Pseudo Cohen–Macaulay and pseudo generalized Cohen–Macaulay modules

a

zed

lay andmodulesply this

entially

rs

tnam.

Journal of Algebra 267 (2003) 156–177

www.elsevier.com/locate/jalgebr

Pseudo Cohen–Macaulay and pseudo generaliCohen–Macaulay modules✩

Nguyen Tu Cuonga,∗ and Le Thanh Nhanb

a Institute of Mathematics, PO Box 631, Boho, 10.000 Hanoi, Viet Namb Thai Nguyen Pedagogical University, Viet Nam

Received 16 April 2002

Communicated by Craig Huneke

Abstract

In this paper we study the structure of two classes of modules called pseudo Cohen–Macaupseudo generalized Cohen–Macaulay modules. We also give a characterization for thesein term of the Cohen–Macaulayness and generalized Cohen–Macaulayness. Then we apresult to prove a cohomological characterization for sequentially Cohen–Macaulay and sequgeneralized Cohen–Macaulay modules. 2003 Elsevier Inc. All rights reserved.

Keywords:Local cohomology; Multiplicity; Generalized fractions; Noetherian; Artinian

1. Introduction

Let (R,m) be a Noetherian local ring andM a finitely generatedR-module withdimM = d . For a system of parametersx = (x1, . . . , xd) of M and a set of positive integen = (n1, . . . , nd), we setx(n) = (x

n11 , . . . , x

ndd ). Consider the differences

IM,x(n) = �(M/x(n)M

) − n1 . . . nd e(x;M),

JM,x(n) = n1 . . .nd e(x;M)− �(M/QM

(x(n)

))✩ This work is supported in part by the National Basis Research Programme in Natural Science of Vie* Corresponding author.

E-mail addresses:[email protected] (N.T. Cuong), [email protected] (L.T. Nhan).

0021-8693/03/$ – see front matter 2003 Elsevier Inc. All rights reserved.doi:10.1016/S0021-8693(03)00225-4

N.T. Cuong, L.T. Nhan / Journal of Algebra 267 (2003) 156–177 157

n

se of all

ial

ded that

l

any

de such–

basicacaulayy (re-caulay

a char-

–

as functions inn1, . . . , nd , wheree(x;M) is the multiplicity ofM with respect tox and

QM(x) =⋃t>0

((xt+1

1 , . . . , xt+1d

)M :xt

1 . . . xtd

).

It was proved in [CK] that�(M/QM(x(n))) is just the length of generalized fractio1/(xn1

1 , . . . , xndd ,1) defined by Sharp and Hamieh [SH]. Therefore, in general,IM,x(n)

andJM,x(n) are not polynomials forn1, . . . , nd large enough (see [GK,CMN]), but it istill nice since they are bounded above by polynomials. Especially, the least degrepolynomials inn bounding aboveIM,x(n) (respectivelyJM,x(n)) is independent of thechoice ofx, and it is denoted byp(M) (respectivelypf(M)). The invariantp(M) is calledthe polynomial typeof M (see [C2,CM]). If we stipulate the degree of the zero polynomis −∞, thenM is a Cohen–Macaulay module if and only ifp(M) = −∞, andM is ageneralized Cohen–Macaulay module if and only ifp(M) � 0. Recall that generalizeCohen–Macaulay modules had been introduced in [CST]. In that paper they showM is generalized Cohen–Macaulay if and only if�(H i

m(M)) < ∞ for all i = 1, . . . , d − 1,whereHi

m(M) is the ith local cohomology module ofM with respect to the maximaidealm. However, little is known about structure ofM whenp(M) > 0.

The purpose of this paper is to study modulesM which satisfypf(M) = −∞ orpf(M) � 0. Note that ifM is Cohen–Macaulay thenpf(M) = −∞, and ifM is generalizedCohen–Macaulay thenpf(M) � 0. However, the converse is not true. There are mmodulesM with pf(M) = −∞, butp(M) is large optionally. We will show that ifM isof pf(M) = −∞ or pf(M) � 0 then the properties ofM are still good and closely relateto that of Cohen–Macaulay modules or generalized Cohen–Macaulay modules. SincmodulesM are, so to speak,pseudo Cohen–Macaulayandpseudo generalized CohenMacaulay, respectively, it seems interesting to clarify such given modules.

The paper is divided into five sections. In Section 2, we first describe someproperties of pseudo Cohen–Macaulay modules and pseudo generalized Cohen–Mmodules. In particular, it follows that a finite direct sum of pseudo Cohen–Macaulaspectively pseudo generalized Cohen–Macaulay modules) is pseudo Cohen–Ma(respectively pseudo generalized Cohen–Macaulay). In the next section, we giveacterization of these modules as follows.

Theorem.LetR be a Noetherian local ring admitting a dualizing complex. Let0 = ⋂Ni ,

whereNi is pi -primary, be a reduced primary decomposition of the submodule0 of M. Set

N =⋂

dimR/pj=d

Nj .

Then the following statements are true.

(i) M is pseudo Cohen–Macaulay if and only ifM/N is a Cohen–Macaulay module.(ii) M is pseudo generalized Cohen–Macaulay if and only ifM/N is a generalized Cohen

Macaulay module.

