On the Chern Number of a Filtration

On the Chern Number of a Filtration

M. E. ROSSI(*) - G. VALLA(*)

1. Introduction.

The Hilbert function of a local ring (A;m) is the function which as- sociates to each non negative integernthe minimal number of generators of mⁿ: It gives deep information on the corresponding singularity and, owing to its geometric significance, has been studied extensively.

In the case of a standard graded algebra, the Hilbert function is well understood, at least in the Cohen-Macaulay case. Instead, very little is known in the local case and we do not have a guess of the shape of the Hilbert function, even in the case of a Cohen-Macalay one dimensional domain.

Due to this lack of information, a long list of papers have been written where constraints on the possible Hilbert functions of a Cohen-Macaulay local ring are described. Usually, restrictions have been found on what are called the Hilbert coefficients of the maximal idealmofA.

The first coefficient,e₀(m), is called the multiplicity ofmand, due to its geometric meaning, has been studied very deeply, see for example [11].

The other coefficients are not as well understood, either geometrically or in terms of how they are related to algebraic properties of the ideal or ring.

The goal of this paper is the study of the second (after the multiplicity) coefficient of the Hilbert polynomial of the local ring (A;m):This integer, e₁;has been recently interpreted in [9] as atrackingnumber of the Rees algebra of A in the set of all such algebras with the same multiplicity.

(*) Indirizzo degli A.: Dipartimento di Matematica, Universita di Genova, Via Dodecaneso 35, I-16146 Genova, Italy.

E-mail: rossim@dima.unige.it valla@dima.unige.it

Research partially supported by the ``Ministero dell'Universita e della Ricerca Scientifica'' in the framework of the National Research Network (PRIN 2005)

``Algebra commutativa, combinatorica e computazionale''.

2000Mathematics Subject Classification. Primary 13H10; Secondary 13H15.

Under various circumstances, it is also called the Chern number or coefficientof the local ring A:An interesting list of questions and con- jectural statements about the values of e₁ for filtrations associated to an m-primary ideal of a local ringAhad been presented in a recent paper by W. Vasconcelos (see [18]).

In the Cohen-Macaulay case, starting from the work of D.G. Northcott in the 60's, several results have been proved which give some relationships between the Hilbert coefficients, thus implying some constrains on the Hilbert function. The way was enlightened by Northcott who proved that, if (A;m) is a Cohen-Macaulay local ring, then

e1(m)e0(m) 1:

Thus, for example, the series1‡2z z²

1 z cannot be the Hilbert series of a Cohen-Macaulay local ring of dimension one because e1(m)ˆ0; while e₀(m)ˆ2.

Several results along this line has been proved in the last years, often extending the framework to Hilbert function associated to a filtration, but always assuming the Cohen-Macaulayness of the basic ring. For a large overview of these kind of results one can see [10].

More recently Goto and Nishida in [5] were able to extend Northcott bound to the case of a primary ideal, avoiding, for the first time in this setting, the assumption that the ring is Cohen-Macayulay. Of course, they need to introduce a correction term which vanishes in the Cohen-Macaulay case.

On the base of this result, Corso in [3] could handle in this wild gen- erality a stronger upper bound fore₁given by Elias and Valla in [4].

Here we will focus, instead, on an upper bound of e₁ which was introduced and studied by Huckaba and Marley in [7]. If qis a primary ideal in the local Cohen-Macaulay ringAandJis an ideal generated by a maximal superficial sequence forq, then Valla proved in [16] that

e0(q)ˆ(1 d)l(A=q)‡l(q=q²)‡l(q²=Jq);

thus extending a classical result of Abhyankar on the maximal ideal to the m-primary case. The above formula shows thatl(q²=Jq) does not depend onJand, for the first time, properties of the Hilbert coefficients are related to superficial sequences. Since then, it was natural to consider the integers

v_j(q):ˆl(q^j‡1=Jq^j)

and to investigate how they are involved in the study of the Hilbert series of

a primary ideal. This has been done by Huckaba and Marley in [7] where they proved that for any ideal filtrationF ˆ fIjg_j0of a Cohen-Macaulay local ringA;one has

e₁(F)X

j0

v_j(F);

with equality if and only if the depth of the associated graded ring is at least d 1;if and only if the Hilbert series is

P_F(z)ˆ

l(A=I0)‡P

j0(v_j(F) v_j‡1(F))z^j‡1

(1 z)^d :

The first main result of this paper, Theorem 2.11, shows that given any good q-filtration M of a module M which is not necessarily Cohen-Ma- caulay, we have

e1(M) e1(N)X

j0

vj(M);

whereNis theJ-adic filtration onM. It is easy to see that, ifMis Cohen- Macaulay, thene_i(N)ˆ0 for everyi1;so that these integers are good correction terms when the Cohen-Macaulay assumption does not hold.

We are able to understand in Theorem 2.13 when equality holds above, at least for the case whenMis them-adic filtration on the local ringA:

Surprisingly enough, we show that the equalitye1(m) e1(N)ˆP

j0v_j(m) forces the ring Ato be Cohen-Macaulay and hence the associated gra- ded ring to have almost maximal depth.

