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Hilbert Functions and Sally Modules
192, 504]523 Ž1996. JA966820
JOURNAL OF ALGEBRA ARTICLE NO.
Hilbert Functions and Sally Modules Maria Vaz PintoU Department of Mathematics, Rutgers Uni¨ ersity, New Brunswick, New Jersey 08903 Communicated by Mel¨ in Hochster Received June 14, 1995
1. INTRODUCTION Let Ž R, m . be a Noetherian local ring of Krull dimension d ) 0 and let I be an m-primary ideal of R. There has been a great deal of interest on two objects associated to the ideal I: Ži. the Hilbert]Samuel function of I, i.e., the numerical function HI Ž n. s lŽ RrI n ., where l stands for length, and Žii. the depth of the associated graded ring of R with respect to I, grI Ž R . s [nG 0 I nrI nq1 Žsee w2]8, 13, 14x.. An approach to their study has consisted in using reductions of I, that is, ideals J : I, with the property that I rq1 s JI r for some integer r Žsee w1x or w9x for this and other basic notions we employ.. This means that the natural inclusion of Rees algebras R w Jt x s
[ J n t n ¨ R w It x s [ I n t n
nG0
nG0
turns Rw It x into a finite Rw Jt x-module. If Rw Jt x has a more accessible character than Rw It x, and if one has good control over the module structure of Rw It x, then one is able to deal with Ži. and Žii. above rather effectively. A setting for obtaining a good Rw Jt x is that of a Cohen]Macaulay local ring Ž R, m . with infinite residue field Rrm. In this case, any minimal reduction J of I is generated by a regular sequence, so that the algebra * The author was partially supported by a JNICT grant. Current address: Department of Mathematics and Computer Science, Montclair State University, Upper Montclair, New Jersey 07043. E-mail address: [email protected]. 504 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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505
Rw Jt x is very well understood. We define the reduction number r J Ž I . of I with respect to J as the smallest integer t such that I tq1 s JI t. One way to deal with the second issue, of the module structure of Rw It x over Rw Jt x, is to define modules where the two algebras interact. In this way Vasconcelos introduced in w15x the so-called Sally module S s S J Ž I . of I with respect to J. It is defined by the exact sequence 0 ª I ? R w Jt x ª I ? R w It x ª S J Ž I . s
[ I nq 1rJ n I ª 0,
nG1
and a motivation for its name is the work of Sally in w10]13x Žsee w15x.. Our aim here is to uncover structural aspects of S J Ž I . in sufficient detail to deal with Ži. and Žii. in several cases of interest. The techniques we introduce are filterings of S J Ž I . by submodules which have Cohen] Macaulay ancestors: to a filtration of the Sally module there corresponds another series of modules Žthe virtual filtration .. When we succeed in proving their equality, or near so, there is an abundance of information about S J Ž I ., which may be used to estimate its depth and to describe its Hilbert function and some of its arithmetical properties like the canonical module. Information about depth, Hilbert coefficients, and Hilbert]Poincare ´ series of the Sally module can be passed back and forth to grI Ž R .. In the case of depth this comes from the above sequence Žsee w16, Proposition 1.2.10 and Corollary 2.1.11x.. For the Hilbert coefficients and Hilbert] Poincare ´ series see w15, Corollary 3.3; 16, Proposition 1.3.3x, respectively. If different from Ž0., S J Ž I . is an Rw Jt x-module of dimension d Žsee w15x., and we write s0 , . . . , s d for the Hilbert coefficients of S J Ž I .; e0 , . . . , e d will denote the Hilbert coefficients of grI Ž R .. We will write HP Ž S, t . and HP ŽgrI Ž R ., t . for the Hilbert]Poincare ´ series of S and grI Ž R ., respectively. We shall now describe our results. The first consequences of the filtration and virtual filtration mentioned above are the following theorem and corollary. THEOREM 1.1. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R and let J be a minimal reduction of I. If s0 is the first Hilbert coefficient of the Sally module S, and r s r J Ž I . is the reduction number of I with respect to J, then ry1
Ý lŽ I nq 1rJI n . .
s0 F
ns1
If equality holds then: Ži. S is Cohen]Macaulay, Žii. for j s 1, . . . , d y 1, the Hilbert coefficients of S are gi¨ en by ry1
sj s
Ý nsj
n l I nq 1rJI n . , j
ž/Ž
Ž 1.
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MARIA VAZ PINTO
and Žiii. the Hilbert]Poincare´ series of S is gi¨ en by HP Ž S, t . s
nq 1 Ý ry1 rJI n . t n ns 1 l Ž I
Ž1 y t .
d
.
Ž 2.
COROLLARY 1.2. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R Ž nq 1rJI n . then: Ži. and let J be a minimal reduction of I. If s0 s Ý ry1 ns1 l I depth ŽgrI Ž R .. G d y 1, Žii. for j s 1, . . . , d, the Hilbert coefficients of grI Ž R . are gi¨ en by ry1
ej s
Ý nsjy1
n l I nq 1rJI n . , jy1
ž /Ž
and Žiii. the Hilbert]Poincare´ series of grI Ž R . is gi¨ en by r
lŽ RrI . q HP Ž grI Ž R . , t . s
Ý
l Ž I nrJI ny1 . y l Ž I nq1rJI n . t n
ns1
Ž1 y t .
d
.
In the same setting as ours, Sam Huckaba proved in w6, Theorem 3.1x Ž nq 1rJI n . if and only if depthŽgrI Ž R .. G d y 1. Since that e1 s Ý ry1 ns0 l I Ž . e1 s s0 q l IrJ w15, Corollary 3.3x, we can restate Huckaba’s result as Ž nq 1rJI n . if and only if depthŽgrI Ž R .. G d y 1. Therefore, s0 s Ý ry1 ns1 l I whenever depthŽgrI Ž R .. G d y 1, Corollary 1.2 gives us a complete description of the Hilbert coefficients and of the Hilbert]Poincare ´ series of grI Ž R .. We observe that the formulas for the Hilbert coefficients were also given by Huckaba in w6, Corollary 2.11x. One would like to understand the structure of the Sally module given the reduction number of I with respect to J and numerical information about some of the components of S. In Sections 3 and 4 several particular cases are studied. Our main result in Section 3 is the following theorem. THEOREM 1.3. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm, let I be an m-primary ideal of R and J a minimal reduction of I. Assume that the reduction number of I with respect to J is two and that I 2rJI has a cyclic socle. Then the Sally module is Cohen]Macaulay. More precisely, S , Ž I 2rJI .w T1 , . . . , Td xwy1x. In particular, depthŽgrI Ž R .. G d y 1. Section 4 is dedicated to cases where the length of I 2rJI is low, namely one, two, or three. One starts with length one and our main result in that case is Theorem 1.4 below, where we present a relation between the first Hilbert coefficient s0 of the Sally module and the reduction number r s r J Ž I . of I with respect to J.
HILBERT FUNCTIONS AND SALLY MODULES
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THEOREM 1.4. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R and J a minimal reduction of I such that lŽ I 2rJI . s 1. Then s0 F r y 1 and Ži. Žii.
s0 s r y 1 implies S Cohen]Macaulay, s0 s r y 2 implies depthŽ S . s d y 1.
When the length of I 2rJI is two, we divide our study into two subcases. One of them corresponds to the situation where I 2rJI is isomorphic to Ž Rrm . 2 . In this case, if the reduction number of I with respect to J is two, Vasconcelos proved in w15, Proposition 2.6x that depthŽ S . G d y 1, and therefore, depthŽgrI Ž R .. G d y 2. In the second situation, I 2rJI is cyclic and one shows in Remark 4.6 that if the reduction number of I with respect to J is two then the Sally module is Cohen]Macaulay. As in the case of length one, the first Hilbert coefficient s0 of S is related to r J Ž I ., and the Sally module is studied in several extremal situations. Finally, if the length of I 2rJI is three, our problem is divided into four subcases, and the structure of the Sally module is well understood in three of them if the reduction number of I with respect to J is two. Throughout this paper Ž R, m . is a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm, I is an m-primary ideal of R, and J is a minimal reduction of I.
2. FILTERING THE SALLY MODULE In this section we shall present a filtration of the Sally module. We start defining, for n G 1, the Rw Jt x-module Cn s
`
[ I iq1rJ iynq1 I n . isn
Notice, in particular, that C1 s S. We define now L n as the Rw Jt x-submodule of Cn generated by the first term of Cn , Ž Cn . n s I nq 1rJI n : L n s R w Jt x ? I nq 1rJI n s
`
[ J iyn I nq1rJ iynq1 I n . isn
We have the short exact sequence of Rw Jt x-modules 0 ª L n ª Cn ª Cnq1 ª 0, where Cnq 1 s CnrL n . If r s r J Ž I . is the reduction number of I with respect to J, then I rq1 s JI r , Cr s 0, and L ry1 , Cry1. We obtain,
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MARIA VAZ PINTO
therefore, the following set of exact sequences of Rw Jt x-modules 0 0
ª ª
0
ª
L1 L2 .. . L ry2
ª ª ª
S C2 .. . Cry2
ª ª
C2 C3 .. .
ª
L ry1
ª ª
0 0
ª
0.
Ž 3.
We shall refer to the factors L n as the reduction modules of I with respect to J. If these factors have a recognizable structure, then we will be able to conclude good properties for the Sally module. To help us recognize the structure of the L n’s, we shall introduce other Rw Jt x-modules, the Dn’s, that will be called the ¨ irtual reduction modules of I with respect to J. For n G 1, let A n s Ann R Ž I nq 1rJI n . and let Bn s RrA nw T1 , . . . , Td x, a d-dimensional Cohen]Macaulay ring with Ass B nŽ Bn . s m Bn4 . Notice that Bn , Rw Jt xrA n Rw Jt x, and since A n ? Rw Jt x : Ann Rw J t xŽ L n ., L n is not only an Rw Jt x-module, but also a Bn-module. We define now, for n g 1, . . . , r y 14 , Dn s Bn
mR Ž I nq 1rJI n . , Ž I nq1rJI n . w T1 , . . . , Td x ,
a maximal Cohen]Macaulay Bn-module, with Ass B nŽ Dn . s m Bn4 . There is an epimorphism of Bn-modules un : Dnwyn x ª L n , that is the identity on I nq 1rJI n and sends each Ti to a generator of Rw Jt x Žclearly, the generators of I nq 1rJI n have degree zero in Dn , but degree n in L n .. If we define K n as the kernel of un , then for n g 1, . . . , r y 14 we have the following exact sequence of Bn-modules un
L n ª 0.
6
0 ª K n ª Dn w yn x
Ž 4.
This sequence and the filtration presented above are the main tools for our study of the Sally module. An immediate consequence of the construction of the reduction modules and the virtual reduction modules of I with respect to J is Theorem 1.1, whose proof we now present: Proof of Theorem 1.1. For i c 0, one has
lŽ Si . s s0
ž
dy 1 iqdy1 iqdy2 y s1 q ??? q Ž y1 . s dy1 . dy1 dy2
/ ž
/
On the other hand, from Ž3. and Ž4.,
lŽ Si . s
ry1
ry1
ns1
ns1
Ý lŽ Ž Ln . i . s Ý Ž lŽ Ž Dn wyn x . i . y lŽ Ž K n . i . . .
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HILBERT FUNCTIONS AND SALLY MODULES
The contribution of lŽŽ Dnwyn x. i . to s0 is exactly lŽ I nq 1rJI n . and we conclude that ry1
s0 F
Ý lŽ I nq 1rJI n . . ns1
Equality in the above expression implies that, for n g 1, . . . , r y 14 , we have no contribution from lŽŽ K n . i . to s0 . And since K n is either Ž0. or has dimension d, we conclude that for n g 1, . . . , r y 14 , K n s 0. This implies L n , Ž I nq 1rJI n .w T1 , . . . , Td xwyn x. By Ž3. and the so-called Depth Lemma Žsee w1, Proposition 1.2.9x., S is Cohen]Macaulay. Now, because of the recognizable structure of the L n’s, we obtain their Hilbert polynomial and Hilbert]Poincare ´ series P Ž L n , i . s l Ž I nq 1rJI n . HP Ž L n , t . s
žŽ
i y n. q d y 1 dy1
l Ž I nq 1rJI n . t n
Ž1 y t .
d
/
and
.
Again by Ž3., we only have to add up these expressions, from n s 1 to r y 1, to conclude that the Hilbert coefficients of S and its Hilbert]Poincare ´ series are given by Ž1. and Ž2., respectively. Corollary 1.2 transfers to the associated graded ring of I the results obtained for the Sally module in Theorem 1.1. Its proof follows from w16, Proposition 1.2.10; 15, Corollary 3.3; 16, Proposition 1.3.3x. COROLLARY 2.1. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R and let J be a minimal reduction of I. The following conditions are equi¨ alent: ry1 Ž nq 1 Ži. s0 s S ns1 l I rJI n .; Žii. S is Cohen]Macaulay; Žiii. depthŽgrI Ž R .. G d y 1.