158 N.T. Cuong, L.T. Nhan / Journal of Algebra 267 (2003) 156–177

giveodules

pect to

notionles istion 4,

notionofCohen–aulayon ofcaulayon ofed-fal rings

t

non-ve the

This result will be shown in Theorem 3.1. As corollaries of the theorem, weproperties of pseudo Cohen–Macaulay and pseudo generalized Cohen–Macaulay mpassing to reducing parameter element, relating to the monomial property, with resthe localization,. . . . The concept ofsequentially Cohen–Macaulay modulewas introducedby Stanley [St] for graded modules. In this paper, by the same way, we introduce thisfor the local case. It follows that the class of sequentially Cohen–Macaulay modustrictly contained in the class of pseudo Cohen–Macaulay modules. Therefore in Secwe are interested in properties of sequentially CM modules. We also introduce theof sequentially generalized Cohen–Macaulay modulesas an extension of the conceptsequentially Cohen–Macaulay modules. Note that the class of pseudo generalizedMacaulay modules also contain strictly all sequentially generalized Cohen–Macmodules. The main result of Section 5 is to give a cohomological characterizatisequentially Cohen–Macaulay modules and sequentially generalized Cohen–Mamodules. This characterization will be shown in Theorems 5.1 and 5.3. The notimodule of deficiencywas studied in [Sch2]. We will show in Proposition 5.6 the unmixness (unmixedness up tom-primary component) of thep(M)-th module of deficiency opseudo Cohen–Macaulay (pseudo generalized Cohen–Macaulay) modules over locadmitting a dualizing complex.

2. Pseudo Cohen–Macaulay and pseudo generalized Cohen–Macaulay modules

Throughout this paper, let(R,m) be a Noetherian local ring andM a finitely generatedR-module with dimM = d . Let x = (x1, . . . , xd) be a system of parameters ofM andn = (n1, . . . , nd) a set of positive integers. Set

IM,x(n) = �(M/

(xn11 , . . . , x

ndd

)M

) − n1 . . .nd e(x;M),

JM,x(n) = n1 . . .nd e(x;M)− �(M/QM

(x(n)

)),

wherex(n) = (xn11 , . . . , x

ndd ) and

QM(x) =⋃t>0

((xt+1

1 , . . . , xt+1d

)M :xn

1 · · ·xtd

).

We considerIM,x(n) and JM,x(n) as functions inn. It has been proved in [CK] tha�(M/QM(x(n))) is just the length of generalized fraction 1/(x

n11 , . . . , x

ndd ,1) defined

by Sharp and Hamieh [SH]. Therefore, in general, bothIM,x(n) and JM,x(n) are notpolynomials forn large enough (see [GK,CMN]), but these functions always takenegative values (see [C2,CM]) and bounded above by polynomials. Moreover, we hafollowing important property.

Theorem 2.1[C2,CM]. The following statements are true.

(i) The least degree of all polynomials inn bounding above the functionIM,x(n) isindependent of the choice ofx.


r

d

ess

s

ralized

.

(ii) The least degree of all polynomials inn bounding above the functionJM,x(n) isindependent of the choice ofx.

The least degree in Theorem 2.1(i) is calledpolynomial typeof M and denoted byp(M).The least degree in Theorem 2.1(ii) is denoted bypf(M).

Definition 2.2.(i) M is called apseudo Cohen–Macaulaymodule (pseudo CM module foshort) if pf(M) = −∞.

(ii) M is called apseudo generalized Cohen–Macaulaymodule (pseudo generalizeCM module for short) ifpf(M) � 0.

We denote byR them-adic completion ofR andM them-adic completion ofM. Thenwe have by [CM, 3.4] thatpf(M) = pf(M). Therefore the pseudo Cohen–Macaulaynand pseudo generalized Cohen–Macaulayness are preserved bym-adic completion.

Lemma 2.3.The following statements are true.

(i) M is pseudo CM if and only if so isM.(ii) M is pseudo generalized CM if and only if so isM.

Lemma 2.4.LetN be a submodule ofM such thatdimN < d . Then we have

(i) M is pseudo CM if and only ifM/N is pseudo CM.(ii) M is pseudo generalized CM if and only ifM/N is pseudo generalized CM.

Proof. Since dimN < d , we have AnnN � p for all p ∈ AssM with dimA/p = d . Thusthere exists a system of parametersx = (x1, . . . , xd) with x1 ∈ AnnN . PutM = M/N .Then it is easy to check thatM/QM(x(n)) ∼= M/QM(x(n)) for all sets of positive integern = (n1, . . . , nd). ThereforeJM,x(n) = JM,x(n). Thus pf(M) = pf(M) and the lemmafollows from the Definition 2.2. ✷Lemma 2.5.The following statements are true.

(i) A direct sum of finitely many pseudo CM modules is pseudo CM.(ii) A direct sum of finitely many pseudo generalized CM modules is pseudo gene

CM.

Proof. (i) It is enough to prove for a direct sum of two modules. LetM = M1⊕M2, whereM1 andM2 are pseudo CM. The case of dimM1 �= dimM2 follows easily from Lemma 2.4Suppose that dimM1 = dimM2. Let x be a system of parameters ofM. Thenx is also asystem of parameters ofM1 andM2. For any set of positive integersn = (n1, . . . , nd), it isclear thate(x(n);M)= e(x(n);M1)+ e(x(n);M2). Moreover, it is easily to check that

QM

(x(n)

) = QM1

(x(n)

) ⊕ QM2

(x(n)

).


e

telyseudo

d CM

y

y

ThereforeJM,x(n) = JM1,x(n) + JM2,x(n). SinceM1 andM2 are pseudo CM, we havJM1,x(n) = 0 andJM2,x(n) = 0, for all n. HenceJM,x(n) = 0, for all n. ThusM is pseudoCM.