Thus we get the Cohen-Macaulayness of the ringAas a consequence of the extremal behavior of the integere1(m):The result can be considered a confirm of the general philosophy of the paper of W. Vasconcelos [18]

where the Chern number is conjectured to be a measure of the distance from the Cohen-Macaulyness ofA:

This main result of the paper is a consequence of a nice and perhaps unexpected property of superficial elements which we prove in Lemma 2.1.

It is essentially a kind of ``Sally machine'' for local rings.

In the last section we describe an application of these results, concerning an upper bound on the multiplicity of the Sally module of a good filtration of a module which is not necessarily Cohen-Macaulay. The bound is given in term of the same ingredients used by Huckaba-Marley fore1:It is an extension to the non Cohen-Macaulay case of a result of Vaz Pinto in [19].

2. An upper bound for the Chern number.

Let (A;m) be a local ring andqa primary ideal form:IfM6ˆ f0gis a finitely generatedA-module, aq-filtration onMis a chain

M: MˆM₀M₁ M_j :::

such thatqMjMj‡1for everyj:In the caseMj‡1ˆqMjforj0;we say that the filtrationMis agoodq-filtration.

IfNis a submodule ofM, it is clear thatf(N‡M_j)=Ng_j0is a goodq- filtration ofM=Nwhich we denote byM=N:

Given the good q-filtration M on M we let gr_q(A):ˆL

j0q^j=q^j‡1; grM(M):ˆL

j0M_j=M_j‡1:We know thatgrM(M) is a gradedgrq(A)-module;

further each elementa2Ahas a natural imageaingr_q(A) which is 0 if aˆ0;isa ˆa2q^t=q^t‡1 ifa2q^tnq^t‡1:

A tool which has been very useful in the study of the depth of blowing- up rings is the so called Valla-Valabrega criterion (see [15]). Given the ideal Iˆ(a1;. . .;ar) in A, the elementsa₁;. . .;a_r form a regular sequence on gr_M(M) if and only if they form a regular sequence onM andIM\M_jˆ

ˆIM_{j 1}for everyj1:

We recall that an elementa2qis called M-superficial for qif there exists a non-negative integercsuch that (Mj‡1:Ma)\Mc ˆMjfor every jc:If we assume thatMhas positive dimension, then every superficial element a for q has order one, that isa2qnq²: Further, since we are assuming that the residue field A=mis infinite, it is well known that su- perficial elements do exist.

A sequence of elements a1;. . .;ar will be called a M-superficial sequence for q if for every jˆ1;. . .;r the element a_j is an (M=(a₁;. . .;a_{j 1})M)-superficial element for q: It is well known that if

a₁;. . .;a_ris aM-superficial sequence forqandJis the ideal they generate,

then we have the following properties which show the relevance of superficial sequences in the study of the depth of blowing-up rings.

(1) fa₁;. . .;a_rg is a regular sequence onM if and only if depth Mr:

(2) fa₁;. . .;a_rgis a regular sequence on gr_M(M)

if and only if depth gr_M(M)r:

If the sequence is maximal then, as in [14], M_n‡1ˆJM_n forn0:

(3)

HenceMis a goodJ-filtration. Further by 8.3.5 in [8] ,J is minimal with respect to this property. It follows that if x1;. . .;xd2J is a maximalM- superficial sequence, thenJˆ(x₁;. . .;x_d):

We recall now that the Hilbert function of the filtration M is the function

HM(j):ˆl(Mj=Mj‡1):

The Hilbert series ofMis PM(z):ˆX

j0

HM(j)z^jˆ hM(z) (1 z)^d

wheredis the dimension ofM:For everyi0 we lete_i(M):ˆh⁽ⁱ⁾_M(1) i! ;then for everyj0 we haveH_M(j)ˆp_M(j) where

p_M(X):ˆX^{d 1}

iˆ0

( 1)ⁱe_i(M) X‡d i 1 d i 1

is a polynomial with rational coefficients and degreed 1 which is called the Hilbert polynomial ofM:Its coefficientsei(M) are theHilbert coeffi- cientsofM:

It is also relevant to consider the numerical function H¹_M(j):ˆl(M=M_j‡1)ˆX^j

iˆ0

H_M(i)

which is called the Hilbert-Samuel function of the filtered moduleM:Its generating function is simply the series

P¹_M(z)ˆ P_M(z)

(1 z)ˆ h_M(z) (1 z)^d‡1: It is clear that the polynomial

p¹_M(X):ˆX^d

iˆ0

( 1)ⁱe_i(M) X‡d i d i

verifies the equalityp¹_M(j)ˆH¹_M(j) for j0:It is called the Hilbert-Sa- muel polynomial ofM:

The first Hilbert coefficiente0 is the multiplicity ofM; by Proposition 11.4 in [2] we know that

e0(M)ˆe0(N) (4)

for every couple of good q-filtrations. Also, if M is artinian, then e0(M)ˆl(M):

The classical case is when the moduleMis the ringAand the filtration is given by the powers of a primary idealIofA. This is called theI-adic filtration onAand the corresponding Hilbert coefficients are simply in- dicated byei(I):

For everya2qone can prove that PM(z)P¹_M=aM(z) (5)

(see [10] and [12]).

