Buchsbaumness in Rees Modules Associated to Ideals of Minimal Multiplicity in the Equi-I-Invariant Case

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1 Journal of Algebra 25, (2002) doi:0.006/jabr Buchsbaumness in Rees Modules Associated to Ideals of Minimal Multiplicity in the Equi-I-Invariant Case Kikumichi Yamagishi Faculty of Econoinformatics, Himeji Dokkyo University, Kamiono 7-2-, Himeji, Hyogo , Japan Communicated by Craig Huneke Received December 8, 2000 The present article gives certain conditions for Rees modules to obtain Buchsbaumness. Suppose a given -primary ideal is of minimal multiplicity in the equi- I-invariant case. Then it is shown that the positively graded submodule of the Rees module must be Buchsbaum and moreover that the Rees module itself is also Buchsbaum if the dimension of a given module is greater than one Elsevier Science (USA) Key Words: Buchsbaumness; Rees module; minimal multiplicity; I-invariant.. INTRODUCTION Let A be a Noetherian local ring and let E be a finitely generated A-module. We denote by R E (resp. by G E the Rees module of E associated to (resp. the associated graded module of E withrespect to ); namely R E = n E and G E = n E/ n+ E n 0 n 0 In the case where E = A we denote it simply by R (resp. G. We denote by the unique homogeneous maximal ideal of R ; i.e., = R +R +. The Rees modules (and also the associated graded modules) play very important roles not only in algebraic geometry but also in commutative algebra, particularly in the theory of Buchsbaum modules. Many authors studied in this field, see, e.g., [Br, G G4, GS, GY, Sc2, St, SV, T] and also [G5, N, SY, Y3] for recent topics. However, we are here interested in their /02 $ Elsevier Science (USA) All rights reserved.

2 24 kikumichi yamagishi ring theoretical behaviours, especially whether they obtain the Buchsbaumness. Recall that E is said to be a Buchsbaum A-module, if the difference l A E/ E e E is an invariant of E, which is so-called the Buchsbaum invariant, not depending on the choice of a parameter ideal of E, where l A and e denote the length and the multiplicity with respect to of an A-module, respectively. The local ring is called a Buchsbaum ring if it is a Buchsbaum module over itself. To avoid complicated terminologies, we simply say that R E (resp. G E is a Buchsbaum R -module, if its localization R E (resp. G E at is so over R. (Readers are referred to the book [SV] for unexplained terminologies on the theory of Buchsbaum modules.) Then our problem is stated as follows. Problem. Let E be a Buchsbaum A-module and let be an ideal of A suchthat l A E/ E <. Then when does the Rees module R E obtain the Buchsbaumness? Necessary and sufficient conditions for Rees modules (associated to -primary ideals) to be Buchsbaum in general have not yet been determined. We should mention that the situations concerning this problem are quite different between the two cases where the dimension of a given module E is equal to one and the others. For instance, in the one-dimensional case (i.e., dim A E = ), we can determine suchconditions completely; see [Y4]. Therefore this article shall be devoted to clarifying the remaining cases. In order to state our result, we shall recall a few more notations. Let H i denote the ithlocal cohomology functor withrespect to. Then we define an invariant of E, written I E and called the I-invariant of E, as s ( ) s I E = h i E i i=0 where we write h i E =l A H i E and s = dim A E. (Notice that there is a canonical isomorphism H i R G E = H i G E, and hence we simply write I G E to have the same meaning as I G E,ifno confusion can be expected.) Any parameter ideal of E is called a minimal reduction of withrespect to E if and r+ E = r E for some integer r 0; cf. [NR]. We denote by ρ E the supremum of the minimal number of generators of E over A among all minimal reductions of withrespect to E; namely ρ E =sup µ A E is a minimal reduction of withrespect to E. Then we say that possesses minimal multiplicity with respect to E if the equality e E =µ A E +l A E/ E ρ E I E holds; see Definition 2.5.

3 buchsbaumness in rees modules 25 Now we are ready to state main results in this article as follows. Theorem.. Let E be a Buchsbaum A-module of dimension s>0 and let be an ideal of A such that l A E/ E <. Suppose that the given ideal fulfills the following two conditions: (i) the equality I G E = I E holds; and (ii) possesses minimal multiplicity with respect to E. Then the positively graded submodule R E + = n>0 n E is a Buchsbaum R module. Moreover, the Rees module R E itself is a Buchsbaum R -module if s 2. Applying Theorem. we immediately get a generalization of Stückrad s theorem [St, Theorem 3]. Namely Corollary.2 (Generalized Stückrad s Theorem). Let E be a Buchsbaum A-module. Then the positively graded part R E + is a Buchsbaum R -module for every parameter ideal of E. For a parameter ideal = a a 2 a d of A (where d = dim A, we denote by the ideal of A suchthat = d i= a â i a d a i +, where the hat ˆ on a i means omit this element a i from the system a a d. (If A is Buchsbaum, then this definition does not depend on the choice of a generating system a a 2 a d for a given parameter ideal.) With this notation we have the following too. Corollary.3 (Nakamura). Let A be a Buchsbaum ring of dimension d>0 and let be an -primary ideal of A. Suppose that for some minimal reduction of the following three conditions are fulfilled: (i) 2 = holds; (ii) holds; and (iii) possesses minimal multiplicity. Then the Rees algebra R is a Buchsbaum ring if 2 d 4. We shall recall several historical remarks concerning our problem. Concerning the Buchsbaumness of associated graded rings/modules after the works given by Goto [G2 G4] the developments in this problem occurred very slowly. Recently, however, we have remarkable progress as follows. In 998, Nakamura [N] generalized Goto s results given in [G2, G3]. In fact, he gave us several equivalent conditions for the associated graded rings of -primary ideals in Buchsbaum rings to be Buchsbaum in the case where the reduction numbers of such -primary ideals are at most one. Following Nakamura s studies, in 2000, the author [Y3] proved a similar argument without any hypotheses about the reduction numbers of -primary ideals. As is well known we have an inequality I G E I E ; see [T], also [GY]. Inspired by the fact that most of the associated graded rings/modules, which are realized to be Buchsbaum, satisfy the equality I G E = I E

