbert]Samuel function, Tokyo J. Math. 8 Ž1985., 439]448. 10. W. Heinzer, B. Johnston, D. Lantz, and K. Shah, Coefficient ideals in and blowups of a commutative Noetherian domain, J. Algebra Ž1993., 355]411. 11. S. Itoh, Hilbert coefficients of integrally closed ideals, J. Algebra 176 Ž1995., 638]652. 12. T. Marley, The coefficients of the Hilbert polynomial and the reduction number of an ideal, J. London Math. Soc. (2) 40 Ž1989., 1]8. 13. M. Nagata, ‘‘Local Rings,’’ Interscience, New York, 1962. 14. M. Narita, A note on the coefficients of Hilbert characteristic functions in semi-regular local rings, Math. Proc. Cambridge Philos. Soc. 59 Ž1963., 269]275. 15. D. G. Northcott, A note on the coefficients of the abstract Hilbert function, J. London Math. Soc. 35 Ž1960., 209]214. 16. A. Ooishi, D-genera and sectional genera of commutative rings, Hiroshima Math. J. 17 Ž1987., 361]372. 17. L. J. Ratliff and D. Rush, Two notes on reductions of ideals, Indiana Uni¨ . Math. J. 27 Ž1978., 929]934. 18. M. E. Rossi and G. Valla, A conjecture of J. Sally, Comm. Algebra 24Ž13., 4249]4261 Ž1996.. 19. I. Swanson, A note on the analytic spread, Comm. Algebra 22 No. 2 Ž1994., 407]411. 20. J. D. Sally, Hilbert coefficients and reduction number 2, J. Algebraic Geom. 1 Ž1992., 325]533. 21. J. D. Sally, Ideals whose Hilbert function and Hilbert polynomial agree at n s 1, J. Algebra 157 Ž1993., 334]547. 22. J. D. Sally, Super regular sequences, Pacific J. Math. 84 No. 2 Ž1979., 475]481. 23. J. D. Sally, Cohen Macaulay local rings of embedding dimension e q d y 2, J. Algebra 83 Ž1983., 393]408. 24. N. V. Trung, D. Q. Viet, and S. Zarzuela, When is the Rees algebra Gorenstein?, preprint, 1993. 25. G. Valla, On form rings which are Cohen]Macaulay, J. Algebra 58 Ž1979., 247]250. 26. D. Q. Viet, A note on the Cohen]Macaulayness of the Rees algebras of filtrations, Comm. Algebra 21 Ž1993., 221]229. 27. P. Valabrega and G. Valla, Form rings and regular sequences, Nago¨ a Math. J. 72 Ž1978., 93]101. 28. O. Zariski and P. Samuel, ‘‘Commutative Algebra,’’ Vol. 2, Van Nostrand, Princeton, 1960.
JOURNAL OF ALGEBRA ARTICLE NO.
Hilbert Coefficients of Hilbert Filtrations A. Guerrieri* Dip. di Matematica, Uni¨ ersita ` di L’Aquila, Via Vetoio Coppito, 67010, L’Aquila, Italy
and M. E. Rossi † Dip. di Matematica, Uni¨ ersita ` di Geno¨ a, Via Dodecanesco, 35-16146, Geno¨ a, Italy Communicated by D. A. Buchsbaum Received June 25, 1996
INTRODUCTION Let Ž A, m. be a d-dimensional local ring with maximal ideal m. Let ` F s Fn4nG 0 be a filtration of A and denote by grF Ž A. s [ns0 FnrFnq1 the associated graded ring of F. If F is a Hilbert filtration, then grF Ž A. is a finite ArF1-module and for all n G 0 we denote by HF Ž n. s lŽ A r F1 .Ž FnrFnq1 . the Hilbert function of the local ring A with respect to F , where lŽ M . denotes the length of the A-module M. The generated series PF Ž z . s
Ý HF Ž n . z n jG0
is called the Hilbert series of F. It is well known that PF Ž z . is rational and that there exists a unique polynomial f F Ž z . g Zw z x with f F Ž1. / 0 such that PF Ž z . s
fF Ž z .
Ž1 y z.
* E-mail address: [email protected]. † E-mail address: [email protected]. 40 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
d
.
COEFFICIENTS OF HILBERT FILTRATIONS
41
For every j G 0 we define ej Ž F . s
f FŽ j. Ž 1 . j!
,
where f FŽ j. Ž1. denotes the jth formal derivative of the polynomial f F Ž z . in z s 1. The integers e j Ž F . are called the Hilbert coefficients of F. In the standard literature on local rings Žsee w7, 8, 13, 20, 21, 28x., only the integers e j Ž F . with 0 F j F d are considered but, as for the m-adic case Žsee w4, 5x., new and simple results can be obtained by considering the integers e j Ž F . also with j G d. The aim of this work is to study the Hilbert function and the Hilbert coefficients for a Hilbert filtration. Let us recall that if I is m-primary, the I-adic filtration is a Hilbert filtration; there are, however, several examples of non-I-adic filtrations, such as the Ratliff]Rush filtration, the filtration of integral closures Žprovided A is analytically unramified., and the filtration of tight closures Žprovided A contains a field.. It is interesting to notice that, in the current literature, many results in the I-adic case are found by using non-I-adic filtrations Žsee w7, 11, 18, 21x.. In the first section of this paper we extend most of the techniques of w4, 5x to the case of Hilbert filtrations and by doing so, we are also able to offer alternative simple proofs of some results of w7x. In the second section, thanks to the setting just described, if A is a local Cohen]Macaulay ring and F a Hilbert filtration, we give a sharp lower bound on e1Ž F . in terms of the multiplicity e0 Ž F ., of the length lŽ ArF1 ., and of a new invariant that extends the concept of embedding codimension of A. If the minimum value is reached, then grF Ž A. is Cohen]Macaulay and the Hilbert function is characterized Žsee Theorem 2.3.. This result was known for the m-adic filtration w4x. Furthermore if J is an ideal generated by a maximal superficial sequence, we find that e0 Ž F . s lŽ F1rF2 . q Ž 1 y d . lŽ ArF1 . q l Ž F2rJF1 . which extends the well known formulas by Abhyankar and Valla for the multiplicity. By using this equality and the bound found on e1Ž F ., we get e1 Ž F . G e0 Ž F . y l Ž ArF1 . G 0, which is well known for the I-adic filtration with I m-primary, w15x. Besides we characterize the Hilbert function when the equality holds Žsee w8, 16x in the m-primary case..