As a consequence of (5), we get some useful properties of superficial elements. In the followingawill be aM-superficial element forqanddthe dimension ofM:We have

dim (M=aM)ˆd 1:

(6)

e_j(M)ˆe_j(M=aM) for every jˆ0;. . .;d 2:

(7)

ed 1(M=aM)ˆed 1(M)‡( 1)^{d 1}l(0:Ma):

(8)

We will also need a property of superficial elements which seems to be neglected in the literature. Recall that if a isM-regular, then M=aMis Cohen-Macaulay if and only if M is Cohen-Macaulay. We prove, as a consequence of the following Lemma, that the same holds for a superficial element, if the dimension of the moduleMis at least two.

In the following we denote byHⁱ_q(M) thei-th local cohomology module ofMwith respect toq:We know thatH⁰_q(M):ˆ S

j0(0:Mq^j)ˆ0:Mq^t for everyt0 and minfijHⁱ_q(M)6ˆ0g ˆgrade(q;M);where grade(q;M) is the common cardinality of all the maximal M-regular sequences of ele- ments inq:

LEMMA 2.1. Let Mbe a goodq-filtration of a module M;a an M-su- perficial element forqand j1. Then grade(q;M)j‡1if and only if grade(q;M=aM)j:

PROOF. Let grade(q;M)j‡1; then depth M>0 so that a isM- regular. This implies grade(q;M=aM)ˆgrade(q;M) 1j‡1 1ˆj:

Let us assume now that grade(q;M=aM)j:Sincej1;this implies H⁰_q(M=aM)ˆ0:HenceH⁰_q(aM)ˆH⁰_q(M);so thatH⁰_q(M)aM:We claim that H⁰_q(M)ˆaH⁰_q(M): If this is the case, then, by Nakayama, we get

H⁰_q(M)ˆ0 which implies depthM>0;so thataisM-regular. Hence grade(q;M)ˆgrade(q;M=aM)‡1j‡1;

as wanted.

Let us prove the claim. Suppose by contradiction that aH⁰_q(M)₄H⁰_q(M)aM;

and letax2H⁰_q(M);x2MnH⁰_q(M):This means that for everyt0 we have

aq^txˆ0 q^tx6ˆ0 : (

We prove that this implies thatais notM-superficial forq:Namely, given a positive integerc;we can find an integertcand an elementd2q^tsuch that adxˆ0 and dx6ˆ0: Since \Miˆ f0g; we have dx2Mj 1nMj for some integer j: Now, d2q^t hence dx2MtMc; which implies jc:

Finally we havedx2Mc;dx2=M_jandadxˆ02M_j‡1;hence (M_j‡1:_M a)\M_c4M_j:

The claim and the Lemma are proved. p

In establishing the properties of the Hilbert coefficients of a filtered module M, it will be convenient to use induction on the dimension of the module. To start the induction we need first to study the one-dimensional case.

Let us be given a goodq-filtrationMˆ fM_jg_j0on a moduleMof di- mension 1. We haveH_M(n)ˆp_M(n)ˆe₀(M) forn0 so that we define for everyj

u_j(M):ˆe0(M) HM(j):

(9)

LEMMA2.2. LetMbe a goodq-filtration of a module M of dimension one. If a is anM-superficial element forq;then for every j0we have

uj(M)ˆl(Mj‡1=aMj) l(0:Mj a):

PROOF. By (6) we have

e0(M)ˆe0(M=aM) l(0:_Ma)ˆl(M=aM) l(0:_Ma)

ˆl(M=aMj) l(aM=aMj) l(0:Ma):

By using the following exact sequence

0!(0:_Ma)=(0:_Mj a)!M=M_j !aM=aM_j!0 we get

e0(M)ˆl(M=aMj) l(M=Mj)‡l((0:Ma)=(0:Mj a)) l(0:Ma) and finally

uj(M)ˆe0(M) l(Mj=Mj‡1)

ˆl(M=aM_j) l(M=M_j) l(0:_Mj a) l(M_j=M_j‡1)

ˆl(M_j‡1=aM_j) l(0:_Mj a):

p It follows that, in the caseMis one dimensional and Cohen-Macaulay, then u_j(M)ˆl(M_j‡1=aM_j) is non negative and we have, for everyj0, H_M(j)ˆe₀(M) l(M_j‡1=aM_j)e₀(M):

It will be useful to write down the Hilbert coefficients through the in- tegersuj(M):

LEMMA2.3. LetMbe a goodq-filtration of a module M of dimension one. Then for every j0we have

e_j(M)ˆ X

kj 1

k j 1

u_k(M):

PROOF. We have

P_M(z)ˆh_M(z)

1 zˆX

j0

H_M(j)z^j:

Hence, if we writehM(z)ˆh0(M)‡h1(M)z‡ ‡hs(M)z^s;then we get for everyk1

h_k(M)ˆH_M(k) H_M(k 1)ˆu_{k 1}(M) u_k(M):

Finally

e_j(M)ˆh⁽_M^j)(1) j! ˆX

kj

k

j h_k(M)ˆX

kj

k

j (u_{k 1}(M) u_k(M))

ˆ X

kj 1

k j 1

u_k(M):

p

If we apply the above Lemma in the case Mis Cohen-Macaulay, and using the fact that the integersuk(M) are non negative, we get

e1(M)ˆX

k0

u_k(M)u0(M)‡u1(M)u0(M):

Since we have u₀(M)ˆe₀(M) l(M=M₁); and u₀(M)‡u₁(M)ˆ

ˆ2e₀(M) l(M=M₂);we trivially get

e1(M)e0(M) l(M=M1);

and e₁(M)2e₀(M) l(M=M₂);

which are the bounds by Northcott and Elias-Valla, in the one dimensional Cohen-Macaulay case.