4 26 kikumichi yamagishi (see [G, G2, N]), we realize a greater necessity to investigate the case where the equality I G E = I E holds; here call it the equi-i-invariant case. In this case, the author showed the following. If E is a Buchsbaum A-module and if = a a 2 a s is a minimal reduction of withrespect to E suchthat r+ E = r E for some integer r 0, then the following three statements are equivalent: (i) G E is a Buchsbaum module such that h p G E = h p E for all 0 p<s; (ii) I G E = I E holds; and (iii) a 2 a2 2 a2 s E n E = a 2 a2 2 a2 s n 2 E for 3 n d + r; see [Y3, Theorem.]. Compared with the progress on associated graded modules, we have seen quite a few results concerning the problem when Rees modules obtain the Buchsbaumness. In 985, Stückrad [St, Theorem 3] (here we call it Stückrad s theorem) showed that the Rees module R E is Buchsbaum over R for any parameter ideal of E, ife is so over A. After Stückrad s work, not much progress concerning this problem was made for a while. But, recently, in 999 Goto [G5] brought us an epoch-making work. He introduced a new notion called an -primary ideal possessing minimal multiplicity in Cohen Macaulay local rings and studied the Buchsbaumness of the Rees algebras (and the associated graded rings and fiber cones also) associated to suchideals. This notion can be naturally extended for Buchsbaum rings; see Section 2 for details. With this extended notion, Nakamura (in January 997) succeeded in generalizing slightly Stückrad s theorem into the case where the dimensions of given Buchsbaum rings are very small; see Corollary.3. In general it is still very difficult to find an answer to our problem: for instance, even if A is a Buchsbaum ring of maximal embedding dimension, it is not known whether the Rees algebra R of maximal ideal is Buchsbaum in general; see Theorem 6. for our considerations. To close this Introduction, we explain the contents of this article. We shall prove Theorem. by following the same procedure established by Stückrad in [St, Theorem 3]. We also mention that the essence of such a procedure already could be found in [GS]. More precisely, we shall divide its proof into these three steps: Step : show the quasi-buchsbaumness of R E + ; Step 2: reduce to the case where depth E>0; Step 3: show the Buchsbaumness of R E + /a tr E +. In Section 2, we shall introduce several improvements (see Lemmas 2.3 and 2.4) to calculations of the Koszul cohomology modules by using double complexes, which were developed in the proof of what is called Stückrad s

5 buchsbaumness in rees modules 27 lemma [St, Lemma 9]. Moreover, the notion of ideals possessing minimal multiplicity shall be generalized for Buchsbaum modules too. After preparing basic facts on the local cohomology of the Rees modules (see Theorem 3.3), we shall show Step in Section 3; (see Theorem 3.7). Section 4 will be devoted to establishing Step 2; see Proposition 4.5. From a technical point of view, the main difficulty in the proof of Theorem. will appear in Step 3 and it will be discussed in Section 5; see Theorem 5.3. After establishing Steps 3, we shall prove Theorem. and corollaries at the end of this section. Finally, we shall discuss several applications and examples of Theorem. in the last section. Throughout this article, let A be a Noetherian local ring, let E be a finitely generated A-module of dimension s>0, and let be an ideal of A suchthat l A E/ E <. For simplicity, we always assume that the residue field A/ of A is infinite. 2. PRELIMINARIES As described above, we shall divide our procedure of proving Theorem. into three steps, Steps 3, which is similar to the proof of Stückrad s theorem [St]. It is routine to obtain the Buchsbaumness of R E + itself from these three steps; see Section 5 for the details. In particular, to show Step 3 (which is the hardest of these three steps) we shall apply Lemma 2. below, under the assumption that s 3 and depth E>0. From this point of view, this lemma explains that the key idea in our procedure consisted of three steps, Steps 3. Namely, Lemma 2.. Let 0 E E E 0 be a short exact sequence of finitely generated A-modules. Suppose that the following two conditions are fulfilled: (a) dim E = dim E = s = dim E ; and (b) the long exact sequence of local cohomology modules is separated into the short exact sequences 0 H p E H p E H p E 0 for 0 p<s. Then the following statements are equivalent. () E is a Buchsbaum A-module. (2) E E are Buchsbaum A-modules and the long exact sequence of Koszul cohomology modules yields the short exact sequence for each 0 p<s. 0 H p E H p E H p E 0

6 28 kikumichi yamagishi Proof. Let v A denote the embedding dimension of A, i.e., v A = µ A. According to [K, Lemma 2.2], we already know the inequality l A H p E p q=0 ( ) v A h q E p q for each0 p s. In fact, by passing the completion and by applying Cohen s structure theorem for complete local rings we may assume that A is a regular local ring; hence our inequality comes immediately from [K]. We also know that the equality as in the above holds for all 0 p<sif and only if E is Buchsbaum over A by [Y2, Theorem 2.]. () (2) Assume that E is Buchsbaum. Notice that our hypothesis (b) means that h p E =h p E +h p E for each0 p<s.bythe remark described above we have l A H p E = p q=0 ( v A p q ) h q E +h q E l A H p E + l A H p E for each0 p<s. The converse of this inequality is obvious. Consequently we get the equality l A H p E = l A H p E + l A H p E and two more equalities l A H p F = p q=0 ( ) v A h q F p q for 0 p<s, where we set F = E and E. Therefore, the sequence of Koszul cohomology modules 0 H p E H p E H p E 0 must be exact, where 0 p<s, and bothmodules E E are Buchsbaum. The converse (2) () is also shown in a routine as the above. Remark 2.2. Unfortunately, we have to find an unexpected difficulty, whenever we try to obtain the Buchsbaumness of the Rees module R E by following the analogy of Stückrad s procedure; namely Step : Step 2 : Step 3 : a = a, show the quasi-buchsbaumness of R E ; reduce to the case where depth E>0; show the Buchsbaumness of R E /at R E, where we put

7 buchsbaumness in rees modules 29 because we may miss the condition (b) of Lemma 2. in general. Assume that E is a Buchsbaum A-module and that the two conditions (i) and (ii) as in Theorem. are satisfied. According to [SY] (see also [KY, Appendix]), we have two typical cases concerning the appearance of homogeneous components of local cohomology modules H p G E as follows: Case I: Case II: H p G E n = 0 for n p 0 p<s ; H p G E n = 0 for n p 0 p<s ; see Lemma 3. in the next section. It is easy to see that, in Case I, the same procedure as that in Stückrad s theorem as Steps 3 still works well; hence we can obtain the Buchsbaumness of the Rees module R E directly. In Case II, however this procedure does not necessarily work in general. More precisely, in this case, we have the short exact sequence 0 U ae U ae /ae t R E /at R E R E/U ae 0 of graded R -modules, where we assume that s 3 and depth E>0 and we put U ae =ae. Then applying the local cohomology functors, we have the following short exact sequences 0 H R E /at R E H R E/U ae H 2 U ae U ae /ae t 0 Since among all of the three local cohomology modules non-zero homogeneous components are concentrated at only the degree 0, this short exact sequence naturally coincides with 0 H E H E H2 E H2 E 0 regarding as graded R -modules via the canonical projection R A. Consequently, the connecting homomorphism H R E/U ae H 2 U ae U ae /ae t is not a zero map if h2 E 0. By these observations, in Case II, we sometimes miss the condition (b) of Lemma 2., and this is exactly the reason we shall consider R E + instead of the Rees module R E itself; see Lemma 5.2. Next, we shall discuss some improvements in Lemmas 2.3 and 2.4 concerning [St, Lemma 9] (called here Stückrad s lemma), which also play very important roles in our arguments, especially in the proof of Theorem 5.3. Let K d d 2 denote a (commutative) double complex of A-modules; namely,eachk p q isana-moduleandd p q K p q K p+ q andd p q 2 K p q K p q+ are A-linear maps suchthat d p+ q d p q p q+ = 0 d2 d p q 2 = 0,

