A characterization for pseudo Buchsbaum modules

Japan. Vol. J. 30, Math. No. 1, 2004 A characterization By for pseudo Buchsbaum modules Nguyen Tu CUONG and Nguyen Thi Hong LOAN (Received April 15, 2002) (Revised October 4, 2002) (from Nagoya Mathematical Journal) Abstract. pseudo erties In this paper Buchsbaum modules for those modules. 1. we study and give the structure of a new a characterization class in term of modules of called Buchsbaum prop Introduction Throughout this paper, R denotes a Noetherian local ring with maximal ideal m and M a finitely generated R-module with dim M=d>1. Let x=(x1,...,xd) be a system of parameters of M. We consider the difference between the multiplicity and the length where be QM(x)=U>0(x',. mentioned example, if M Therefore by [5] by [14], also for is supx for that every if system above module or to pseudo a natural JM(x) to is Let take to study 0=•¿Ni 1)M lot all M are is a systems x true in general. or supx JM(x)<•‡. generalized value modules be which a reduced for all primary systems called It should of Further, in [8] one studied M. For by [11]. In of value of the of modules Cohen-Macaulay Thus be parameters. [3] we all M of known a constant structure pseudo is Therefore respectively. modules decomposition [14]. converse the it of pseudo takes the called it module. JM(x) module, structure are M. fraction then They the of Unfortunately, In M. structure Cohen-Macaulay Cohen-Macaulay what x of generalized M. of the QM(x)=(x1,...,xd)M module of a submodule on parameters length Buchsbaum is then a generalized not a constant xd informations of the is parameters question: : of module if M of JM(x)=0 a is just JM(x)<•‡ showed gives a Cohen-Macaulay that l(M/QM(x)) satisfying paper xd JM(x) JM(x)=0 statements rise t1+, that should The purpose it gives if it fores of this 0 in M, Buchsbaum. of the submodule 2000 Mathematics Subject Classification. 13H10, 13D45, 13C99. Key words and Phrases. Buchsbaum, pseudo Buchsbaum, local cohomology, multiplicity. 166 NGUYEN where Ni is pi-primary. where M is the m-adic TU From CUONG and NGUYEN THI HONG LOAN now on we set completion of M and where Him(M) stands for the ith local cohomologymodule of M with respect to the maximal ideal m. Note that this invariant J(M) is finite if and only if M is a generalized Cohen-Macaulay module. But it was proved in [8] that M is pseudo Cohen-Macaulayor pseudo generalized Cohen-Macaulay if and only if M is a Cohen-Macaulay module or a generalized Cohen-Macaulay module over the m-adic completion R of R, respectively. The main result of this paper is to prove the followingtheorem. THEOREM 1.1. M is a pseudo Buchsbaum module if and only if M is a Buchsbaum R-module. Moreover, in this case we have JM(x)=J(M), for every system of parameters x=(x1,...,xd) of M. Note that the MonomialConjecture posed by M. Hochster (see [13])says that, (x1...xd)t _??_(xt+11,...,xt+1d)Rfor every system of parameters x=(x1,...,xd) of the ring R (d=dim R) and for all positive integers t. It is well-knownthat the MonomialConjecture is true for Buchsbaum rings (see [9]). Therefore the following result is an immediate COROLLARY1.2. Monomial Conjecture. consequence of Theorem 1.1. If R is a pseudo Buchsbaum ring, then R satisfies the The paper is divided into 4 sections. In Section 2, we shall outline some prop erties of JM(x) which will be needed later. Some properties of pseudo Buchsbaum modules will be given in Section 3. In the last section, we shall give the proof of Theorem 1.1 and corollaries. Characterization 2. The function for pseudo Buchsbaum modules 167 JM(x(n)) Let x=(x1,...,xd) be a system of parameters of M and n=(n1,...,nd) a d-tuple of positive integers. Set