(ii) Follows similarly by the proof of (i). ✷The following result of [CM, 3.6] gives us some vanishing (respectively fini

generated) properties of local cohomology modules for pseudo CM (respectively pgeneralized CM) modules.


(i) If M is pseudo CM thenHim(M) = 0, for all i = p(M)+ 1, . . . , d − 1.

(ii) If M is pseudo generalized CM then�(H im(M)) < ∞, for all i = p(M)+1, . . . , d −1.

3. A characterization of pseudo CM and pseudo generalized CM modules

The following characterization of pseudo CM modules and pseudo generalizemodules is the main result of this section.

Theorem 3.1.Let R be a Noetherian local ring admitting a dualizing complex andM

a finitely generatedR-module. Let0 = ⋂Ni , whereNi ispi -primary, be a reduced primar

decomposition of the submodule0 of M. Set

N =⋂

dimR/pj=d

Nj .


(i) M is pseudo CM if and only ifM/N is a Cohen–Macaulay module.(ii) M is pseudo generalized CM if and only ifM/N is a generalized Cohen–Macaula

module. Moreover, in this case

JM,x(n) = JM/N,x (n) =d−1∑i=1

(d − 1

i − 1

)�(Hi

m(M/N))

for all systems of parametersx andn � 0.

Proof. (ii) Suppose thatM is pseudo generalized CM. Since dimN < d , we have byLemma 2.4(ii) thatM/N is pseudo generalized CM. Therefore�(H i

m(M/N)) < ∞ forall i = p(M/N) + 1, . . . , d − 1 by Lemma 2.6(ii). Setai = Ann(H i

m(M/N)) for i =1, . . . , d − 1, a = a1 . . .ad−1, andp = p(M/N). We need to show thatp � 0. Supposethatp > 0. Then we obtain by [C2, 3.1] and [Sch1, 2.4.6] that

p = dimR/a = dimR/ap.


ely,.

ce

ters

est

ple,le 2]

t

t is

on

On the other hand, it is clear thatM/N is equidimensional, i.e. dimR/p = dimM/N , forall minimal prime idealsp ∈ SuppM/N . Moreover, for allp ∈ SuppM/N , we have

depthRp(M/N)p � min

{dimRp

(M/N)p,1}.

So,M/N satisfies Serre’s condition(S1). Therefore dimR/ap � p− 1 by [Sch1, 3.2.1]. Itgives a contradiction. Thusp � 0, i.e.M/N is generalized Cohen–Macaulay. Converssuppose thatM/N is generalized Cohen–Macaulay. ThenM/N is pseudo generalized CMBecause dimN < d , we have by Lemma 2.4(ii) thatM is pseudo generalized CM. SinM/N is generalized Cohen–Macaulay, the formula follows by [SH, 3.7].

(i) The case whered � 1 is trivial. Let d > 1. Suppose thatM is pseudo CM. ThenM/N is generalized Cohen–Macaulay by (ii). Therefore, for any system of paramexof M/N , we get

JM/N,x (n) =d−1∑i=1

(d − 1

i − 1

)�(Hi

m(M/N))

for all n � 0. SinceM/N is pseudo CM,JM/N,x(n) = 0 for all n. So,Him(M/N) = 0

for all i = 1, . . . , d − 1. Moreover, it is clear thatH 0m(M/N) = 0. Thus,M/N is Cohen–

Macaulay. Conversely, suppose thatM/N is Cohen–Macaulay. ThenM/N is pseudo CM.Because dimN < d , we have by Lemma 2.4(i) thatM is pseudo CM. ✷Remark 3.2. (i) The submoduleN of M defined in Theorem 3.1 is exactly the largsubmodule of dimension strictly less thand of M (see Lemma 4.4(i) for more details).

(ii) Theorem 3.1 is not true ifR does not possess a dualizing complex. For examlet R be the 2-dimension local domain considered by Nagata [N, Appendix, Examp(see also [FR]). It follows by [Sch2, 6.1] thatR/I is Cohen–MacaulayR-module, whereI is the largest submodule ofR of dimension at most 1. ThereforeR is pseudo CMby Lemma 2.4(i) and hence so isR by Lemma 2.3(i). Since AssR = {0}, the largestsubmodule ofR of dimension at most 1 is the zero ideal. But it is well known thaRis not a Cohen–Macaulay ring.

Since the complete ringR always admits a dualizing complex, the following resulan immediate consequence of Theorem 3.1.

Corollary 3.3. Let0 = ⋂Ni , whereNi is pi-primary, be a reduced primary decompositi

of the submodule0 of R-moduleM . Let

N =⋂

dimR/pj=d

Ni .


(i) M is pseudo CM if and only ifM/N is a Cohen–MacaulayR-module.


y

uchs-

eter

for all

).

(ii) M is pseudo generalized CM if and only ifM/N is a generalized Cohen–MacaulaR-module.

The notion of reducing parameter element was introduced by Auslander–Bbaum [AB]: a parameter elementx of M is calledreducingif x /∈ p, for all p ∈ AssM withdimR/p � d − 1. Note that ifM is generalized Cohen–Macaulay then every paramelement ofM is reducing. Moreover, ifM is Cohen–Macaulay then so isM/xM and ifM is generalized Cohen–Macaulay then so isM/xM for all parameter elementx of M. Inthe case of pseudo CM or pseudo generalized CM modules, this property still holdreducing parameter elements.