Proof. Ži. implies Žii. follows from Theorem 1.1 and Žii. implies Žiii. from w16, Proposition 1.2.10x. Now, as we have seen in Section 1, w6, Ž nq 1rJI n . if and only if Theorem 3.1x can be restated as s0 s Ý ry1 ns1 l I depthŽgrI Ž R .. G d y 1. This gives us the equivalence of Žiii. and Ži.. Remark 2.2. When we have equality in Theorem 1.1, and so, for n s 1, . . . , r y 1, L n , Ž I nq 1rJI n .w T1 , . . . , Td xwyn x, the canonical module v L n of L n can be explicitly computed. For n s 1, . . . , r y 1, k
v L n , Ž I nq1rJI n . w T1 , . . . , Td xw yd q n x , where Ž I nq 1rJI n . ks Hom R Ž I nq1rJI n, E ., R s RrA n , m s mrA n , and E is the injective envelope of Rrm Žsee w1x for more details..
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MARIA VAZ PINTO
3. REDUCTION NUMBER TWO Let M be a module of finite length over a Noetherian local ring Ž R, m .. We start this section defining the Loewy ascending chain of submodules of M 0 s M0 : M1 : M2 : ??? : Mty1 : Mt s M, where M0 s 0 and Mi s Ž Miy1: M m . for i G 1. Each MirMiy1 s Ž Miy1: M m .rMiy1 s Ž0: M r M iy 1 m ., being the socle of MrMiy1 , is not only an R-module but also an Rrm-vector space. If lŽ MirMiy1 . s n i , then MirMiy1 , Ž Rrm . n i . This construction shows that we can associate to M a partition Ž n1 , . . . , n t . of lŽ M . s n. Partitions of the form Ž1, n 2 , . . . , n t . correspond to cases where the socle M1 of M is cyclic. Let L s Ann R Ž M ., R s RrL, and m s mrL. Using the notation introduced in Section 2, we define M ks Hom R Ž M, E ., where E is the injective envelope of Rrm. LEMMA 3.1. Let Ž R, m . be a Noetherian local ring and let M be an R-module of finite length. Then the minimal number of generators of M k is equal to the minimal number of generators of the socle of M. Proof. This lemma is a special case of Matlis duality and its proof can be found in w1, Proposition 3.2.12x. The module of finite length we are interested in is I 2rJI. In this section we will study the Sally module of I with respect to J, fixing r s r J Ž I . equal to two, and letting lŽ I 2rJI . be any number n. There are 2 ny 1 possible chains of submodules associated to I 2rJI, and in all the cases where its socle is cyclic, the Sally module will have a nice structure. This is exactly the statement of Theorem 1.3. Before its proof we need the following lemma. LEMMA 3.2. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0, let A be an m-primary ideal of R, B s RrAw T1 , . . . , Td x, and v B the canonical module of B. Assume that M is a finitely generated B-module such that Ass B Ž M . : m B4 . Then M : Hom B Ž Hom B Ž M, v B . , v B . . Proof. We will show that the standard map w : M ª Hom B ŽHom B Ž M, v B ., v B . is one]one. The exact sequence of B-modules w
Hom B Ž Hom B Ž M, v B . , v B . ª C ª 0,
6
0ªKªM
HILBERT FUNCTIONS AND SALLY MODULES
511
where K s KerŽ w . and C s CokerŽ w ., tensored with Bm B , yields 0 ª K m B ª Mm B ª Hom B m B Ž Hom B m B Ž Mm B , v B m B . , v B m B . ª Cm B ª 0. Now, since Bm B is Artinian, v B m B is isomorphic to the injective envelope of the residue field of Bm B Žsee w1, Chap. 3x for more details.. At the same time the Artinian ring Bm B is complete, so by Matlis duality w1, Theorem 3.2.13x, Hom B m BŽHom B m BŽ Mm B , v B m B ., v B m B . , Mm B . We conclude that K m B s 0 and Cm B s 0. But Ass B Ž K . : Ass B Ž M . : m B4 , and so K s 0. This shows that w is one]one. Proof of Theorem 1.3. Our notation and definitions are the ones of Section 2. For simplicity we write A for A1 s Ann R Ž I 2rJI . and B for B1 s RrA1w T1 , . . . , Td x in this proof. Since I 3 s JI 2 , one has L2 s Rw Jt x ? I 3rJI 2 s 0, and we conclude from Ž3. that S , L1. In particular, S is a B-module, and since Ass Rw J t xŽ S . s m Rw Jt x4 Žsee w15x., we have Ass B Ž S . s m B4 . From Ž4. one has the exact sequence of B-modules 0 ª K 1 ª Ž I 2rJI . w T1 , . . . , Td xw y1 x ª S ª 0, and our aim is to prove that K 1 s 0. Let R s RrA, m s mrA, and let v R and v B be the canonical modules of R and B, respectively. Since R is Artinian, v R is the injective envelope E of Rrm, and v B , v R w T1 , . . . , Td x , Ew T1 , . . . , Td x Žsee w1, Chap. 3x for more details.. Applying Hom B Ž ], v B . to the exact sequence above yields the long exact sequence 0 ª Hom B Ž S, v B . ª Hom B Ž I 2rJI w T1 , . . . , Td x , v B . ª Hom B Ž K 1 , v B . ª Ext 1B Ž S, v B . ª Ext 1B Ž I 2rJI w T1 , . . . , Td x , v B . ª ??? . Since B is a Cohen]Macaulay local ring of dimension d and I 2rJI w T1 , . . . , Td x is a finitely generated Cohen]Macaulay B-module of dimension d, one has Ext 1B Ž I 2rJI w T1 , . . . , Td x, v B . s 0 Žsee w1, Corollary 3.5.11x.. Let F s Ž I 2rJI . k and observe that Hom B Ž I 2rJI w T1 , . . . , Td x, v B . , F w T1 , . . . , Td x. The fact that the socle of I 2rJI is cyclic implies, by Lemma 3.1, that F is a cyclic R-module, and one can write F , RrAnn R Ž F .. But R is Artinian, and therefore complete. By Matlis duality Žsee w1, Theorem 3.2.13x., Ž F . k, I 2rJI, as R-modules Žand so as R-modules.. Therefore A s Ann R Ž I 2rJI . s Ann R Ž F . and F , RrA s R. We conclude that Hom B ŽŽ I 2rJI .w T1 , . . . , Td x, v B . , Rw T1 , . . . , Td x s B. The above long exact sequence can be rewritten as 0 ª Hom B Ž S, v B . ª B ª Hom B Ž K , v B . ª Ext 1B Ž S, v B . ª 0. Ž 5 .
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MARIA VAZ PINTO
Observe that Hom B Ž S, v B . is just an ideal of B and there are two cases to be analyzed: Ž1. Hom B Ž S, v B . : m B and Ž2. Hom B Ž S, v B . m B. Case Ž1.. Hom B Ž S, v B . : m B. Since Ass B Ž S . s m B4 , we have S : Hom B ŽHom B Ž S, v B ., v B . by Lemma 3.2. Therefore, Ann R Ž m B . : Ann R Ž Hom B Ž S, v B . . : Ann R Ž Hom B Ž Hom B Ž S, v B . , v B . . : Ann R Ž S . : Ann R Ž I 2rJI . s 0, and one concludes that Ann R Ž m B . s 0. But m B g Ass B Ž B . and then it is easy to prove that Ann R Ž m B . / 0. This gives us a contradiction. Case Ž2.. Hom B Ž S, v B . m B. Factoring the exact sequence Ž5., one obtains the short exact sequence of B-modules 0 ª BrHom B Ž S, v B . ª Hom B Ž K 1 , v B . ª Ext 1B Ž S, v B . ª 0, from which we conclude that Supp Ž Hom B Ž K 1 , v B . . : Supp Ž BrHom B Ž S, v B . . j Supp Ž Ext 1B Ž S, v B . . . We know that m B f SuppŽ BrHom B Ž S, v B .. because Hom B Ž S, v B . m B. Now, we have ŽExt 1B Ž S, v B .. m B , Ext 1B m BŽ Sm B , v B m B .. Since Bm B is Artinian, v B m B is the injective envelope of the residue field of Bm B , an injective Bm B-module. Therefore, Ext 1B m BŽ Sm B , v B m B . s 0, and m B f SuppŽExt 1B Ž S, v B ... We conclude that mB f SuppŽHom B Ž K 1 , v B .. and so Ann B ŽHom B Ž K 1 , v B .. m B. Let x g Ann B ŽHom B Ž K 1 , v B .., x f m B. Since K 1 : D 1 wy1x, Ass B Ž K 1 . : m B4 . It follows from Lemma 3.2 that K 1 : Hom B ŽHom B Ž K 1 , v B ., v B .. Now, x ? Hom B Ž K 1 , v B . s 0 implies that x ? Hom B Ž Hom B Ž K 1 , v B . , v B . s 0, and thus xK 1 s 0. On the other hand, x f m B and Ass B Ž D 1 . s m B4 implies that x is regular on D 1 , and therefore, x is regular on K 1. Since xK 1 s 0, one has K 1 s 0, as wanted. Remark 3.3. We have seen in Theorem 1.3 that S , Ž I 2rJI .w T1 , . . . , Td xwy1x, and in this situation the Hilbert polynomial of S is P Ž S, i . s l Ž I 2rJI .
žŽ
i y 1. q d y 1 . dy1
/
HILBERT FUNCTIONS AND SALLY MODULES
513
In particular, the first Hilbert coefficient of S is exactly s0 s lŽ I 2rJI . s ry1 Ž nq 1 S ny rJI n .. Therefore, when in the hypotheses of Theorem 1.3, 1l I Theorem 1.1 and Corollary 1.2 give us very simply expressions for the Hilbert coefficients and Hilbert]Poincare ´ series of S J Ž I . and grI Ž R ., respectively.
4. LOW LENGTHS In this section we are interested in studying the Sally module of I with respect to J when the lengths of I 2rJI are either 1, 2, or 3. Length One If lŽ I 2rJI . s 1, I 2rJI is a cyclic R-module and is equal to its socle. There is only one possible chain of submodules for I 2rJI, as described in Section 3, namely Ž0. : I 2rJI. We need to observe some simple facts and our notation will be that of Section 2. Since I 2rJI is a cyclic R-module, we must have I nq 1rJI n cyclic for all n G 1. We can write I nq 1rJI n s R ? a n , RrA n , for some a n g I nq 1rJI n, where A n s Ann R Ž I nq 1rJI n .. Since A n : A nq1 for all n G 1, and since A1 s m, we conclude that, for n g 1, . . . , r y 14 , A n s m, I nq 1rJI n , Rrm, and lŽ I nq 1rJI n . s 1. In particular, Bn s Rrm w T1 , . . . , Td x. For simplicity, we will write B s Bn , for n g 1, . . . , r y 14 , in this subsection. Observe also that for n g 1, . . . , r y 14 , L n is a cyclic Rw Jt x-module: L n s R w Jt x ? I nq 1rJI n s R w Jt x ? a n , R w Jt x rAnn Rw J t x Ž L n . w yn x . We know that m Rw Jt x s A1 Rw Jt x : Ann Rw J t xŽ L1 .. On the other hand, Ass Rw J t xŽ L1 . : Ass Rw J t xŽ S . s m Rw Jt x4 , and therefore, Ann Rw J t xŽ L1 . : m Rw Jt x. We conclude that m Rw Jt x s Ann Rw J t xŽ L1 . and L1 , R w Jt x rm R w Jt xw y1 x , Ž Rrm . w T1 , . . . , Td xw y1 x s B w y1 x . For n g 2, . . . , r y 14 , we also have m Rw Jt x s A n Rw Jt x : Ann Rw J t xŽ L n ., but now we cannot say that Ass Rw J t xŽ L n . : m Rw Jt x4 , and so we cannot conclude that L2 , . . . , L ry1 are isomorphic to B. What can be said is that for n g 2, . . . , r y 14 , L n is a cyclic B-module: L n s B ? a n , BrAnn B Ž L n . w yn x . Remark 4.1. If the reduction number of I with respect to J is two and lŽ I 2rJI . s 1, Theorem 1.3 tells us that S , Ž Rrm .w T1 , . . . , Td xwy1x.