x(x)=(xn1,...,xndd). Then the differencebetween multiplicities can be and considered Lemma 3.1]) ni>mi, as and alized n JM(x(n)) proved Question (n>>0 is the in n. i.e., Note for in that for bounded following of short)? above function [14] By by the is non-negative ([7, m=(m1,...,md) ([3, defined 1.2 this n=(n1,...,nd), Corollary is just by Sharp as follows: Is [3, Lemma 4.1, polynomial with 4.3]). [5] that l(M/QM(x(n))) M(1/(xn11,...,xndd,1)) describe enough function JM(x(n))>JM(x(m)) it was fraction can have a ascending, i=1,...,d, Moreover, we lengths and the length Hamieh JM(x(n)) of [14]. Therefore, a polynomial (ii)], we see n1•cndJM(x). that for the More gener large function generally, we theorem. THEOREM2.1 ([7, Theorem 3.2]). The least degree of all polynomials in n bounding above the function JM(x(n)) is independent of the choiceof the system of parameters The nomial x. numerical type convenience, of we invariant fractions of stipulate of M given M that and the in the denoted degree by of the above pf theorem (M) ([7, is called Definition zero-polynomial is the poly 3.3]). equal to For -•‡. REMARK2.2 ([7, Lemma 3.4]). The followingstatements are true: (i) Let M be the m-adic completion of M. Then (ii) Let x be a parameter element of M with dim(0:x)M<d-2. (iii) Suppose that dim M=d>1. Then Then pf(M)<d-2. It was shown in [6]that the function JM(x(n)) is not a polynomial in n for n>>0 in general. Therefore the polynomial type of fractions pf(M) plays an important role in the study of the function JM(x(n)). The followingdefinition was introduced in [8, Definition 2.2]. DEFINITION 2.3. (i) M is said to be a pseudo Cohen-Macaulay module pf(M)=-•‡. (ii) 0. M is said to be a pseudo generalized Cohen-Macaulay module if pf(M)< if 168 NGUYEN TU CUONG and NGUYEN Till HONG LOAN By Theorem 2.1, the first statement is equivalentto saying that there exists a system of parameters x=(x1,...,xd) of M such that JM(x)=0 and the second is equivalent to the existence of a system of parameters x=(x1,...,xd) of M such that JM(x(n)) is bounded above by a constant for all n>>0. By Remark 2.2, (i), M is a pseudo Cohen-Macaulay (pseudo generalized Cohen-Macaulay) module if and only if so is the m-adic completion M of M. It should be mentioned that every Cohen-Macaulay module is pseudo Cohen-Macaulay and every generalized Cohen-Macaulay module is pseudo generalized Cohen-Macaulay. Next, we recall some characterizations of these modules. Note that for a module M we often use in this paper the followingmodules where UM(0)=•¿dim R/pj=d Nj derived from a reduced primary decomposition of the submodule 0 of M. THEOREM2.4 ([8, Theorem 3.1]). Suppose that R admits a dualizing com plex. Then the followingstatements are true. (i) M is a pseudo Cohen-Macaulaymodule if and only if M is a Cohen - Macaulay R-module. (ii) M is a pseudo generalizedCohen-Macaulaymodule if and only if M is a generalized Cohen-MacaulayR-module. Let (x1,...,xn) sequence a ƒÂ of matrix n B. obtain a which is is injective. [16, of that R, Corollary elements in (y1,...,yn)R •º 1<i, j<n 5.1.15], we m. Let (x1,..., such that (y1,...,yn) xn)R. be Then yi=‡”nj=1 have ƒÂQM(x) •º another there bijxj. QM(y). Therefore, exists Put we homomorphism independent LEMMA map sequence bij •¸ By determinantal systems a such B=(bij), =det two be elements of the choice of the matrix B. This map •º is called the map. 2.5 of ([3, parameters Lemma 3.1]). of M Let such that x=(x1,...,xd) (y)R •º and (x)R. Then y=(y1,...,yd) the determinantal be Characterization for pseudo Buchsbaum modules 169 The followingproperties of JM(x) were proved in [3]. LEMMA (i) 2.6. ([3, Lemma parameters x (ii) of ([3, systems of The