Hilbert Functions and Sally Modules

Transcription

1 JOURNAL OF ALGEBRA 9, ARTCLE NO. JA96680 Hilbert Functions and Sally Modules Maria Vaz Pinto Department of Mathematics, Rutgers Uniersity, New Brunswick, New Jersey Communicated by Melin Hochster Received June 4, 995. NTRODUCTON Let Ž R,. be a Noetherian local ring of Krull dimension d 0 and let be an -primary ideal of R. There has been a great deal of interest on two objects associated to the ideal : Ž. i the HilbertSamuel function of, i.e., the numerical function H Ž n. ŽR n., where stands for length, and Ž ii. the depth of the associated graded ring of R with respect to n n,gr R Žsee 8, 3, 4.. n 0 An approach to their study has consisted in using reductions of, that r r is, ideals J, with the property that J for some integer r Žsee or 9 for this and other basic notions we employ.. This means that the natural inclusion of Rees algebras n n n n RJt J t Rt t n0 turns Rt into a finite RJt-module. f RJt has a more accessible character than Rt, and if one has good control over the module structure of Rt, then one is able to deal with Ž. i and Ž ii. above rather effectively. A setting for obtaining a good RJt is that of a CohenMacaulay local ring Ž R,. with infinite residue field R. n this case, any minimal reduction J of is generated by a regular sequence, so that the algebra * The author was partially supported by a JNCT grant. Current address: Department of Mathematics and Computer Science, Montclair State University, Upper Montclair, New Jersey address: pinto@math.rutgers.edu $5.00 Copyright 997 by Academic Press All rights of reproduction in any form reserved. 504 n0

2 HLBERT FUNCTONS AND SALLY MODULES 505 RJt is very well understood. We define the reduction number r Ž. J of with respect to J as the smallest integer t such that t J t. One way to deal with the second issue, of the module structure of Rt over RJt,is to define modules where the two algebras interact. n this way Vasconcelos introduced in 5 the so-called Sally module S S Ž. J of with respect to J. t is defined by the exact sequence n n 0RJt Rt SJŽ. J 0, and a motivation for its name is the work of Sally in 03 Žsee 5.. Our aim here is to uncover structural aspects of S Ž. J in sufficient detail to deal with Ž. i and Ž ii. in several cases of interest. The techniques we introduce are filterings of S Ž. J by submodules which have Cohen Macaulay ancestors: to a filtration of the Sally module there corresponds another series of modules Ž the virtual filtration.. When we succeed in proving their equality, or near so, there is an abundance of information about S Ž. J, which may be used to estimate its depth and to describe its Hilbert function and some of its arithmetical properties like the canonical module. nformation about depth, Hilbert coefficients, and HilbertPoincaré series of the Sally module can be passed back and forth to gr Ž R.. n the case of depth this comes from the above sequence Žsee 6, Proposition..0 and Corollary.... For the Hilbert coefficients and Hilbert Poincare series see 5, Corollary 3.3; 6, Proposition.3.3, respectively. f different from Ž. 0, S Ž. is an RJt-module of dimension d Žsee 5. J, and we write s,...,s for the Hilbert coefficients of S Ž. 0 d J ; e 0,...,ed will denote the Hilbert coefficients of gr Ž R.. We will write HPŽ S, t. and HPŽgr Ž R., t. for the HilbertPoincare series of S and gr 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.. Let Ž R,. be a CohenMacaulay local ring of dimension d 0 and infinite residue field R. Let be an -primary ideal of R and let J be a minimal reduction of. f s0 is the first Hilbert coefficient of the Sally module S, and r r Ž. is the reduction number of with respect to J, then J n r n n s0 Ý Ž J.. n f equality holds then: Ž. i S is CohenMacaulay, Ž ii. for j,...,d, the Hilbert coefficients of S are gien by r Ý / n n n sj ž J, Ž j. nj