Corollary 3.4. Let x be a reducing parameter element ofM. Then we have

(i) If M is pseudo CM then so isM/xM.(ii) If M is pseudo generalized CM then so isM/xM.

Proof. Let N be the largest submodule ofM of dimension at mostd − 1. LetM = M/N .Then we have:

M/xM ∼= M

N + xM∼= M/xM

(N + xM)/xM. (∗)

Sincex is also a reducing parameter element ofM , it follows thatx is M-regular and ifdimN = d − 1 thenx is a parameter element ofN . It implies thatN ∩ xM = xN anddimN/xN = d − 2. Therefore we have

dim(N + xM

)/xM = dimN/

(N ∩ xM

) = dimN/xN = d − 2.

On the other hand, it is clear that dim(N + xM)/xM � d − 2, if dimN � d − 2. Thus inany case we have

dim(N + xM

)/xM � d − 2. (∗∗)

Now we prove (i). Suppose thatM is pseudo CM. ThenM is pseudo CM by Lemma 2.3(iThereforeM is Cohen–Macaulay by Theorem 3.1(i). It implies thatM/xM is Cohen–Macaulay. Therefore we have by(∗) that

M/xM

(N + xM)/xM

is Cohen–Macaulay and hence it is pseudo CM. Since dim((N + xM)/xM) � d − 2 by(∗∗), it follows by Lemma 2.4(i) thatM/xM is pseudo CM and hence so isM/xM byLemma 2.3(i).

(ii) Follows similarly to the proof of (i). ✷


ofofy doesexists

reter

s.

le

y

to

A system of parametersx = (x1, . . . , xd) of M is said to havethe monomial propertyif

xt1 · · ·xt

dM �(xt+1

1 , . . . , xt+1d

)M

for all t > 0. Note thatx has the monomial property if and only if�(M/QM(x)) �= 0.Therefore if M is pseudo CM then the monomial property holds for all systemparameters ofM. In [H], Hochster has conjectured that any system of parametersRhas the monomial property. He also showed that in general the monomial propertnot hold for modules, but it holds for high powers of system of parameters, i.e. therefor each system of parametersx = (x1, . . . , xd) of M a positive integern(x), which ingeneral depends onx, such that

(xn11

)t · · · (xndd

)tM �

((xn11

)t+1, . . . ,

(xndd

)t+1)M

for all n1, . . . , nd � n(x) andt � 0. Therefore it seems to be interesting to find a concuniform bound for such numbern(x). The following result is to solve this problem fopseudo generalized CM modules. Letq is anm-primary ideal ofR. A system of parameter(x1, . . . , xd) of M is said to be aweakq-sequenceif

(x1, . . . , xi−1)M :Mxi ⊆ (x1, . . . , xi−1)M :Mq for i = 1, . . . , d.

It was proved in [SV] that ifM is generalized Cohen–Macaulay then there exists anm-primary ideal q such that every system of parameters ofM is a weakq-sequence.

Corollary 3.5. Suppose thatM is pseudo generalized CM. LetN be the largest submoduof M of dimension at mostd − 1 andx = (x1, . . . , xd) a system of parameters ofM. Thenwe haveM/N is generalized Cohen–Macaulay. Letq be am-primary ideal such that eversystem of parameters ofM/N is a weakq-sequence. Ifxi ∈ mq for somei, thenx has themonomial property. In particular, ifM is Buchsbaum andxi ∈ m2 for somei, thenx hasthe monomial property. IfM is generalized Cohen–Macaulay andxi ∈ mI (M)+1 for somei,thenx has the monomial property, where

I (M) =d−1∑1=0

(d − 1

i

)�(Hi

m(M)).

Proof. Note that�(M/QM(x)) = �(M/QM(x)). Therefore we can assume thatR iscomplete andM = M, N = N . Let M = M/N . Then for any system of parametersx

of M, we have a surjection

f :M/Q(x;M)→ M/Q(x;M)

defined byf (m + Q(x;M)) = m + Q(x;M). Therefore, by Theorem 3.1, it is enoughprove for the case whereM is generalized Cohen–Macaulay. For allt � 0, we have by [T,3.5] that


entsbaum

nd al

al

seudoresult

,

(xt+1

1 , . . . , xt+1d

)M :Mxt

1 · · ·xtd = (x1, . . . , xd)M

+d∑

i=1

(x1, . . . , xi−1, xi+1, . . . , xd)M :Mq.

We claim that(x1, . . . , xd)M :q �= M. In fact, setM ′ = M/(x1, . . . , xi−1, xi+1, . . . , xd)M.

Then dimM ′ = 1. Suppose in contradiction that(x1, . . . , xd)M :q = M. Then qM ⊆(x1, . . . , xd)M. It implies thatqM ′ ⊆ xiM

′ ⊆ qmM ′. ThereforemqM ′ = qM ′ and henceqM ′ = 0 by Nakayama Lemma. So, dimM ′ � 0, a contradiction. It follows by the claimthat�(M/QM(x)) �= 0 and hencex has the monomial property. The remaining statemfollows from the well-known facts that every system of parameters of a Buchsmodule (respectively of a generalized Cohen–Macaulay module) is a weakm-sequence(respectively a weakmI (M)-sequence). ✷Remark 3.6. To get the monomial property for modules, in general, we can not fipower less strictly than that given in Corollary 3.5. In fact, letS = k[x, y] be the polynomiaring of two variables over a fieldk. Let m = (x, y)S, R = Sm andM = (x, y)R. Thenwe haveH 0

m(M) = 0, H 1m(M) ∼= k. ThereforeI (M) = 1 and henceM is Buchbaum. It

follows thatM is pseudo generalized CM and every system of parameters ofM is a weakm-sequence. However, the system of parameters(x, y) of M does not have the monomiproperty.