following 4.1]) statements are true. JM(x)=JM(x)=JM/H0m(M)(x), for every system of M. Lemma 4.2]) parameters of Let M x=(x1,...,xd) such that and (y)R •º (x)R. y=(x1,...,xd-l, Then yd) be two JM(x)<JM(y). Let x=(x1,...,xd) be a system of parameters of M. Put M1=M/x1M, x'=(x2,...,xd). For any (d-1)-tuple of positive integers n'=(n2,...,nd), we set x'(n')=(xn22,...xndd) and x(n1)=(x1,xn22,...,xndd). LEMMA2.7. With the same notations as above, if the function JM(x(rn)) is constant for all n1,...,nd>1 and dim(0M:x1)<d-1, so is the function PROOF. Let and be the epimorphisms defined in [3, Lemma 2.2] by for any u •¸ M, u1 the image of u in M1. Then we have the following commutative diagram where ƒÂ, ƒÂ1 ƒÂ1 are are determinantal injective by Lemma such that l (Ker ƒÕ)<l(Ker ƒÕn') non-negative =e(x'(n'); for k M1). 2.5, Therefore all maps and the induced get rows are exact sequences. homomorphism ƒÂ' n2,...,nd>1. JM(x(n))=k we the By for all the n1,...,nd>1 is hypothesis Since ƒÂ, injective. there and e(x(n'); Thus exists M) a 170 NGUYEN This implies that JM1(x'(n')) and lemma is LEMMA 1, of where k1, M of such that N. by parameters that Tm all HONG LOAN n2,...,nd>1. JM1(x')=JM1(x'(n')) x=(x1,...,xd) we Since the all n2,...,nd>1 for of 2.6, (i). z=(z1,...,zd) function be JM(y(n))=k2 for two all systems of n1,...,nd> k1=k2. generality we From of y=(y1,...,yd) and have loss Lemma and JM(x(n))=k1 Then Without M>0 NGUYEN for it follows Let k2 •¸ PROOF. depth and proved. • 2.8. parameters CUONG JM1(x')>JM1(x(n')) is ascending, the TU [16, M and may assume Proposition positive that 8.2.5], integers AnnR there M=0 exists r1,...,rd and a system such that By Lemma 2.6, (ii) we have Hence k1>k2. The Similarly, following we lemma get will k2>k1. be Therefore used often in k1= the k2. • sequel. LEMMA2.9. Let M be a pseudo generalized Cohen-Macaulaymodule and x a system of parameters of M. Then we have (i) JM(x(n))=J(M) for n>>0. Therefore JM(z)<J(M) holds for every system of parameters z of M. (ii) If R admits a dualizing complex then JM(x(n))=J(M) for n>>0 and JM(z)<J(M) holds for every system of parameters z of M. PROOF. - ule. Then the complete Macaulay (i) ring by a is difficult R Theorem system of 2.4, check of . Then, (ii). Assume in Theorem 2.4, by same the a pseudo (ii) method Lemma M J(M)<•‡. is Since of M such for follows 2.6, Theorem there y1UM(0)=0. It n1,...,nd>1, 2.1, Since Cohen UM(0)<d, for from (i). generalized dim that all mod where n>>0 Lemma by [14, and the again by 2.8 JM(x(n)). that M by complex, JM(y(n))=JM(y(n))=J(M) function addition Cohen-Macaulay n1,...,nd>1 dualizing JM(y(n))=JM(y(n)) statement the all a generalized y=(y1,...,yd) that the of is Therefore Thus Now property (ii) virtue admits parameters to 3.7]. ascending M for always y(n)=(yn11,...,yndd). Theorem that JM(x(n))=JM(x(n)) exists not Suppose R admits is generalized as above we a dualizing complex. Cohen-Macaulay and get JM(x(n))=J(M), Thus, therefore for J(M)<•‡ n>>0 and Characterization JM(z)<J(M) for 3. every Pseudo We begin with system for pseudo Buchsbaum modules of parameters Buchsbaum modules the following definition. z of 171 M. • DEFINITION3.1. An R-module M is called a pseudo Buchsbaum module if there exists a constant K such that JM(x)=K for every system of parameters x of M. R is called a pseudo Buchsbaum ring if it is a pseudo Buchsbaum module as a module over itself. Recall that the notion of standard system of parameters is an important tool for studying generalized Cohen-Macaulay modules. A system of parameters x= (x1,...,xd) of M is called a standard system of parameters if Clearly, if x is a standard system of parameters of M, so is x(n) for all n1,...,nd> 