3 506 MARA VAZ PNTO and Ž iii. the HilbertPoincare series of S is gien by Ý r n n J n t n HP Ž S, t.. Ž d. Ž t. COROLLARY.. Let Ž R,. be a CohenMacaulay local ring of dimension d 0 and infinite residue field R. Let be an -primary ideal of R r and let J be a minimal reduction of. f s Ý Ž n J n. then: Ž i. 0 n depth Žgr Ž R.. d, Ž ii. for j,...,d,the Hilbert coefficients of gr Ž R. are gien by r Ý / n n n ej ž J, j nj and Ž iii. the HilbertPoincare series of gr Ž R. is gien by r Ý R J J t n HP Ž grž R., t. d Ž t. n n n n n n the same setting as ours, Sam Huckaba proved in 6, Theorem 3. r that e Ý Ž n J n. if and only if depthžgr Ž R.. n0 d. Since e s Ž J. 5, Corollary 3.3 0, we can restate Huckaba s result as r Ž n s Ý J n. if and only if depthžgr Ž R.. 0 n d. Therefore, whenever depthžgr Ž R.. d, Corollary. gives us a complete description of the Hilbert coefficients and of the HilbertPoincare series of gr Ž R.. We observe that the formulas for the Hilbert coefficients were also given by Huckaba in 6, Corollary.. One would like to understand the structure of the Sally module given the reduction number of with respect to J and numerical information about some of the components of S. n Sections 3 and 4 several particular cases are studied. Our main result in Section 3 is the following theorem. THEOREM.3. Let Ž R,. be a CohenMacaulay local ring of dimension d 0 and infinite residue field R, let be an -primary ideal of R and J a minimal reduction of. Assume that the reduction number of with respect to J is two and that J has a cyclic socle. Then the Sally module is CohenMacaulay. More precisely, S Ž J.T,...,T. d n particular, depthžgr Ž R.. d. Section 4 is dedicated to cases where the length of J is low, namely one, two, or three. One starts with length one and our main result in that case is Theorem.4 below, where we present a relation between the first Hilbert coefficient s0 of the Sally module and the reduction number rr Ž. of with respect to J. J.

4 HLBERT FUNCTONS AND SALLY MODULES 507 THEOREM.4. Let Ž R,. be a CohenMacaulay local ring of dimension d 0 and infinite residue field R. Let be an -primary ideal of R and J a minimal reduction of such that Ž J.. Then s r and Ž. i s0 r implies S CohenMacaulay, Ž ii. s r implies depthž S. d. 0 When the length of J is two, we divide our study into two subcases. One of them corresponds to the situation where J is isomorphic to Ž R.. n this case, if the reduction number of with respect to J is two, Vasconcelos proved in 5, Proposition.6 that depthž S. d, Ž. and therefore, depth gr R d. n the second situation, J is cyclic and one shows in Remark 4.6 that if the reduction number of with respect to J is two then the Sally module is CohenMacaulay. As in the case of length one, the first Hilbert coefficient s of S is related to r Ž. 0 J, and the Sally module is studied in several extremal situations. Finally, if the length of J 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 with respect to J is two. Throughout this paper Ž R,. is a CohenMacaulay local ring of dimension d 0 and infinite residue field R, is an -primary ideal of R, and J is a minimal reduction of. 0. FLTERNG THE SALLY MODULE n this section we shall present a filtration of the Sally module. We start defining, for n, the RJt-module i in n Cn J. in Notice, in particular, that C S. We define now L as the RJt-submod- n n n ule of C generated by the first term of C, C J : n n n n n n in n in n L RJt J J J. n in We have the short exact sequence of RJt-modules 0 L C C 0, n n n where C C L. f rr Ž. n n n J is the reduction number of with respect to J, then r J r, Cr 0, and Lr C r. We obtain,

5 508 MARA VAZ PNTO therefore, the following set of exact sequences of RJt-modules 0 L S C 0 0 L C C Lr Cr Lr 0. Ž 3. We shall refer to the factors Ln as the reduction modules of with respect to J. f 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 s, we shall introduce other RJt-modules, n the D n s, that will be called the irtual reduction modules of with respect to J. For n, let A Ann Ž n J n. and let B RA T,...,T n R n n d,a d-dimensional CohenMacaulay ring with Ass Ž B. B 4 B n n. Notice that n B RJtA RJt, and since A RJt Ann Ž L. n n n RJt n, Ln is not only an RJt-module, but also a Bn-module. We define now, for n,...,r, 4 n n n D B Ž J. Ž J n. T,...,T, n n R d a maximal CohenMacaulay B -module, with Ass Ž D. B 4 n B n n. There n is an epimorphism of B -modules : D n n n n L n, that is the identity on n n J and sends each T to a generator of RJt Ž i clearly, the generators n n of J have degree zero in D, but degree n in L. n n. f we define K n as the kernel of, then for n,...,r4 n we have the following exact sequence of Bn-modules 0K D n n L 0. Ž 4. n n n 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 with respect to J is Theorem., whose proof we now present: Proof of Theorem.. For i 0, one has ž / ž / i d i d d Ž Si. s0 s s d. d d Ž. Ž. On the other hand, from 3 and 4, r r Ž S. Ž Ž L.. Ž Ž D n.. Ž Ž K... Ý i Ý Ž i i. i n n n n n