If we do not assume thatMis Cohen-Macaulay, the integersu_k(M) can be negative and the above formulas do not hold anymore.

At this point we are going to describe the correction term which will appear in our upper bound ofe1:

Given a good q-filtration M of the module M of dimension d; let

a1;. . .;a_db e aM-superficial sequence forq:We denote byJthe ideal they

generate and consider the J-adic filtration of the module M. This is by definition the filtration

N:ˆ fJ^jMg_j0

which is clearly a goodJ-filtration. By (3)Mis also a goodJ-filtration, so that, by (4),e₀(M)ˆe₀(N):

In the caseMis Cohen-Macaulay, the elementsa₁;. . .;a_dform a reg- ular sequence onMso thatJⁱM=J^i‡1M'(M=JM)…^d‡ii1†:This implies that the Hilbert Series ofNisPN(z)ˆl(M=JM)

(1 z)^d and thusei(N)ˆ0 for every i1:This proves that these integers give a good measure of howMdiffers from being Cohen-Macaulay. In the caseMˆA;Vasconcelos in [17] and [18] conjectured that if A is not Cohen-Macaulay, thene₁(J)50 and he proved it for large classes of local rings.

First we prove thate1(N)0 in the one dimensional case where the in- tegere₁(N) can be easily related with the 0-th local cohomology module ofM:

LEMMA2.4. LetMbe a goodq-filtration of a module M of dimension one. If a is anM-superficial element forq, then for every t0we have

e1(N)ˆ l(0:M a^t):

PROOF. We have p¹_N(X)ˆe0(N)(X‡1) e1(N)ˆe0(M)(X‡1) e1(N):On the other hand we have short exact sequences

0!(0:Ma^k)=0:aMa^k!M=aM!^a a^kM=a^k‡1M!0 0!0:_Ma!0:_Ma^i‡1!^a 0:_aMaⁱ!0 which give

H¹_N(j)ˆX^j

iˆ0

HN(i)ˆX^j

iˆ0

l(aⁱM=a^i‡1M)ˆX^j

iˆ0

l(M=aM) l(0:Maⁱ=0:aMaⁱ)

ˆ(j‡1)l(M=aM) l(0:Ma) X^j

iˆ1

l(0:Ma^i‡1) l(0:aMaⁱ)

‡l(0:Ma^j‡1)

ˆ(j‡1)[l(M=aM) l(0:Ma)]‡l(0:Ma^j‡1)ˆ(j‡1)e0(M)‡l(0:Ma^j‡1):

The conclusion follows. p

In this section we denote by W the 0-th local cohomology module H⁰_m(M) ofMwith respect tom:Recall that

W :ˆ[

j0

(0:Mm^j):

WhenMis a module of dimension one andaanM-superficial element forq;

thenM=aMhas finite length so that 0:_A(M=aM) is a primary ideal. Now it is easy to see that 0:_A(M=aM) 

(a)‡0:_AM

p ;and thus (a)‡0:_AMis a primary ideal too. This means that a power ofmis contained in (a)‡0:_AM, saym^s (a)‡0:_AM:Hence, fort0

Wˆ0:Mm^t0:Ma^t0:Mm^tsˆW: The above lemma and this remark give

l(W)ˆ e₁(N) (10)

as was already proved in [5], Lemma 2.4.

Given a good q-filtration of the module M, we consider now the cor- responding filtration of the saturated module M^sat:ˆM=W: This is the filtration

M^sat:ˆM=W ˆ fM_n‡W=Wg_n0:

SinceWhas finite length and\Miˆ f0g;we haveMi\Wˆ f0gfor every i0:This impliespM(X)ˆp_M^sat(X):

Further, it is clear that for everyj0 we have an exact sequence 0!W=(M_j‡1\W)!M=M_j‡1!M=(M_j‡1‡W)!0

so that for everyj0 we have

l(M=M_j‡1)ˆl[M=(M_j‡1‡W)]‡l(W) which implies

p¹_M(X)ˆp¹_Msat(X)‡l(W):

(11)

This proves the following result:

PROPOSITION2.5. Let Mbe a goodq-filtration of the module M and W :ˆH⁰_m(M):If we let d:ˆdim (M)andM^sat:ˆM=W;then

e_i(M)ˆe_i(M^sat) 0id 1; e_d(M)ˆe_d(M^sat)‡( 1)^dl(W):

We remark that, if dim (M)1, the moduleM=W has always positive depth. This is the reason why, sometimes, we move our attention from the moduleMto the moduleM=W:This will be the strategy of the proof of the next proposition which gives, in the one dimensional case, the promised upper bound fore₁:

PROPOSITION 2.6. Let M be a good q-filtration of a module M of dimension one. If a is a M-superficial element forqandNthe(a)-adic filtration on M;then

e1(M) e1(N)X

j0

l(Mj‡1=aMj):

If W M1and equality holds above, then M is Cohen-Macaulay.