1. M is a generalized Cohen-Macaulaymodule if and only if M admits a standard system of parameters. Note that standard systems of parameters are also used to characterize Buchsbaum modules (see [15]). A module M is Buchsbaum if and only if every system of parameters of M is standard. The followingresult will be used often in this paper. THEOREM3.2 ([3, Theorem 5.1 and Corollary 5.2]). The following state ments are true. (i) Let M be a generalized Cohen-Macaulaymodule and x=(x1,...,xd) a standard system of parameters of M. Then JM(x)=J(M). (ii) Let M be a Buchsbaummodule. Then JM(x)=J(M), for any system of parameters x of M, The and D.A. reducing dim notion of Buchsbaum system R/p>d-i reducing in of and system [1]. parameters A of parameters system if xi _??_ p of for was parameters all p •¸ introduced x=(x1,...,xd) Ass(M/(x1,...,xi-1)M) by M. Auslander is called a with i=1,...,d. LEMMA 3.3. Thefollowingstatementsare true. (i) M is a pseudoBuchsbaummoduleif and onlyif so is M/H0m(M). (ii) Let M be a pseudoBuchsbaummoduleand x=(x1,...,xd) a reducing systemofparametersof M. ThenM/(x1,...,xi)M is a pseudoBuchsbaummodule for i=1,...,d. PROOF. (i) Since JM(x)=JM/H0m(M)(x) by Lemma 2.6, (i), the statement is clearly true. 172 NGUYEN TU CUONG and NGUYEN THI HONG LOAN (ii) By induction on i, it suffies to show for the case i=1. Since x1 is a reducing element, dim(0M:x1)<d-1. Therefore the statement follows from M Lemma 2.7. • PROPOSITION3.4. M is a pseudo Buchsbaum module if and only if the m adic completion M of M is a pseudo Buchsbaum module over R. PROOF. system ideal M, of Let M parameters be a of b=(x1,...,xd)R such that e(x; pseudo M and of q, a pseudo by Lemma J(M). M)=e(q; is a pseudo Since the 2.9, converse 4. (i). On the Buchsbaum every We begin Then Thus there 2.5. other hand, by Lemma M is of Since M and reduction pseudo Therefore of xM •º yM, Since hence a parameters JM(x)<Ja(y). so is (i). be a minimal system M). Therefore since 2.6, a M)=e(y; module, y=(y1,...,yd) exists is e(x; Lemma Let M JM(y)<J(M) Buchsbaum, JM(x)= JM(y)=J(M) and M of M, module. system statement Proof R. Cohen-Macaulay JM(x)=J(M) module. x=(x1,...,xd) M). by generalized Thus q=yR •¿ where l (M/QM(x))>l(M/QM(y)) is Buchsbaum of is parameters an of Theorem with some of immediate M is also consequence a system of Lemma of parameters 2.6, (i). • 1.1 auxiliary results as follows. LEMMA4.1. Let M be a pseudo Buchsbaum module. Suppose that M is generalized Cohen-Macaulay. Then we have mHim(M)=0 for i=1,...,d-1. PROOF. Since Him(M/H0m(M))_??_Him(M) for all i>0, by Lemma 3.3, (i) we may assume without loss of generality that depth M>0. We prove the lemma by induction on d. For the case d=1 there is nothing to prove. Now we assume that d=2. Let x be a parameter element of M. Then there are by [15, Proposition 1.9, Ch. I] the elements y1,...,yt in m such that (y1,...,yt) is a system of generators of m and yi is a parameter element of M/xM for every i=1,...,t. Since M is generalized Cohen-Macaulay,the element yi satisfies the hypothesis of Lemma 2.3 in [4]for all i=1,...,t. Therefore we obtain by [4, Lemma 2.3] the followingexact sequence for n>>0, where Mi:=M/yiM. Then we have Since dim Mi=1, JMi((xn))=0 by [14, Proposition 3.1]. Thus JM((yi,xn))= l(H1m(M)/yiH1m(M)).On the other hand, for n>>0 JM((yni,xn))=l(H1m(M)) Characterization for pseudo Buchsbaum modules 173 by [14, Theorem 3.7]. Therefore l(H1m(M)/yiH1m(M))=l(H1m(M))since M is a pseudo Buchsbaum module. This implies that yiH1m(M)=0 for all i=1,...,t. Thus mH1m(M)=0. Let d>3. M>0, Since M is a generalized i=0,...,d-1. From the exact sequence sequence we get the following short exact for Since M/xM i=1,...,d-2. pseudo