6 HLBERT FUNCTONS AND SALLY MODULES 509 The contribution of ŽŽ D n.. to s is exactly Ž n J n. n i 0 and we conclude that r n n s0 Ý Ž J.. n Equality in the above expression implies that, for n,...,r,we 4 have no contribution from ŽŽ K.. to s. And since K is either Ž 0. n i 0 n or has dimension d, we conclude that for n,...,r,k 4 n 0. This implies L Ž n J n.t,...,t n. By Ž. n d 3 and the so-called Depth Lemma Žsee, Proposition..9., S is CohenMacaulay. Now, because of the recognizable structure of the L n s, we obtain their Hilbert polynomial and HilbertPoincare series n n Ž i n. d PŽ L n,i. Ž J. and ž d / n J n t n HP Ž L n, t.. d Ž t. Again by Ž. 3, we only have to add up these expressions, from n to r, to conclude that the Hilbert coefficients of S and its HilbertPoincare series are given by Ž. and Ž., respectively. Corollary. transfers to the associated graded ring of the results obtained for the Sally module in Theorem.. ts proof follows from 6, Proposition..0; 5, Corollary 3.3; 6, Proposition.3.3. COROLLARY.. Let R, be a CohenMacaulay local ring of dimension d 0 and infinite residue field R. Let be an -primary ideal of R and let J be a minimal reduction of. The following conditions are equialent: Ž. i r s Ž n J n. 0 n ; Ž ii. S is CohenMacaulay; Ž iii. depthžgr Ž R.. d. Proof. Ž. i implies Ž ii. follows from Theorem. and Ž ii. implies Ž iii. from 6, Proposition..0. Now, as we have seen in Section, 6, r Theorem 3. can be restated as s Ý Ž n J n. 0 n if and only if depthžgr Ž R.. d. This gives us the equivalence of Ž iii. and Ž i.. Remark.. When we have equality in Theorem., and so, for n,...,r, L Ž n J n.t,...,t n n d, the canonical module of L can be explicitly computed. For n,...,r, Ln n n n Ž J. T,...,T dn, Ln d Ž n n. Ž n n. R n n where J Hom J, E, R RA, A, and Eis the injective envelope of R Žsee for more details..

7 50 MARA VAZ PNTO 3. REDUCTON NUMBER TWO Let M be a module of finite length over a Noetherian local ring R,. We start this section defining the Loewy ascending chain of submodules of M 0 M M M M M M, 0 t t where M 0 and M Ž M :. for i. Each M M Ž 0 i i M i i M i: M. M Ž 0:. i M M, being the socle of MM i, is not only an i R-module but also an R-vector space. f Ž M M. i i n i, then MM Ž R. n i i i. This construction shows that we can associate to M a partition Ž n,...,n. of Ž M. n. Partitions of the form Ž, n,...,n. t t correspond to cases where the socle M of M is cyclic. Let L Ann Ž M. R, R RL, and L. Using the notation introduced in Section, we define M Hom Ž M, E. R, where E is the injective envelope of R. LEMMA 3.. Let Ž R,. be a Noetherian local ring and let M be an R-module of finite length. Then the minimal number of generators of M 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, Proposition 3... The module of finite length we are interested in is J. n this section we will study the Sally module of with respect to J, fixing r r Ž. J equal Ž. n to two, and letting J be any number n. There are possible chains of submodules associated to J, 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.3. Before its proof we need the following lemma. LEMMA 3.. Let Ž R,. be a CohenMacaulay local ring of dimension d0, let A be an -primary ideal of R, B RAT,...,T d, and B the canonical module of B. Assume that M is a finitely generated B-module such that Ass Ž M. B 4. Then B M Hom Hom Ž M,.,. B B B B Proof. We will show that the standard map : M Hom ŽHom Ž B B M,.,. is oneone. The exact sequence of B-modules B B 0KM Hom Hom Ž M,., C 0, B B B B

8 HLBERT FUNCTONS AND SALLY MODULES 5 where K Ker and C Coker, tensored with B, yields B Ž. 0 K M Hom Hom M,, C 0. B B B B B B B B B B B B Now, since B B is Artinian, B B is isomorphic to the injective envelope of the residue field of B Žsee, Chap. 3 for more details. B. At the same time the Artinian ring B is complete, so by Matlis duality B, Theorem 3..3, Hom ŽHom Ž M,.,. B B B B B M B. We con- B B B B clude that K 0 and C 0. But Ass Ž K. Ass Ž M. B 4 B B B B, and so K 0. This shows that is oneone. Proof of Theorem.3. Our notation and definitions are the ones of Section. For simplicity we write A for A Ann Ž J. R and B for 3 B RA T,...,T in this proof. Since J, one has L RJt d 3 J 0, and we conclude from Ž. 3 that S L. n particular, S is a B-module, and since Ass Ž S. RJt 4 Žsee 5., we have Ass Ž S. R Jt B B 4. From Ž. 4 one has the exact sequence of B-modules 0K Ž J. T,...,T S0, d and our aim is to prove that K 0. Let R RA, A, and let R and B be the canonical modules of R and B, respectively. Since R is Artinian, R is the injective envelope E of R, and T,...,T ET,...,T Žsee, Chap. 3 B R d d for more details.. Applying Hom Ž,. B B to the exact sequence above yields the long exact sequence 0Hom Ž S,. Hom J T,...,T, Hom Ž K,. B B B d B B B Ext S, Ext J T,...,T,. B B B d B Since B is a CohenMacaulay local ring of dimension d and J T,...,T d is a finitely generated CohenMacaulay B-module of dimension d, one has Ext Ž JT,...,T,. 0 Žsee, Corollary B d B. Let FŽ J. and observe that Hom Ž JT,...,T,. FT,..., B d B T d. The fact that the socle of J is cyclic implies, by Lemma 3., that F is a cyclic R-module, and one can write F RAnn Ž F. R. But R is Artinian, and therefore complete. By Matlis duality Žsee, Theorem. 3..3, F J, as R-modules Ž and so as R-modules.. Therefore Ž AAnn J. Ann Ž F. R R and F RA R. We conclude that ŽŽ Hom J.T,...,T,. RT,...,T B d B d B. The above long exact sequence can be rewritten as 0 Hom S, B Hom K, Ext S, 0. 5 B B B B B B