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Proof of Theorem 1.4. We only need to consider here r ) 2, because the case r s 2 has been done in greater generality, as observed in Remark 4.1. We have seen that lŽ I nq 1rJI n . s 1 for n g 1, . . . , r y 14 . The fact that s0 F r y 1 is therefore a consequence of Theorem 1.1. When s0 s r y 1, S is Cohen]Macaulay also by Theorem 1.1. One needs to look now at the second case, when s0 s r y 2. The idea is to show that, for n g 1, . . . , r y 24 , L n , Bwyn x an L ry1 , BrŽ f .wyŽ r y 1.x, where f / 0 is a regular element on B. From this it follows that, for n g 1, . . . , r y 24 , the depth of L n is d, and L ry1 has depth G d y 1. Going back to our filtration of the Sally module in Section 2, and using the Depth Lemma, we conclude that depthŽ S . G d y 1. But the depth of S cannot be d when s0 s r y 2, by Corollary 2.1. Therefore, in this case, depthŽ S . s d y 1. Exactly as in the proof of Theorem 1.1, one concludes from Ž3. and Ž4. that, for n g 1, . . . , r y 14 , the contribution of lŽŽ Dnwyn x. i . to s0 is exactly lŽ I nq 1rJI n . s 1. Since s0 s r y 2, we must have r y 2 of the K n’s equal to zero, and one K n different from zero; i.e., we must have r y 2 of the L n’s isomorphic to B and one L n isomorphic to BrAnn B Ž L n ., where Ann B Ž L n . / 0. But since for n G 1, Ann B Ž L n . : Ann B Ž L nq1 ., we conclude that the r y 2 L n’s isomorphic to B have to be exactly the first r y 2. And so, for n g 1, . . . , r y 24 , L n , Bwyn x, and L ry1 , Br Ann B Ž L ry1 .wyŽ r y 1.x, where Ann B Ž L ry1 . / 0. Our aim now is to prove that Ann B Ž L ry1 . is a principal ideal of B, and in order to do this, one shows that it has height one and is unmixed. Since B is a UFD, we will get our result. Let P be a prime ideal of B, heightŽ P . G 2, and one wants P f Ass B Ž L ry1 .. Now, B s Rw Jt xrm Rw Jt x, and so P s prm Rw Jt x, where p is a prime ideal of Rw Jt x. Applying the functor Hom Rw J t xŽ Rw Jt xrp, ] . to the first exact sequence of Ž3. yields the long exact sequence 0 ª Hom Rw J t x Ž R w Jt x rp , L1 . ª Hom Rw J t x Ž R w Jt x rp , S . ª Hom Rw J t x Ž R w Jt x rp , C2 . ª Ext 1Rw J t x Ž R w Jt x rp , L1 . ª ??? . Ass Rw J t xŽ S . s m Rw Jt x4 and since m Rw Jt x « p, there exists in p a regular element on S. Therefore, Hom Rw J t xŽ Rw Jt xrp, S . s 0. Also, Ext 1Rw J t xŽ Rw Jt xr p, L1 . s 0. The reason for this is that since heightŽ P . G 2 and B is Cohen]Macaulay, there exists Ž x, y . : P, a regular sequence on B; therefore, Ž x, y . : p is a regular sequence on B, where x, y are the lifts of x and y to p. Hence, Ext 1Rw J t xŽ Rw Jt xrp, B . s 0. But L1 , B, and we have Ext 1Rw J t xŽ Rw Jt xrp, L1 . s 0. From the above long exact sequence we conclude that Hom Rw J t xŽ Rw Jt xrp, C2 . s 0.
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Applying now successively Hom Rw J t xŽ Rw Jt xrp, ] . to the other sequences of Ž3., and using the fact that L2 , ??? , L ry2 , B, one obtains Hom Rw J t xŽ Rw Jt xrp, L ry1 . s 0. Therefore, p f Ass Rw J t xŽ L ry1 . and so, P f Ass B Ž L ry1 .. We have proved that if P is a prime of B of height G 2 then P f Ass B Ž L ry1 .. But Ž0. f Ass B Ž L ry1 ., since Ann B Ž L ry1 . / 0. Therefore, the only primes of B that belong to Ass B Ž L ry1 . are primes of height 1. This means that Ann B Ž L ry1 . has height 1 in B and is unmixed. Remark 4.2. When lŽ I 2rJI . s 1 and s0 s r y 1, Theorem 1.1 and Corollary 1.2 give us very simple expressions for the Hilbert coefficients and Hilbert]Poincare ´ series of S and grI Ž R ., respectively. It also follows from Corollary 1.2 that depthŽgrI Ž R .. G d y 1. Remark 4.3. When lŽ I 2rJI . s 1 and s0 s r y 2, since depthŽ S . s d y 1, it follows from w16, Proposition 1.2.10x that depthŽgrI Ž R .. G d y 2. But the depth of grI Ž R . cannot be G d y 1 when s0 s r y 2, by Corollary 2.1. Therefore, when lŽ I 2rJI . s 1, if d ) 1 and s0 s r y 2, then depthŽgrI Ž R .. s d y 2. Another observation is that in the case s0 s r y 2 our method does not allow us to compute the Hilbert coefficients or the Hilbert]Poincare ´ series of either the Sally module or the associated graded ring. Indeed, since the degree of the element f is not known, we cannot compute the Hilbert coefficients or the Hilbert]Poincare ´ series of L ry1. COROLLARY 4.4. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm, let I be an m-primary ideal of R and J a minimal reduction of I. Assume that lŽ I 2rJI . s 1 and s0 s r y 1. If v t denotes the degree t component of the canonical module v S of the Sally module, then
vt s
~¡
if t - d y r q 1,
0,
¢Ž Rrm . ž 1 Ý ry js 1
kq dyj dy 1
/,
if t s k q d y r q 1, k G 0.
Proof. Let v Rw J t x , v L n, and vC n denote the canonical modules of Rw Jt x, L n , and Cn , respectively. It follows from Remark 2.2 that
v L n , Rrm w T1 , . . . , Td xw yd q n x
Ž 6.
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MARIA VAZ PINTO
in our situation. Applying now the functor Hom Rw J t xŽ ], v Rw J t x . to each sequence of Ž3. yields the set of exact sequences ª
ª
v L ry 1 .. . vC 3
ª
vC 2
0
ª
0 0
ª
ª
vC ry 2 .. . vC 2
ª
0
ª
v L ry 2 .. . vL2
ª
0
ª
vS
ª
v L1
ª
0.
Ž 7.
Our result follows from Ž6. and Ž7.. Length Two We are now interested in studying the Sally module when lŽ I 2rJI . s 2. According to the structure of ascending chains of submodules described in Section 3, we have two possible cases to analyze. In the first one I 2rJI and its socle are both cyclic and different from each other. In the second case I 2rJI is equal to its socle and isomorphic to Ž Rrm . 2 . For the rest of this subsection we will focus on the first case, starting with reduction number two. Then, as in the case of length 1, we will relate the first Hilbert coefficient of S to the reduction number of I with respect to J, and we will analyze the Sally module in several extremal situations. One shows in the following proposition that the first reduction module L1 of I with respect to J and the first virtual reduction module D 1 of I with respect to J are isomorphic. PROPOSITION 4.5. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R, J a minimal reduction of I, lŽ I 2rJI . s 2, and I 2rJI a cyclic R-module. Then the first reduction module L1 of I with respect to J is Cohen]Macaulay. In fact, L1 , Ž RrA1 .w T1 , . . . , Td xwy1x. Proof. I 2rJI is a cyclic R-module, so one can write I 2rJI s R ? a1 s RrA1 , for some a1 g I 2rJI, A1 s Ann R Ž I 2rJI .. In this case, since lŽ RrA1 . s 2 and lŽ Rrm . s 1, A1 / m. Let R s RrA1 , m s mrA1 , and notice that lŽ m . s lŽ R . y lŽ Rrm . s 1. So m is cyclic, and we can write m s Ž s ., for some s g mrA1 , s / 0. Observe at this point that since Ž m . 2 / m, we must have Ž m . 2 s 0. In the notation of Section 2, let B1 s RrA1w T1 , . . . , Td x. From Ž4. we have the exact sequence of B1-modules u1
L1 ª 0.
6
0 ª K 1 ª B1 w y1 x
Our aim is to show that K 1 s 0, so that L1 , B1wy1x. Let K denote K 1wq1x and we will prove that the ideal K of B1 vanishes.
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517
One has Ass Rw J t xŽ L1 . : Ass Rw J t xŽ S . s m Rw Jt x4 , and so Ass B1Ž L1 . s m B14 . This tells us that dim B Ž L1 . s d; but also dim B Ž B1 . s d. There1 1 fore, from the exact sequence above one concludes that heightŽ K . s 0. Now, we know that m B1 is the only height zero prime ideal of B1 , and so, K : m B1 s s ? B1. We can write K s sK X , where K X is an ideal of B1. Notice that K X / B1; for if K X s B1 , then K s sB1 , and so, s g K; in fact, s g Ž K . 0 ; but this gives us a contradiction, since u 1 is an isomorphism in degree 1, and so Ž K . 0 s Ž K 1 .1 s 0. Therefore K X / B1. If K X : m B1 , then K s sK X : m 2 B1 s Ž m . 2 B1 s 0, and our proof is complete. Assume then that K X m B1. One has L1 , B1rK , B1rsK X , as B1-modules. Now, as we have seen above, Ass B1Ž L1 . s m B14 ; but we will show that B1rsK X has associated primes in B1 of height G 1. This is a contradiction, and this case K X m B1 is not possible. One has s g B1 , s f K s sK X , so s is nonzero in B1rsK X and K X s s 0. Therefore, there eixsts a prime P of B1 , such that K X : Ann B1Ž s . : Ann B1Ž x . s P, where x is a nonzero element of B1rsK X . So, one has P g Ass B1Ž B1rsK X . and height B1Ž P . G height B1Ž K X .; but since we are in the case where K X m B1 , we must have height B1Ž K X . G 1, and therefore, height B1Ž P . G 1. Remark 4.6. Notice that if the reduction number of I with respect to J is two, lŽ I 2rJI . s 2, and I 2rJI is a cyclic R-module, then S , Ž RrA1 .w T1 , . . . , Td xwy1x. This result follows from Proposition 4.5 together with the fact that S , L1 when r J Ž I . s 2. Our result follows also from Theorem 1.3, because when lŽ I 2rJI . s 2 and I 2rJI is cyclic, the socle of I 2rJI must be cyclic. PROPOSITION 4.7. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R and let J be a minimal reduction of I. If lŽ I 2rJI . s 2 and I 2rJI is a cyclic R-module, then s0 F 2Ž r y 1.. In case of equality, S is Cohen]Macaulay. Proof. Since I 2rJI is a cyclic R-module, we must have I nq 1rJI n cyclic for all n G 1. Therefore, we can write I nq 1rJI n , RrA n , A n s Ann R Ž I nq 1rJI n .. But A n : A nq1 for all n G 1, and by hypotheses, lŽ RrA1 . s lŽ I 2rJI . s 2. We conclude then that lŽ I nq 1rJI n . F 2, for all n G 1. This result together with Theorem 1.1 gives us ry1
s0 F
Ý lŽ I nq 1rJI n . F 2 Ž r y 1. . ns1
Ž nq 1rJI n ., and so S is If s0 s 2Ž r y 1., then, in particular, s0 s Ý ry1 ns1 l I Cohen]Macaulay by Theorem 1.1.
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Remark 4.8. If s0 s 2Ž r y 1. then lŽ I nq 1rJI n . s 2 for n g 1, . . . , r y 14 , and so I nq 1rJI n , RrA1; from the proof of Theorem 1.1, L n , RrA1w T1 , . . . , Td xwyn x for n g 1, . . . , r y 14 . We also observe that when s0 s 2Ž r y 1., we may use the second part of Theorem 1.1 and Corollary 1.2. In particular, the associated graded ring of I has depth G d y 1, and we obtain simple expressions for the Hilbert coefficients and Hilbert] Poincare ´ series of S and grI Ž R .. The last result of Proposition 4.7 can be generalized to other extremal cases. Before doing it, we need some observations. L1 , RrA1w T1 , . . . , Td xwy1x by Proposition 4.5. The question now becomes: what happens to the other reduction modules of I with respect to J ? Do they have a nice structure so that good properties for the Sally module can be concluded? We will look first at L2 . As observed before, since I 2rJI is a cyclic R-module, one has I 3rJI 2 also cyclic, and so I 3rJI 2 , RrA 2 , where A 2 s Ann R Ž I 3rJI 2 .. If B2 s RrA 2 w T1 , . . . , Td x, we have from Ž4. the exact sequence of B2-modules 0 ª K 2 ª B2 w y2 x ª L2 ª 0.
Ž 8.