42
GUERRIERI AND ROSSI
In the last section we pursue the point of view that the extremal behaviour of the Hilbert coefficients forces the Hilbert function and, at the same time, gives information on the depth of the associated graded ring. Employing the results of the previous sections in the case of the I-adic filtration with I an m-primary ideal of a local Cohen]Macaulay ring, we characterize the Hilbert function when e1Ž I . y e0 Ž I . q lŽ ArI . s 1. In this case, we prove that the associated graded ring, denoted by grI Ž A., is Cohen]Macaulay if, and only if, I 2 l J s IJ with J an ideal generated by a maximal superficial sequence Žsee Proposition 3.1.. Similar results were obtained by Itoh w11x for I integrally closed, by Elias and Valla w4x for the m-adic filtration, and, if d G 2 and e2 Ž I . / 0, by Sally w20x. When I is integrally closed and e1Ž I . y e0 Ž I . q lŽ ArI . s 2 we completely describe the behavior of the Hilbert function and we give full information on the depth of the associated graded ring Žsee Theorem 3.2.. The latter completes, if I is integrally closed, a result proved by Sally in w21x in the particular case e2 Ž I . s 2 and d G 2. This result was obtained by Elias, Rossi, and Valla w5x in the case of the m-adic filtration. & If I is m-primary and F s I n4nG 0 is the Ratliff]Rush filtration, it is well known that e i Ž F . s e i Ž I . for i F d. We give a formula that relates e dq 1Ž F . and e dq1Ž I .. As a consequence we find a lower bound for e3 Ž I . in the 2-dimensional case. An example given by Narita w14x shows that this bound is sharp. 1. PRELIMINARIES In this section we start by introducing the concept of Hilbert filtration. After having recalled several well known concepts related to this kind of filtration, we extend, to this setting, various past results and find some new ones. Let Ž A, m. be a local ring of dimension d and maximal ideal m. DEFINITION 1.1. A set of ideals F s Fi 4i G 0 of A is called a filtration if F0 s A, F1 / A and for all indexes i and j one has Fiq1 : Fi and Fi Fj : Fiqj . Given a filtration F one can construct the Rees algebra and the graded ring associated to it. Namely R F Ž A . s A [ F1 t [ F2 t 2 [ ??? grF Ž A . s ArF1 [ F1rF2 [ ??? .
COEFFICIENTS OF HILBERT FILTRATIONS
43
F is said to be a Noetherian filtration if R F Ž A. is a Noetherian ring. If F is Noetherian, then dim grF Ž A. s d Žsee w7, Sect. 2x.. It is well known that if Ž A, m. is a local ring, then each of the above rings has exactly one maximal homogeneous ideal and the residue ring of this ideal is a field. If M is the unique homogeneous maximal ideal of grF Ž A., by depth grF Ž A. we mean the depth of the local ring grF Ž A.M . Let us remark that if F is the I-adic filtration for some ideal I we denote, as usual, R F Ž A. and grF Ž A. by R I Ž A. and grI Ž A.. DEFINITION 1.2. A filtration F s Fi 4i G 0 is said to be a Hilbert filtration if F1 is m-primary and R F Ž A. is a finite R F1Ž A.-module. As a reference on Hilbert filtrations, see Hoa, Trung, Viet, and Zarzuela w6, 24, 26x. It is also useful to recall that it is possible to prove Žsee Theorem III.3.1.1 and Corollary III.3.1.4 of w3x. that R F Ž A. is a finite R F1Ž A.-module if, and only if, there exists an integer k such that Fn : Ž F1 . ny k for all n G k. Clearly, if I is an m-primary ideal and F is the I-adic filtration, then F is a Hilbert filtration. There are however several examples of non-I-adic Hilbert filtrations Žsee w7, 6x.. In particular, if I is an ideal that contains a non-zero divisor in a local ring Ž A, m., Ratliff and Rush w17x introduced the ideal I˜s
D Ž I kq 1 : I k . s x g A : xI h : I hq1 for some h4 .
kG1
&
&
Since I ns D k G 1Ž I kqn & : I k ., then I ns I n for all n large enough. If I is m-primary, then F s I n4 is a Hilbert filtration and I˜ is the largest ideal with the same Hilbert polynomial as I. We call this filtration the Ž A. ) 0 Žsee w7, Lemma Ratliff]Rush filtration. In this case, depth gr &F 4.12x., while depth grI Ž R . ) 0 if, and only if, I ns I n for all n ) 0. Another example of Hilbert filtration is F s I n4nG 0 where I n denotes the integral closure of I n in an analytically unramified local ring A. If F is a Hilbert filtration, then grF Ž A. is a finite ArF1-module and, for every n G 0, we denote by HF Ž n. [ lŽ FnrFnq1 . the Hilbert function of F where lŽ M . denotes the length of the A-module M. The higher iterated Hilbert functions are defined as
HFi
¡H Ž n. , ¢Ý H
Ž n . s~
if i s 0
F
n
iy1 F
js0
Ž n. ,
if i ) 0
for every n G 0.
44
GUERRIERI AND ROSSI
The Hilbert series of F is PF Ž z . s
HF Ž n . z n ,
Ý nG0
and the iterated Hilbert series are PFi Ž z . s
HFi Ž n . z n .
Ý nG0
Since HFi Ž n. y HFi Ž n y 1. s HFiy1 Ž n. it is easy to see that PFi Ž z . s Ž1 y z . PFiq1 Ž z . for every i G 0. Since grF Ž A. is a finite ArF1-graded module it is well known Žsee w2, Corollary 4.1.8x. that there exists a unique polynomial f F Ž z . g Zw z x with f F Ž1. / 0 such that PF Ž z . s
fF Ž z .
Ž1 y z.
d
.
sŽ F . Clearly, if f F Ž z . s Ý is0 a i Ž F . z i with a i Ž F . g Z, then
a0 Ž F . s l Ž ArF1 .
and
a1 Ž F . s l Ž F1rF2 . y d lŽ ArF1 . .
As in the case of the m-adic filtration, we call Ž a0 Ž F ., . . . , a sŽ F .Ž F .. the h-¨ ector of F , f F Ž z . the h-polynomial of F , and we denote by sŽ F . the degree of the h-polynomial of F. For the higher iterated Hilbert series we have PFi Ž z . s
fF Ž z .
Ž1 y z.
dqi
.
For every i G 0, HFi Ž n. is a polynomial function for large n and we denote by h iF Ž x . g Qw x x its associated polynomial. This is a polynomial of degree d q i y 1 and it is called the i-Hilbert polynomial of A with respect to F. As in w2, Proposition 4.1.9x, it can be proved that dqiy1
h iF Ž x . s
Ý Ž y1. j e j Ž F .
js0
ž
xqdqiyjy1 , dqiyjy1
/
where, if we denote by f FŽ j. the jth formal derivative of the h-polynomial of F , ej Ž F . s
f FŽ j. Ž 1 . j!