PROOF. By Proposition 2.5 and (10) we have

e1(M)ˆe1(M^sat) l(W)ˆe1(M^sat)‡e1(N) so that we need to prove thate1(M^sat)P

j0l(M_j‡1=aM_j):

Now M=W is Cohen-Macaulay and a is regular on M=W; hence by Lemma 2.3 and Lemma 2.2, we get

e1(M^sat)ˆX

j0

u_j(M^sat)ˆX

j0

l(M_j‡1^sat=aM^sat_j )

ˆX

j0

l M_j‡1‡W aMj‡W

ˆX

j0

l M_j‡1

aMj‡Mj‡1\W

X

j0

l(Mj‡1=aMj):

The first assertion follows. In particular equality holds if and only if Mj‡1\WaMj for every j0: Let as assume WM1 and equality above; then we haveWˆW\M₁aM:Now recall thatWˆ0:_Ma^tfor t0; hence if c2W then cˆam with a^tcˆa^t‡1mˆ0: This implies m20:_Ma^t‡1ˆW so thatWaWand, by Nakayama,Wˆ0: p In the last section we need to compare the Hilbert coefficients of theJ- adic filtrationNonMwith those of the following filtration. Given a goodq- filtrationMonM and the idealJ generated by a maximalM-superficial sequence forq;we let

E(J;M):ˆMM1JM1J²M1 J^{n 1}M1. . .

When there is no ambiguity we simply writeEinstead ofE(J;M):

It is clear that this is a goodJ-filtration so that e₀(M)ˆe₀(E)ˆe₀(N):

By considering the leading coefficients of p¹_M(X) p¹_E(X) and p¹_E(X) p¹_N(X);sinceJ^n‡1MJⁿM1Mn‡1it follows that

e1(M)e1(E)e1(N):

PROPOSITION 2.7. Let M be a good q-filtration of the module M; if dimMˆ1and Jˆ(a)with aM-superficial forq, then

e₁(E) e₁(N)e₀(M) h₀(M):

PROOF. It is clear that for everyn0 we have p¹_E(n)ˆl(M=aⁿM1)ˆe0(M)(n‡1) e1(E);

p¹_N(n 1)ˆl(M=aⁿM)ˆe₀(M)n e₁(N):

It follows that

e1(E) e1(N)ˆe0(M)‡l(M=aⁿM) l(M=aⁿM1)e0(M) l(aⁿM=aⁿM1):

The conclusion follows because the map M=M₁!^an aⁿM=aⁿM₁ is

surjective. p

We come now to the higher dimensional case; a natural strategy will be to lower dimension by using superficial elements.

The following Lemma is the key for doing the job. It is due to David Conti.

LEMMA2.8. Let M1;. . .;Mr be A-modules of dimension d;J am-pri- mary ideal of A andM1ˆ fM1;jg;. . .;Mrˆ fMr;jggood J-filtrations of

M₁;. . .;Mrrespectively. Then we can find elements a₁;. . .;a_dwhich are

M_i-superficial for J for every iˆ1;. . .;r:

If d2and M₁ˆ ˆM_rˆM;then for every1ijr we have e1(Mi) e1(Mi=a1M)ˆe1(Mj) e1(Mj=a1M):

PROOF. Let M^r

iˆ1

M_i:ˆ M^r

iˆ1

M_iM^r

iˆ1

M_i;1 M^r

iˆ1

M_i;2 M^r

iˆ1

M_i;j It is clear that this is a goodJ-filtration onL^r

iˆ1M_i:Let us choose aL^r

iˆ1M_i- superficial sequence fa₁;. . .;a_dg for J. Then it is easy to see that

fa₁;. . .;a_dg is a sequence of M_i-superficial elements for J for every

iˆ1;. . .;r:This proves the first assertion. As for the second one, by (7) and

(8) we have

e₁(M_i) e₁(M_i=a₁M)ˆ l(0:Ma1) if dˆ2

0 if d3

from which the conclusion follows. p

We first extend to the higher dimensional case the result of Proposition 2.7. We will see that the following result is a strengthened version of the classical inequality due to Northcott.

PROPOSITION2.9. LetMbe a goodq-filtration of the module M and let J be the ideal generated by a maximal sequence ofM-superficial elements forq:Then we have

e₁(E) e₁(N)e₀(M) h₀(M):

PROOF. If dimMˆ1 we use Proposition 2.7. Let dimM2; by the above Lemma we can find elementsa₁;. . .;a_d2Jwhich are superficial for Jwith respect toEandNat the same time. Further

e1(E) e1(E=a1M)ˆe1(N) e1(N=a1M) andJˆ(a₁;. . .;a_d) by (3).