9 5 MARA VAZ PNTO Observe that Hom Ž S,. B B is just an ideal of B and there are two cases to be analyzed: Ž. Hom Ž S,. B and Ž. Hom Ž S,. B. B B B B Case Ž.. Hom Ž S,. B B B. Since Ass Ž S. B 4, we have S Hom ŽHom Ž S,.,. B B B B B by Lemma 3.. Therefore, Ž. Ann Ž B. Ann Hom Ž S,. Ann Hom Hom Ž S,., R R B B R B B B B Ann R S Ann R J 0, and one concludes that Ann Ž B. 0. But B Ass Ž B. R B and then it is easy to prove that Ann Ž B. 0. This gives us a contradiction. R Case Ž.. Hom Ž S,. B B B. Factoring the exact sequence Ž. 5, one obtains the short exact sequence of B-modules 0 BHom BŽ S, B. Hom BŽ K, B. Ext BŽ S, B. 0, from which we conclude that Ž B B. Ž B B. Ž B B. Supp Hom Ž K,. Supp BHom Ž S,. Supp Ext Ž S,.. We know that B SuppŽBHom Ž S,.. because Hom Ž S,. B B B B Ž. B. Now, we have Ext S, Ext Ž S,. B B B B B B. Since BB is B B Artinian, B is the injective envelope of the residue field of B B,an B injective B -module. Therefore, Ext Ž S,. B B B B 0, and B B B SuppŽExt Ž S,... We conclude that mb SuppŽHom Ž K,.. B B B B and so Ann ŽHom Ž K,.. B B B B. Let x Ann ŽHom Ž K,.., x B. Since K D, Ass Ž K. B B B B B 4. t follows from Lemma 3. that K Hom ŽHom Ž K,.,. B B B B. Now, x Hom Ž K,. 0 implies that B B x Hom Hom Ž K,., 0, B B B B and thus xk 0. On the other hand, x B and Ass Ž D. B4 B implies that x is regular on D, and therefore, x is regular on K. Since xk 0, one has K 0, as wanted. Remark 3.3. We have seen in Theorem.3 that S Ž J.T,..., T, and in this situation the Hilbert polynomial of S is d Ž i. d P S,i J. d ž /

10 HLBERT FUNCTONS AND SALLY MODULES 53 n particular, the first Hilbert coefficient of S is exactly s Ž J. 0 r Ž n J n. n. Therefore, when in the hypotheses of Theorem.3, Theorem. and Corollary. give us very simply expressions for the Hilbert coefficients and HilbertPoincare series of S Ž. and gr Ž R. J, respectively. 4. LOW LENGTHS n this section we are interested in studying the Sally module of with respect to J when the lengths of J are either,, or 3. Length One Ž. f J, J is a cyclic R-module and is equal to its socle. There is only one possible chain of submodules for J, as described in Ž. Section 3, namely 0 J. We need to observe some simple facts and our notation will be that of Section. Since J is a cyclic R-module, we must have n J n cyclic for all n. We can write n J n R a n n n RA, for some a J, where A Ann Ž n J n. n n n R. Since AnAn for all n, and since A, we conclude that, for n 4 n n,...,r, A, J R, and Ž n J n. n. n par- ticular, Bn R T,...,T d. For simplicity, we will write B B n, for n,...,r 4, in this subsection. Observe also that for n,...,r,l 4 is a cyclic RJt-module: n n L RJt J RJt a RJtAnn Ž L. n. n n RJt n We know that RJt ARJt Ann Ž L. R Jt. On the other hand, Ass Ž L. Ass Ž S. RJt 4, and therefore, Ann Ž L. R Jt RJt RJt RJt. We conclude that RJt Ann Ž L. and n R Jt L RJtRJt Ž R. T,...,T B. d For n,...,r 4, we also have RJt A RJtAnn Ž L. n RJt n, but now we cannot say that Ass Ž L. RJt 4 R Jt n, and so we cannot conclude that L,...,Lr are isomorphic to B. What can be said is that for n,...,r,l 4 is a cyclic B-module: n L Ba BAnn Ž L. n. n n B n Remark 4.. f the reduction number of with respect to J is two and Ž J., Theorem.3 tells us that S Ž R.T,...,T. d