It is natural to ask that whether the pseudo Cohen–Macaulayness and pgeneralized Cohen–Macaulayness are preserved by localization? The followinggives a partial answer to this question.

Proposition 3.7.Suppose thatM is quasi-unmixed(i.e. M is equidimensional). Then thefollowing statements are true.

(i) If M is pseudo CM thenMq is pseudo CM for allq ∈ SuppM.(ii) If M is pseudo generalized CM thenMq is pseudo CM for allq ∈ SuppM\{m}.

Proof. It is clear that (i) follows from (ii). So we need only to prove (ii). LetN be thelargest submodule ofM of dimension at mostd − 1. Let q ∈ SuppM\{mR}. BecauseM ispseudo generalized CM, so isM . So, we get by Theorem 3.1(ii) thatM/N is generalizedCohen–Macaulay. Therefore,Mq/Nq is Cohen–Macaulay. SinceM is equidimensionalany minimal prime ideal of AssR M does not belong to AssR N . Therefore dimNq <

dimMq. SinceMq/Nq is also Cohen–Macaulay,Nq is the largest ofMq of dimensionat most dimMq − 1. So,Mq is pseudo CM by Theorem 3.1(i). Now letq ∈ SuppM\{m}and q an element of Ass(R/qR) such that dimR/q = dimR/q. Let f :Rq → Rq be thenatural homomorphism. Sincef is faithfully flat and dim(Mq) = dim(Mq), we can checkthatpf(Mq) = pf(Mq). Thus,Mq is pseudo CM. ✷


n

y

f

Corollary 3.7 is not true, even in the case thatR is a complete ring, whenM is notequidimensional.

Example 3.8.Let k � 1 be an integer. Then there exists a pseudo CM moduleM anda prime idealp ∈ SuppM such thatpf(Mp) = k. In this case,Mp is neither pseudo CMnor pseudo generalized CM.

Proof. First we assume that there exist finitely generatedR-modulesA, B with thefollowing properties:

(i) A is Cohen–Macaulay.(ii) B is of dimension at most dimA − 1.(iii) There existsp ∈ SuppB andp /∈ SuppA such thatpf(Bp) = k.

Then we setM = A ⊕ B. It follows by Lemma 2.4(i) thatM is pseudo CM. Sincep ∈ SuppB, p ∈ SuppM. Since p /∈ SuppA, we haveAp = 0 and henceMp = Bp.Thereforepf(Mp) = k > 0. Now we show the existence ofA andB as above. Letd � k+2be an integer andK a field. LetR be the formal power series ringK❏x1, . . . , xd, y, z, t❑of d + 3 variables overK. Let A = R/yR. ThenA is Cohen–Macaulay of dimensiod + 2. Let C = R/(z, t)R. ThenC is Cohen–Macaulay of dimensiond + 1. Let B =(x1, . . . , xd−k)C. ThenB is of dimensiond + 1. Let p = (x1, . . . , xd, z, t)R. Thenp ∈SuppB and p /∈ SuppA. We will prove thatpf(Bp) = k. SinceC is Cohen–Macaulaand ht(p/(z, t)R) = d , Cp is Cohen–Macaulay of dimensiond . It is clear thatBp =(x1, . . . , xd−k)Cp. ThereforeBp is of dimensiond andCp/Bp is Cohen–Macaulay odimensionk. From the exact sequence ofRp-modules

0 → Bp → Cp → Cp/Bp → 0,

we have

HipRp

(Bp) =

0, if i �= k + 1, i �= d;Hk

pRp(Cp/Bp), if i = k + 1;

HdpRp

(Cp), if i = d.

Therefore depth(Bp) = k + 1. Moreover, we have by [C1, 1.1] that

p(Bp) = maxi=0,...,d−1

{dim

(Rp/AnnRp

(Hi

pRp(Bp)

))} = k.

So, depth(Bp) > p(Bp). Thuspf(Bp) = k by [CM, 3.5]. ✷


t,se.

ration

se,

up

y

4. Sequentially Cohen–Macaulay and sequentially generalized Cohen–Macaulaymodules

The concept ofsequentially Cohen–Macaulay modulewas introduced by Stanley [Sp. 87] for graded modules (see also [HS]). Here we define this notion for the local ca

Definition 4.1.(i) A filtration 0 = M0 ⊂ M1 ⊂ · · · ⊂ Mt = M of submodules ofM is calledthedimension filtrationof M if Mi−1 is the largest submodule ofMi which has dimensionstrictly less than dimMi for all i = 1, . . . , t .

(ii) A filtration 0 = N0 ⊂ N1 ⊂ · · · ⊂ Nt = M of submodules ofM is said to be aCohen–Macaulay filtrationif

(a) Each quotientNi/Ni−1 is Cohen–Macaulay.(b) dimN1/N0 < dimN2/N1 < · · · < dimNt/Nt−1.

Definition 4.2. We say thatM is a sequentially Cohen–Macaulay module(sequentiallyCM module for short) if there exists a Cohen–Macaulay filtration ofM.

Similarly, we introduce the following notion.