The moduleM=a₁M has dimensiond 1 and M=a₁M is a good J-fil- tration on it. Furthera2;. . .;adis a maximalM=a1M-superficial sequence forJ:If we letK:ˆ(a2;. . .;ad);then theK-adic filtration onM=a1Mis

given by

fK^j(M=a₁M)g_j0ˆ f(K^jM‡a₁M)=a₁Mg_j0ˆ f(J^jM‡a₁M)=a₁Mg_j0 and thus coincides withN=a₁M:On the other hand the filtration E(K;M=a₁M):ˆM=a₁MM₁‡a₁M

a₁M KM₁‡a₁M

a₁M K²M₁‡a₁M a₁M :::

coincides with the filtrationE=a₁MbecauseK^j M₁‡a₁M

a₁M ˆJ^jM₁‡a₁M a₁M : By induction we get

e1(E) e1(N)ˆe1(E=a1M) e1(N=a1M)

e₀(M=a₁M) h₀(M=a₁M)ˆe₀(M) h₀(M)

as desired. p

Since e1(M)e1(E) we obtain Nothcott's inequality which had been extended to non Cohen-Macaulay case by Goto and Nishida in [5].

COROLLARY2.10. LetMbe a goodq-filtration of the module M and let J be the ideal generated by a maximal sequence of M-superficial elements forq:Then we have

e1(M) e1(N)e0(M) h0(M):

Given a good q-filtration Mˆ fM_jg_j0 of the A-module M and an ideal J generated b y a maximal sequence of M-superficial elements for q, we let for every j0:

v_j(M):ˆl(M_j‡1=JM_j):

THEOREM 2.11. Let M be a good q-filtration of a module M of di- mension d1;J an ideal generated by a maximal sequence ofM-super- ficial elements forqandNthe J-adic filtration of M:Then we have

e₁(M) e₁(N)X

j0

v_j(M):

PROOF. We prove the Theorem by induction ond:Ifdˆ1 we can apply Proposition 2.6; hence letd2. As in the above Proposition, we can find a system of generatorsa₁;. . .;a_dofJwith the properties that are superficial forJwith respect toMandN;and verify

e1(N) e1(N=a1M)ˆe1(M) e1(M=a1M)

by (3). Also, if we letK:ˆ(a2;. . .;ad);thenK is generated by a maximal M=a1M-superficial sequence forJ and theK-adic filtration onM=a1Mis N=a₁M:Thus, by induction, we get

e1(M) e1(N)ˆe1(M=a1M) e1(N=a1M)X

j0

v_j(M=a1M)

ˆX

j0

l(M_j‡1‡a₁M)=(KM_j‡a₁M)

ˆX

j0

l(M_j‡1‡a₁M)=(JM_j‡a₁M)

ˆX

j0

lM_j‡1=(JM_j‡(a₁M\M_j‡1)) X

j0

l(M_j‡1=JM_j):

We would like to study when the equality in the above Theorem holds.

In the caseMis Cohen-Macaulay, we have a solution for a general filtra- tion which broadly extends previous results of [7] and [6].

THEOREM2.12. LetMˆ fMjg_j0be a goodq-filtration of the Cohen- Macaulay module M of dimension d1and J an ideal generated by a maximal sequence ofM-superficial elements forq:Then we have

a) e₁(M)P

j0v_j(M) b) e2(M)P

j0j v_j(M):

c) The following conditions are equivalent 1. depth gr_M(M)d 1:

2. ei(M)ˆ P

ji 1

j i 1

vj(M)for every i1:

3. e1(M)ˆP

j0v_j(M):

4. e₂(M)ˆP

j0j v_j(M):

5. PM(z)ˆ

l(M=M1)‡P

j0(vj(M) vj‡1(M))z^j‡1

(1 z)^d :

PROOF. Let Jˆ(a₁; ;a_d) and Kˆ(a₁; ;a_{d 1}); we first re- mark that, by Valabrega-Valla, depthgr_M(M)d 1 if and only if

Mj‡1\KMˆKMj for every j0: Further we have v_j(M)ˆv_j(M=KM)‡l(M_j‡1\KM‡JM_j=JM_j);

hence v_j(M)v_j(M=KM) and equality holds if and only if M_j‡1\

\KMJM_j:This is certainly the case whenM_j‡1\KMˆKM_j;hence, if depthgrM(M)d 1; then vj(M)ˆvj(M=KM) for every j0: By induction on j; we can prove that the converse holds. Namely M₁\KMˆKMand, ifj1;then we have

M_j‡1\KMJM_j\KMˆ(KM_j‡a_dM_j)\KM

ˆKMj‡(adMj\KM)KMj‡ad(Mj\KM)

ˆKM_j‡a_dKM_{j 1}ˆKM_j

wherea_dM_j\KMa_d(M_j\KM) becausea_dis regular moduloKM;while M_j\KMˆKM_{j 1}follows by induction.

SinceM=KMis Cohen-Macaulay of dimension one, we get e₁(M)ˆe₁(M=KM)ˆX

j0

v_j(M=KM)X

j0

v_j(M):

Equality holds if and only if depthgr_M(M)d 1:This, once more, proves a) and moreover gives the equivalence between 1 and 3 in c). By using Lemma 2.2 this also gives the equivalence between 1. and 5. in c).