45
COEFFICIENTS OF HILBERT FILTRATIONS
for j ) 0 and e0 Ž F . s f F Ž1.. In fact, by using the Taylor expansion in z s 1 one gets t
fF Ž z . s
Ý
f FŽ j. Ž 1 . j!
js0
t
j Ž z y 1. s
Ý e j Ž F . Ž z y 1. j js0
for every t G sŽ F . where e j Ž F . s 0 if j ) sŽ F .. Consequently PFi Ž z . s
Ýtjs0 e j Ž F . Ž z y 1 .
Ž1 y z.
j
dqi
dqiy1
s
Ý js0
j Ž y1. e j Ž F .
Ž1 y z.
dqiyj
q Qi Ž z . ,
where Q i Ž z . g Qw z x. Thus PFi Ž z . s
Ý
HFi Ž n . z n
nG0 dqiy1
Ý Ž y1. j e j Ž F . Ý
s
js0
nG0
dqiy1
s
Ý nG0
ž ž ž
žÝ
j Ž y1. e j Ž F .
js0
nqdqiyjy1 n z q Qi Ž z . dqiyjy1 nqdqiyjy1 dqiyjy1
/ / //
z n q Qi Ž z .
and hence for n 4 0 dqiy1
Ý Ž y1. j e j Ž F .
HFi Ž n . s
js0
ž
nqdqiyjy1 . dqiyjy1
/
If we define for every integer q G 0
ž
Ž x q q . Ž x q q y 1 . ??? Ž x q 1 . xqq s q q!
/
then we have dqiy1
h iF Ž x . s
Ý Ž y1. j e j Ž F .
js0
ž
xqdqiyjy1 . dqiyjy1
/
The integers ej Ž F . s
f FŽ j. Ž 1 . j!
sŽ F .
s
Ý ksj
k a F j kŽ .
ž/
46
GUERRIERI AND ROSSI
are called Hilbert coefficients of A with respect to F. In particular sŽ F . e0 Ž F . s f F Ž1. s Ý ks0 a k Ž F . is called the multiplicity with respect to F. It is clear that knowing the h-vector of F one knows the Hilbert coefficients and vice versa. Whenever F is an I-adic filtration with I m-primary we use the notation e i Ž I ., PIi Ž z ., HIi Ž n., sŽ I ., a i Ž I ., etc. If L is an ideal of A, we denote by FrL the filtration Ž Fn q L.rL4nG 0 . If F is a Hilbert filtration, then FrL is a Hilbert filtration too. In the following F will always be a Hilbert filtration. If x g F1 , it is easy to see that the following extension of Singh’s equality holds for all n G 0 HF Ž n . s HF1 Ž n . y l Ž Ž Fnq1 : x . rFn . , where F s FrŽ x .. DEFINITION 1.3. element x g F1 is integer c such that
Let Ž A, m. be a local ring of positive dimension d. An called a superficial element for F if there exists an Ž Fn : x . l Fc s Fny1 for all n ) c.
It is easy to see that a superficial element x is not in F2 and that x is superficial for F if, and only if, the initial form x* in F1rF2 does not belong to the relevant associated primes of grF Ž A.. Moreover if grade F1 G 1 and x is superficial for F , then x is a regular element of A, Ž Fnq 1 : x . s Fn for all n 4 0, and we have h F s h F1 where F s FrŽ x .. 1 Consequently e j Ž F . s e j Ž F . for every j s 0, . . . , d y 1. Last, note that if the residue field Arm is infinite, then a superficial element exists. The result below, proved in w7, Proposition 3.5x, is a generalization of a very well known result of Valabrega and Valla w27, Corollary 2.7x. PROPOSITION 1.4. Let F be a Noetherian filtration and x s x 1 , . . . , x k elements of F1. Then xU1 , . . . , xUk is a regular sequence if, and only if, x is a regular sequence and Ž x . l Fn s Ž x . Fny1 for all n G 1. The next result extends Proposition 1.2 in w5x to the case of Hilbert filtrations. PROPOSITION 1.5. filtration F. Then
Let x be a regular element superficial for a Hilbert
d d Ž y1. e d Ž F . s Ž y1. e d Ž F . y
Ý l Ž Ž Fjq1 : x . rFj . , jG0
where F s FrŽ x .. Furthermore the following facts are equi¨ alent: Ž1. Ž2.
e d Ž F . s e d Ž F ., e j Ž F . s e j Ž F . for all j G 0,
COEFFICIENTS OF HILBERT FILTRATIONS
Ž3. Ž4.
47
x* is a regular element in grF Ž A., PF Ž z . s Ž1 y z . PF Ž z ..
Proof. Since HF Ž n. s HF1 Ž n. y lŽŽ Fnq1 : x .rFn . for all n G 0, it is easy to see that n
HFiq1 Ž n . s HFiq2 Ž n . y
Ý js0
ž
nyjqi l Ž Ž Fjq1 : x . rFj . i
/
for all positive integers n, i. Hence for n 4 0 dqi
Ý Ž y1. j e j Ž F .
js0
ž
nqdyjqi dyjqi
dqi
s
Ý Ž y1. j e j Ž F .
js0
y
Ý jG0
ž
ž
/
nqdyjqi dyjqi
/
nyjqi l Ž Ž Fjq1 : x . rFj . . i
/
Since e j Ž F . s e j Ž F . for j s 1, . . . , d y 1, we get dqi
Ý Ž y1. j e j Ž F .
jsd
ž
nqdyjqi dyjqi
dqi
s
Ý Ž y1. j e j Ž F .
jsd
y
Ý jG0
ž
ž
/
nqdyjqi dyjqi
/
nyjqi l Ž Ž Fjq1 : x . rFj . . i
/
If i s 0 one has d d Ž y1. e d Ž F . s Ž y1. e d Ž F . y
Ý l Ž Ž Fjq1 : x . rFj . . jG0
From the previous formulas one obtains the following equivalent facts. First of all notice that e d Ž F . s e d Ž F . if, and only if, Ž Fjq1 : x . s Fj for all j G 0, or equivalently if, and only if, x* is regular in grF Ž A.. From the above equalities it is simple to see that Ž Fjq1 : x . s Fj for all j G 0 is also equivalent to e j Ž F . s e j Ž F . for all j G 0. Thus Ž1., Ž2., and Ž3. are all equivalent. Moreover, by Singh’s formula, x* is regular in grF Ž A. if, and only if, HF Ž n. s HF1 Ž n. for all n G 0 or equivalently if, and only if, PF Ž z . s PF1 Ž z . s PF Ž z .rŽ1 y z .. Then Ž3. and Ž4. are equivalent.