esis Buchsbaum for module M/xM to mHim(M)=0 that Hence by obtain for Recall i•‚d. Cohen-Macaulay module and depth there always exists a non-zero divisor x E m such that xHim(M)=0, that is of local cohomology a generalized Lemma 3.3, mHim (M/xM)=0 modules Cohen-Macaulay (ii), we can apply for module the and inductive a hypoth i=1,...,d-2. Therefore i=1,...,d-1. • M is Lemma COROLLARY called 4.1 a leads 4.2. quasi-Buchsbaum module immediately Let M be to a pseudo the if mHim(M)=0, following for all consequence. Buchsbaum module. Suppose that M is generalized Cohen-Macaulaywith mH0m(M)=0. Then M is a quasi-Buchsbaum module. Let where 0=•¿Ni Ni a generalized of Theorem is be a reduced pi-primary. primary Note that Cohen-Macaulay decomposition of UM(0)=•¿dimR/pj=d module. Thus the submodule 0 Nj=H0m(M) the following in when result is a M, M special is case 1.1. LEMMA4.3. The followingstatments are equivalent: (i) M/H0m(M) is a Buchsbaum module. (ii) M is generalized Cohen-Macaulayand pseudo Buchsbaum. PROOF. (i)•Ë(ii) (ii)•Ë(i): erality the that case d=1. non-zero the condition Buchsbaum By depth is virtue of M>0. of (ii). module. Then M/xM: <m> M. It Put Theorem 3.3, show d>2. follows by Lemma We Suppose divisor clear Let M'= from the x the (i), we lemma be M/xM. 3.2, any (ii) and may by assume Then dim Let is a Buchsbaum module since on element M'=d-1 hypothesis 3.3, without induction parameter inductive Lemma loss d. It of M. and that (i). is Then M' M'/H0m(M') of gen trivial for x is a satisfies is a 174 On NGUYEN TU CUONG and NGUYEN Tin HONG LOAN the module other by hand [15, PROOF 3.4 by 2.23, OF THEOREM by Proposition Theorem mH1m(M)=0 Proposition that Lemma Ch. 1.1. I] the Without R=R. Since 1.1 is now an immediate 4.1. and Therefore lemma is M is any loss of generality, R always consequence admits of the a Buchsbaum proved. • we may a dualizing following assume complex, lemma. LEMMA 4.4. Suppose that R admits a dualizing complex. Then M is a pseudo Buchsbaum module if and only if M=M/UM(0) is a Buchsbaum R-module. Moreover, in this case we have JM(x)=J(M) for every system of parameters x=(x1,...,xd) of M. PROOF. The last statement of the lemma followsimmediately from Lemma 2.9, (ii). So we need only to prove the first one. a) Sufficient condition. Assume that M is a Buchsbaum module. The state ment is trivial for the case d=1. Suppose now that d>1. First we prove the followingclaim. CLAIM. PROOF only need OF to equality of For any system of parameters the CLAIM. prove claim the It is clear converse. is just that Note QM(x). x of M and t>>0, the that inclusion •º for is t>>0 the true left for all t. submodule So in we the Let Since M is a Buchsbaum module, we get by [10, Theorem 2.3] that for all t>1, M with aixi •¸ where •È means deleting the [(X1,...,xi,...,xd)M+UM(0)] For any positive integers n1,...,nd>1, item. Therefore, such we have that there a=‡”di=1ai. are a1,...,ad in Characterization for pseudo Buchsbaum modules 175 Hence On the orem other 2.4, follows hand, (ii). that So e(x; M), epimorphism it implies u+QM(x(n)) n>>0. is a pseudo Lemma generalized 2.9, JM(x(n))=JM(x(n)), the for M using by Cohen-Macaulay we get Theorem module JM(x(n))=J(M) 3.2, (ii) for for n>>0. that l(M/QM(x(n)))=l(M/QM(x(n))) for M/QM(x(n))•¨M/QM(x(n)) for Thus (ii) any we u •¸ M defined with u being the image of by The n>>0. Since e(x; n>>0. It M)= Therefore by ƒÕ(u+QM(x(Z)))= u in M is an isomorphism have for n>>0 and k>>0. Further, since M is a Buchsbaum module, the right term of the equality above is just equal by [10, Theorem 2.3] to for all n1,...,nd>1 and k>1. for n>>0 and k>>0. Combining for n>>0 and k>>0. Thus for This t>>0. finishes the proof Therefore these of the we obtain facts we have claim. • Now we continue to prove the sufficient condition of Lemma 4.4. Let x be any system of parameters of M. It followsby Claim that M/QM(x)_??