Further, ifais the ideal generated bya1; ;ad 2;then, as before, we get

e₂(M)ˆe₂(M=aM)e₂(M=KM)ˆX

j1

j v_j(M=KM)X

j1

j v_j(M):

This proves b) and 4ˆ)1. To complete the proof of the Theorem, we need only to show that 1ˆ)2. If depthgr_M(M)d 1;then MandM=KM have the sameh-polynomial; this implies that for everyi1 we have

e_i(M)ˆe_i(M=KM)ˆ X

ji 1

j i 1

v_j(M=KM)ˆ X

ji 1

j i 1

v_j(M):

p If we do not assume thatM is Cohen-Macaulay, we are able to handle the problem only for them-adic filtration onA:

THEOREM2.13. Let(A;m)be a local ring of dimension d1and J the ideal generated by a maximal m-superficial sequence. The following

conditions are equivalent:

1. e₁(m) e₁(J)ˆP

j0v_j(m):

2. A is Cohen-Macaulay and depth gr_m(A)d 1:

PROOF. IfAis Cohen-Macaulay, thene1(J)ˆ0 and, by the above re- sult, we get that 2) implies 1).

We prove now that 1) implies 2) by induction on d: If dˆ1; the result follows by Proposition 2.6 sinceW m:Letd2; by Lemma 2.8 we can find a minimal basisfa₁;. . .;a_dgofJsuch thata₁isJ-superficial,

fa₁;. . .;a_dg is an m-superficial sequence and e₁(m) e₁(J)ˆ

ˆe1(m=(a1)) e1(J=(a1)): Now A=(a1) is a local ring of dimensiond 1 and J=(a1) is generated by a maximal m=(a1)-superficial sequence. We can then apply Theorem 2.11 to get

X

j0

vj(m)ˆe1(m) e1(J)ˆe1(m=(a1)) e1(J=(a1))X

j0

vj(m=(a1))X

j0

vj(m):

This implies

e₁(m=(a₁)) e₁(J=(a₁))ˆX

j0

v_j(m=(a₁))

which, by the inductive assumption, implies A=(a₁) is Cohen-Macaulay.

By Lemma 2.1 A is Cohen-Macaulay so that e1(J)ˆ0 and then e1(m)ˆP

j0vj(m); this implies depthgrm(A)d 1 and the Theorem is

proved. p

As a trivial consequence of the above result we have the following COROLLARY2.14. Let(A;m)be a local ring of dimension d1;J the ideal generated by a maximalm-superficial sequence. If e1(J)0;then

e₁(m)X

j0

v_j(m):

Moreover, the following conditions are equivalent:

1. e1(m)ˆP

j0vj(m):

2. A is Cohen-Macaulay and depth grm(A)d 1:

We notice that the conditione1(J)0 is satisfied if, for example,Ais Buchsbaum, see [13] Proposition 2.7. It is verified as well if depth Ad 1; namely, if this is the case, given a J-superficial sequence

a1;. . .;ad 1, we havee1(J)ˆe1(J=(a1;. . .;ad 1)):SinceA=(a1;. . .;ad 1) is one dimensional, by Lemma 2.4 we get e1(J=(a1;. . .;ad 1))0: More in general Vasconcelos conjectured in [17] thate₁(J)50 wheneverA is not Cohen-Macaulay. This is proved for local integral domain essentially of finite type over a field.

3. Application to the Sally module.

Given a primary idealqin the local ring (A;m) and a minimal reduction Jofq;Vasconcelos introduced in [17] the so-calledSally'smoduleS_J(q) of qwith respect toJ:It is defined by the exact sequence

0!qA[Jt]!qA[qt]!S_J(q)ˆM

n1

q^n‡1=Jⁿq!0:

We know that, if different from zero,SJ(q) is anA[Jt]-module of the same dimensiondasA:

The Hilbert function of this graded module is H_SJ(q)(n):ˆl(q^n‡1=Jⁿq);

and its Hilbert series is

PSJ(q)(z)ˆX

n1

l(q^n‡1=Jⁿq)zⁿ:

We writeei(SJ(q)) for the corresponding Hilbert coefficients.

In [19] it has been proved that ifAis Cohen-Macaulay then e₀(S_J(q))X

j1

l(q^j‡1=Jq^j) and equality holds if and only if depthgr_q(A)d 1:

In this section we extend the definition of the Sally module to any good q-filtration of anA-moduleMand prove that the inequality above holds in this generality and without the assumption that M is Cohen-Macaulay.

Further, we study when equality holds in the special case of the q-adic filtration.

LetMbe a goodq-filtration of theA-moduleMof dimensiondand letJ be the ideal generated by a maximalM-superficial sequence forq:

We let

S_J(M):ˆM

n1

(M_n‡1=JⁿM₁)

and call it the Sally's module ofMwith respect toJ:We remark thatSJ(M) is not a graded module associated to a filtration, but if we consider the fil- tration

E:ˆMM1JM1J²M1. . .JⁿM1. . .

then S_J(M) is related togrM(M) and grE(M) by the following two short exact sequences:

0!J^{n 1}M1=JⁿM1!Mn=JⁿM1!Mn=J^{n 1}M1 !0 0!M_n‡1=JⁿM₁!M_n=JⁿM₁!M_n=M_n‡1!0:

SinceMn=J^{n 1}M1ˆ(SJ(M)( 1))_n;by standard facts it follows that depthgrM(M)minfdepth S_J(M) 1;depthgrE(M)g (12)

Moreover we get

PSJ(M)( 1)(z)‡PE(z)ˆPM(z)‡PSJ(M)(z) so that

(z 1)PSJ(M)(z)ˆPM(z) PE(z) (13)

If dimSJ(M)ˆd;this implies that for everyi0 we have e_i(S_J(M))ˆe_i‡1(M) e_i‡1(E) (14)

Now, by Proposition 2.9, we know that

e₁(E) e₁(N)e₀(M) h₀(M);

so that

e0(SJ(M))ˆe1(M) e1(E)e1(M) e1(N) e0(M)‡h0(M):

Notice that, for the special case of theq-adic filtration onA, this result has been proved in [3].