48
GUERRIERI AND ROSSI
DEFINITION 1.6. A sequence x 1 , . . . , x k is called a superficial sequence for F if x 1 is superficial for F and x i is superficial for FrŽ x 1 , . . . , x iy1 . for 2 F i F k. As we proved for one superficial element in Proposition 1.5, it is easy to see that if x 1 , . . . , x k is a superficial sequence for F , then xU1 , . . . , xUk is a regular sequence in grF Ž A. if, and only if, PF Ž z . s Ž1 y z . k PF Ž z . where F s FrŽ x 1 , . . . , x k .. If grade F1 G k and x 1 , . . . , x k is a superficial sequence for F , then e i Ž F . s e i Ž F . for 0 F i F d y k where F s FrŽ x 1 , . . . , x k . Žsee w9, Lemma 4x.. Recall that if A is a d-dimensional Cohen]Macaulay ring and F a Hilbert filtration, then there always exists a superficial sequence x 1 , . . . , x d for F. In w19x is proved that if x 1 , . . . , x d is a superficial sequence for the I-adic filtration with I m-primary, then Ž x 1 , . . . , x d . is a minimal reduction of I, that is, I n s Ž x 1 , . . . , x d . I ny 1 for all n 4 0. In the following we denote by J the ideal generated by a maximal superficial sequence x 1 , . . . , x d for F. If d s 0, we let J s Ž0.. We now recall a property of superficial sequences, shown by Huckaba and Marley in w7, Lemma 2.2x, that will be fundamental in the following. PROPOSITION 1.7. Let F be a Noetherian filtration and x 1 , . . . , x k a superficial sequence for F. If depth grF rŽ x 1 , . . . , x k .Ž ArŽ x 1 , . . . , x k .. ) 0, then depth grF Ž A. G k q 1. In w23; 22, Theorem 2.3, Corollary 2.4x. Sally already introduced the above ‘‘machine’’ in the m-adic case for k s d y 1 and she used it as a tool to reduce to dimension 1 the problem of proving the Cohen] Macaulayness of grmŽ A.. LEMMA 1.8. If A is a 1-dimensional Cohen]Macaulay local ring, F a Hilbert filtration, e0 Ž F . the multiplicity with respect to F , then HF Ž n . s e0 Ž F . y lŽ Fnq1rxFn . , where x is a superficial element for F. Proof. For this it is enough to notice that for each n G 0 one has HF Ž n . s lŽ FnrFnq1 . s lŽ ArFnq1 . y lŽ ArFn . s lŽ ArxFn . y l Ž Fnq1rxFn . y l Ž ArFn . s lŽ ArxA . q lŽ xArxFn . y lŽ Fnq1rxFn . y lŽ ArFn . . In fact e0 Ž F . s lŽ ArxA. and since x is regular lŽ xArxFn . s lŽ ArFn ..
49
COEFFICIENTS OF HILBERT FILTRATIONS
The previous lemma allows us to obtain, in a very simple way, some results due to Huckaba and Marley w7, Proposition 4.6, Lemma 4.12x. PROPOSITION 1.9. Let F be a Hilbert filtration of a Cohen]Macaulay local ring of positi¨ e dimension d. If depth grF Ž A. G d y 1, then ek Ž F . s
ž
Ý jGky1
j r ky1 j
;k G 1,
/
where r j s lŽ Fjq1rJFj .. Proof. Since depth grF Ž A. G d y 1, xU1 , . . . , xUdy1 is a regular sequence in grF Ž A., and e k Ž F . s e k Ž F . for all k G 0 where F s FrŽ x 1 , . . . , x dy1 . s Fn4nG 0 . Using Proposition 1.4 one obtains Fn Fnq 1rx d Fn ( Fnq1r Ž x d Fn q Fnq1 l Ž x 1 , . . . , x dy1 . . s Fnq1rJF and we can reduce ourselves to the 1-dimensional case. Thus we have PF Ž z . s
Ý sjs0 a j Ž F . z j
Ž1 y z.
and by Lemma 1.8 we deduce that for all n G 0 one has HF Ž n. s e0 Ž F . y rn s Ý nj G 0 a j Ž F . where a j Ž F . s 0 for j ) s. It follows that a j Ž F . s r jy1 y r j for every j G 1. Hence for every k G 1 ek Ž F . s
Ý jGk
j a ŽF. s k j
ž/
Ý jGk
j k
ž /Ž
r jy1 y r j . s
Ý jGky1
ž
j r. ky1 j
/
COROLLARY 1.10. Let Ž A, m. be a 2-dimensional Cohen]Macaulay local & ring and I an m-primary ideal of A. If F s I n4nG 0 is the Ratliff]Rush filtration, then e1 Ž I . s
&
&
Ý l ž I nq 1rJI n /
nG0
e2 Ž I . s
&
&
Ý n l ž I nq 1rJI n / .
nG1
Proof. Since, as previously noted, depth grF Ž A. ) 0, by Proposition 1.9 we have e1 Ž F . s
&
&
Ý l ž I nq 1rJI n /
nG0
e2 Ž F . s
&
&
Ý n l ž I nq 1rJI n / .
nG1
50
GUERRIERI AND ROSSI
&
However, I ns I n for n 4 0 implies that h1I s h1F and we conclude that e1 Ž I . s e1 Ž F . s
&
&
Ý l ž I nq 1rJI n /
nG0
e2 Ž I . s e2 Ž F . s
&
&
Ý n l ž I nq 1rJI n / .