8 220 kikumichi yamagishi 2 for p q 0. Considering only the dif- ) we can define complexes in the usual sense,, as and d p+ q 2 d p q p q+ = d d p q ferentiation d (resp. d denoted by K q d q 2 (resp. Kp d p 2 K q 0 K 0 q d K n q d K n+ q K p 0 K p 0 d 2 K p n d 2 K p n+ for each p q. Moreover we define the cohomology module H withrespect to the differentiation d (resp. d 2 )as H p q = Ker d p q /Im d p q and H p q 2 = Ker d p q 2 /Im d (resp. H 2 ) p q 2 for each p q. Thus H q and H p 2 coincide with the complexes of cohomology modules of K q and K p, respectively. The associated total complex of K d d 2 denoted by Tot K Tot K is defined as Tot K n = p+q=n Kp q and n Tot K Tot K n Tot K n+ suchthat n Tot K = d p q + p d p q 2 p+q=n for each n 0. We denote by H Tot K the complex of its cohomology modules. Then we have the following. Lemma denote a double complex of A-modules described as in the above. Let a be an element in A, which annihilates all cohomology modules H H 2, and H Tot K, and let α be an element of Kp q, where p q 0, such that d p q α =0 and d p q 2 α =0. Then there exists an element β K p q such that Let K d d a p+q+ p q α = d2 d p q β Proof. This lemma follows by the same way as that in the first half of the proof of Stückrad s lemma [St, Lemma 9]. But we shall here prove it quickly. If p = 0 (resp. q = 0), we may consider α H 0 q (resp. α H p 0 2 ); hence we see aα = 0 by our assumption. So we can choose β = 0inK p q in this case. Next, let p q > 0. By our assumptions on α and a, we can find elements γ K p q and δ K p q so that aα = d p q p q γ =d2 δ.by induction we easily find elements γ j K p j q+j j p and δ k K p+k q k k q suchthat and p j q+j ad2 γ j =d p j q+j γ j+ ad p+k q k p+k q k δ k =d2 δ k+

9 buchsbaumness in rees modules 22 where we put γ p+ = 0 and δ q+ = 0. We define b j = p + p p j for j p and also c = 0 c k = p p p + k for 2 k q. Consider the element ε = ε 0 ε p ε p ε p+q Tot K p+q defined by ε p j = b j a p+q j γ j for j p and ε p+k = c k a p+q k δ k for k q. Then it is routine to check that p+q Tot K ε = 0. Since ah Tot K = 0 there exists an element ζ Tot K p+q 2 suchthat aε = p+q 2 Tot K ζ. Write ζ = ζ 0 ζ ζ p+q 2, where ζ m K m p+q 2 m for each0 m p + q 2. In particular, we have a p+q δ = d p q ζ p + p p q 2 d2 ζ p Now put β = ζ p. Consequently, we have a p+q+ α = a p+q p q p q d2 δ =d2 d p q β and this finishes the proof of Lemma 2.3. We shall apply Lemma 2.3 to a double complex defined by the Koszul (co-)complexes as follows. Let be proper ideals of A suchthat and l A E/ E <. Choose two systems of elements in A, say b = b b 2 b r and c = c c 2 c t, suchthat (i) b b 2 b r is a minimal system of generators of ; (ii) c c 2 c t is a minimal system of generators of ; (iii) any s-elements of b b r c c t form a system of parameters of E. Then we denote by K E d d 2 the double complex of A-modules such that each K p q E is given by K p q E =K p b b 2 b r K q c c 2 c t E and whose differentiations d p q K p q E K p+ q E and d p q 2 K p q E K p q+ E are induced from the differentiations of Koszul (co-)complexes K p b K p+ b and K q c E K q+ c E, respectively. Namely, the complex K q E coincides withthe Koszul complex K b over the A-module K q c E and the complex K p E coincides withthe direct sum of ( r p) -copies of the Koszul complex K c E. Recall a few more notations. We denote by i j the set of integers n such that i n j. Of course we set i j = if i>j. Moreover the notation I means the number of elements of a set I. IfI is a set of integers and i is an integer, we denote by I i the number of elements i in I suchthat i <i; i.e., I i = i I i <i.

10 222 kikumichi yamagishi Let fj I K E as I r J t be a free basis of K E and represent K E = E fj I Kp q E = E fj I I J I =p J =q Let α be an element of K p q E and assume that α = d p q γ for some γ K p q p q E (resp. α = d2 δ for some δ K p q E. Write α, γ, and δ as α = α I J f J I I J γ = γj K f J K and δ = δ I L f L I K J I L where I K r, J L t suchthat I =p, J =q, K =p, L =q, and α I J γk J δi L E. Then we have α I J = ( I i b i γ I\ i J resp. α I J = ) i I j J J j c j δ I J\ j for each I J. For a short exact sequence of A-modules, say 0 E E E 0, there naturally exists an exact sequence of double complexes 0 K E K E K E 0 Moreover, as is well known, there exists an isomorphism of complexes between the Koszul (co-)complex over E generated by the system b b r c c t and the associated total complex Tot K E of K E ; i.e., K b b r c c t E = Tot K E Now we can state our improvement of the latter half of Stückard s lemma [St, Lemma 9]. In his lemma he needed the assumption that depth E>0, but here we shall try to omit it in our arguments. Lemma 2.4. Let E be a Buchsbaum A-module of dimension s 2 and let K E d d 2 be the double complex introduced as above. Let α be an element of K p q E, where p q > 0 and p + q s, and express α as α = I J α I J f I J where α I q J E I =p and J =q. Suppose that dp α =0 and d p q 2 α = 0. Then the following statements are true. () α I J b i i I E + H 0 E for each I J. (2) If α I J b i i I E for all I J with I =p and J =q, then there exists an element δ K p q p q E such that α = d2 δ.