11 54 MARA VAZ PNTO Proof of Theorem.4. We only need to consider here r, because the case r has been done in greater generality, as observed in Remark 4.. We have seen that Ž n J n. for n,...,r 4. The fact that s0 r is therefore a consequence of Theorem.. When s0 r, S is CohenMacaulay also by Theorem.. One needs to look now at the second case, when s r. The idea is to show that, for n 0,...,r, 4 L Bn an L BŽ f.ž r. n r, where f 0 is a regular element on B. From this it follows that, for n,...,r 4, the depth of Ln is d, and Lr has depth d. Going back to our filtration of the Sally module in Section, and using the Depth Lemma, we conclude that depthž S. d. But the depth of S cannot be d when s0 r, by Corollary.. Therefore, in this case, depthž S. d. Exactly as in the proof of Theorem., one concludes from Ž. 3 and Ž. 4 that, for n,...,r 4, the contribution of ŽŽ D n.. n i to s0 is exactly Ž n J n.. Since s0 r, we must have r of the K n s equal to zero, and one K n different from zero; i.e., we must have r of the L s isomorphic to B and one L isomorphic to BAnn Ž L. n n B n, where Ann Ž L. 0. But since for n, Ann Ž L. Ann Ž L. B n B n B n, we conclude that the r L n s isomorphic to B have to be exactly the first r. And so, for n,...,r, 4 L Bn, n and Lr B Ann Ž L.Ž r., where Ann Ž L. B r B r 0. Our aim now is to prove that Ann Ž L. B r 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 be a prime ideal of B, heightž., and one wants Ass Ž L.. Now, B RJtRJt, and so RJt, B r where is a prime ideal of RJt. Applying the functor Hom ŽRJt,. R Jt to the first exact sequence of Ž. 3 yields the long exact sequence 0Hom Ž RJt, L. Hom Ž RJt,S. RJt RJt Hom Ž RJt,C. Ext Ž RJt, L.. RJt RJt Ass Ž S. RJt 4 and since RJt R Jt, there exists in a regular Ž. element on S. Therefore, Hom RJt,S 0. Also, Ext ŽRJt RJt RJt, L. 0. The reason for this is that since heightž. and B is CohenMacaulay, there exists Ž x, y., a regular sequence on B; therefore, Ž x, y. is a regular sequence on B, where x, y are the lifts of x and y to. Hence, Ext Ž RJt,B. R Jt 0. But L B, and we have Ext ŽRJt,L. R Jt 0. From the above long exact sequence we conclude that Hom R Jt Ž RJt,C. 0.

12 HLBERT FUNCTONS AND SALLY MODULES 55 Applying now successively Hom Ž RJt,. R Jt to the other sequences of Ž. 3, and using the fact that L Lr B, one obtains Hom Ž RJt,L. 0. Therefore, Ass Ž L. R Jt r RJt r and so, Ass Ž L. B r. We have proved that if is a prime of B of height then Ass Ž L.. But Ž 0. Ass Ž L., since Ann Ž L. B r B r B r 0. Therefore, the only primes of B that belong to Ass Ž L. B r are primes of height. This means that Ann Ž L. has height in B and is unmixed. B r Remark 4.. When Ž J. and s0 r, Theorem. and Corollary. give us very simple expressions for the Hilbert coefficients and HilbertPoincare series of S and gr Ž R., respectively. t also follows from Corollary. that depthžgr Ž R.. d. Remark 4.3. When Ž J. and s r, since depthž S. 0 d, it follows from 6, Proposition..0 that depthžgr Ž R.. d. But the depth of gr Ž R. cannot be d when s0 r, by Corollary.. Therefore, when Ž J., if d and s0 r, then depthžgr Ž R.. d. Another observation is that in the case s0 r our method does not allow us to compute the Hilbert coefficients or the HilbertPoincare series of either the Sally module or the associated graded ring. ndeed, since the degree of the element f is not known, we cannot compute the Hilbert coefficients or the HilbertPoincare series of L r. COROLLARY 4.4. Let Ž R,. be a CohenMacaulay local ring of dimension d 0 and infinite residue field R, let be an -primary ideal of R and J a minimal reduction of. Assume that Ž J. and s0 r. f t denotes the degree t component of the canonical module S of the Sally module, then 0, if t d r, t r kdj Ý Ž R. jž d /, if t k d r, k 0. Proof. Let,, and denote the canonical modules of RJt, RJt Ln Cn L n, and C n, respectively. t follows from Remark. that RT,...,T dn Ž 6. Ln d