nG1
Note that if x 1 , . . . , x k is a superficial sequence for the Ratliff]Rush filtration, then it is also a superficial sequence for the I-adic filtration. As a consequence of this setting we obtain the following result which is a generalization of Proposition 2.1 of w5x and of most of the results of Marley w12x concerning the properties of the Hilbert coefficients with respect to the I-adic filtration when I is m-primary and depth grI Ž A. G d y 1. Using Proposition 1.9 the proof is as in w5x. PROPOSITION 1.11. Let F be a Hilbert filtration of a Cohen]Macaulay local ring of positi¨ e dimension d and suppose depth grF Ž A. G d y 1. Then Ž1. e k Ž F . G 0 for e¨ ery k G 0. Ž2. If e k Ž F . s 0 for some k G 1, then e j Ž F . s 0 for e¨ ery j G k. Ž3. Žy1. k wŽÝ kjs 0 Žy1. j e j Ž F .. y lŽ ArF1 .x G 0, for e¨ ery k s 0, . . . , sŽ F .. Ž4. If Ý kjs0 Žy1. j e j Ž F . s lŽ ArF1 . for some k G 0, then sŽ F . F k. 2. BOUNDS ON e0 Ž F . AND e1Ž F . Let Ž A, m. be a d-dimensional Cohen]Macaulay local ring. In w4, Theorem 2.1x, Elias and Valla showed that if we denote by h [ dim k Ž mrm2 . y d the embedding codimension of A, then e1 Ž m . G 2 e 0 Ž m . y Ž h q 2 . . They proved that, if the equality holds, not only grmŽ A. is Cohen]Macaulay, but also the Hilbert series is characterized. Notice that, in this case, the embedding codimension h coincides with a1Ž m., the second term of the h-vector. It is also worth mentioning that the previous bound extends the well known inequality of Northcott, e1Ž m. G e0 Ž m. y 1 by using the well known inequality e0 Ž m. G h q 1, see w1x. Let us introduce the following extension of the concept of embedding codimension by defining the integer h Ž F . [ lŽ F1rF2 . y d lŽ ArF1 . q l Ž Ž F2 l J . rJF1 . , where J is the ideal introduced in the previous section.
COEFFICIENTS OF HILBERT FILTRATIONS
51
Clearly, hŽ F . coincides with the usual concept of embedding codimension in the case of the m-adic filtration. When F2 l J s JF1 Že.g., in the case of the I-adic filtration with I an integrally closed m-primary ideal., then hŽ F . s a1Ž F .. In general, as shown below ŽProposition 2.1., hŽ F . s a1Ž FrJ .. Using hŽ F . and the methods described in Section 1, we obtain a lower bound for e1Ž F . which generalizes the bound in w4x to the case of Hilbert filtrations and we characterize the minimal case. PROPOSITION 2.1. Let Ž A, m. be a Cohen]Macaulay local ring of dimension d and F s Fn4nG 0 a Hilbert filtration. Then e1 Ž F . G 2 e0 Ž F . y Ž h Ž F . q 2 l Ž ArF1 . . . Proof. Clearly, e1 Ž FrJ . s
Ý ja j Ž FrJ . jG1
s2
žÝ
a j Ž FrJ . y 2 a0 Ž FrJ . y a1 Ž FrJ .
jG0
q
/
Ý Ž j y 2. a j Ž FrJ . . jG3
Since ArJ is an Artinian ring, the a j Ž FrJ .’s are all non-negative. Thus e1 Ž FrJ . G 2 e0 Ž FrJ . y 2 l Ž ArF1 . y l Ž Ž F1 q J . rJr Ž F2 q J . rJ . s 2 e0 Ž F . y 2 lŽ ArF1 . y l Ž F1r Ž F2 q J . . . If d s 0 the result is proved. Let now d G 1 and notice that
l Ž F1r Ž F2 q J . . s l Ž F1rF2 . y l Ž Jr Ž F2 l J . . s lŽ F1rF2 . y l Ž JrJF1 . q l Ž Ž F2 l J . rJF1 . , where actually lŽ JrJF1 . s d lŽ ArF1 ... It is now clear that a1Ž FrJ . s lŽ F1rŽ F2 q J .. s hŽ F .. Since e1 Ž F . s e1 Ž Fr Ž x 1 , . . . , x dy1 . . G e1 Ž FrJ . one may eventually deduce that e1 Ž F . G 2 e0 Ž F . y 2 lŽ ArF1 . y l Ž F1r Ž F2 q J . . s 2 e0 Ž F . y Ž h Ž F . q 2 lŽ ArF1 . . .
52
GUERRIERI AND ROSSI
Let us remark that when F is the m-adic filtration, then, from Proposition 2.1, one obtains the bound, already given in w4x, e1 Ž m . G 2 e Ž m . y Ž h q 2 . . THEOREM 2.2. Let Ž A, m. be a Cohen]Macaulay local ring of dimension d and F s Fn4n G 0 a Hilbert filtration. Then the following are equi¨ alent Ž1. Ž2.
e1Ž F . s 2 e0 Ž F . y Ž hŽ F . q 2 lŽ ArF1 .. F2 l J s JF1 and sŽ F . F 2.
If either of the abo¨ e conditions holds, then grF Ž A. is a Cohen]Macaulay ring. Proof. We have e1Ž F . s Ý j G 1 ja j Ž F . and, clearly, Ž2. implies Ž1. because hŽ F . s a1Ž F ., a i Ž F . s 0 for every i G 3 and a2 Ž F . s e0 Ž F . y hŽ F . y lŽ ArF1 .. To prove that Ž1. implies Ž2. one must observe that in proving Proposition 2.1 we obtained the following particular inequality e1 Ž F . G e1 Ž FrJ . s 2 e0 Ž F . y Ž h Ž F . q 2 l Ž ArF1 . . q
Ý Ž j y 2. a j Ž FrJ . . jG3
Thus when e1Ž F . attains its minimal value, then a j Ž FrJ . s 0 for j G 3. If d s 0, the result follows; if d G 1, then e1 Ž F . s e1 Ž Fr Ž x 1 , . . . , x dy1 . . s e1 Ž FrJ . and Proposition 1.5 guarantees that depth gr F r Ž x 1 , . . . , x d y 1 . Ž ArŽ x 1 , . . . , x dy1 .. ) 0. Now by Proposition 1.7 one has that grF Ž A. is a Cohen]Macaulay ring. In particular we get that PF Ž z .Ž1 y z . d s PF r J Ž z . and F2 l J s JF1. Now the result follows from the 0-dimensional case. Let us further remark that if e1Ž F . is minimal, then, being grF Ž A. Cohen]Macaulay, F2 l J s JF1. Therefore hŽ F . s a1Ž F . and e1 Ž F . s 2 e0 Ž F . y Ž a1 Ž F . q 2 l Ž ArF1 . . . Actually the above condition by itself is not sufficient to guarantee the Cohen]Macaulayness of grF Ž A.. Take, for example, A s k w x, y x, I s Ž x 6 , x 4 y, xy 5, y 6 . A and F the I-adic filtration. In this case J s Ž x 4 y, x 6 q y 6 . is a minimal reduction of I. Then, as shown in w10, Example 6.1x, I / I˜ and depth grI Ž A. s 0. However, using a computer algebra program as COCOA or MACAULAY, one can check that e0 Ž I . s 30, e1Ž I . s 10, e2 Ž I . s 3, and lŽ IrI 2 . s 50, so that e1Ž I . s 2 e0 Ž I . y lŽ IrI 2 . s 2 e0 Ž I . y Ž a1Ž I . q 2 lŽ ArI .. while lŽŽ I 2 l J .rIJ . s 3.