_M/QM(x). Therefore JM(x)=JM(x).But M is a Buchsbaum module by the hypothesis, JM(x)=J(M). Therefore M is a pseudo Buchsbaum module. 176 NGUYEN TU CUONG and NGUYEN THI HONG LOAN In order to prove the necessary condition of Lemma 4.4 we need the following lemma. LEMMA 4.5. (y1,...,yd) exists for a system a part all M be a of parameters of a system generalized of M, of parameters Cohen-Macaulay and i a positive (x1,...,xi) of M module, integer, such i<d. that y= Then xj-yj •¸ there AnnR M j=1,...,i. PROOF. Suppose for Let We we M there prove already such that exists a this have result for xj-yj •¸ parameter by induction i>1 a AnnR M element part xi+1 for of on of the i. Let system i=1, of j=1,...,i. Mi We with we choose x1=y1. parameters x1,...,xi have xi+1-yi+1 •¸ to show AnnR M, that where Mi=M/(x1,...,xi)M. Set where If for a module L=_??_, we condition N we denote choose of the xi+1=yi+1 lemma. and Suppose the that sequence x1,...,xi+1 L•‚_??_. Note satisfies the required that Since by the choice of x1,...,xi Let such z •¸ m be R such (x1,...,xi)R •º there that that that Since M/(x1,...,xi)M Therefore it follows exist is a generalized Cohen-Macaulay module, we get a positive integer k zk=ai+1+‡”ij=1rjxj. p for all and elements Since p •¸ Assh Mi, zk •¸ ai+1 •¸ •¿p•¸Tp ai+1 •¸ AnnR M, r1,...,ri •¸ (•¿p•¸Tp)•_(•¾q•¸Lq) and ai+1 _??_ •¾q•¸Lq. and We Characterization choose now lemma is xi+1=yi+1+ai+1. therefore for pseudo Buchsbaum modules Then xi+1 is a parameter 177 element of Mi and the proved. • b) Necessary condition of Lemma 4.4. Assume that M is a pseudo Buchs baum module. Then M is a generalized Cohen-Macaulay module by Theorem 2.4, (ii). Let y=(y1,...,yd) be any system of parameters of M. Then there exists by Lemma 4.5 a system of parameters x=(x1,...,xd) of M such that e(y; M)=e(x; M) and l(M/QM(y))=l(M/QM(x)). Therefore Since the homomorphism ƒÕ: l(M/QM(x)). M/QM(x)•¨M/QM(x) is surjective, l(M/QM(x))< Hence by Lemma 2.9, (ii). Therefore we get So M is pseudo a Buchsbaum therefore module the proof PROOF Then OF Moreover, by Lemma of Theorem is ring. 4.3. 1.2. the [9], l(R/QR(x))•‚0. This depth M>0, finishes Suppose a Buchsbaum Since since the it follows proof that of Lemma M is 4.4 and 1.1 is complete. COROLLARY R=R/UR(0) Buchsbaum by Buchsbaum. that R-module Monomial R by Conjecture is a pseudo Theorem holds Therefore l(R/QR(x))•‚0 true and Buchsbaum 1.1. Therefore for Buchsbaum the corollary ring. R is a rings is proved. • For the next consequence of Theorem 1.1 we need to recall the polynomial type of a module defined in [2] as follows. Set It was proved function the only In in [2] IM(x(n)) polynomial type if p(M)=-•‡ general, we that the does not of M and have COROLLARY least depend and M degree on denoted is generalized M. all the by 4.6. Let M polynomials choice p(M). of Then in x. This M Cohen-Macaulay pf(M)<p(M)<d-1 by be a pseudo mHim(M)=0 for i=p(M)+1,...,d-1, the module of n bounding least above degree is the called is Cohen-Macaulay if and only if and if p(M)<0. [7]. Buchsbaum module. Then we have wherep(M) is the polynomial type of 178 NGUYEN TU CUONG and NGUYEN THI HONG LOAN PROOF. 0. Since The M is is pseudo p(M)=p(M) Since by M Lemma 3.6, hand, we (i)]. from get from case p(M)<0 pseudo [2], we exact dim for only without Lemma if 4.1. so loss is of by M Assume by that Proposition generality for UM(0)<p(M) that p(M)> 3.4 R is [2, Theorem and complete. i=p(M)+1,...,d-1 i=p(M)+1,...,d-1. 