Since A1 : A 2 : m, lŽ Rrm . s 1, and lŽ RrA1 . s 2, either A 2 s A1 or A 2 s m; in the first case, B2 s RrA1 w T1 , . . . , Td x s B1 , and in the second one, B2 s Rrm w T1 , . . . , Td x; in each of these cases, we can have K 2 s 0 or K 2 / 0. Therefore, there are four possibilities for L2 : L2 , RrA1 w T1 , . . . , Td xw y2 x , L2 , Ž RrA1 w T1 , . . . , Td xw y2 x . rK 2 , K 2 / 0, m Ž RrA1 w T1 , . . . , Td xw y2 x . K 2 , L2 , Rrm w T1 , . . . , Td xw y2 x , L2 , Ž Rrm w T1 , . . . , Td xw y2 x . rK 2 ,
K 2 / 0.
Suppose that L2 , RrA1w T1 , . . . , Td xwy2x. In this case L2 is Cohen] Macaulay. Observe that, for all i G 0, lŽŽ L2 . i . s lŽ RrA1 .Ž Ž i y 2d.yq1d y 1 .. From the filtration of the Sally module in Section 2, one concludes that the contribution of lŽŽ L2 . i . to s0 is 2. Suppose next that L2 , Ž RrA1 w T1 , . . . , Td xwy2x.rK 2 , K 2 / 0. One has Ass B 2Ž K 2 . : Ass B 2Ž B2 . s m B2 4 and since K 2 / 0, Ass B 2Ž K 2 . s m B2 4 . This tells us that dim B 2Ž K 2 . s d, and we can write, for i c 0, lŽŽ K 2 . i . s dy 1. dy 2. k 0 Ž i qd y y k 1Ž i qd y q ??? , where k 0 / 0 Ž k 0 is called the multiplicity 1 2 of K 2 .. It follows from Ž8. and from the filtration of the Sally module presented in Section 2, that the contribution of lŽŽ L2 . i . to s0 is, in this case, 2 y k 0 Žand since k 0 / 0, this number can be either 1 or zero..
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HILBERT FUNCTIONS AND SALLY MODULES
In the third case, L2 , Rrm w T1 , . . . , Td xwy2x, which is Cohen]Macaulay. Now, lŽŽ L2 . i . s lŽ Rrm .Ž Ž i y 2d.yq1d y 1 . for all i G 0 and the contribution of lŽŽ L2 . i . to s0 is 1. Finally, suppose that L2 , Ž Rrm w T1 , . . . , Td xwy2x.rK 2 , K 2 / 0. Since Rrm w T1 , . . . , Td x is a domain, heightŽ K 2 . ) 0, dimŽ L2 . - d, and the contribution of lŽŽ L2 . i . to s0 is zero. In a similar way we could analyze the other L n’s Ž L3 , . . . , L ry1 .. If A ny 1 s A1 , then L n has exactly the same four possibilities that we saw for A 2 . If A ny1 s m, then A n s m Žsince A ny1 : A n : m ., and there are only two possibilities for L n : L n , Rrm w T1 , . . . , Td xwyn x or L n , Ž Rrm w T1 , . . . , Td xwyn x.rK n , K n / 0. We conclude that if the contribution of lŽŽ L n . i . to s0 is 2, then we must have L n , RrA1w T1 , . . . , Td xwyn x; if the contribution of lŽŽ L n . i . to s0 is 1, L n , Rrm w T1 , . . . , Td xwyn x or L n , Ž RrA1w T1 , . . . , Td xwyn x.rK n , K n / 0, and the multiplicity of K n is 1; if the contribution of lŽŽ L n . i . to s0 is zero, then L n , Ž Rrm w T1 , . . . , Td xwyn x.rK n , K n / 0, or L n , Ž RrA1w T1 , . . . , Td xwyn x.rK n , K n / 0, and the multiplicity of K n is 2. An interesting consequence of the considerations just made is that for n g 1, . . . , r y 14 , the contribution of lŽŽ L n . i . to s0 is nonincreasing. We present now two theorems where we get results for the depth of the Sally module in several extremal cases. THEOREM 4.9. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R, J a minimal reduction of I, lŽ I 2rJI . s 2, and I 2rJI a cyclic R-module. Suppose we are in one of the following situations: Contribution of lŽŽ L1 . i . to s0
Contribution of lŽŽ L2 . i . to s0
2 2 .. .
2 2 .. .
??? ???
2 1 .. .
2
1
???
1
???
Contribution of lŽŽ L ry1 . i . to s0
where all the 1’s come from L n , Rrm w T1 , . . . , Td xwyn x. Then the Sally module is Cohen]Macaulay. Proof. In all the cases above, L n , RrA1w T1 , . . . , Td xwyn x or L n , Rr m w T1 , . . . , Td xwyn x, as described before. In any situation, L n is Cohen] Macaulay for all n g 1, . . . , r y 14 , and our result follows from the filtration of S presented in Section 2.
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Remark 4.10. Notice that in each of the cases of Theorem 4.9, we are Ž nq 1rJI n .. Therefore, in the situation of Theorem 1.1, where s0 s Ý ry1 ns1 l I we may use the second part of Theorem 1.1 and Corollary 1.2. In particular, depthŽgrI Ž R .. G d y 1. THEOREM 4.11. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R, J a minimal reduction of I, lŽ I 2rJI . s 2, and I 2rJI a cyclic R-module. Suppose we are in one of the following situations: Contribution of lŽŽ L1 . i . to s0
Contribution of lŽŽ L2 . i . to s0
2 2 .. .
2 2 .. .
2
1
Contribution of lŽŽ L ry2 . i . to s0
Contribution of lŽŽ L ry1 . i . to s0
??? ???
2 1 .. .
0 0 .. .
???
1
0
???
where all the 1’s come from L n , Rrm w T1 , . . . , Td xwyn x, and the zeros on the last column come from L ry1 , Ž Rrm w T1 , . . . , Td xwyŽ r y 1.x.rK ry1 , K ry1 / 0. Then, depthŽ S . G d y 1. Proof. By the structure described before, L1 , . . . , L ry2 all have depth d. An argument similar to the one used in the proof of Theorem 1.4 gives us depthŽ L ry1 . G d y 1. Therefore, depthŽ S . G d y 1. COROLLARY 4.12. y 2.
In the hypotheses of Theorem 4.11, depthŽgrI Ž R .. G d
Length Three We will now study the Sally module when lŽ I 2rJI . s 3. According to the structure of ascending chains of submodules described in Section 3, we have four possible cases to analyze: in the first one, I 2rJI and its socle are both cyclic, and therefore they are different from each other; in the second case, I 2rJI is not cyclic but it has a cyclic socle; in the third case, I 2rJI is a cyclic R-module, but its socle is not cyclic; finally, in the last case, I 2rJI is equal to its socle and isomorphic to Ž Rrm . 3. In the first two cases, the socle of I 2rJI is cyclic, and so, if the reduction number of I with respect to J is two, we conclude from Theorem 1.3 that the Sally module is Cohen]Macaulay. We will look here at the third case and the result we have tells that, if the reduction number of I with respect to J is two, then the Sally module has depth bigger or equal than d y 1. In fact, we will only assume that lŽ I 2rJI . s 3 and that I 2rJI is cyclic, so our argument will also include the first case Žwhere we already have a better result..
HILBERT FUNCTIONS AND SALLY MODULES
521
Using the notation and definitions of Section 2, we have A1 s Ann R Ž I 2rJI . and B1 s RrA1 w T1 , . . . , Td x; from Ž4., one has the following exact sequence of B1-modules 0 ª K 1 ª D 1 w y1 x ª L1 ª 0.
Ž 9.
Before our main theorem, we need the following lemma. LEMMA 4.13. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm, let I be an m-primary ideal of R, and J a minimal reduction of I. Then K 1 satisfies property S2 of Serre. Proof. Property S2 of Serre says that for all p g Supp B1Ž K 1 ., depth Ž Ž K 1 . p . G min 2, dim Ž Ž K 1 . p . 4 . Assume that K 1 / 0; since m B1 is the only height zero prime of B1 and Ass B1Ž K 1 . s m B14 , K 1 satisfying S2 means that for all p, depthŽŽ K 1 . p . G min 2, htŽ p .4 . If htŽ p . s 0, there is nothing to show. Let now p be such that htŽ p . G 1; in this case m B1 « p, so there exists r g p _ m B1. Since L1 : S, Ass B1Ž L1 . s m B14 , and therefore r is regular on L1. One has depthŽŽ L1 . p . G 1. Tensoring Ž9. with Ž B1 . p yields 0 ª Ž K 1 . p ª Ž D 1 . p w y1 x ª Ž L1 . p ª 0. Since D 1 is a Cohen]Macaulay B1-module with Ass B1Ž D 1 . s m B14 , depthŽŽ D 1 . p . s dimŽŽ D 1 . p . s htŽ p .. Now, the Depth Lemma on the last exact sequence tells us that depthŽŽ K 1 . p . G min 1 q 1, htŽ p .4 , and this proves that K 1 has S2 . THEOREM 4.14. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R, J a minimal reduction of I, lŽ I 2rJI . s 3, and I 2rJI a cyclic R-module. If the reduction number of I with respect to J is two, then depthŽ S . G d y 1. Proof. Since the reduction number of I with respect to J is two, one has by Ž3. that S , L1. Since I 2rJI is cyclic, we can write I 2rJI , RrA1 , where A1 s Ann R Ž I 2rJI .; B1 s RrA1w T1 , . . . , Td x, and in our case the exact sequence Ž9. becomes 0 ª K 1 ª B1 w y1 x ª S ª 0.
Ž 10 .
If K 1 s 0, our proof is finished. Let us then assume that K 1 / 0, and since Ass B1Ž K 1 . : Ass B1Ž B1 . s m B1 4 , one has Ass B1Ž K 1 . s m B1 4 and dim B1Ž K 1 . s d. There exist k 0 , k 1 , . . . , k dy1 g Z such that for i c 0,
lŽ Ž K 1 . i . s k 0
ž
dy 1 iqdy1 iqdy2 y k1 q ??? q Ž y1 . k dy1 , dy1 dy2
/
ž
/
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MARIA VAZ PINTO
where k 0 / 0. From Ž10., k 0 q s0 s 3, and so k 0 can be either 1, 2, or 3. If k 0 s 3, then s0 s 0, and, in particular, S s 0 Žsee w15x.. Then lŽ I 2rJI . s 0. But since we are in the case where lŽ I 2rJI . s 3, we conclude that k 0 / 3. If k 0 s 2, then s0 s 1, and it follows from w15, Proposition 3.5x that m S s 0; in particular, m Ž I 2rJI . s 0. But this is a contradiction because I 2rJI is cyclic and lŽ I 2rJI . s 3. One concludes that k 0 / 2. We are left with the only possible case: k 0 s 1. Now, K 1 is not only a B1-module, but also an Rw Jt x-module with Ass Rw J t xŽ K 1 . s m Rw Jt x4 . Therefore m K 1 s 0, as in w15, Proposition 3.5x. This implies that K 1 is also a B-module, where B s Rw Jt xrm Rw Jt x , Rrm w T1 , . . . , Td x, and the structure of K 1 as an Rw Jt x-module, as a B-module, or as a B1-module is the same. Again as in w15, Proposition 3.5x, K 1 , K Žas B-modules., where K is an ideal of B. By Lemma 4.13, K satisfies property S2 of Serre. It is not easy to show that as an ideal of B, K has height one and is unmixed. Since B is a UFD, K has to be principal, i.e., K s B ? f, for some f / 0, f g K. In particular, f is regular on B, and so B ? f , B as B-modules. Therefore, as B-modules, K 1 , B and depthŽ K 1 . s d. We conclude that depthŽ S . G d y 1, using the Depth Lemma on Ž10.. COROLLARY 4.15. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R, J a minimal reduction of I, lŽ I 2rJI . s 3, and I 2rJI a cyclic R-module. If the reduction number of I with respect to J is two, then depthŽgrI Ž R .. G d y 2.
REFERENCES 1. W. Bruns and J. Herzog, ‘‘Cohen]Macaulay Rings,’’ Cambridge Univ. Press, Cambridge, UK, 1993. 2. J. Elias, M. E. Rossi, and G. Valla, On the coefficients of the Hilbert polynomial, preprint. 3. J. Elias and G. Valla, Rigid Hilbert functions, J. Pure Appl. Algebra 71 Ž1991., 19]41. 4. A. Guerrieri, On the depth of the associated graded ring of an m-primary ideal of a Cohen]Macaulay local ring, J. Algebra 167 Ž1994., 745]757. 5. L. T. Hoa, Two notes on the coefficients of the Hilbert-Samuel polynomial, preprint. 6. S. Huckaba, A d-dimensional extension of a lemma of Huneke’s and formulas for the Hilbert coefficients, Proc. Amer. Math. Soc., in press. 7. S. Huckaba and T. Marley, Hilbert coefficients and the depths of associated graded rings, preprint. 8. T. Marley, The coefficients of the Hilbert polynomial and the reduction number of an ideal, J. London Math. Soc. Ž 2 . 40 Ž1989., 1]8. 9. H. Matsumura, ‘‘Commutative Ring Theory,’’ Cambridge Univ. Press, Cambridge, UK, 1986.