53
COEFFICIENTS OF HILBERT FILTRATIONS
The next result gives the extension to filtrations of Abhyankar’s and Northcott’s inequalities w1, 15x. PROPOSITION 2.3. Let Ž A, m. be a Cohen]Macaulay local ring of dimension d and F a Hilbert filtration. Then Ž1. Ž2.
e0 Ž F . G lŽ F1rF2 . q Ž1 y d . lŽ ArF1 . e1Ž F . G e0 Ž F . y lŽ ArF1 ..
Proof. To prove Ž1. we just extend to filtrations a well known formula of Valla w25, Lemma 1x. In fact considering the following short exact sequences 0 ª F1rJ ª ArJ ª ArF1 ª 0, 0 ª F2rJF1 ª F1rJF1 ª F1rF2 ª 0 0 ª JrJF1 ª F1rJF1 ª F1rJ ª 0 and recalling that lŽ JrJF1 . s d lŽ ArF1 ., one immediately obtains e0 Ž F . s l Ž F1rF2 . q Ž 1 y d . l Ž ArF1 . q lŽ F2rJF1 . .
Ž ).
Note that e0 Ž F . s a0 Ž F . q a1Ž F . q lŽ F2rJF1 .. In order to show Ž2. it is enough to use the inequality in Proposition 2.1 and Ž).. In fact we get e1 Ž F . y e0 Ž F . q l Ž ArF1 . G l Ž F2rJF1 . y l Ž Ž F2 l J . rJF1 . G 0.
Ž )).
Now we are interested in characterizing the Hilbert functions in the minimal case in both previous inequalities. In the case of the maximal ideal see w23, Theorem 1; 4, Corollary 2.2 Ž2.x; in the more general case of the m-primary ideal see w8, Theorem 2.1; 16x. Eventually, in the case of the Ratliff]Rush filtration, the following result is described in w11, Proposition 8x with d G 2. COROLLARY 2.4. Let Ž A, m. be a Cohen]Macaulay local ring of dimension d and F a Hilbert filtration. The following are equi¨ alent Ž1. Ž2.
e1Ž F . y e0 Ž F . q lŽ ArF1 . s 0, sŽ F . F 1.
If either of the abo¨ e conditions holds then grF Ž A. is Cohen]Macaulay. Proof. Clearly, Ž2. implies Ž1.. If Ž1. holds we have that e1Ž F . is minimal also with respect to the bound in Proposition 2.1. Thus Theorem
54
GUERRIERI AND ROSSI
2.2 implies that grF Ž A. is Cohen]Macaulay, F2 l J s JF1 and sŽ F . F 2. Hence Ž1. and Ž)). force F2 s JF1 and we get e0 Ž F . s a0 Ž F . q a1Ž F ., thus a2 Ž F . s 0. Next we recover a well known result for the I-adic filtration with I m-primary. COROLLARY 2.5. Let Ž A, m. be a Cohen]Macaulay local ring of dimension d and I an m-primary ideal. The following are equi¨ alent Ž1. Ž2. Ž3.
e1Ž I . y e0 Ž I . q lŽ ArI . s 0, e0 Ž I . s lŽ IrI 2 . q Ž1 y d . lŽ ArI ., sŽ I . F 1.
If either of the abo¨ e conditions holds then grI Ž A. is Cohen]Macaulay. Proof. In fact it is enough to notice that by Corollary 2.4, Ž1. is equivalent to Ž3. and that if Ž2. holds, then e0 Ž I . s a0 Ž I . q a1Ž I . and I 2 s JI. Consequently grI Ž A. is Cohen]Macaulay and Ž3. follows easily. It is now natural to ask if the result above described holds for Hilbert filtrations. The problem is to characterize the Hilbert functions when e0 Ž F . is minimal with respect to the bound given by Ž1. in Proposition 2.3. In particular we do not know if the condition F2 s JF1 Žequivalent to e0 Ž F . s lŽ F1rF2 . q Ž1 y d . lŽ ArF1 .. forces the associated graded ring to be Cohen]Macaulay. This problem does not exist for the I-adic filtration thanks to its multiplicative structure. Actually we are able to give a positive answer for those Hilbert filtrations that are definitely F1-adic, i.e., Fn s F1n
for all n 4 0.
We call these filtrations the F1-Hilbert filtrations. Clearly, for such filtrations h1F s h1F1. Examples of this type are given by the Ratliff]Rush filtration or F s Ž I nq 1 : Ž x 1 , . . . , x k ..4nG 0 where x 1 , . . . , x k is a superficial sequence with respect to the I-adic filtration. PROPOSITION 2.6. Let Ž A, m. be a Cohen]Macaulay local ring of dimension d and F a F1-Hilbert filtration. The following are equi¨ alent Ž1. Ž2. Ž3.
e1Ž F . y e0 Ž F . q lŽ ArF1 . s 0, e0 Ž F . s lŽ F1rF2 . q Ž1 y d . lŽ ArF1 ., sŽ F . F 1.
If either of the abo¨ e conditions holds then grF Ž A. is Cohen]Macaulay. Proof. Part Ž1. is equivalent to Ž3. by Corollary 2.4. Thus to conclude it is enough to show that Ž2. implies Ž3. since the converse is trivial. If Ž2.
COEFFICIENTS OF HILBERT FILTRATIONS
55
holds then F2 s JF1 and F12 : F2 s JF1 : F12 . We get F12 s F2 s JF1 and, in particular, e0 Ž F1 . s lŽ F1rF12 . q Ž1 y d . lŽ ArF1 .. By Corollary 2.5 we get sŽ F1 . F 1. However, a0 Ž F1 . s a0 Ž F . and a1Ž F1 . s a1Ž F . because F12 s F2 . Since e0 Ž F . s e0 Ž F1 ., if d s 0 we conclude. If d G 1 we have e1Ž F . s e1Ž F1 . since h1F s h1F1. Thus e1Ž F . y e0 Ž F . q lŽ ArF1 . s e1Ž F1 . y e0 Ž F1 . q lŽ ArF1 . s 0 by Corollary 2.5. Now the result follows by Corollary 2.4.