11 buchsbaumness in rees modules 223 Proof. Though this follows by an argument similar to that in the latter half of the proof of Stückrad s lemma, we here discuss it more precisely. At the beginning we choose one more element a so that any s-elements of a b b r c c t form a system of parameters for E. () Since a by our choice, this element a annihilates all cohomology modules H E H 2 E, and H Tot K E. By Lemma 2.3, there exists an element β K p q E suchthat a p+q+ p q α = d2 d p q β. Write β as β = K L β K L f L K, where βk L E K =p, and L =q. By our definitions of differentiations d p q p q and d2 it is easy to see that a p+q+ α I J = ±b i c j β I\ i J\ j I J E i I j J for each I J with I =p and J =q, where we put I = b i i I and J = c j j J. Recall that p q > 0 and p + q s. According to [St, Lemma 4], we finally get α I J I J E a p+q+ I E + H 0 E for each I J. (2) Let K c E denote the Koszul (co-)complex over E generated by the system c = c c 2 c t.let e J J t be a free basis of K c E and express K c E as K c E = E e J K q c E = E e J J J =q For each I r with I =p, we define the element α I K q c E as α I = α I J e J J =q Since d p q 2 α =0 by our assumption, we have d q α I =0 for each I via the differentiation d q K q c E K q+ c E. Now assume that α I J IE for all I J with I =p and J =q, where we still put I = b i i I. By this assumption, we may regard α I as an element in K q c I E. According to [St, Lemma 8], the canonical map H q c I E H q c E is a zero map, because p + q s and I =p>0. Thus there exists δ I K q c E such that α I = d q δ I. Write δ I as δ I = L =q δ I L e L, where δ I L E and L t, and consider the element δ K p q E given by δ = δ I L f L I I =p L =q Then it is routine to check that α = d p q 2 δ. This completes the proof of Lemma 2.4.

12 224 kikumichi yamagishi To close this section we explain the generalized notion on ideals, called possessing minimal multiplicity, for Buchsbaum modules. Recall that, in 999 Goto [G5] introduced a notion called an -primary ideal possessing minimal multiplicity in Cohen Macaulay local rings. That is, an -primary ideal of a Cohen Macaulay local ring A is said to possess minimal multiplicity if the equality e A =µ A +l A A/ dim A holds. This notion can be naturally extended for Buchsbaum rings. Namely, for an -primary ideal in a Buchsbaum ring A, we say that possesses minimal multiplicity if the equality e A =µ A +l A A/ dim A I A holds. Notice that the ring-hand side makes a lower bound of the multiplicity of in general. Since we have I A =0 in the case where A is Cohen Macaulay, this is naturally realized as a generalization of the notion in Cohen Macaulay rings above. Moreover, it is easy to see that this is equivalent to saying that and hence = holds for some (hence every) minimal reduction of. Using this notion Nakamura (in January 997) succeeded in generalizing slightly the Stückrad s theorem stated in Corollary.3. To state our result, however, we need to extend once more the notion of ideals possessing minimal multiplicity from ring cases into (Buchsbaum) module cases. Let E be a Buchsbaum A-module (of dimension s>0). We denote by ρ E the supremum of the minimal number of generators of E, where runs all minimal reduction of withrespect to E; namely, ρ E =sup µ A E is a minimal reduction of withrespect to E Then, choosing as a minimal reduction of withrespect to E, wehave e E =e E =l A E/ E I E = µ A E +l A E/ E µ A E I E +l A E/ E µ A E +l A E/ E ρ E I E because l A E/ E 0 and µ A E ρ E by our definition. Hence we can introduce the following new notion. Definition 2.5. Let E be a Buchsbaum A-module. Then we say that possesses minimal multiplicity with respect to E if the equality holds, e E =µ A E +l A E/ E ρ E I E

13 buchsbaumness in rees modules 225 In the case where E = A it holds that ρ A =dim A; hence this definition is a natural generalization of the notion for Buchsbaum rings. For this reason, we simply say that possesses minimal multiplicity, when E = A. Since the difference between the multiplicity e E and the lower bound described above is just equal to l A E/ E + ρ E µ A E, this new notion is naturally characterized as follows. Proposition 2.6. Let E be a Buchsbaum A-module. Then possesses minimal multiplicity with respect to E if and only if E = E and µ A E =ρ E for some (hence every) minimal reduction of with respect to E. Let K E denote the Koszul complex over E generated by a minimal system of generators of in the usual sense. Moreover, Z E B E, and H E denote the cycle, boundary, and homology of the Koszul complex, respectively. Since B E is contained in K E and hence in K E, we have the exact sequence H E K E / K E E/ E 0 According to [Su, Sc], the Buchsbaumness of E implies that the length of the first Koszul homology module H E becomes a constant value; thus we conclude that s ( ) s s µ A E h i E ρ E s µ i + A E i=0 Let us recall one more useful notation. For a parameter ideal = a a 2 a s of E, we define the submodule of E, say a a s E, as a a s E = s a â i a s E a i + E i= where the hatˆon a i means omit this element a i from the system a a s. Since E is Buchsbaum, this definition of the submodule a a s E is determined by a given parameter ideal ; in particular it does not depend on the choice of a generating system a a 2 a s for. So we usually denote it simply by E. In the case where E = A, we usually omit the letter A from our original notation as instead of A ; see Corollary.3. Then we have the following. Lemma 2.7. Let E be a Buchsbaum A-module. Then () ρ E =s µ A E holds if and only if E E for some minimal reduction of with respect to E.

14 226 kikumichi yamagishi (2) ρ E = s µ A E ( s s i=0 i+) h i E holds if and only if B E =Z E K E for any minimal reduction of with respect to E. Proof. By the exact sequence above, it is easy to see that ρ E = s µ A E holds if and only if the canonical epimorphism K E / K E E/ E is an isomorphism for some minimal reduction of withrespect to E. According to [KY, Theorem 6 in Appendix], this is equivalent to saying that E E holds. The second assertion is also shown in the same way. Example 2.8. following. Let E be a Buchsbaum A-module. Then we have the () Assume that E is free over A (hence notice that A must be Buchsbaum in this case). Then it is easy to see that E E for any parameter ideal of E. By Lemma 2.7 we consequently have ρ E =s µ A E Therefore, possesses minimal multiplicity if and only if it does so with respect to E. (2) Assume that E is also a Buchsbaum A-module. (For example, in the case where E is Cohen Macaulay or s = this assumption is automatically satisfied.) Then choose a parameter ideal = a a 2 a s of E. By putting E = E/H 0 E, since the module E is also Buchsbaum, this system a a 2 a s forms a u.s.d-sequence on E.By[KY,Theorem 6 in Appendix] we get that µ A E =s µ A E. Combining this observation and the fact that µ A E =µ A E we conclude that ρ E =s µ A E =µ A E +I E I E Therefore possesses minimal multiplicity withrespect to E if and only if the equality holds, e E =µ A E +l A E/ E µ A E I E (3) Assume that E is a linear maximal Buchsbaum A-module; i.e., the equality µ A E =e E +I E holds, which is introduced by Yoshida in [Yo]. We know that this is equivalent to saying that E = E for any minimal reduction of withrespect to E. Hence µ A E =µ A E for any such, and this implies that ρ E =µ A E by our definition of ρ E. Moreover, notice that K E = K E holds; hence according to the relative regular sequences of the system of parameters for a Buchsbaum module E we know that Z E K E =B E ;