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10. J. D. Sally, On the associated graded ring of a local Cohen]Macaulay ring, J. Math. Kyoto Uni¨ . 17 Ž1977., 19]21. 11. J. D. Sally, Cohen]Macaulay local rings of maximal embedding dimension, J. Algebra 56 Ž1979., 168]183. 12. J. D. Sally, Tangent cones at Gorenstein singularities, Compositio Math. 40 Ž1980., 167]175. 13. J. D. Sally, Hilbert coefficients and reduction number 2, J. Algebraic Geom. 1 Ž1992., 325]333. 14. J. D. Sally, Ideals whose Hilbert function and Hilbert polynomial agree at n s 1, J. Algebra 157 Ž1993., 534]547. 15. W. V. Vasconcelos, Hilbert functions, analytic spread, and Koszul homology, Contemp. Math. 159 Ž1994., 401]422. 16. M. Vaz Pinto, ‘‘Structure of Sally Modules and Hilbert Functions,’’ Ph.D. dissertation, Rutgers University, 1995.
JOURNAL OF ALGEBRA ARTICLE NO.
Hilbert Functions and Sally Modules Maria Vaz PintoU Department of Mathematics, Rutgers Uni¨ ersity, New Brunswick, New Jersey 08903 Communicated by Mel¨ in Hochster Received June 14, 1995
1. INTRODUCTION Let Ž R, m . be a Noetherian local ring of Krull dimension d ) 0 and let I be an m-primary ideal of R. There has been a great deal of interest on two objects associated to the ideal I: Ži. the Hilbert]Samuel function of I, i.e., the numerical function HI Ž n. s lŽ RrI n ., where l stands for length, and Žii. the depth of the associated graded ring of R with respect to I, grI Ž R . s [nG 0 I nrI nq1 Žsee w2]8, 13, 14x.. An approach to their study has consisted in using reductions of I, that is, ideals J : I, with the property that I rq1 s JI r for some integer r Žsee w1x or w9x for this and other basic notions we employ.. This means that the natural inclusion of Rees algebras R w Jt x s
[ J n t n ¨ R w It x s [ I n t n
nG0
nG0
turns Rw It x into a finite Rw Jt x-module. If Rw Jt x has a more accessible character than Rw It x, and if one has good control over the module structure of Rw It x, then one is able to deal with Ži. and Žii. above rather effectively. A setting for obtaining a good Rw Jt x is that of a Cohen]Macaulay local ring Ž R, m . with infinite residue field Rrm. In this case, any minimal reduction J of I is generated by a regular sequence, so that the algebra * The author was partially supported by a JNICT grant. Current address: Department of Mathematics and Computer Science, Montclair State University, Upper Montclair, New Jersey 07043. E-mail address: [email protected]. 504 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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Rw Jt x is very well understood. We define the reduction number r J Ž I . of I with respect to J as the smallest integer t such that I tq1 s JI t. One way to deal with the second issue, of the module structure of Rw It x over Rw Jt x, is to define modules where the two algebras interact. In this way Vasconcelos introduced in w15x the so-called Sally module S s S J Ž I . of I with respect to J. It is defined by the exact sequence 0 ª I ? R w Jt x ª I ? R w It x ª S J Ž I . s
[ I nq 1rJ n I ª 0,
nG1
and a motivation for its name is the work of Sally in w10]13x Žsee w15x.. Our aim here is to uncover structural aspects of S J Ž I . in sufficient detail to deal with Ži. and Žii. in several cases of interest. The techniques we introduce are filterings of S J Ž I . by submodules which have Cohen] Macaulay ancestors: to a filtration of the Sally module there corresponds another series of modules Žthe virtual filtration .. When we succeed in proving their equality, or near so, there is an abundance of information about S J Ž I ., which may be used to estimate its depth and to describe its Hilbert function and some of its arithmetical properties like the canonical module. Information about depth, Hilbert coefficients, and Hilbert]Poincare ´ series of the Sally module can be passed back and forth to grI Ž R .. In the case of depth this comes from the above sequence Žsee w16, Proposition 1.2.10 and Corollary 2.1.11x.. For the Hilbert coefficients and Hilbert] Poincare ´ series see w15, Corollary 3.3; 16, Proposition 1.3.3x, respectively. If different from Ž0., S J Ž I . is an Rw Jt x-module of dimension d Žsee w15x., and we write s0 , . . . , s d for the Hilbert coefficients of S J Ž I .; e0 , . . . , e d will denote the Hilbert coefficients of grI Ž R .. We will write HP Ž S, t . and HP ŽgrI Ž R ., t . for the Hilbert]Poincare ´ series of S and grI Ž R ., respectively. We shall now describe our results. The first consequences of the filtration and virtual filtration mentioned above are the following theorem and corollary. THEOREM 1.1. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R and let J be a minimal reduction of I. If s0 is the first Hilbert coefficient of the Sally module S, and r s r J Ž I . is the reduction number of I with respect to J, then ry1
Ý lŽ I nq 1rJI n . .
s0 F
ns1
If equality holds then: Ži. S is Cohen]Macaulay, Žii. for j s 1, . . . , d y 1, the Hilbert coefficients of S are gi¨ en by ry1
sj s
Ý nsj
n l I nq 1rJI n . , j
ž/Ž
Ž 1.
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MARIA VAZ PINTO
and Žiii. the Hilbert]Poincare´ series of S is gi¨ en by HP Ž S, t . s
nq 1 Ý ry1 rJI n . t n ns 1 l Ž I
Ž1 y t .
d
.
Ž 2.
COROLLARY 1.2. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R Ž nq 1rJI n . then: Ži. and let J be a minimal reduction of I. If s0 s Ý ry1 ns1 l I depth ŽgrI Ž R .. G d y 1, Žii. for j s 1, . . . , d, the Hilbert coefficients of grI Ž R . are gi¨ en by ry1
ej s
Ý nsjy1
n l I nq 1rJI n . , jy1
ž /Ž
and Žiii. the Hilbert]Poincare´ series of grI Ž R . is gi¨ en by r
lŽ RrI . q HP Ž grI Ž R . , t . s
Ý
l Ž I nrJI ny1 . y l Ž I nq1rJI n . t n
ns1
Ž1 y t .
d
.
In the same setting as ours, Sam Huckaba proved in w6, Theorem 3.1x Ž nq 1rJI n . if and only if depthŽgrI Ž R .. G d y 1. Since that e1 s Ý ry1 ns0 l I Ž . e1 s s0 q l IrJ w15, Corollary 3.3x, we can restate Huckaba’s result as Ž nq 1rJI n . if and only if depthŽgrI Ž R .. G d y 1. Therefore, s0 s Ý ry1 ns1 l I whenever depthŽgrI Ž R .. G d y 1, Corollary 1.2 gives us a complete description of the Hilbert coefficients and of the Hilbert]Poincare ´ series of grI Ž R .. We observe that the formulas for the Hilbert coefficients were also given by Huckaba in w6, Corollary 2.11x. One would like to understand the structure of the Sally module given the reduction number of I with respect to J and numerical information about some of the components of S. In Sections 3 and 4 several particular cases are studied. Our main result in Section 3 is the following theorem. THEOREM 1.3. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm, let I be an m-primary ideal of R and J a minimal reduction of I. Assume that the reduction number of I with respect to J is two and that I 2rJI has a cyclic socle. Then the Sally module is Cohen]Macaulay. More precisely, S , Ž I 2rJI .w T1 , . . . , Td xwy1x. In particular, depthŽgrI Ž R .. G d y 1. Section 4 is dedicated to cases where the length of I 2rJI is low, namely one, two, or three. One starts with length one and our main result in that case is Theorem 1.4 below, where we present a relation between the first Hilbert coefficient s0 of the Sally module and the reduction number r s r J Ž I . of I with respect to J.
HILBERT FUNCTIONS AND SALLY MODULES
507
THEOREM 1.4. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R and J a minimal reduction of I such that lŽ I 2rJI . s 1. Then s0 F r y 1 and Ži. Žii.
s0 s r y 1 implies S Cohen]Macaulay, s0 s r y 2 implies depthŽ S . s d y 1.
When the length of I 2rJI is two, we divide our study into two subcases. One of them corresponds to the situation where I 2rJI is isomorphic to Ž Rrm . 2 . In this case, if the reduction number of I with respect to J is two, Vasconcelos proved in w15, Proposition 2.6x that depthŽ S . G d y 1, and therefore, depthŽgrI Ž R .. G d y 2. In the second situation, I 2rJI is cyclic and one shows in Remark 4.6 that if the reduction number of I with respect to J is two then the Sally module is Cohen]Macaulay. As in the case of length one, the first Hilbert coefficient s0 of S is related to r J Ž I ., and the Sally module is studied in several extremal situations. Finally, if the length of I 2rJI is three, our problem is divided into four subcases, and the structure of the Sally module is well understood in three of them if the reduction number of I with respect to J is two. Throughout this paper Ž R, m . is a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm, I is an m-primary ideal of R, and J is a minimal reduction of I.
2. FILTERING THE SALLY MODULE In this section we shall present a filtration of the Sally module. We start defining, for n G 1, the Rw Jt x-module Cn s
`
[ I iq1rJ iynq1 I n . isn
Notice, in particular, that C1 s S. We define now L n as the Rw Jt x-submodule of Cn generated by the first term of Cn , Ž Cn . n s I nq 1rJI n : L n s R w Jt x ? I nq 1rJI n s
`
[ J iyn I nq1rJ iynq1 I n . isn
We have the short exact sequence of Rw Jt x-modules 0 ª L n ª Cn ª Cnq1 ª 0, where Cnq 1 s CnrL n . If r s r J Ž I . is the reduction number of I with respect to J, then I rq1 s JI r , Cr s 0, and L ry1 , Cry1. We obtain,
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MARIA VAZ PINTO
therefore, the following set of exact sequences of Rw Jt x-modules 0 0
ª ª
0
ª
L1 L2 .. . L ry2
ª ª ª
S C2 .. . Cry2
ª ª
C2 C3 .. .
ª
L ry1
ª ª
0 0
ª
0.
Ž 3.
We shall refer to the factors L n as the reduction modules of I with respect to J. If these factors have a recognizable structure, then we will be able to conclude good properties for the Sally module. To help us recognize the structure of the L n’s, we shall introduce other Rw Jt x-modules, the Dn’s, that will be called the ¨ irtual reduction modules of I with respect to J. For n G 1, let A n s Ann R Ž I nq 1rJI n . and let Bn s RrA nw T1 , . . . , Td x, a d-dimensional Cohen]Macaulay ring with Ass B nŽ Bn . s m Bn4 . Notice that Bn , Rw Jt xrA n Rw Jt x, and since A n ? Rw Jt x : Ann Rw J t xŽ L n ., L n is not only an Rw Jt x-module, but also a Bn-module. We define now, for n g 1, . . . , r y 14 , Dn s Bn
mR Ž I nq 1rJI n . , Ž I nq1rJI n . w T1 , . . . , Td x ,
a maximal Cohen]Macaulay Bn-module, with Ass B nŽ Dn . s m Bn4 . There is an epimorphism of Bn-modules un : Dnwyn x ª L n , that is the identity on I nq 1rJI n and sends each Ti to a generator of Rw Jt x Žclearly, the generators of I nq 1rJI n have degree zero in Dn , but degree n in L n .. If we define K n as the kernel of un , then for n g 1, . . . , r y 14 we have the following exact sequence of Bn-modules un
L n ª 0.
6
0 ª K n ª Dn w yn x
Ž 4.
This sequence and the filtration presented above are the main tools for our study of the Sally module. An immediate consequence of the construction of the reduction modules and the virtual reduction modules of I with respect to J is Theorem 1.1, whose proof we now present: Proof of Theorem 1.1. For i c 0, one has
lŽ Si . s s0
ž
dy 1 iqdy1 iqdy2 y s1 q ??? q Ž y1 . s dy1 . dy1 dy2
/ ž
/
On the other hand, from Ž3. and Ž4.,
lŽ Si . s
ry1
ry1
ns1
ns1
Ý lŽ Ž Ln . i . s Ý Ž lŽ Ž Dn wyn x . i . y lŽ Ž K n . i . . .