1.1 that of Buchsbaum pseudo generalized - Macaulay by and by 3.1]. On the [7, other So the statement follows 1.1. • We see by Theorem the class assume proved sequence Him(M)_??_Him(M) Theorem exist may if Buchsbaum, l(Him(M))<•‡ Therefore the is Buchsbaum the class of pseudo modules. Buchsbaum Here modules Buchsbaum we give some which are not examples modules to show Buchsbaum modules, Cohen-Macaulay modules. Moreover, there exist modules which are not pseudo Buchsbaum modules. contains that there not generalized even Cohen EXAMPLE. (1) Let k[[X1,...,X4]] be the formal power series ring. Put Then It is clear that, A is not a generalized Cohen-Macaulay ring. Since A/UA(0) is a Buchsbaum ring (see [15, Example 6, Introduction]), A is a pseudo-Buchsbaum ring by Theorem 1.1. Moreover,it is easy to see that JA(x)=1 for every x of A. (2) Let R=k[[X1,...,X7]] (n>2) be the formal power series ring and M=(X21,X2,...,Xn)R. From the exact sequence we have M is 0, M Him(M)=0, a generalized is a Buchsbaum not for i•‚1, n Cohen-Macaulay a Buchsbaum module module. which implies and H1m(M)_??_R/(X21,X2,...,Xn)R. module. On Moreover, that M the UM(0)=0, is not other Therefore hand, hence a pseudo-Buchsbaum as mH1m(M)•‚ M/UM(0) is not module. (3) Let M=_??_ni=1Mi, where Mi is a R-module which dim Mi=dim M for i=1,...,n. Then M is a pseudo Buchsbaum module if Mi is so for every i=1,...,n. Indeed, it sufficesto prove this for n=2. Clearly, x=(x1,...xd) is a system of parameters of M if and only if it is a system of parameters of M1 and M2. For any set of positive integers n=(n1,...,nd), it is clear that Characterization for pseudo Buchsbaum modules 179 Therefore It follows baum that M is a pseudo Buchsbaum module if M1 and M2 are pseudo Buchs modules. Nowweconsiderthe Mathsdual Dd(M)=HomR(Hdm(M),E)of the dth-local cohomologymoduleHdm(M),whereE denotesthe injectivehullof the residuefield R/m of R. It is knownthat Dd(M) is a finitelygeneratedR-module.Moreover,if M has a canonicalmodulethen (see [12]) as R-module. Next, we are interested in the Buchsbaum property for the module Dd(M) of a pseudo Buchsbaum module. COROLLARY 4.7. Let M be a pseudo Buchsbaummodule. Then Dd(M) is a BuchsbaumR-module. PROOF. Since dim UM(0)<d-1, an isomorphism other Hdm(M)_??_Hdm(M). hand, since a Buchsbaum M COROLLARY KM. We of M that Then denote with for finitely module (see [15, corollary 4.8. to be the by 4.9, Ch. immediately minimal maximal generated a pseudo R-module Theorem follows ('7')1(H(M)). R KM_??_KM is a Buchsbaum the [17]) Therefore Let M be a pseudo KM by ƒÊR(M) respect any Let Buchsbaum by where I(M)=® baum a R-module The following module is we get from the exact sequence as Theorem R-module. 1.1, On KM_??_Dd(M) is II]. • from the Buchsbaum corollary module above. having a canonical module. number ideal m R-module we Then of generators is denoted of M. by The multiplicity e(M). It is well Linear Maximal ring. known have M is called a Buchs when ƒÊR(M)=e(M)+I(M). Buchsbaum the Then we have by Theorem 1.1 that 180 NGUYEN TU CUONG and NGUYEN THI HONG LOAN for any system of parameters x=(x1,...,xd) of R, where R=R/UR(0) and d= dim R. Furthermore, since R satisfies the Monomial Conjecture by Corollary 1.2, e(x;R)>1+J(R), for any system of parameters x of R. It followsthat e(R)> 1+J(R). When e(R)=1+J(R), we say that R has the minimal multiplicity. COROLLARY4.9. Let R be a pseudo Buchsbaum ring. Then the following statments are equivalent: (i) R has the minimal multiplicity. (ii) Dd(R) is a Linear Maximal Buchsbaum R-module. PROOF. e(R). Thus Then we is (i)•Ë(ii): R can a Linear is by show as in Maximal (ii)•Ë(i): Maximal Since Theorem First by the we proof of minimal have as Corollary above 3.1]. is the minimal multiplicity. that [17, that R with 4.7 by Hence Proposition multiplicity, 1+J(R)=e(R)= ring R-module R-module. [17, the a Buchsbaum Buchsbaum Buchsbaum multiplicity R has 1.1 Proposition 3.1]. KR_??