Definition 4.3. (i) A filtration 0 = N0 ⊂ N1 ⊂ · · · ⊂ Nt = M of submodules ofM is saidto be ageneralized Cohen–Macaulay filtrationif

(a) Each quotientNi/Ni−1 is generalized Cohen–Macaulay.(b) dimN1/N0 < dimN2/N1 < · · · < dimNt/Nt−1.

(ii) We say that M is asequentially generalized Cohen–Macaulay module(sequentiallygeneralized CM module for short) if there exists a generalized Cohen–Macaulay filtof M.


(i) The dimension filtration always exists and it is unique. Moreover, let0 = M0 ⊂ M1 ⊂· · · ⊂ Mt = M be a dimension filtration ofM with dimMi = di . Then we have

Mi =⋂

dimR/pj>di

Nj ,

for all i = 1, . . . , t − 1, where0 = ⋂nj=1Nj is a reduced primary decomposition of0

in M with Nj is pj -primary for j = 1, . . . , n.(ii) Suppose thatM has a Cohen–Macaulay filtration. Then it is unique and in this ca

it is exactly the dimension filtration ofM.(iii) Suppose thatM has a generalized Cohen–Macaulay filtration. Then it is unique

to m-primary components, i.e. if0 = M0 ⊂ M1 ⊂ · · · ⊂ Mt ′ = M is the dimensionfiltration of M and0 = N0 ⊂ N1 ⊂ · · · ⊂ Nt = M is a generalized Cohen–Macaula


e

rian

ener-ohen–

mples

tially

filtration thent = t ′ and�(Mi/Ni) < ∞ for all i = 1, . . . , t −1. Therefore in this casthe dimension filtration is also a generalized Cohen–Macaulay filtration.

Proof. (i) The unique existence of the dimension filtration follows from the Noetheproperty ofM. Then we get the formula by [Sch2, 2.2].

(ii) Let 0 = M0 ⊂ M1 ⊂ · · · ⊂ Mt ′ = M be the dimension filtration ofM and 0= N0 ⊂N1 ⊂ · · ·Nt = M a Cohen–Macaulay filtration ofM. Since

dimN1/N0 < dimN2/N1 < · · · < dimNt/Nt−1,

we have dimNi−1 < dimNi , for all i = 1, . . . , t . ThereforeMt ′−1 ⊇ Nt−1. SinceM/Nt−1is Cohen–Macaulay, every submodule ofM/Nt−1 is zero or is of dimensiond . Thus, sincedim(Mt ′−1/Nt−1) < d, Mt ′−1 = Nt−1. Similarly, Mt ′−2 = Nt−2,Mt ′−3 = Nt−3, . . . .

Thereforet = t ′ andMi = Ni for all i = 0,1, . . . , t .(iii) Since M/Nt−1 is generalized Cohen–Macaulay, every submodule ofM/Nt−1 is

either of dimensiond or of finite length. Thus, since dim(Mt ′−1/Nt−1) < d , we have�(Mt ′−1/Mt−1) < ∞. Therefore, from the exact sequence

0 → Nt−1/Nt−2 → Mt ′−1/Nt−2 → Mt ′−1/Nt−1 → 0

and the notice thatNt−1/Nt−2 is generalized Cohen–Macaulay, we obtain thatMt ′−1/Nt−2is also generalized Cohen–Macaulay. Therefore every submodule ofMt ′−1/Nt−2 is eitherof dimension dimMt ′−1 or of finite length. Thus, since dim(Mt ′−2/Nt−2) < dimMt ′−1,�(Mt ′−2/Nt−2) < ∞. Continue this process, aftert steps we gett ′ = t, �(Mi/Ni) < ∞andMi/Ni−1 is generalized Cohen–Macaulay for alli = 1, . . . , t . Now, for all i = 1, . . . , t ,from the exact sequence

0 → Mi−1/Ni−1 → Mi/Ni−1 → Mi/Mi−1 → 0

with the notice thatMi/Ni−1 is generalized Cohen–Macaulay, it follows thatMi/Mi−1 is ageneralized Cohen–Macaulay module. So 0= M0 ⊂ M1 ⊂ · · · ⊂ Mt = M is a generalizedCohen–Macaulay filtration as required.✷

The simple examples of sequentially CM modules (respectively sequentially galized CM modules) are Cohen–Macaulay modules (respectively generalized CMacaulay modules). Especially, it follows by [G, 1.1,(1) ⇔ (4)] that any approximatelyCohen–Macaulay ring is sequentially CM. The following lemma produces many exaof sequentially CM modules and sequentially generalized CM modules.

Proposition 4.5.The following statements are true.

(i) A direct sum of finitely many sequentially CM modules is sequentially CM.(ii) A direct sum of finitely many sequentially generalized CM modules is sequen

generalized CM.


CMet

y

Proof. (i) By induction, it is enough to prove for a direct sum of two sequentiallymodules. LetM = M ′ ⊕ M ′′, whereM ′ and M ′′ are sequentially CM modules. LdimM = d . We prove by induction ond thatM is sequentially CM. Ifd � 1, it is trivial.