15 buchsbaumness in rees modules 227 see [SV, Section in Chap. II] and also [GY, Corollary (.5)], and thus by Lemma 2.7 again we conclude that s ( ) s ρ E =µ A E =s µ A E h i E i + Thus we get e E =µ A E I E =µ A E +l A E/ E ρ E I E hence possesses minimal multiplicity withrespect to E. (4) Finally, assume that A is a Buchsbaum ring of maximal embedding dimension; i.e., v A =e A +dim A + I A holds [G2]. Since ρ A =dim A holds (see () above), we have i=0 e A =µ A +l A A/ ρ A I A thus possesses minimal multiplicity in this case. 3. QUASI-BUCHSBAUMNESS OF R E + Throughout the rest of this article, we usually use the following notation: R = n 0 n, the Rees algebra of ; R E = n 0 n E, the Rees module of E associated to ; G E = n 0 n E/ n+ E, the associated graded module of E with respect to ; R E + = n>0 n E, the positively graded R-submodule of R E ; = R + R +, the unique homogeneous maximal ideal of R. We say that E is a quasi-buchsbaum A-module [Su2] if all the local cohomology modules H E p p s of E are annihilated by the maximal ideal ; i.e., H E p = 0 holds for all p s. The local ring is called a quasi-buchsbaum ring if it is a quasi-buchsbaum module over itself. We also say that the graded R-modules R E R E +, etc., are quasi-buchsbaum if the localizations at are so over R. We refer to the paper [Su2] for unexplained terminology on the quasi-buchsbaumness. We usually regard the Rees algebra R and the Rees module R E as the graded A-subalgebra of A t, which is the polynomial ring over A withan indeterminate t and the graded submodule of E t = E A A t, respectively; namely R = n t n A t and R E = n E t n E t n 0 n 0

16 228 kikumichi yamagishi Let us recall several basic facts from the theory of graded modules; see [GW] for unexplained terminologies on the graded rings/modules. Let W = n W n be a graded module (over R). We sometimes denote by W n the homogeneous component of degree n instead of W n.letr be an integer. We denote by W r the graded R-submodule of W defined by W r = n r W n. In particular we usually denote this by W + instead of W. Moreover, we denote by W r the shifted module of W of degree r, which is a graded R-module withthe graduation defined by W r n = W n+r for all n. For an A-module F we denote by F the graded R-module which coincides with F itself as the underlying module and whose graduation is given by F 0 = F and F n = 0 for n 0. From here to the end of this section, we always assume that E is a Buchsbaum A-module (of dimension s>0) and that the following two conditions are satisfied: (i) the equality I G E = I E holds; and (ii) 2 E = E holds for some minimal reduction of withrespect to E. We mention that the condition (ii) described above is weaker than the condition (ii) in Theorem.; see Lemma 4. below for more details. For further discussion, we recall one more notation. We denote by a G E the a-invariant of G E (see [GW]); i.e., a G E = max n H s G E n 0. Then we naturally claim the following too. Lemma 3.. The following statements are true. () G E is a Buchsbaum R-module such that H p G E n = 0 n p p for 0 p<sand a G E s. (2) For any minimal reduction of with respect to E, say = b b 2 b s, the equalities 2 E = E and b n i i i I E n E = b n i i n n i E i I hold, where I s n i > 0, and n. Proof. According to [Y3, Theorem.] the hypothesis (i) just means that G E is a Buchsbaum module over G and hence over R too. Moreover, by [Y3, Corollary 2.7 and Proposition 3.3] and [SY, Proposition 3.4], the hypothesis (ii) immediately leads to the rest of assertions in Lemma 3.. First, we calculate the 0th local cohomology modules of R E and R E +. Lemma 3.2. The following statements are true.

17 buchsbaumness in rees modules 229 () H 0 R E n = n E H 0 E holds for all n 0. Hence H 0 R E = 0 holds. (2) H 0 R E + =H 0 R E + holds. Hence, H 0 R E + = 0 holds too. (3) There are two short exact sequences of graded R-modules 0 H 0 R E R E R E/H0 E 0 0 H 0 R E + R E + R E/H 0 E + 0 Proof. Since H 0 R E = H0 R E as graded A-modules via the graded canonical inclusion A R, we have the assertion () at once. Other assertions follow in routinely. Theorem 3.3. The following statements are true. () H H 0 R E 0 E n = 0 n = E H 0 E n = 0 else. (2) { H H R E n = G E 0 n = 0 0 else. (3) H 2 R E = 0 if s 2. (4) If 3 p s, then H H p R E p E n 3 p n = H p G E p n = 2 p 0 else. (5) { H 0 E H R E 0 + n = E n = 0 else. (6) H R E + = 0 holds. (7) If s 2, then { H 2 H R E + n = G E n = 0 0 else. (8) If 3 p s, then H H p R E p E n 3 p 0 + n = H p G E p n = 2 p 0 else.