Now we can use Theorem 2.11, to get e₀(S_J(M))X

j0

v_j(M) e₀(M)‡h₀(M):

Thus we have proved the following

THEOREM3.1. LetMbe a good q-filtration of the A-module M of di- mension d and let J be the ideal generated by a maximal M-superficial sequence forq:IfdimS_J(M)ˆd;then

e₀(S_J(M))X

j0

v_j(M) e₀(M)‡h₀(M):

It is clear that if the equality holds above then e1(M) e1(N)ˆ

ˆP

j0vj(M): Hence, in the case of the m-adic filtration on A, we can apply Theorem 2.13 and we get that A is Cohen-Macaulay and depth gr_m(A)d 1:

In this way we get the following result which completes Theorem 2.13.

THEOREM3.2. Let(A;m)be a local ring of dimension d1and J the ideal generated by a maximalm-superficial sequence. If dimS_J(m)ˆd;

then e0(SJ(m))P

j0vj(m) e0(m)‡1:Moreover the following conditions are equivalent:

1.e0(S_J(m))ˆP

j0v_j(m) e0(m)‡1 2.e1(m) e1(J)ˆP

j0vj(m)

3.A is Cohen-Macaulay and depth gr_m(A)d 1:

We finish the paper with the following Theorem which adds some more equivalent conditions, involving the Sally module ofM, to those of Theo- rem 2.12 in the caseM is Cohen-Macaulay. The result extends to a con- siderable extent a series of results proved in [19] for the special case of the q-adic filtration onA:

First we remark that ifMis Cohen-Macaulay, thengr_E(M) is Cohen- Macaulay with minimal multiplicity and hence

PE(z)ˆh₀(M)‡(e₀(M) h₀(M))z

(1 z)^d :

In particulare₁(E)ˆe₀(M) h₀(M):

By using (14), (12), (13) we get

1. If dimS_J(M)ˆd;thene₀(S_J(M))ˆe₁(M) e₀(M)‡h₀(M) 2. depthgrM(M)depthSJ(M) 1

3. (z 1)PSJ(M)(z)ˆPM(z) l(M=M₁)‡(e₀(M) (l(M=M₁))z

(1 z)^d :

IfM is the q-adic filtration on M then it is easy to see that the as- sumption dimSJ(M)ˆd is equivalent to SJ(M)6ˆ0: In fact, by (13), we have that dimSJ(M)ˆd if and only if e1(M)>e1(E)ˆe0(M) h0(M):

This is equivalent toM₂6ˆJM₁ and henceS_J(M)6ˆ0:

THEOREM3.3. Let M be a Cohen-Macaulay A-module of dimension d1;qa primary ideal in A;Mtheq-adic filtration on M and J the ideal generated by a maximal sequence of M-superficial elements for q:The following conditions are equivalent:

1.e₀(S_J(M))ˆP

j1v_j(M) 2.PSJ(M)(z)ˆ

P

j1v_j(M)z^j (1 z)^r 3.SJ(M)is Cohen-Macaulay

and each of them is equivalent to the equivalent conditions of The- orem 2.12.

PROOF. We have e₀(S_J(M))ˆe₁(M) e₀(M)‡h₀(M): Hence, by Theorem 2.12, we get

e₀(S_J(M))X

j0

v_j(M) e₀(M)‡h₀(M)ˆX

j1

v_j(M):

By Theorem 2.12, the equality holds if and only ife1(M)ˆP

j0vj(M):Hence 1. is equivalent to 1., 2., 3., 4., 5. of Theorem 2.12. Because 1. is equivalent to

PM(z)ˆ

l(M=M₁)‡P

j0(v_j(M) v_j‡1(M))z^j‡1 (1 z)^d

and we know that

(z 1)P_SJ(M)(z)ˆPM(z) l(M=M1)‡(e0(M) (l(M=M1))z

(1 z)^d ;

it is easy to see that 1. is also equivalent to 2. Now, since depthgr_M(M) depthSJ(M) 1;we get that 3. implies depthgrM(M)d 1 which is equivalent to 1.

We have only to prove that 2. implies 3. We may assume S_J(M) of dimensiondand recall thatS_J(M) is aR(J)ˆA[JT]-module and we have S_J(M)=JTS_J(M)ˆL

n1M_n‡1=JM_n: By 2. we deduce that P_SJ(M)(z)ˆ

ˆ 1

(1 z)^dP_{SJ(M)=JTSJ(M)}(z):ThenJTis generated by a regular sequence of lenghtdˆdimS_J(M) and henceS_J(M) is Cohen-Macaulay. p

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Manoscritto pervenuto in redazione il 5 febbraio 2008.