3. THE I-ADIC FILTRATION In this section we want to use the previously developed techniques in order to gain information on the I-adic case. We characterize the Hilbert function when e1Ž I . attains values close to the minimal one with respect to Nothcott’s inequality. As in w4x the point of view is that the extremal behaviour of the Hilbert coefficients forces the Hilbert function and gives information on the depth of the associated graded ring. According with the previous notation J s Ž x 1 , . . . , x d . denotes the ideal generated by a superficial sequence of I; if d s 0, we let J s Ž0.. PROPOSITION 3.1. Let Ž A, m. be a Cohen]Macaulay local ring of dimension d and I an m-primary ideal such that I 2 l J s IJ. The following are equi¨ alent: Ž1. e1Ž I . y e0 Ž I . q lŽ ArI . s 1, Ž2. e 0 Ž I . s lŽ IrI 2 . q Ž1 y d . lŽ ArI . q 1 and gr I Ž A . is a Cohen]Macaulay ring, Ž3. e1Ž I . s lŽ IrI 2 . q Ž1 y d . lŽ ArI . q 2, e2 Ž I . s 1, e i Ž I . s 0 ; i G 3, Ž4. PI Ž z . s Ž lŽ ArI . q Ž lŽ IrI 2 . y d lŽ ArI .. z q z 2 .rŽ1 y z . d. Proof. Note that Ž4. implies Ž1. trivially. In the same way it is clear that Ž3. and Ž4. are equivalent. Thus we just need to show that Ž1. implies Ž2. which implies Ž4.. If Ž1. is true, then e1 Ž I . s e0 Ž I . y l Ž ArI . q 1 G 2 e0 Ž I . y Ž l Ž IrI 2 . q Ž 2 y d . Ž ArI . . . It follows that e0 Ž I . F lŽ IrI 2 . q Ž1 y d . lŽ ArI . q 1. However, by Ž). in Proposition 2.3 applied to the case of the I-adic filtration, we have e0 Ž I . s l Ž IrI 2 . q Ž 1 y d . lŽ ArI . q l Ž I 2rIJ . .
56
GUERRIERI AND ROSSI
Thus by comparing the two expressions we get lŽ I 2rIJ . F 1. If it were lŽ I 2rIJ . s 0 we would have e0 Ž I . s lŽ IrI 2 . q Ž1 y d . lŽ ArI . and by Corollary 2.5 this would be equivalent to e1Ž I . y e0 Ž I . q lŽ ArI . s 0 which is not our case. We are forced to conclude that Ž1. implies lŽ I 2rIJ . s 1, e0 Ž I . s lŽ IrI 2 . q Ž1 y d . lŽ ArI . q 1, and e1Ž I . s 2 e0 Ž I . y Ž lŽ IrI 2 . q Ž2 y d . lŽ ArI ... Since by hypothesis I 2 l J s IJ, we can apply Theorem 2.2 to the case of the I-adic filtration to conclude that grI Ž A. is Cohen]Macaulay, so Ž1. implies Ž2.. Let now grI Ž A. be a Cohen]Macaulay ring and e0 Ž I . s lŽ IrI 2 . q Ž1 y . d lŽ ArI . q 1. By Ž)., we get lŽ I 2 rIJ . s 1. Since gr I Ž A . is Cohen]Macaulay all the terms a j Ž I ., 0 F j F sŽ I ., in the h-vector are positive. As known a0 Ž I . s lŽ ArI . and a1Ž I . s lŽ IrI 2 . y d lŽ ArI ., thus, since
l Ž IrI 2 . q Ž 1 y d . lŽ ArI . q 1 s e0 Ž I . s a0 Ž I . q ??? qa sŽ I . Ž I . we deduce that a2 Ž I . q ??? qa sŽ I .Ž I . s 1. Then a2 Ž I . s 1 and a j Ž I . s 0 for all j G 3. In w20, Theorem 1.4x, Sally showed that if d G 2, e2 Ž I . / 0, and e1Ž I . y Ž e0 I . q lŽ ArI . s 1 then depth grI Ž A. G d y 1. Itoh recently obtained the result described in Proposition 3.1 in the case of integrally closed ideals Žsee w11, Corollary 14x.. We note that in order to get the Cohen]Macaulayness of the associated graded ring is enough to have I 2 l J s IJ. This condition is automatically satisfied when I is the maximal ideal of A and in fact, precisely in this case, the previous result had already been obtained by Elias and Valla in w4, Corollary 2.2 Ž2.x. Let us remark that the assumption I 2 l J s IJ is essential and indeed, as illustrated in the following example due to Sally w20, Example 3.3x, the condition e1Ž I . s e0 Ž I . y lŽ ArI . q 1 alone does not guarantee the Cohen]Macaulayness of grI Ž A.. Let k be a field, A s w < t 4 , t 5, t 6 , t 7
57
COEFFICIENTS OF HILBERT FILTRATIONS
Elias, Rossi, and Valla w5, Proposition 2.4x, found that, in the case of the maximal ideal, the possible values for e2 Ž I . are either 2 or 3. Moreover they determined the Hilbert function in both cases. In the result below we extend the results in w5x to the case of an integrally closed m-primary ideal. THEOREM 3.2. Let Ž A, m. be a Cohen]Macaulay local ring of dimension d and I an integrally closed m-primary ideal. Suppose that e1Ž I . s e0 Ž I . y lŽ ArI . q 2. Then either Ž1. PI Ž z . s Ž lŽ ArI . q Ž lŽ IrI 2 . y d lŽ ArI .. z q 2 z 2 .rŽ1 y z . d and grI Ž A. is Cohen]Macaulay, or Ž2. PI Ž z . s Ž lŽ ArI . q Ž lŽ IrI 2 . y d lŽ ArI .. z q z 3 .rŽ1 y z . d and depth grI Ž A. s d y 1 holds. Proof. Note that by our assumption we have I 2 l J s IJ. Since e1 Ž I . s e0 Ž I . y l Ž ArI . q 2 G 2 e0 Ž I . y Ž l Ž IrI 2 . q Ž 2 yd . l Ž ArI . . , it follows that e0 Ž I . F l Ž IrI 2 . q Ž 1 y d . lŽ ArI . q 2. If the equality holds, then e1Ž I . s 2 e0 Ž I . y Ž lŽ IrI 2 . q Ž2 y d . lŽ ArI .. and we get Ž1. because of Theorem 2.3. Let us now assume e0 Ž I . - lŽ IrI 2 . q Ž1 y d . lŽ ArI . q 2. This means that e0 Ž I . s l Ž IrI 2 . q Ž 1 y d . lŽ ArI . q l Ž I 2rIJ . F l Ž IrI 2 . q Ž 1 y d . l Ž ArI . q 1 which implies lŽ I 2rIJ . F 1. However, lŽ I 2rIJ . / 0 since otherwise e0 Ž I . s lŽ IrI 2 . q Ž1 y d . lŽ ArI . which, by Corollary 2.5, would imply e1Ž I . s e0 Ž I . y lŽ ArI .. Thus we have lŽ I 2rIJ . s 1 and e0 Ž I . s lŽ IrI 2 . q Ž1 y d . lŽ ArI . q 1. Then, by Proposition 3.1, we deduce that grI Ž A. is not Cohen]Macaulay and, in particular, d G 1. If d s 1, then H F Ž n . s e 0 Ž I . y lŽ I nq 1 rxI n . and e 1Ž I . s Ý nG 0 lŽ I nq 1rxI n . where x is a superficial element in A. It follows that Ý nG 1 lŽ I nq 1rxI n . s 2. Hence lŽ I 2rxI . s lŽ I 3rxI 2 . s 1, and PI Ž z . s
lŽ ArI . q Ž l Ž IrI 2 . y lŽ ArI . . z q z 3
Ž1 y z.