13 56 MARA VAZ PNTO in our situation. Applying now the functor Hom Ž,. R Jt RJt to each sequence of Ž. 3 yields the set of exact sequences 0 L C L 0 r r r... 0 C 3 C L 0 0 C S L 0. Ž 7. Ž. Ž. Our result follows from 6 and 7. Length Two Ž. We are now interested in studying the Sally module when J. According to the structure of ascending chains of submodules described in Section 3, we have two possible cases to analyze. n the first one J and its socle are both cyclic and different from each other. n the second case J is equal to its socle and isomorphic to R. 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, we will relate the first Hilbert coefficient of S to the reduction number of 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 L of with respect to J and the first virtual reduction module D of with respect to J are isomorphic. PROPOSTON 4.5. Let Ž R,. be a CohenMacaulay local ring of dimension d 0 and infinite residue field R. Let be an -primary ideal of R, Ž. J a minimal reduction of, J, and J a cyclic R-module. Then the first reduction module L of with respect to J is CohenMacaulay. n fact, L Ž RA.T,...,T. d Proof. J is a cyclic R-module, so one can write J R a RA, for some a J, A Ann Ž J. R. n this case, since Ž RA. and Ž R., A. Let R RA, A, and notice that Ž. Ž R. Ž R.. So is cyclic, and we can write Ž s., for some s A, s 0. Observe at this point that since, we must have 0. n the notation of Section, let B RA T,...,T. From Ž. d 4 we have the exact sequence of B-modules 0K B L 0. Our aim is to show that K 0, so that L B. Let K denote K and we will prove that the ideal K of B vanishes.

14 HLBERT FUNCTONS AND SALLY MODULES 57 One has Ass Ž L. Ass Ž S. RJt 4, and so Ass Ž L. R Jt RJt B B 4. This tells us that dim Ž L. d; but also dim Ž B. B B d. There- fore, from the exact sequence above one concludes that heightž K. 0. Now, we know that B is the only height zero prime ideal of B, and so, KB sb. We can write K sk, where K is an ideal of B. Notice that K B ; for if K B, then K sb, and so, s K; in fact, s Ž K. ; but this 0 gives us a contradiction, since is an isomorphism in degree, and so Ž K. Ž K Therefore K B. f K B, then K sk B B 0, and our proof is complete. Assume then that K B. One has L B K B sk,as B -modules. Now, as we have seen above, Ass Ž L. B 4; but we will B show that B sk has associated primes in B of height. This is a contradiction, and this case K B is not possible. One has s B, sksk,so sis nonzero in BsK and Ks0. Therefore, there eixsts a prime of B, such that K Ann B Ž s. Ann B Ž x., where x is a nonzero element of B sk. So, one has Ass ŽB sk. B and Ž height height K. B B ; but since we are in the case where K B, we must have height Ž K., and therefore, height Ž.. B B Remark 4.6. Notice that if the reduction number of with respect to J Ž. is two, J, and J is a cyclic R-module, then S Ž RA.T,...,T d. This result follows from Proposition 4.5 together with the fact that S L when r Ž. J. Our result follows also from Ž. Theorem.3, because when J and J is cyclic, the socle of J must be cyclic. PROPOSTON 4.7. Let Ž R,. be a CohenMacaulay local ring of dimension d 0 and infinite residue field R. Let be an -primary ideal of R Ž. and let J be a minimal reduction of. f J and J is a cyclic R-module, then s Ž r.n. case of equality, S is CohenMacaulay. 0 Proof. Since J is a cyclic R-module, we must have n J n cyclic for all n. Therefore, we can write n J n RA n, An Ann Ž n J n. R. But An An for all n, and by hypotheses, Ž RA. Ž J.. We conclude then that Ž n J n., for all n. This result together with Theorem. gives us r n n s0 Ý Ž J. Ž r.. n r f s r, then, in particular, s Ý Ž n J n. 0 0 n, and so S is CohenMacaulay by Theorem..

15 58 MARA VAZ PNTO Remark 4.8. f s Ž r. then Ž n J n. for n 0,...,r 4 n n, and so J RA ; from the proof of Theorem., Ln RA T,...,T n for n,...,r 4 d. We also observe that when s Ž r. 0, we may use the second part of Theorem. and Corollary.. n particular, the associated graded ring of has depth d, and we obtain simple expressions for the Hilbert coefficients and Hilbert Poincare series of S and gr Ž R.. The last result of Proposition 4.7 can be generalized to other extremal cases. Before doing it, we need some observations. L RA T,...,T d by Proposition 4.5. The question now becomes: what happens to the other reduction modules of 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 L. As observed before, since J is a cyclic R-module, one has 3 J also cyclic, and so 3 J RA, where AAnn Ž 3 J..f B RA T,...,T, we have from Ž. 4 the exact R d sequence of B -modules 0K B L 0. Ž 8. Since A A, Ž R., and Ž RA., either A A or A ; in the first case, B RA T,...,T d B, and in the second one, B RT,...,T d ; in each of these cases, we can have K 0or K 0. Therefore, there are four possibilities for L : L RA T,...,T, d L Ž RA T,...,T. K, d L RT,...,T, d K 0, Ž RA T,...,T. K, d L Ž RT,...,Td. K, K0. Suppose that L RA T,...,T d. n this case L is Cohen Ž. Ž Ž i. d Macaulay. Observe that, for all i 0, L RA. i. d From the filtration of the Sally module in Section, one concludes that the contribution of ŽŽ L.. i to s0 is. Suppose next that L Ž RA T,...,T. d K, K0. One has Ass Ž K. Ass Ž B. B 4 and since K 0, Ass Ž K. B 4 B B B. This tells us that dim Ž K. d, and we can write, for i 0, ŽŽ K.. B i Ž i d. Ž i d k k., where k 0 Ž 0 d d 0 k0 is called the multiplicity of K.. t follows from Ž 8. and from the filtration of the Sally module presented in Section, that the contribution of ŽŽ L.. i to s0 is, in this case, k Ž and since k 0, this number can be either or zero.. 0 0