509
HILBERT FUNCTIONS AND SALLY MODULES
The contribution of lŽŽ Dnwyn x. i . to s0 is exactly lŽ I nq 1rJI n . and we conclude that ry1
s0 F
Ý lŽ I nq 1rJI n . . ns1
Equality in the above expression implies that, for n g 1, . . . , r y 14 , we have no contribution from lŽŽ K n . i . to s0 . And since K n is either Ž0. or has dimension d, we conclude that for n g 1, . . . , r y 14 , K n s 0. This implies L n , Ž I nq 1rJI n .w T1 , . . . , Td xwyn x. By Ž3. and the so-called Depth Lemma Žsee w1, Proposition 1.2.9x., S is Cohen]Macaulay. Now, because of the recognizable structure of the L n’s, we obtain their Hilbert polynomial and Hilbert]Poincare ´ series P Ž L n , i . s l Ž I nq 1rJI n . HP Ž L n , t . s
žŽ
i y n. q d y 1 dy1
l Ž I nq 1rJI n . t n
Ž1 y t .
d
/
and
.
Again by Ž3., we only have to add up these expressions, from n s 1 to r y 1, to conclude that the Hilbert coefficients of S and its Hilbert]Poincare ´ series are given by Ž1. and Ž2., respectively. Corollary 1.2 transfers to the associated graded ring of I the results obtained for the Sally module in Theorem 1.1. Its proof follows from w16, Proposition 1.2.10; 15, Corollary 3.3; 16, Proposition 1.3.3x. COROLLARY 2.1. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R and let J be a minimal reduction of I. The following conditions are equi¨ alent: ry1 Ž nq 1 Ži. s0 s S ns1 l I rJI n .; Žii. S is Cohen]Macaulay; Žiii. depthŽgrI Ž R .. G d y 1.
Proof. Ži. implies Žii. follows from Theorem 1.1 and Žii. implies Žiii. from w16, Proposition 1.2.10x. Now, as we have seen in Section 1, w6, Ž nq 1rJI n . if and only if Theorem 3.1x can be restated as s0 s Ý ry1 ns1 l I depthŽgrI Ž R .. G d y 1. This gives us the equivalence of Žiii. and Ži.. Remark 2.2. When we have equality in Theorem 1.1, and so, for n s 1, . . . , r y 1, L n , Ž I nq 1rJI n .w T1 , . . . , Td xwyn x, the canonical module v L n of L n can be explicitly computed. For n s 1, . . . , r y 1, k
v L n , Ž I nq1rJI n . w T1 , . . . , Td xw yd q n x , where Ž I nq 1rJI n . ks Hom R Ž I nq1rJI n, E ., R s RrA n , m s mrA n , and E is the injective envelope of Rrm Žsee w1x for more details..
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MARIA VAZ PINTO
3. REDUCTION NUMBER TWO Let M be a module of finite length over a Noetherian local ring Ž R, m .. We start this section defining the Loewy ascending chain of submodules of M 0 s M0 : M1 : M2 : ??? : Mty1 : Mt s M, where M0 s 0 and Mi s Ž Miy1: M m . for i G 1. Each MirMiy1 s Ž Miy1: M m .rMiy1 s Ž0: M r M iy 1 m ., being the socle of MrMiy1 , is not only an R-module but also an Rrm-vector space. If lŽ MirMiy1 . s n i , then MirMiy1 , Ž Rrm . n i . This construction shows that we can associate to M a partition Ž n1 , . . . , n t . of lŽ M . s n. Partitions of the form Ž1, n 2 , . . . , n t . correspond to cases where the socle M1 of M is cyclic. Let L s Ann R Ž M ., R s RrL, and m s mrL. Using the notation introduced in Section 2, we define M ks Hom R Ž M, E ., where E is the injective envelope of Rrm. LEMMA 3.1. Let Ž R, m . be a Noetherian local ring and let M be an R-module of finite length. Then the minimal number of generators of M k is equal to the minimal number of generators of the socle of M. Proof. This lemma is a special case of Matlis duality and its proof can be found in w1, Proposition 3.2.12x. The module of finite length we are interested in is I 2rJI. In this section we will study the Sally module of I with respect to J, fixing r s r J Ž I . equal to two, and letting lŽ I 2rJI . be any number n. There are 2 ny 1 possible chains of submodules associated to I 2rJI, and in all the cases where its socle is cyclic, the Sally module will have a nice structure. This is exactly the statement of Theorem 1.3. Before its proof we need the following lemma. LEMMA 3.2. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0, let A be an m-primary ideal of R, B s RrAw T1 , . . . , Td x, and v B the canonical module of B. Assume that M is a finitely generated B-module such that Ass B Ž M . : m B4 . Then M : Hom B Ž Hom B Ž M, v B . , v B . . Proof. We will show that the standard map w : M ª Hom B ŽHom B Ž M, v B ., v B . is one]one. The exact sequence of B-modules w
Hom B Ž Hom B Ž M, v B . , v B . ª C ª 0,
6
0ªKªM
HILBERT FUNCTIONS AND SALLY MODULES
511
where K s KerŽ w . and C s CokerŽ w ., tensored with Bm B , yields 0 ª K m B ª Mm B ª Hom B m B Ž Hom B m B Ž Mm B , v B m B . , v B m B . ª Cm B ª 0. Now, since Bm B is Artinian, v B m B is isomorphic to the injective envelope of the residue field of Bm B Žsee w1, Chap. 3x for more details.. At the same time the Artinian ring Bm B is complete, so by Matlis duality w1, Theorem 3.2.13x, Hom B m BŽHom B m BŽ Mm B , v B m B ., v B m B . , Mm B . We conclude that K m B s 0 and Cm B s 0. But Ass B Ž K . : Ass B Ž M . : m B4 , and so K s 0. This shows that w is one]one. Proof of Theorem 1.3. Our notation and definitions are the ones of Section 2. For simplicity we write A for A1 s Ann R Ž I 2rJI . and B for B1 s RrA1w T1 , . . . , Td x in this proof. Since I 3 s JI 2 , one has L2 s Rw Jt x ? I 3rJI 2 s 0, and we conclude from Ž3. that S , L1. In particular, S is a B-module, and since Ass Rw J t xŽ S . s m Rw Jt x4 Žsee w15x., we have Ass B Ž S . s m B4 . From Ž4. one has the exact sequence of B-modules 0 ª K 1 ª Ž I 2rJI . w T1 , . . . , Td xw y1 x ª S ª 0, and our aim is to prove that K 1 s 0. Let R s RrA, m s mrA, and let v R and v B be the canonical modules of R and B, respectively. Since R is Artinian, v R is the injective envelope E of Rrm, and v B , v R w T1 , . . . , Td x , Ew T1 , . . . , Td x Žsee w1, Chap. 3x for more details.. Applying Hom B Ž ], v B . to the exact sequence above yields the long exact sequence 0 ª Hom B Ž S, v B . ª Hom B Ž I 2rJI w T1 , . . . , Td x , v B . ª Hom B Ž K 1 , v B . ª Ext 1B Ž S, v B . ª Ext 1B Ž I 2rJI w T1 , . . . , Td x , v B . ª ??? . Since B is a Cohen]Macaulay local ring of dimension d and I 2rJI w T1 , . . . , Td x is a finitely generated Cohen]Macaulay B-module of dimension d, one has Ext 1B Ž I 2rJI w T1 , . . . , Td x, v B . s 0 Žsee w1, Corollary 3.5.11x.. Let F s Ž I 2rJI . k and observe that Hom B Ž I 2rJI w T1 , . . . , Td x, v B . , F w T1 , . . . , Td x. The fact that the socle of I 2rJI is cyclic implies, by Lemma 3.1, that F is a cyclic R-module, and one can write F , RrAnn R Ž F .. But R is Artinian, and therefore complete. By Matlis duality Žsee w1, Theorem 3.2.13x., Ž F . k, I 2rJI, as R-modules Žand so as R-modules.. Therefore A s Ann R Ž I 2rJI . s Ann R Ž F . and F , RrA s R. We conclude that Hom B ŽŽ I 2rJI .w T1 , . . . , Td x, v B . , Rw T1 , . . . , Td x s B. The above long exact sequence can be rewritten as 0 ª Hom B Ž S, v B . ª B ª Hom B Ž K , v B . ª Ext 1B Ž S, v B . ª 0. Ž 5 .
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MARIA VAZ PINTO
Observe that Hom B Ž S, v B . is just an ideal of B and there are two cases to be analyzed: Ž1. Hom B Ž S, v B . : m B and Ž2. Hom B Ž S, v B . m B. Case Ž1.. Hom B Ž S, v B . : m B. Since Ass B Ž S . s m B4 , we have S : Hom B ŽHom B Ž S, v B ., v B . by Lemma 3.2. Therefore, Ann R Ž m B . : Ann R Ž Hom B Ž S, v B . . : Ann R Ž Hom B Ž Hom B Ž S, v B . , v B . . : Ann R Ž S . : Ann R Ž I 2rJI . s 0, and one concludes that Ann R Ž m B . s 0. But m B g Ass B Ž B . and then it is easy to prove that Ann R Ž m B . / 0. This gives us a contradiction. Case Ž2.. Hom B Ž S, v B . m B. Factoring the exact sequence Ž5., one obtains the short exact sequence of B-modules 0 ª BrHom B Ž S, v B . ª Hom B Ž K 1 , v B . ª Ext 1B Ž S, v B . ª 0, from which we conclude that Supp Ž Hom B Ž K 1 , v B . . : Supp Ž BrHom B Ž S, v B . . j Supp Ž Ext 1B Ž S, v B . . . We know that m B f SuppŽ BrHom B Ž S, v B .. because Hom B Ž S, v B . m B. Now, we have ŽExt 1B Ž S, v B .. m B , Ext 1B m BŽ Sm B , v B m B .. Since Bm B is Artinian, v B m B is the injective envelope of the residue field of Bm B , an injective Bm B-module. Therefore, Ext 1B m BŽ Sm B , v B m B . s 0, and m B f SuppŽExt 1B Ž S, v B ... We conclude that mB f SuppŽHom B Ž K 1 , v B .. and so Ann B ŽHom B Ž K 1 , v B .. m B. Let x g Ann B ŽHom B Ž K 1 , v B .., x f m B. Since K 1 : D 1 wy1x, Ass B Ž K 1 . : m B4 . It follows from Lemma 3.2 that K 1 : Hom B ŽHom B Ž K 1 , v B ., v B .. Now, x ? Hom B Ž K 1 , v B . s 0 implies that x ? Hom B Ž Hom B Ž K 1 , v B . , v B . s 0, and thus xK 1 s 0. On the other hand, x f m B and Ass B Ž D 1 . s m B4 implies that x is regular on D 1 , and therefore, x is regular on K 1. Since xK 1 s 0, one has K 1 s 0, as wanted. Remark 3.3. We have seen in Theorem 1.3 that S , Ž I 2rJI .w T1 , . . . , Td xwy1x, and in this situation the Hilbert polynomial of S is P Ž S, i . s l Ž I 2rJI .
žŽ
i y 1. q d y 1 . dy1
/
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513
In particular, the first Hilbert coefficient of S is exactly s0 s lŽ I 2rJI . s ry1 Ž nq 1 S ny rJI n .. Therefore, when in the hypotheses of Theorem 1.3, 1l I Theorem 1.1 and Corollary 1.2 give us very simply expressions for the Hilbert coefficients and Hilbert]Poincare ´ series of S J Ž I . and grI Ž R ., respectively.
4. LOW LENGTHS In this section we are interested in studying the Sally module of I with respect to J when the lengths of I 2rJI are either 1, 2, or 3. Length One If lŽ I 2rJI . s 1, I 2rJI is a cyclic R-module and is equal to its socle. There is only one possible chain of submodules for I 2rJI, as described in Section 3, namely Ž0. : I 2rJI. We need to observe some simple facts and our notation will be that of Section 2. Since I 2rJI is a cyclic R-module, we must have I nq 1rJI n cyclic for all n G 1. We can write I nq 1rJI n s R ? a n , RrA n , for some a n g I nq 1rJI n, where A n s Ann R Ž I nq 1rJI n .. Since A n : A nq1 for all n G 1, and since A1 s m, we conclude that, for n g 1, . . . , r y 14 , A n s m, I nq 1rJI n , Rrm, and lŽ I nq 1rJI n . s 1. In particular, Bn s Rrm w T1 , . . . , Td x. For simplicity, we will write B s Bn , for n g 1, . . . , r y 14 , in this subsection. Observe also that for n g 1, . . . , r y 14 , L n is a cyclic Rw Jt x-module: L n s R w Jt x ? I nq 1rJI n s R w Jt x ? a n , R w Jt x rAnn Rw J t x Ž L n . w yn x . We know that m Rw Jt x s A1 Rw Jt x : Ann Rw J t xŽ L1 .. On the other hand, Ass Rw J t xŽ L1 . : Ass Rw J t xŽ S . s m Rw Jt x4 , and therefore, Ann Rw J t xŽ L1 . : m Rw Jt x. We conclude that m Rw Jt x s Ann Rw J t xŽ L1 . and L1 , R w Jt x rm R w Jt xw y1 x , Ž Rrm . w T1 , . . . , Td xw y1 x s B w y1 x . For n g 2, . . . , r y 14 , we also have m Rw Jt x s A n Rw Jt x : Ann Rw J t xŽ L n ., but now we cannot say that Ass Rw J t xŽ L n . : m Rw Jt x4 , and so we cannot conclude that L2 , . . . , L ry1 are isomorphic to B. What can be said is that for n g 2, . . . , r y 14 , L n is a cyclic B-module: L n s B ? a n , BrAnn B Ž L n . w yn x . Remark 4.1. If the reduction number of I with respect to J is two and lŽ I 2rJI . s 1, Theorem 1.3 tells us that S , Ž Rrm .w T1 , . . . , Td xwy1x.