18 230 kikumichi yamagishi Proof. By Lemma 3. we already know that G E is a Buchsbaum R- module suchthat H p G E n = 0 n p p for 0 p<sand a G E s hold. Thus, the Rees module R E and hence R E + too have finite local cohomology; i.e., the local cohomology modules H p R E and Hp R E + are finite lengthfor all 0 p s; see [T, Proposition 6.; GY, Theorem 7.8; KY, Remark 2.3] for more related topics. If we look at the following two short exact sequences of graded R-modules 0 R E + R E E 0 (3.) 0 R E + R E G E 0 (3.2) our Theorem 3.3 follows at once routinely. As consequences of Theorem 3.3 we have Corollary 3.4. The following statements are true. () H R E = 0 holds if s 2. (2) R E + is a Buchsbaum R-module if s =. (3) H 2 R E + = 0 holds if s 2. Hence R E + is a quasi- Buchsbaum R-module if s = 2. Corollary 3.5. modules Let 3 p s. Then the sequence of local cohomology 0 H p E H p R E + H p R E 0 induced from (3.) is exact. We here collect several short exact sequences on R E +, which are very useful for calculating the local cohomology modules of it. We write = a a 2 a s. Then we have the following. Lemma 3.6. Suppose that depth E>0 and put a = a. Then there are the short exact sequences of graded R-modules, a 0 R E + R E + R E + /ar E G E R E + /ar E + R E/aE R E + R E at + R E + /at R E E R E + /at R E + R E/aE + 0

19 buchsbaumness in rees modules 23 Proof. By (2) of Lemma 3. we get that ae n E = a n E for all n, and hence we obtain these short exact sequences in a routine. Now we are ready to state our main result in this section. Theorem 3.7. R E + is a quasi-buchsbaum R-module; i.e., H p R E + = 0 for all 0 p s. Moreover, the Rees module R E is also a quasi-buchsbaum R-module if s 2. Proof. Before starting our arguments, we mention that there is no explicit statement on the quasi-buchsbaumness of the Rees modules even in the case where is a parameter ideal of E (see [St]) and that the basic idea of the proof described here is essentially the same as the arguments established in [GY, Chap. 4]. Notice that, by () of Lemma 3.2, (3) of Theorem 3.3, () of Corollary 3.4, and Corollary 3.5, it is easy to see that the quasi-buchsbaumness of the Rees module R E follows from the same property of R E +. Hence we prove sufficiently that R E + is a quasi-buchsbaum R-module. Use induction on s (the dimension of E). For s 2 our assertion is obvious by Corollary 3.4. So we may assume that s 3 and that our assertion is true for s. By Corollary 3.4 again, we may deal withonly the case where 3 p s. Recall that both E and G E are Buchsbaum and that H p G E n = 0 n p p for 0 p<sand a G E s; see () of Lemma 3.. By Theorem 3.3, these facts imply that H p R E + = 0 (3.3) for all 3 p s. In order to lead the quasi-buchsbaumness of R E +,it is enough to show that R H p R E + = 0 (3.4) for 3 p s. By (4) of Lemma 3.2 we may further assume that depth E>0. Look at the short exact sequence a 0 R E + R E + R E + /ar E + 0 introduced in Lemma 3.6. Applying the local cohomology functor we have the exact sequence H p ξ R E +/ar E + H p R E + H p R E + (3.5) Since the element a annihilates H p R E + by the observation (3.3) above, we know that the map ξ becomes a graded R-epimorphism. Then we claim the following. Claim p s. R H p R E +/ar E + n = 0 for all n 3 p, where a

20 232 kikumichi yamagishi Proof of Claim 3 8 Let b and let α H p R E +/ar E + n = 0 for n 3 p. From the short exact sequence given in Lemma G E R E + /ar E + R E/aE + 0 we have the exact sequence of local cohomology modules H p G E ε H p R E δ +/ar E + H p R E/aE + Since R E/aE + is quasi-buchsbaum by the hypothesis of induction on s, we see that δ bt α =bt δ α =0 Thus there exists an element β H p G E n+ suchthat bt α = ε β. On the other hand, we already know that H p G E m = 0 for m p 2 p; hence H p G E n+ = H p G E n = 0 for all n 3 p. This means that β = 0 consequently. Thus we get bt α = 0 and Claim 3.8 is now established. By Claim 3.8 and the graded R-epimorphism ξ in (3.5), we see that R H p R E + n = 0 for all n 3 p So the rest of our assertions are to show that R H p R E + 2 p = 0 cf. (8) of Theorem 3.3. To do it, we consider the short exact sequence (3.2). Then we have the exact sequence of graded R-modules H p G E φ H p R E + H p R E Let b again and let γ H p R E + 2 p. Then we may regard γ H p R E + p. Since H p R E p = 0 by (4) of Theorem 3.3, there exists a homogeneous element ζ H p G E p suchthat γ = φ ζ. Then it is clear that bt ζ = 0inH p G E by the Buchsbaumness of G E, and hence bt γ = φ bt ζ =0 Therefore this completes the proof of Theorem 3.7.

21 buchsbaumness in rees modules 233 Remark 3.9. In the case where s =, to get the quasi-buchsbaumness of R E we need an additional assumption. In fact, in this case we have that H R E 0 = H G E 0 = E/aE + E H 0 E and hence H R E is annihilated by if and only if E ae + E H 0 E holds. To close this section we shall state one more fact, which will be needed to show Theorem 5.3 later. Recall that we put = a a 2 a s. Then choose elements a s+ a s+2 a u (where u = µ A ) of suchthat a = a a 2 a u forms an E-basis of consisting of minimal reductions; that is, it is a (minimal) system of generators of and its s-elements form a minimal reduction of withrespect to E. In fact, we can choose such elements by applying a method similar to that in Lemma 4.2. So, by (2) of Lemma 3. the equality 2 E = a i i I E holds for any I u with I =s. Then we have the following. Proposition 3.0. Let at = a t a 2 t a u t and let a = a. Then () H p at R E + n = 0 for all n p and p 0. (2) H p at R E + /at R E + n = 0 for all n p and p 0. Proof. Put E = E/H 0 E and put R E + = R E + /at R E +.At first, looking at the short exact sequence as in (4) of Lemma H 0 R E + R E + R E + 0 and tensoring R/at R to it, we have the short exact sequence of graded R-modules 0 H 0 R E + R E + R E + 0 Applying the Koszul cohomology functors to these short exact sequences, we have the exact sequences and H p at H 0 R E + H p at R E + H p at R E + H p at H 0 R E + H p at R E + H p at R E + Since H 0 R E + m = 0 for m by (5) of Theorem 3.3, we know that H p at H 0 R E + n = 0 for all n p and p 0. Combining these observations we may assume that depth E>0.