16 HLBERT FUNCTONS AND SALLY MODULES 59 n the third case, L RT,...,T d, which is CohenMacaulay. Ž. Ž Ž i. d Now, L R. i for all i 0 and the contribution of d ŽŽ L.. i to s0 is. Finally, suppose that L Ž RT,...,T. d K, K0. Since RT,...,T is a domain, heightž K. 0, dimž L. d d, and the contribution of ŽŽ L.. i to s0 is zero. n a similar way we could analyze the other L s Ž L,...,L. n 3 r. f AnA, then Ln has exactly the same four possibilities that we saw for A.f A, then A Ž since A A. n n n n, and there are only two possibilities for L : L RT,...,T n n n d or Ln ŽRT,...,T n. d K n, Kn0. We conclude that if the contribution of ŽŽ L.. n i to s0 is, then we must have L RA T,...,T n ; if the contribution of ŽŽ L.. n d n i to s0 is, L RT,...,T n or L ŽRA T,...,T n. n d n d K n, Kn0, and the multiplicity of K is ; if the contribution of ŽŽ L.. n n i to s0 is zero, then L ŽRT,...,T n. K, K 0, or L ŽRA n d n n n T,..., T n. d K n, K n 0, and the multiplicity of K n is. An interesting consequence of the considerations just made is that for n,...,r 4, the contribution of ŽŽ 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,. be a CohenMacaulay local ring of dimension d 0 and infinite residue field R. Let be an -primary ideal of R, Ja Ž. minimal reduction of, J, and J a cyclic R-module. Suppose we are in one of the following situations: Contribution of Contribution of Contribution of ŽŽ L.. to s ŽŽ L.. to s ŽŽ L.. to s i 0 i 0 r i 0... where all the s come from L RT,...,T n n d. Then the Sally module is CohenMacaulay. Proof. n all the cases above, L RA T,...,T n n d or Ln R T,...,T n d, as described before. n any situation, Ln is Cohen Macaulay for all n,...,r 4, and our result follows from the filtration of S presented in Section.

17 50 MARA VAZ PNTO Remark 4.0. Notice that in each of the cases of Theorem 4.9, we are r in the situation of Theorem., where s Ý Ž n J n. 0 n. Therefore, we may use the second part of Theorem. and Corollary.. n particular, depthžgr Ž R.. d. THEOREM 4.. Let Ž R,. be a CohenMacaulay local ring of dimension d 0 and infinite residue field R. Let be an -primary ideal of R, Ž. J a minimal reduction of, J, and J a cyclic R-module. Suppose we are in one of the following situations: Contribution of Contribution of Contribution of Contribution of ŽŽ L.. to s ŽŽ L.. to s ŽŽ L.. to s ŽŽ L.. to s i 0 i 0 r i 0 r i where all the s come from L RT,...,T n n d, and the zeros on the last column come from L Ž RT,...,T Ž r.. r d K r, Kr 0. Then, depthž S. d. Proof. By the structure described before, L,...,Lr all have depth d. An argument similar to the one used in the proof of Theorem.4 gives us depthž L. d. Therefore, depthž S. d. r COROLLARY 4.. n the hypotheses of Theorem 4., depthžgr Ž R.. d. Length Three We will now study the Sally module when Ž J. 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, J and its socle are both cyclic, and therefore they are different from each other; in the second case, J is not cyclic but it has a cyclic socle; in the third case, J is a cyclic R-module, but its socle is not cyclic; finally, in the last case, J is equal to its socle and isomorphic to Ž R. 3. n the first two cases, the socle of J is cyclic, and so, if the reduction number of with respect to J is two, we conclude from Theorem.3 that the Sally module is CohenMacaulay. We will look here at the third case and the result we have tells that, if the reduction number of with respect to J is two, then the Sally module has depth bigger or equal than d. n fact, we will only assume that Ž. J 3 and that J is cyclic, so our argument will also include the first case Ž where we already have a better result..