_KR_??_Dd(R) a Buchsbaum Therefore R ring has the is with the minimal a Linear minimal multiplicity. • Let now M be a pseudo Buchsbaum module, then JM(x)=J(M) for all system of parameters x of M. Hence e(x;M)>J(M) for all system of parameters of M. Therefore e(M)>J(M). In the case the equality holds, we get the following result. COROLLARY statments are (i) M (ii) is M ~II Suppose that dim M=dim R>2. Then the following a pseudo is a Buchsbaum Linear module Maximal with e(M)=J(M). Buchsbaum R-module and ƒÊR(M)= (H1(Mi)). PROOF. xR is we get It 4.10. equivalent: follows module (i)•Ë(ii): a minimal that Suppose reduction of e(M)=J(M). follows that Since x is a system M Therefore and ƒÊR(M)=>IIO' (ii)•Ë(i) m. (t(H(M)) ) from Theorem is M m by 1.1 of parameters a Buchsbaum and is a [17, the Linear by Maximal Proposition fact of module e(M)=J(M). • 3.5]. M such that Theorem 1.1, Buchsbaum R Characterization ACKNOWLEDGEMENT. useful The for pseudo Buchsbaum modules authors would like to thank 181 the referee for his suggestions. References [1] M. Auslander and D.A. Buchsbaum, Codimension and multiplicity, Ann. of Math., 68 (1958), 625-657. [2] N.T. Cuong, On the least degree of polynomials bounding above the differences between lengths and multiplicities of certain systems of parameters in local rings, Nagoya Math. J., 125 (1992), 105-114. [3] N.T. Cuong, N.T. Hoa and N.T.H. Loan, On certain length functions associated to a system of parameters in local rings, Vietnam J. Math., 27 (3) (1999), 259-272. [4] N.T. Cuong, N.T. Hoa and L.T. Nhan, On modules whose local cohomology modules have generalized Cohen-Macaulay Maths deals, East-West J. Math., 3 (2) (2001), 109-123. [5] N.T. Cuong and V.T. Khoi, Modules whose local cohomology modules have Cohen - Macaulay Maths deals, In Commutative Algebra, Algebraic Geometry, and Computa tional Methods (Editor D. Eisenbud), Spinger 1999, 223 231. [6] N.T. Cuong, M. Morales and L.T. Nhan, On the length of generalized fractions, J. Algebra, 265 (2003), 100-113. [7] N.T. Cuong and ND. Minh, Lengths of generalized fractions of modules having small polynomial type, Math. Proc. Cambridge Philos. Soc., 128 (2000), 269-282. [8] N.T. Cuong and L.T. Nhan, Pseudo Cohen-Macaulay and pseudo generalized Cohen - Macaulay modules, J. Algebra, 267 (2003), 156-177. [9] S. Goto, On the associated graded rings of parameter ideals in Buchsbaum rings, J. Algebra, 85 (1983), 490-534. [10] S. Goto and K. Yamagishi, The theory of unconditioned strong d-sequences and modules of finite local cohomology, (Preprint, 1986). [11] R. Hartshorne, Property of A-sequence, Bull. Soc. Math. France, 4 (1966), 61-66. [12] J. Herzog, M. E. Kunz et al., Der kanonische Modul eines Cohen-Macaulay-Rings, Lecture Notes in Math., 238, Spinger-Verlag, Berlin-Heidelberg-New York, 1971. [13] M. Hochster, Contraced ideals from integral extensions of regular rings, Nagoya Math. J., 51 (1973), 25-43. [14] R.Y. Sharp and M.A. Hamieh, Lengths of certain generalized fractions, J. Pure Appl. Al gebra, 38 (1985), 323-336. [15] J. Stuckrad and W. Vogel, Buchsbaum rings and applications, Spinger-Verlag, Berlin - Heidelberg-New York, 1986. [16] J.R. Strooker. Homological questions in local algebra, London Math. Lecture Notes Series, Vol. 145, Cambridge University Press, 1990. [17] K. Yoshida, On linear maximal Buchsbaum modules and the syzygy modules, Comm. in Algebra, 23 (3) (1995), 1085-1130. INSTITUTEOF MATHEMATICS 18 HOANG QUOC VIET ROAD 10307 HANOI, VIETNAM E-mail:[email protected]