22 234 kikumichi yamagishi Assume that s =. Using induction on r, where r u, we shall show two formulas and H p a t a 2 t a r t R E + n = 0 (3.6) H p a t a 2 t a r t R E + n = 0 (3.7) for all n p and p 0. After showing these formulas (3.6) and (3.7), the assertions () and (2) are obviously true for s =. In fact, since 2 E = a E we know that R E + = E t = E are graded R-modules, and hence we have H 0 a t R E + H 0 R E + = 0 H a t R E + =R E + and H 0 a t R E + =R E + H a t R E + =R E + Thus, by (5) of Theorem 3.3, both formulas (3.6) and (3.7) hold for r =. So we may assume that r 2 and the formulas (3.6) and (3.7) are true for r. Put at = a t a 2 t a r t. Consider the exact sequences of graded Koszul cohomology modules H p at R E + H p a t a 2 t a r t R E + H p at R E + and H p at R E + H p a t a 2 t a r t R E + H p at R E + for each p 0. Applying the hypothesis of induction on r, our formulas (3.6) and (3.7) follow immediately from these two exact sequences. Next let us assume that s 2 and that the assertions () and (2) are true for s. Look at the short exact sequence given in Lemma E R E + R E/aE + 0 From this we have the next exact sequence H p at E H p at R E + H p at R E/aE + (3.8) At first, it is easy to see that H p at R E/aE + n = 0 for all n p. (In fact, after replacing the given system a = a a 2 a u by a new minimal system of generators of, say b = b b 2 b u, so that any s -elements of b form a minimal reduction of withrespect to E/aE, we get this by the hypothesis of induction on s, because H p at =H p bt.) By (3.8), we have H p at R E + n = 0 for all n p, and thus we get the assertion (2).

23 buchsbaumness in rees modules 235 Finally, consider the short exact sequence 0 R E + at R E + R E + 0 given in Lemma 3.6 again. From this we have a graded R-monomorphism as H p at R E + H p at R E + Thus the assertion () follows immediately from the assertion (2), and this completes the proof of Proposition REDUCTION STEP TO THE CASE WHERE depth E>0 Throughout this section, we always assume that E is a Buchsbaum A-module (of dimension s>0) and that the following two conditions are fulfilled: (i) the equality I G E = I E holds; and (ii) possesses minimal multiplicity withrespect to E. Namely the equality e E =µ A E +l A E/ E ρ E I E holds; Definition 2.5. Then we begin with the following. Lemma 4.. E E and 2 E = E hold for any minimal reduction of with respect to E. Proof. Since possesses minimal multiplicity, we already know by Proposition 2.6 that E = E. According to [Y3, Theorem.], the equality I G E = I E implies that E 2 E = E. Consequently this means that 2 E = E 2 E = E 2 E E 2 E = E As in the previous section we still write u = µ A, and we further write v = v A, the embedding dimension of A. Moreover, for a set S we still denote by S the number of elements in S and for integers i, j we denote by i j the set of integers n suchthat i n j. Then we have the following. Lemma 4.2. The following statements are true. () There exists an E-basis, say a a 2 a u,of consisting of minimal reductions; i.e., this is equivalent to saying that = a a 2 a u and there exists an integer r 0 such that r+ E = a i i I r E for all I u with I =s.

24 236 kikumichi yamagishi (2) Let a a 2 a u be an E-basis of consisting of minimal reductions as above. Then there exists a minimal system x x 2 x v of generators of such that any s-elements of the system a a u, x x v form a system of parameters for E. Proof. Applying [SV, Proposition.9 in Chap. I], we can find a system of elements of, say a a 2 a u, suchthat via R G the image of elements a t a 2 t a u t into G forms a G E -basis of the ideal G + in the sense of Stückrad and Vogel; see [SV, Definition.7 in Chap. I] for details. Then according to the method in [SV, Lemma 2.4 of Chap. IV], this system is extended to the required one immediately; see also [Y, Remark 5 of Section, p. 454]. Now let a a 2 a u be an E-basis of consisting of minimal reductions, as in () of Lemma 4.2. By Lemma 4. we already know that E a i i I E for all I u with I =s. Furthermore, let x = x x 2 x v be a minimal system of generators of satisfying the condition as in (2) of Lemma 4.2. We put at = a t a 2 t a u t in R, homogeneous elements of degree one. Let K at x R E be the Koszul (co-)complex generated (over R) by the joint system at x withrespect to R E. Since at x is a minimal system of generators of, this Koszul (co-)complex K at x R E is uniquely determined by the ideal up to isomorphisms not depending on the particular choice of a minimal system of generators; cf. [SV, Section of Chap. 0, p. 27]. Hence we denote it simply by K R E. Moreover p K p R E K p+ R E denotes its pthdifferentiation. Notice that the Koszul complex K R E is a complex consisting of direct sums of copies of a graded R-module R E. Hence we have an expression of it as K R E = R E e I J Kp R E = R E e I J I u J v I + J =p where e I J I u J v is the graded free basis with deg e I J = I. Let ξ be an element of K p R E and assume that ξ = p η for some element η of K p R E. Write ξ, η as ξ = ξj I ei J η = η P Q ep Q I + J =p where ξj I, ηp Q R E. Then we have P + Q =p ξj I = I i a i t η I\ i J + i I j J I +J j x j η I J\ j (4.)

25 buchsbaumness in rees modules 237 where I i denotes the number of elements of the set i I i <i ; i.e., I i = i I i <i, and J j is also defined in the same. Furthermore assume that ξ, η are homogeneous and write deg ξ = deg η = n. Since ξj I R E n+ I and η P Q R E n+ P, these elements ξj I, η P Q are expressed as ξ I J = bi J tn+ I η P Q = cp Q tn+ P where b I J n+ I E and c P Q n+ P E for each I, J and P, Q. Then by (4.) we have b I J = I i a i c I\ i J + i I j J I +J j x j c I J\ j (4.2) in n+ I E, for each I, J. Look at the short exact sequences given in Lemma 3.2 as 0 H 0 R E τ R E R E/H 0 E 0 (4.3) 0 H 0 R E + τ + R E + R E/H 0 E + 0 (4.4) Then we have the following. Lemma 4.3. Suppose that s 2. Then the following statements are true. () For each 0 p s, the canonical map τ p H p H 0 R E Hp R E induced by τ is injective. (2) For each s n, the homogeneous component of τ s+ of degree n is injective. τ s+ n H s+ H 0 R E n H s+ R E n Proof. We look for the commutative diagram 0 K p H 0 R E Kp R E 0 p 0 K p H 0 R E Kp R E Choose a homogeneous element ξ K p H 0 R E withdeg ξ = n, where n, and assume that there exists a homogeneous element η