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Proof of Theorem 1.4. We only need to consider here r ) 2, because the case r s 2 has been done in greater generality, as observed in Remark 4.1. We have seen that lŽ I nq 1rJI n . s 1 for n g 1, . . . , r y 14 . The fact that s0 F r y 1 is therefore a consequence of Theorem 1.1. When s0 s r y 1, S is Cohen]Macaulay also by Theorem 1.1. One needs to look now at the second case, when s0 s r y 2. The idea is to show that, for n g 1, . . . , r y 24 , L n , Bwyn x an L ry1 , BrŽ f .wyŽ r y 1.x, where f / 0 is a regular element on B. From this it follows that, for n g 1, . . . , r y 24 , the depth of L n is d, and L ry1 has depth G d y 1. Going back to our filtration of the Sally module in Section 2, and using the Depth Lemma, we conclude that depthŽ S . G d y 1. But the depth of S cannot be d when s0 s r y 2, by Corollary 2.1. Therefore, in this case, depthŽ S . s d y 1. Exactly as in the proof of Theorem 1.1, one concludes from Ž3. and Ž4. that, for n g 1, . . . , r y 14 , the contribution of lŽŽ Dnwyn x. i . to s0 is exactly lŽ I nq 1rJI n . s 1. Since s0 s r y 2, we must have r y 2 of the K n’s equal to zero, and one K n different from zero; i.e., we must have r y 2 of the L n’s isomorphic to B and one L n isomorphic to BrAnn B Ž L n ., where Ann B Ž L n . / 0. But since for n G 1, Ann B Ž L n . : Ann B Ž L nq1 ., we conclude that the r y 2 L n’s isomorphic to B have to be exactly the first r y 2. And so, for n g 1, . . . , r y 24 , L n , Bwyn x, and L ry1 , Br Ann B Ž L ry1 .wyŽ r y 1.x, where Ann B Ž L ry1 . / 0. Our aim now is to prove that Ann B Ž L ry1 . is a principal ideal of B, and in order to do this, one shows that it has height one and is unmixed. Since B is a UFD, we will get our result. Let P be a prime ideal of B, heightŽ P . G 2, and one wants P f Ass B Ž L ry1 .. Now, B s Rw Jt xrm Rw Jt x, and so P s prm Rw Jt x, where p is a prime ideal of Rw Jt x. Applying the functor Hom Rw J t xŽ Rw Jt xrp, ] . to the first exact sequence of Ž3. yields the long exact sequence 0 ª Hom Rw J t x Ž R w Jt x rp , L1 . ª Hom Rw J t x Ž R w Jt x rp , S . ª Hom Rw J t x Ž R w Jt x rp , C2 . ª Ext 1Rw J t x Ž R w Jt x rp , L1 . ª ??? . Ass Rw J t xŽ S . s m Rw Jt x4 and since m Rw Jt x « p, there exists in p a regular element on S. Therefore, Hom Rw J t xŽ Rw Jt xrp, S . s 0. Also, Ext 1Rw J t xŽ Rw Jt xr p, L1 . s 0. The reason for this is that since heightŽ P . G 2 and B is Cohen]Macaulay, there exists Ž x, y . : P, a regular sequence on B; therefore, Ž x, y . : p is a regular sequence on B, where x, y are the lifts of x and y to p. Hence, Ext 1Rw J t xŽ Rw Jt xrp, B . s 0. But L1 , B, and we have Ext 1Rw J t xŽ Rw Jt xrp, L1 . s 0. From the above long exact sequence we conclude that Hom Rw J t xŽ Rw Jt xrp, C2 . s 0.
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Applying now successively Hom Rw J t xŽ Rw Jt xrp, ] . to the other sequences of Ž3., and using the fact that L2 , ??? , L ry2 , B, one obtains Hom Rw J t xŽ Rw Jt xrp, L ry1 . s 0. Therefore, p f Ass Rw J t xŽ L ry1 . and so, P f Ass B Ž L ry1 .. We have proved that if P is a prime of B of height G 2 then P f Ass B Ž L ry1 .. But Ž0. f Ass B Ž L ry1 ., since Ann B Ž L ry1 . / 0. Therefore, the only primes of B that belong to Ass B Ž L ry1 . are primes of height 1. This means that Ann B Ž L ry1 . has height 1 in B and is unmixed. Remark 4.2. When lŽ I 2rJI . s 1 and s0 s r y 1, Theorem 1.1 and Corollary 1.2 give us very simple expressions for the Hilbert coefficients and Hilbert]Poincare ´ series of S and grI Ž R ., respectively. It also follows from Corollary 1.2 that depthŽgrI Ž R .. G d y 1. Remark 4.3. When lŽ I 2rJI . s 1 and s0 s r y 2, since depthŽ S . s d y 1, it follows from w16, Proposition 1.2.10x that depthŽgrI Ž R .. G d y 2. But the depth of grI Ž R . cannot be G d y 1 when s0 s r y 2, by Corollary 2.1. Therefore, when lŽ I 2rJI . s 1, if d ) 1 and s0 s r y 2, then depthŽgrI Ž R .. s d y 2. Another observation is that in the case s0 s r y 2 our method does not allow us to compute the Hilbert coefficients or the Hilbert]Poincare ´ series of either the Sally module or the associated graded ring. Indeed, since the degree of the element f is not known, we cannot compute the Hilbert coefficients or the Hilbert]Poincare ´ series of L ry1. COROLLARY 4.4. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm, let I be an m-primary ideal of R and J a minimal reduction of I. Assume that lŽ I 2rJI . s 1 and s0 s r y 1. If v t denotes the degree t component of the canonical module v S of the Sally module, then
vt s
~¡
if t - d y r q 1,
0,
¢Ž Rrm . ž 1 Ý ry js 1
kq dyj dy 1
/,
if t s k q d y r q 1, k G 0.
Proof. Let v Rw J t x , v L n, and vC n denote the canonical modules of Rw Jt x, L n , and Cn , respectively. It follows from Remark 2.2 that
v L n , Rrm w T1 , . . . , Td xw yd q n x
Ž 6.
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MARIA VAZ PINTO
in our situation. Applying now the functor Hom Rw J t xŽ ], v Rw J t x . to each sequence of Ž3. yields the set of exact sequences ª
ª
v L ry 1 .. . vC 3
ª
vC 2
0
ª
0 0
ª
ª
vC ry 2 .. . vC 2
ª
0
ª
v L ry 2 .. . vL2
ª
0
ª
vS
ª
v L1
ª
0.
Ž 7.
Our result follows from Ž6. and Ž7.. Length Two We are now interested in studying the Sally module when lŽ I 2rJI . s 2. According to the structure of ascending chains of submodules described in Section 3, we have two possible cases to analyze. In the first one I 2rJI and its socle are both cyclic and different from each other. In the second case I 2rJI is equal to its socle and isomorphic to Ž Rrm . 2 . For the rest of this subsection we will focus on the first case, starting with reduction number two. Then, as in the case of length 1, we will relate the first Hilbert coefficient of S to the reduction number of I with respect to J, and we will analyze the Sally module in several extremal situations. One shows in the following proposition that the first reduction module L1 of I with respect to J and the first virtual reduction module D 1 of I with respect to J are isomorphic. PROPOSITION 4.5. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R, J a minimal reduction of I, lŽ I 2rJI . s 2, and I 2rJI a cyclic R-module. Then the first reduction module L1 of I with respect to J is Cohen]Macaulay. In fact, L1 , Ž RrA1 .w T1 , . . . , Td xwy1x. Proof. I 2rJI is a cyclic R-module, so one can write I 2rJI s R ? a1 s RrA1 , for some a1 g I 2rJI, A1 s Ann R Ž I 2rJI .. In this case, since lŽ RrA1 . s 2 and lŽ Rrm . s 1, A1 / m. Let R s RrA1 , m s mrA1 , and notice that lŽ m . s lŽ R . y lŽ Rrm . s 1. So m is cyclic, and we can write m s Ž s ., for some s g mrA1 , s / 0. Observe at this point that since Ž m . 2 / m, we must have Ž m . 2 s 0. In the notation of Section 2, let B1 s RrA1w T1 , . . . , Td x. From Ž4. we have the exact sequence of B1-modules u1
L1 ª 0.
6
0 ª K 1 ª B1 w y1 x
Our aim is to show that K 1 s 0, so that L1 , B1wy1x. Let K denote K 1wq1x and we will prove that the ideal K of B1 vanishes.
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One has Ass Rw J t xŽ L1 . : Ass Rw J t xŽ S . s m Rw Jt x4 , and so Ass B1Ž L1 . s m B14 . This tells us that dim B Ž L1 . s d; but also dim B Ž B1 . s d. There1 1 fore, from the exact sequence above one concludes that heightŽ K . s 0. Now, we know that m B1 is the only height zero prime ideal of B1 , and so, K : m B1 s s ? B1. We can write K s sK X , where K X is an ideal of B1. Notice that K X / B1; for if K X s B1 , then K s sB1 , and so, s g K; in fact, s g Ž K . 0 ; but this gives us a contradiction, since u 1 is an isomorphism in degree 1, and so Ž K . 0 s Ž K 1 .1 s 0. Therefore K X / B1. If K X : m B1 , then K s sK X : m 2 B1 s Ž m . 2 B1 s 0, and our proof is complete. Assume then that K X m B1. One has L1 , B1rK , B1rsK X , as B1-modules. Now, as we have seen above, Ass B1Ž L1 . s m B14 ; but we will show that B1rsK X has associated primes in B1 of height G 1. This is a contradiction, and this case K X m B1 is not possible. One has s g B1 , s f K s sK X , so s is nonzero in B1rsK X and K X s s 0. Therefore, there eixsts a prime P of B1 , such that K X : Ann B1Ž s . : Ann B1Ž x . s P, where x is a nonzero element of B1rsK X . So, one has P g Ass B1Ž B1rsK X . and height B1Ž P . G height B1Ž K X .; but since we are in the case where K X m B1 , we must have height B1Ž K X . G 1, and therefore, height B1Ž P . G 1. Remark 4.6. Notice that if the reduction number of I with respect to J is two, lŽ I 2rJI . s 2, and I 2rJI is a cyclic R-module, then S , Ž RrA1 .w T1 , . . . , Td xwy1x. This result follows from Proposition 4.5 together with the fact that S , L1 when r J Ž I . s 2. Our result follows also from Theorem 1.3, because when lŽ I 2rJI . s 2 and I 2rJI is cyclic, the socle of I 2rJI must be cyclic. PROPOSITION 4.7. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R and let J be a minimal reduction of I. If lŽ I 2rJI . s 2 and I 2rJI is a cyclic R-module, then s0 F 2Ž r y 1.. In case of equality, S is Cohen]Macaulay. Proof. Since I 2rJI is a cyclic R-module, we must have I nq 1rJI n cyclic for all n G 1. Therefore, we can write I nq 1rJI n , RrA n , A n s Ann R Ž I nq 1rJI n .. But A n : A nq1 for all n G 1, and by hypotheses, lŽ RrA1 . s lŽ I 2rJI . s 2. We conclude then that lŽ I nq 1rJI n . F 2, for all n G 1. This result together with Theorem 1.1 gives us ry1
s0 F
Ý lŽ I nq 1rJI n . F 2 Ž r y 1. . ns1
Ž nq 1rJI n ., and so S is If s0 s 2Ž r y 1., then, in particular, s0 s Ý ry1 ns1 l I Cohen]Macaulay by Theorem 1.1.
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Remark 4.8. If s0 s 2Ž r y 1. then lŽ I nq 1rJI n . s 2 for n g 1, . . . , r y 14 , and so I nq 1rJI n , RrA1; from the proof of Theorem 1.1, L n , RrA1w T1 , . . . , Td xwyn x for n g 1, . . . , r y 14 . We also observe that when s0 s 2Ž r y 1., we may use the second part of Theorem 1.1 and Corollary 1.2. In particular, the associated graded ring of I has depth G d y 1, and we obtain simple expressions for the Hilbert coefficients and Hilbert] Poincare ´ series of S and grI Ž R .. The last result of Proposition 4.7 can be generalized to other extremal cases. Before doing it, we need some observations. L1 , RrA1w T1 , . . . , Td xwy1x by Proposition 4.5. The question now becomes: what happens to the other reduction modules of I with respect to J ? Do they have a nice structure so that good properties for the Sally module can be concluded? We will look first at L2 . As observed before, since I 2rJI is a cyclic R-module, one has I 3rJI 2 also cyclic, and so I 3rJI 2 , RrA 2 , where A 2 s Ann R Ž I 3rJI 2 .. If B2 s RrA 2 w T1 , . . . , Td x, we have from Ž4. the exact sequence of B2-modules 0 ª K 2 ª B2 w y2 x ª L2 ª 0.