Let d > 1. Denote byN,N ′, andN ′′ respectively are the largest submodule ofM,M ′,andM ′′ which has dimension strictly less thand . ThenM ′/N ′ andM ′′/N ′′ are zero orCohen–Macaulay. We first claim thatN = N ′ ⊕ N ′′. In fact, since dimN ′ ⊕ N ′′ < d , wehaveN ⊇ N ′ ⊕N ′′. Let a ∈ N . Thena = b + c, whereb ∈ M ′ andc ∈ M ′′. If dimRb = d

then there existsp ∈ AssM ′ such that dimR/p = d andp = Ann(rb) for somer ∈ R.Thereforep ⊇ Ann(ra) and hence dimRa � d . It gives a contradiction becausea ∈ N .Thus dimRb < d . Similarly, dimRc < d . ThereforeRb ⊕ Rc ⊆ N ′ ⊕ N ′′. It follows thata ∈ N ′ ⊕ N ′′ and henceN = N ′ ⊕N ′′. The claim is proved.

Next, we proveM/N is Cohen–Macaulay. For a system of parametersx of M, sincedimN < d , dimN ′ < d , dimN ′′ < d , we have

e(x;M/N) = e(x;M) = e(x;M ′)+ e(x;M ′′) = e(x;M ′/N ′)+ e(x;M ′′/N ′′).

We have the exact sequence

0 → Ker(f ) → (M ′ ⊕ M ′′)/(

(xM ′ ⊕ xM ′′)+ N) f−→ (M ′′/N ′′)/x(M ′′/N ′′) → 0,

wheref (b + c) = c + N ′′, for all b ∈ M ′, c ∈ M ′′. Therefore

�((M ′ ⊕M ′′)

/((xM ′ ⊕ xM ′′)+ N

))� �

((M ′′/N ′′)/x(M ′′/N ′′)

) + �(Kerf ).

It is clear that Kerf = (M ′ ⊕ (xM ′′ + N ′′))/((xM ′ ⊕ xM ′′) + N). Moreover, we have asurjection

p : (M ′/N ′)/x(M ′/N ′) → (M ′ ⊕ (xM ′′ + N ′′)

)/((xM ′ ⊕ xM ′′)+ N

)which is defined byp(b + N ′) = b + 0, for all b ∈ M ′. Therefore we have

�((M/N)/x(M/N)

) = �((M ′ ⊕M ′′)

/((xM ′ ⊕ xM ′′)+ N

))� �

((M ′/N ′)/x(M ′/N ′)

) + �((M ′′/N ′′)/x(M ′′/N ′′)

)= e(x;M ′/N ′)+ e(x;M ′′/N ′′)

= e(x;M/N).

ThusM/N is Cohen–Macaulay. SinceN = N ′ ⊕ N ′′, N ′ andN ′′ are also sequentiallCM modules and dimN < d , we can apply induction hypothesis toN , and we get thatNis sequentially CM. ThereforeM is sequentially CM.

(ii) follows similarly to the proof of (i). ✷Lemma 4.6.LetM be a sequentially generalized CM module. ThenSuppM is a catenarysubset ofSpecR.


ii).

it

d

plex

Proof. It is clear that

SuppM =t⋃

i=1

SuppMi/Mi−1.

SinceMi/Mi−1 is generalized Cohen–Macaulay, it follows by [CST] that SuppMi/Mi−1is catenary for alli = 1, . . . , t . Therefore so is SuppM. ✷Proposition 4.7.The following statements are true.

(i) If M is sequentially CM then so isMp for all p ∈ SuppM.(ii) If M is sequentially generalized CM then for allp ∈ SuppM\{m}, Mp is sequen-

tially CM.

Proof. It is clear that (i) follows immediately by (ii). Therefore it is enough to prove (Let 0 = M0 ⊂ M1 ⊂ · · · ⊂ Mt = M be the dimension filtration ofM. Then, it followsfrom Lemma 4.4(iii) thatMi/Mi−1 is generalized Cohen–Macaulay for alli = 1, . . . , t .Let p ∈ SuppM\{m}. We claim that(Mt−1)p = Mp or dim(Mt−1)p < dimMp. In fact,suppose that(Mt−1)p �= Mp. Thenp ∈ SuppM/Mt−1. Therefore there existsq ∈ AssMsuch thatq ⊆ p and dimR/q = d . SinceM/Mt−1 is generalized Cohen–Macaulay,follows by Lemma 4.6 that

dim(Mt−1)p � dimMt−1 − dimR/p < d − dimR/p

= dimR/q − dimR/p = ht (p/q)� dimMp.

Continue this process we obtain(Mi−1)p = (Mi)p or dim(Mi−1)p < dim(Mi)p for alli = 1, . . . , t . Thus, from the family{(Mi)p}i=0,1,...,t of submodules ofMp, we can choosea Cohen–Macaulay filtration of submodules ofMp

0= (Mi0)p ⊂ (Mi1)p ⊂ · · · ⊂ (Mit1)p = Mp.

ThusMp is sequentially CM. ✷

5. Cohomological characterizations of sequentially CM and sequentially generalizeCM modules

Suppose thatR possesses a dualizing complex. Then there is a bounded comD•

R of injectiveR-modulesDiR whose cohomology modulesHi(D•

R), i ∈ Z, are finitelygeneratedR-modules. For a finitely generatedR-moduleM of dimM = d , the homologymodule

Ki(M) := H−i(Hom

(M,D•

R

))


1,

al

y

–

at

ll

tt

is also a finitely generatedR-module, for alli = 0,1, . . . , d . Note that the moduleKd(M)

is just the canonical module ofM. Following [Sch1], fori = 0,1, . . . , d − 1, the moduleKi(M) is calledith module of deficiencyof M. Moreover, by the local duality (see [Sch1.1]) there are following isomorphisms:

Him(M) ∼= Hom

(Ki(M),E

), ∀i,

whereE is the injective hull ofR/m. The two following theorems give a cohomologiccharacterization of sequentially CM and sequentially generalized CM modules.