26 238 kikumichi yamagishi K p R E suchthat ξ = p η in K p R E. Our goal is to prove ξ = 0. So write ξ, η as ξ = ξj I ei J and ξj I = bi J tn+ I (4.5) I J η = P Q η P Q ep Q and η P Q = cp Q tn+ P (4.6) where b I J n+ I E and cq P n+ P E for each I, J and P, Q. So, in order to get ξ = 0, it is enough to show that ξj I = 0, and hence bi J = 0inE, for all I, J with I + J =p. () Let 0 p s. Since ξ K p H 0 R E, this is equivalent to saying that ξj I H0 R E, and hence bi J H0 R E n+ I = n+ I E H 0 E for all I, J by Lemma 3.2. By the formula (4.2), we see b I J a i x j i I j J E H 0 E = 0 because a i x j i I j J (recall that I + J =p) is a subsystem of parameters for E. Therefore we obtain that the map τ p is injective. (2) Now let us consider the case where p = s +. We may assume that s n. If n + I 0, then b I J = 0, because H0 R E n+ I = 0 ; see Lemma 3.2. Hence we may assume that n + I =0. Since s n and I + J =s + we naturally see that I, J s. At first, assume that n + I =0. Then deg η I\ i J =, and hence c I\ i J = 0 easily, because R E has no homogeneous component of negative degree. Thus, by the formula (4.2) we have b I J = j J I +J j x j c I J\ j x j j J E H 0 E = 0 because of J s. Next, assume that n + I =. In this case it consequently follows that c I J\ j R E = E. Then choose a subset I of u so that I =s and I I. Since E a i i I E by Lemma 4., we naturally see that x j c I J\ j a i i I E. By (4.2), we finally obtain that b I J a i i I E H 0 E = 0 in this case too. Therefore the map τ s+ n must be injective and this completes the proof of Lemma 4.3. Lemma 4.4. The canonical map induced by τ + τ p + H p H 0 R E + H p R E + is injective for all 0 p s +.

27 buchsbaumness in rees modules 239 Proof. Consider the commutative diagram with graded R-monomorphisms H 0 R E τ R E σ 0 σ H 0 R E + R E τ + + Since H 0 R E is a vector space, the vertical map σ 0 must split. Consequently, the induced map by σ 0 given in the commutative diagram, say σ p 0, splits too, H p H 0 R E τ p Hp R E σ p 0 σ p H p H 0 R E + τ p + H p R E + Therefore the injectivity of τ p leads us to the same property of τ p + for all 0 p s. The rest of our assertion, the injectivity of τ+ s+, is shown by direct calculation. Namely, look for the commutative diagram with exact rows 0 K s+ H 0 R E + K s+ R E + 0 s 0 K s H 0 R E + K s R E + Choose a homogeneous element ξ K s+ H 0 R E + withdeg ξ = n, where n, and assume that there exists a homogeneous element η K s R E + suchthat ξ = s η in K s+ R E +. Our goal is to prove that ξ = 0. First, if n< s, then K s R E + n = 0 ; hence this naturally means that η = 0. So we may assume that n s. Now write ξ, η as in (4.5) and (4.6). Since ξ K s+ H 0 R E +, it is equivalent to saying that b I J n+ I E H 0 E for all I, J with I + J =s +. If n + I, then b I J = 0, because H0 R E + m = 0 for m by Lemma 3.2. Hence we may consider only the case where n + I =. Recall that we already assume that n s; hence by these observations we know that I = n s. Now choose a subset I of u suchthat I =s and I I. Since E a i i I E by Lemma 4., we see that x j c I J\ j a i i I E. Consequently, by the formula (4.2), we obtain b I J a i i I E H 0 E = 0 Thus the map τ+ s+ is injective too and this completes the proof of Lemma 4.4.

28 240 kikumichi yamagishi Recall that a finitely generated graded R-module W is a Buchsbaum R-module if and only if the canonical map φ p W Hp W H p W is surjective for all 0 p<dim W ; see [SV, Theorem 2.5 in Chap. I]. Now, applying Lemma 4.4, we can establish the reduction step to the case where depth A E>0 as follows. Proposition 4.5. Let E = E/H 0 E. Then the following statements are true. () Suppose that s 2. Then, for each p s, φ p R E is surjective if and only if φ p R E is so. Hence R E is a Buchsbaum R-module if and only if R E is so. (2) For each p s, φ p R E + is surjective if and only if φ p R E + is so. Hence R E + is a Buchsbaum R-module if and only if R E + is so. Proof. By (4) of Lemma 3.2 it is enough to check only the if part of our assertion. () Let s 2 and let p s. We look at the commutative diagram withexact rows τ p+ H p R E H p R E H p+ H R E 0 H p+ R E φ p R E φ p R E H R E p H R E p = First, consider the case where p<s. By () of Lemma 4.3, τ p+ is injective, and hence φ p R E is also surjective if φp R E is so. Next, recall that H s R E m = 0 for m 2 s. By this fact we sufficiently check the surjectivity only of homogeneous components φ s R E n for 2 s n. By (2) of Lemma 4.3, however, τ s+ n is injective, and hence φ s R E n for 2 s n is obviously surjective if φ s R E n is so. (2) By Lemma 4.4 this is shown in the same way as the assertion () and Proposition 4.5 is now established. We mention that Nakamura showed an argument similar to that in () of Proposition 4.5 under the same assumption as that in Corollary.3. As stated as in our proposition 4.5 above, however, such an argument is true under more general situations. As a consequence of Proposition 4.5 we have the following corollary. Corollary 4.6. The following are true. () Suppose that s 2. Then φ p R E is surjective for p 2. Hence R E is a Buchsbaum R-module if s = 2.

29 buchsbaumness in rees modules 24 (2) φ p R E + is surjective for p 2. Hence R E + is a Buchsbaum R-module if s 2. Combining Corollary 3.5 and () of Corollary 4.6, we have the following. Corollary 4.7. Suppose that s 3. Then, for each 3 p s, φ p R E is surjective if φ p R E + is so. Hence R E is a Buchsbaum R-module if R E + is so. 5. PROOF OF THEOREM. Throughout this section, we always assume that E is a Buchsbaum A-module (of dimension s>0) and that a a 2 a u (where u = µ A is an E-basis of consisting of minimal reductions; see () of Lemma 4.2 above. Moreover the following two conditions are still fulfilled: (i) the equality I G E = I E holds; and (ii) possesses minimal multiplicity withrespect to E. By Lemma 4., we already know that E a i i I E and 2 E = a i i I E hold for all I u with I =s. Moreover, we have the following. Lemma 5.. Suppose that s 2. Puta = a and put U ae =ae. Then () E U ae is a Buchsbaum A-module of dimension s such that E H 0 E p = 0 H E p U ae = U ae /H 0 E + E U ae p = H E p (else). (2) E/ E U ae is a Buchsbaum A-module of dimension s such that { U ae / E U ae p = 0 H E/ E p U ae = H E/aE p (else). Proof. () This is shown by an argument similar to that in [GS, Theorem 2.5] (see also the proof of [St, Theorem 3]). First of all, after passing through the completion of A, we may assume without loss of generality that A is a regular local ring. Look at the short exact sequence of A-modules 0 ae E U ae E U ae /ae 0