Ž 8.
Since A1 : A 2 : m, lŽ Rrm . s 1, and lŽ RrA1 . s 2, either A 2 s A1 or A 2 s m; in the first case, B2 s RrA1 w T1 , . . . , Td x s B1 , and in the second one, B2 s Rrm w T1 , . . . , Td x; in each of these cases, we can have K 2 s 0 or K 2 / 0. Therefore, there are four possibilities for L2 : L2 , RrA1 w T1 , . . . , Td xw y2 x , L2 , Ž RrA1 w T1 , . . . , Td xw y2 x . rK 2 , K 2 / 0, m Ž RrA1 w T1 , . . . , Td xw y2 x . K 2 , L2 , Rrm w T1 , . . . , Td xw y2 x , L2 , Ž Rrm w T1 , . . . , Td xw y2 x . rK 2 ,
K 2 / 0.
Suppose that L2 , RrA1w T1 , . . . , Td xwy2x. In this case L2 is Cohen] Macaulay. Observe that, for all i G 0, lŽŽ L2 . i . s lŽ RrA1 .Ž Ž i y 2d.yq1d y 1 .. From the filtration of the Sally module in Section 2, one concludes that the contribution of lŽŽ L2 . i . to s0 is 2. Suppose next that L2 , Ž RrA1 w T1 , . . . , Td xwy2x.rK 2 , K 2 / 0. One has Ass B 2Ž K 2 . : Ass B 2Ž B2 . s m B2 4 and since K 2 / 0, Ass B 2Ž K 2 . s m B2 4 . This tells us that dim B 2Ž K 2 . s d, and we can write, for i c 0, lŽŽ K 2 . i . s dy 1. dy 2. k 0 Ž i qd y y k 1Ž i qd y q ??? , where k 0 / 0 Ž k 0 is called the multiplicity 1 2 of K 2 .. It follows from Ž8. and from the filtration of the Sally module presented in Section 2, that the contribution of lŽŽ L2 . i . to s0 is, in this case, 2 y k 0 Žand since k 0 / 0, this number can be either 1 or zero..
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HILBERT FUNCTIONS AND SALLY MODULES
In the third case, L2 , Rrm w T1 , . . . , Td xwy2x, which is Cohen]Macaulay. Now, lŽŽ L2 . i . s lŽ Rrm .Ž Ž i y 2d.yq1d y 1 . for all i G 0 and the contribution of lŽŽ L2 . i . to s0 is 1. Finally, suppose that L2 , Ž Rrm w T1 , . . . , Td xwy2x.rK 2 , K 2 / 0. Since Rrm w T1 , . . . , Td x is a domain, heightŽ K 2 . ) 0, dimŽ L2 . - d, and the contribution of lŽŽ L2 . i . to s0 is zero. In a similar way we could analyze the other L n’s Ž L3 , . . . , L ry1 .. If A ny 1 s A1 , then L n has exactly the same four possibilities that we saw for A 2 . If A ny1 s m, then A n s m Žsince A ny1 : A n : m ., and there are only two possibilities for L n : L n , Rrm w T1 , . . . , Td xwyn x or L n , Ž Rrm w T1 , . . . , Td xwyn x.rK n , K n / 0. We conclude that if the contribution of lŽŽ L n . i . to s0 is 2, then we must have L n , RrA1w T1 , . . . , Td xwyn x; if the contribution of lŽŽ L n . i . to s0 is 1, L n , Rrm w T1 , . . . , Td xwyn x or L n , Ž RrA1w T1 , . . . , Td xwyn x.rK n , K n / 0, and the multiplicity of K n is 1; if the contribution of lŽŽ L n . i . to s0 is zero, then L n , Ž Rrm w T1 , . . . , Td xwyn x.rK n , K n / 0, or L n , Ž RrA1w T1 , . . . , Td xwyn x.rK n , K n / 0, and the multiplicity of K n is 2. An interesting consequence of the considerations just made is that for n g 1, . . . , r y 14 , the contribution of lŽŽ L n . i . to s0 is nonincreasing. We present now two theorems where we get results for the depth of the Sally module in several extremal cases. THEOREM 4.9. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R, J a minimal reduction of I, lŽ I 2rJI . s 2, and I 2rJI a cyclic R-module. Suppose we are in one of the following situations: Contribution of lŽŽ L1 . i . to s0
Contribution of lŽŽ L2 . i . to s0
2 2 .. .
2 2 .. .
??? ???
2 1 .. .
2
1
???
1
???
Contribution of lŽŽ L ry1 . i . to s0
where all the 1’s come from L n , Rrm w T1 , . . . , Td xwyn x. Then the Sally module is Cohen]Macaulay. Proof. In all the cases above, L n , RrA1w T1 , . . . , Td xwyn x or L n , Rr m w T1 , . . . , Td xwyn x, as described before. In any situation, L n is Cohen] Macaulay for all n g 1, . . . , r y 14 , and our result follows from the filtration of S presented in Section 2.
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Remark 4.10. Notice that in each of the cases of Theorem 4.9, we are Ž nq 1rJI n .. Therefore, in the situation of Theorem 1.1, where s0 s Ý ry1 ns1 l I we may use the second part of Theorem 1.1 and Corollary 1.2. In particular, depthŽgrI Ž R .. G d y 1. THEOREM 4.11. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R, J a minimal reduction of I, lŽ I 2rJI . s 2, and I 2rJI a cyclic R-module. Suppose we are in one of the following situations: Contribution of lŽŽ L1 . i . to s0
Contribution of lŽŽ L2 . i . to s0
2 2 .. .
2 2 .. .
2
1
Contribution of lŽŽ L ry2 . i . to s0
Contribution of lŽŽ L ry1 . i . to s0
??? ???
2 1 .. .
0 0 .. .
???
1
0
???
where all the 1’s come from L n , Rrm w T1 , . . . , Td xwyn x, and the zeros on the last column come from L ry1 , Ž Rrm w T1 , . . . , Td xwyŽ r y 1.x.rK ry1 , K ry1 / 0. Then, depthŽ S . G d y 1. Proof. By the structure described before, L1 , . . . , L ry2 all have depth d. An argument similar to the one used in the proof of Theorem 1.4 gives us depthŽ L ry1 . G d y 1. Therefore, depthŽ S . G d y 1. COROLLARY 4.12. y 2.
In the hypotheses of Theorem 4.11, depthŽgrI Ž R .. G d
Length Three We will now study the Sally module when lŽ I 2rJI . s 3. According to the structure of ascending chains of submodules described in Section 3, we have four possible cases to analyze: in the first one, I 2rJI and its socle are both cyclic, and therefore they are different from each other; in the second case, I 2rJI is not cyclic but it has a cyclic socle; in the third case, I 2rJI is a cyclic R-module, but its socle is not cyclic; finally, in the last case, I 2rJI is equal to its socle and isomorphic to Ž Rrm . 3. In the first two cases, the socle of I 2rJI is cyclic, and so, if the reduction number of I with respect to J is two, we conclude from Theorem 1.3 that the Sally module is Cohen]Macaulay. We will look here at the third case and the result we have tells that, if the reduction number of I with respect to J is two, then the Sally module has depth bigger or equal than d y 1. In fact, we will only assume that lŽ I 2rJI . s 3 and that I 2rJI is cyclic, so our argument will also include the first case Žwhere we already have a better result..
HILBERT FUNCTIONS AND SALLY MODULES
521
Using the notation and definitions of Section 2, we have A1 s Ann R Ž I 2rJI . and B1 s RrA1 w T1 , . . . , Td x; from Ž4., one has the following exact sequence of B1-modules 0 ª K 1 ª D 1 w y1 x ª L1 ª 0.
Ž 9.
Before our main theorem, we need the following lemma. LEMMA 4.13. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm, let I be an m-primary ideal of R, and J a minimal reduction of I. Then K 1 satisfies property S2 of Serre. Proof. Property S2 of Serre says that for all p g Supp B1Ž K 1 ., depth Ž Ž K 1 . p . G min 2, dim Ž Ž K 1 . p . 4 . Assume that K 1 / 0; since m B1 is the only height zero prime of B1 and Ass B1Ž K 1 . s m B14 , K 1 satisfying S2 means that for all p, depthŽŽ K 1 . p . G min 2, htŽ p .4 . If htŽ p . s 0, there is nothing to show. Let now p be such that htŽ p . G 1; in this case m B1 « p, so there exists r g p _ m B1. Since L1 : S, Ass B1Ž L1 . s m B14 , and therefore r is regular on L1. One has depthŽŽ L1 . p . G 1. Tensoring Ž9. with Ž B1 . p yields 0 ª Ž K 1 . p ª Ž D 1 . p w y1 x ª Ž L1 . p ª 0. Since D 1 is a Cohen]Macaulay B1-module with Ass B1Ž D 1 . s m B14 , depthŽŽ D 1 . p . s dimŽŽ D 1 . p . s htŽ p .. Now, the Depth Lemma on the last exact sequence tells us that depthŽŽ K 1 . p . G min 1 q 1, htŽ p .4 , and this proves that K 1 has S2 . THEOREM 4.14. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R, J a minimal reduction of I, lŽ I 2rJI . s 3, and I 2rJI a cyclic R-module. If the reduction number of I with respect to J is two, then depthŽ S . G d y 1. Proof. Since the reduction number of I with respect to J is two, one has by Ž3. that S , L1. Since I 2rJI is cyclic, we can write I 2rJI , RrA1 , where A1 s Ann R Ž I 2rJI .; B1 s RrA1w T1 , . . . , Td x, and in our case the exact sequence Ž9. becomes 0 ª K 1 ª B1 w y1 x ª S ª 0.
Ž 10 .
If K 1 s 0, our proof is finished. Let us then assume that K 1 / 0, and since Ass B1Ž K 1 . : Ass B1Ž B1 . s m B1 4 , one has Ass B1Ž K 1 . s m B1 4 and dim B1Ž K 1 . s d. There exist k 0 , k 1 , . . . , k dy1 g Z such that for i c 0,
lŽ Ž K 1 . i . s k 0
ž
dy 1 iqdy1 iqdy2 y k1 q ??? q Ž y1 . k dy1 , dy1 dy2
/
ž
/
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MARIA VAZ PINTO
where k 0 / 0. From Ž10., k 0 q s0 s 3, and so k 0 can be either 1, 2, or 3. If k 0 s 3, then s0 s 0, and, in particular, S s 0 Žsee w15x.. Then lŽ I 2rJI . s 0. But since we are in the case where lŽ I 2rJI . s 3, we conclude that k 0 / 3. If k 0 s 2, then s0 s 1, and it follows from w15, Proposition 3.5x that m S s 0; in particular, m Ž I 2rJI . s 0. But this is a contradiction because I 2rJI is cyclic and lŽ I 2rJI . s 3. One concludes that k 0 / 2. We are left with the only possible case: k 0 s 1. Now, K 1 is not only a B1-module, but also an Rw Jt x-module with Ass Rw J t xŽ K 1 . s m Rw Jt x4 . Therefore m K 1 s 0, as in w15, Proposition 3.5x. This implies that K 1 is also a B-module, where B s Rw Jt xrm Rw Jt x , Rrm w T1 , . . . , Td x, and the structure of K 1 as an Rw Jt x-module, as a B-module, or as a B1-module is the same. Again as in w15, Proposition 3.5x, K 1 , K Žas B-modules., where K is an ideal of B. By Lemma 4.13, K satisfies property S2 of Serre. It is not easy to show that as an ideal of B, K has height one and is unmixed. Since B is a UFD, K has to be principal, i.e., K s B ? f, for some f / 0, f g K. In particular, f is regular on B, and so B ? f , B as B-modules. Therefore, as B-modules, K 1 , B and depthŽ K 1 . s d. We conclude that depthŽ S . G d y 1, using the Depth Lemma on Ž10.. COROLLARY 4.15. Let Ž R, m . be a Cohen]Macaulay local ring of dimension d ) 0 and infinite residue field Rrm. Let I be an m-primary ideal of R, J a minimal reduction of I, lŽ I 2rJI . s 3, and I 2rJI a cyclic R-module. If the reduction number of I with respect to J is two, then depthŽgrI Ž R .. G d y 2.
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