18 HLBERT FUNCTONS AND SALLY MODULES 5 Using the notation and definitions of Section, we have A Ann Ž J. and B RA T,...,T ; from Ž. R d 4, one has the following exact sequence of B-modules 0K D L 0. Ž 9. Before our main theorem, we need the following lemma. LEMMA 4.3. Let R, be a CohenMacaulay local ring of dimension d 0 and infinite residue field R, let be an -primary ideal of R, and J a minimal reduction of. Then K satisfies property S of Serre. Proof. Property S of Serre says that for all Supp Ž K., B Ž. Ž. 4 depth Ž K. min, dim Ž K.. Assume that K 0; since B is the only height zero prime of B and Ass Ž K. B 4, K satisfying S means that for all, depthžž K.. B min, htž.4.fht 0, there is nothing to show. Let now be such that htž. ; in this case B, so there exists r B. Since L S, Ass Ž L. B 4 B, and therefore r is regular on L. One has depthžž L... Tensoring Ž 9. with Ž B. yields 0 Ž K. Ž D. Ž L. 0. Since D is a CohenMacaulay B -module with Ass Ž D. B 4 B, depthžž D.. dimžž D.. htž.. Now, the Depth Lemma on the last exact sequence tells us that depthžž K.. min, htž.4, and this proves that K has S. THEOREM 4.4. Let Ž R,. be a CohenMacaulay local ring of dimension d 0 and infinite residue field R. Let be an -primary ideal of R, Ž. J a minimal reduction of, J 3, and J a cyclic R-module. f the reduction number of with respect to J is two, then depthž S. d. Proof. Since the reduction number of with respect to J is two, one Ž. has by 3 that S L. Since J is cyclic, we can write J RA, where A Ann Ž J.; B RA T,...,T R d, and in our case the exact sequence Ž. 9 becomes 0K B S0. Ž 0. f K 0, our proof is finished. Let us then assume that K 0, and since Ass Ž K. Ass Ž B. B 4, one has Ass Ž K. B 4 B B B and dim Ž K. d. There exist k, k,...,k such that for i 0, B 0 d id id d Ž Ž K.. k0 k k d, d d i ž / ž /

19 5 MARA VAZ PNTO where k 0. From Ž 0. 0, k0 s0 3, and so k0 can be either,, or 3. f k 3, then s 0, and, in particular, S 0 Žsee 5.. Then Ž J But since we are in the case where Ž J. 3, we conclude that k0 3. f k, then s, and it follows from 5, Proposition that S0; in particular, Ž J. 0. But this is a contradiction because J is cyclic and Ž J. 3. One concludes that k0. We are left with the only possible case: k0. Now, K is not only a B -module, but also an RJt-module with Ass Ž K. RJt 4 RJt. Therefore K 0, as in 5, Proposition 3.5. This implies that K is also a B-module, where B RJtRJt RT,...,T d, and the structure of K as an RJt-module, as a B-module, or as a B-module is the same. Again as in 5, Proposition 3.5, K K Ž as B-modules., where K is an ideal of B. By Lemma 4.3, K satisfies property S of Serre. t 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 B f, for some f 0, f K. n particular, f is regular on B, and so B f B as B-modules. Therefore, as B-modules, K B and depthž K. d. We conclude that depthž S. d, using the Depth Lemma on Ž 0.. COROLLARY 4.5. Let Ž R,. be a CohenMacaulay local ring of dimension d 0 and infinite residue field R. Let be an -primary ideal of R, Ž. J a minimal reduction of, J 3, and J a cyclic R-module. f the reduction number of with respect to J is two, then depthžgr Ž R.. d. REFERENCES. W. Bruns and J. Herzog, CohenMacaulay Rings, Cambridge Univ. Press, Cambridge, UK, 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 7 Ž 99., A. Guerrieri, On the depth of the associated graded ring of an m-primary ideal of a CohenMacaulay local ring, J. Algebra 67 Ž 994., 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. 40 Ž 989., H. Matsumura, Commutative Ring Theory, Cambridge Univ. Press, Cambridge, UK, 986.

20 HLBERT FUNCTONS AND SALLY MODULES J. D. Sally, On the associated graded ring of a local CohenMacaulay ring, J. Math. Kyoto Uni. 7 Ž 977., 9.. J. D. Sally, CohenMacaulay local rings of maximal embedding dimension, J. Algebra 56 Ž 979., J. D. Sally, Tangent cones at Gorenstein singularities, Compositio Math. 40 Ž 980., J. D. Sally, Hilbert coefficients and reduction number, J. Algebraic Geom. Ž 99., J. D. Sally, deals whose Hilbert function and Hilbert polynomial agree at n, J. Algebra 57 Ž 993., W. V. Vasconcelos, Hilbert functions, analytic spread, and Koszul homology, Contemp. Math. 59 Ž 994., M. Vaz Pinto, Structure of Sally Modules and Hilbert Functions, Ph.D. dissertation, Rutgers University, 995.