.
Notice also that, in this case, e2 Ž I . s 3. Assume now d ) 1. To prove Ž2. it is enough to show that depth grI Ž A. G d y 1. By Proposition 1.7 we just need to reduce ourselves to the
58
GUERRIERI AND ROSSI
2-dimensional case and to prove that, under the same assumptions, depth grI Ž A. ) 0. By Proposition 1.5, e2 Ž I . F 3 s e2 Ž IrŽ x 1 .. where x 1 is a superficial element in A. The result follows if, and only if, e2 Ž I . s 3. We now prove that e2 Ž I . - 3 does not occur. By Corollary 1.10 e1 Ž I . s
&
&
Ý l ž I nq 1rJI n /
e2 Ž I . s
and
nG0
&
&
Ý n l ž I nq1rJI n / .
nG1
Since I is integrally closed, then I˜s I and &
&
Ý l ž I nq 1rJI n / s e1Ž I . y lŽ IrJ . s 2.
nG1
If e2 Ž I . F 2, then e2 Ž I . s 2 and depth grI Ž A. s 0. This however is impossible by w21, Theorem 3.1x. Let us remark that, as already noted in the case of the maximal ideal, both the Hilbert functions given in Theorem 3.2 are realizable. Next we want to compare the Hilbert coefficients coming from the & I-adic filtration with those coming from the filtration F s I n4nG 0 . LEMMA 3.3. Let Ž A, m. be&a Noetherian local ring of dimension d ) 0, I an m-primary ideal, and F s I n4nG 0 . Then e dq 1 Ž I . s e dq1 Ž F . q Ž y1 .
dq 1
&
Ý l ž I irI i / .
iG1
&
Proof. Let N be the least integer for which I ns I n for all n ) N. In particular take n ) N big enough so that one also has HI2 Ž n. s h 2I Ž n. and HF2 Ž n. s h 2F Ž n.. Clearly, we have dq1
h2I Ž n . y h 2F Ž n . s
Ý Ž y1. i Ž ei Ž I . y ei Ž F . .
is0
Ý is0
nqdq1yi dq1yi
/
&
Ny1
s
ž
l I iq1rI iq1 .
ž
/
&
Being I ns I n for n 4 0 it is clear that h1I Ž n. s h1F Ž n. for all n and one has e i Ž I . s e i Ž F . for all 0 F i F d. Thus we immediately get
Ž y1.
dq 1
Ny1
&
Ž e dq 1Ž I . y e dq1Ž F . . s Ý l ž I iq1rI iq1 / . is0
In the following J denotes, as before, the ideal generated by a maximal superficial sequence.
59
COEFFICIENTS OF HILBERT FILTRATIONS
PROPOSITION 3.4. Let Ž A, m. be a 2-dimensional Cohen]Macaulay local, I m-primary. Then
e3 Ž I . s e2 Ž I . y e1 Ž I . q e0 Ž I . y l Ž ArI . q
Ý jG3
y
& & jy2 l I jrJI jy1 2
ž /
ž
/
&
Ý l ž I jrI j / .
jG2
&
Proof. Since when F s I n4nG 0 one has depth grF Ž A. G 1, then, by Proposition 1.9, one has
ek Ž F . s
Ý jGky1
ž
& j l I jq1rJI˜j ky1
/
ž
/
for all k&G 1. As by Lemma 3.4 we have that e3 Ž I . y e3 Ž F . s yÝ i G 1 lŽI irI i ., we may conclude that e3 Ž I . s e3 Ž F . y
&
Ý l ž I irI i / s Ý
iG1
jG3
& & jy1 l I jrJI jy1 2
ž /
ž
/
y Ý l Ž I˜irI i . . iG1
Moreover, using Corollary 1.10, we have that e2 Ž I . y e1 Ž I . q e0 Ž I . y l Ž ArI . s e2 Ž F . y e1 Ž F . q e0 Ž F . y l Ž ArI . s
&
&
Ý j l ž I jq1rJI˜j / y Ý l ž I jq1rJI˜j / q lŽ ArJ . y lŽ ArI .
jG1
s
jG0
&
˜ . q lŽ IrJ . Ý Ž j y 1. l ž I jq1rJI˜j / y l Ž IrJ
jG1
s
&
˜ .. Ý Ž j y 2. l ž I˜jrJI jy1 / y l Ž IrI
jG3
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GUERRIERI AND ROSSI
Now we can conclude by noticing the identities e3 Ž I . s
jG3
s
& jy1 l I˜jrJI jy1 y 2
ž /ž / Ý žž / ž // Ýž / ž˜ / Ý
jy2 jy2 q 2 1
jG3
s
jG3
&
l I˜jrJI jy1 y
ž
& jy2 l I jrJI jy1 q 2
˜ .y y l Ž IrI
Ý l Ž I˜jrI j . jG1
/
Ý l Ž I˜jrI j . jG1
&
Ý Ž j y 2. l ž I˜jrJI jy1 /
jG3
Ý l Ž I˜jrI j . jG2
s e2 Ž I . y e1 Ž I . q e0 Ž I . y l Ž ArI . q
Ý jG3
y
Ý l Ž I˜jrI j . .
& jy2 l I˜jrJI jy1 2
ž /
ž
/
B
jG2
PROPOSITION 3.5. Let Ž A, m. be a 2-dimensional Cohen]Macaulay local ring and I an m-primary ideal of A. Then e3 Ž I . G y
&
Ý l ž I jrI j / .
jG1
Proof. This comes simply from the fact that by Lemma 3.4 and by Corollary 1.10 e3 Ž I . s e3 Ž F . y
&
Ý l ž I irI i / s Ý
iG1
jG3
& & jy1 l I jrJI jy1 2
ž /
ž
/
&
y Ý l I irI i . iG1
ž
/
The following example due to Narita w14x shows that the bound is sharp. Let A s k w < x, y, z
COEFFICIENTS OF HILBERT FILTRATIONS
61
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