Theorem 5.1.Let 0 = M0 ⊂ M1 ⊂ · · · ⊂ Mt = M be the dimension filtration ofM anddi = dimMi for i = 1, . . . , t . Suppose thatR possesses a dualizing complex.

(a) The following statements are equivalent:(i) M is sequentially CM.(ii) Mi is pseudo CM for alli = 1, . . . , t .(iii) For all j = 0,1, . . . , d the modulesKj(M) are either zero or Cohen–Macaula

of dimensionj .(iv) For all j = 0,1, . . . , d − 1 the modulesKj(M) are either zero or Cohen

Macaulay of dimensionj .(b) Suppose that M satisfies the equivalent conditions above. Then

di−1 = dimMi−1 = p(Mi)

for all i = 1, . . . , t , wherep(Mi) is the polynomial type ofMi .

Proof. (a): (i) ⇔ (ii). Let M be sequentially CM. Then we get by Lemma 4.4(ii) thMi/Mi−1 is Cohen–Macaulay for alli = 1, . . . , t . Therefore, by Lemma 2.4(i),Mi ispseudo CM for alli = 1, . . . , t . The converse follows immediately by Theorem 3.1(i).

(i) ⇒ (iii ). Let M be sequentially CM. ThenMi/Mi−1 is Cohen–Macaulay for ai = 1, . . . , t . It can be derived from the exact sequence

0 → Mt−1 → M → M/Mt−1 → 0

thatKd(M) ∼= Kd(M/Mt−1),Kj (M) = 0 for all j = dt−1 + 1, . . . , d − 1, andKj(M) ∼=

Kj(Mt−1) for all j = 0, . . . , dt−1. SinceM/Mt−1 is Cohen–Macaulay, it follows thaKd(M) is Cohen–Macaulay of dimensiond = dt . Similarly, by applying to the exacsequence

0→ Mt−2 → Mt−1/Mt−2 → 0

with notice thatMt−1/Mt−2 is Cohen–Macaulay, we haveKj(M) ∼= Kj(Mt−2) for allj = 0, . . . , dt−2, Kj (M) = 0 for all j = dt−2 + 1, . . . , dt−1 − 1, andKdt−1(M) is Cohen–Macaulay of dimensiondt−1. Continuing this process, we get the result.

(iii ) ⇒ (iv) is obvious.


ll

xact

ve

m 5.1ces andraded

les ofing

ory

someAny

f

(iv) ⇒ (i). We prove by induction ond that Mi/Mi−1 is Cohen–Macaulay for ai = 1, . . . , t . If d = 1, it is trivial. Letd > 1. It follows by [CK, 1.1] thatM is pseudo CM.So M/Mt−1 is Cohen–Macaulay by Theorem 3.1(i). Therefore we get from the esequence

0 → Mt−1 → M → M/Mt−1 → 0

that Ki(M) ∼= Ki(Mt−1) for all i = 1, . . . ,dimMt−1. It follows thatMt−1 satisfies thehypothesis of (iv). Since dimMt−1 < d , by applying the inductive assumption toMt−1,we obtain thatMi/Mi−1 is Cohen–Macaulay for alli = 1, . . . , t − 1. ThereforeM issequentially CM.

(b) Setaj = Ann(H jm(Mi)) for i = 1, . . . , t and j = 0, . . . , di − 1. We have by the

proof of (a),(i) ⇒ (iii ), thatKj(Mi) = 0 for all j = di−1 + 1, . . . , di − 1 andKdi−1(Mi)

is Cohen–Macaulay of dimensiondi−1. Therefore, by [C1, 1.1] and [Sch1, 2.2.4], we ha

p(Mi) = maxj=0,...,di−1

dimR/aj = maxj=0,...,di−1

dimKj(Mi) = di−1

for all i = 1, . . . , t . ✷It should be noted that the equivalences of statements (i), (iii), and (iv) of Theore

has been shown by P. Schenzel [Sch2, Theorem 5.5] by using spectral sequen(i) ⇔ (iii ) have been proved by Herzog–Sbarra [HS, Theorem 1.4] for a standard gCohen–Macaulayk-algebraR.

The following consequence gives us the structure of local cohomology modua sequentially CM module. Note that in this corollaryR does need to possess a dualizcomplex.

Corollary 5.2. Let M be a sequentially CM module and0 = M0 ⊂ M1 ⊂ · · · ⊂ Mt = M

the dimension filtration ofM. Setdi = dimMi andaj = Ann(H jm(M)) for j = 0,1, . . . , d .

ThenHjm(M) = 0 if and only ifj /∈ {d1, . . . , dt} anddimR/aj = j for all j ∈ {d1, . . . , dt}.

Proof. Denote byN the m-adic completion of a moduleN . Then 0= M0 ⊂ M1 ⊂· · · ⊂ Mt = M is clearly a Cohen–Macaulay filtration ofM . It follows by applyingTheorem 5.1(b) forM that Hj

m(M) = 0 and thereforeHjm(M) = 0 if j /∈ {d1, . . . , dt}.

On the other hand, by the proof of(i) ⇒ (iii ) in Theorem 5.1, we get for allj = 1, . . . , d

thatHdjm (M) ∼= H

djm (Mj ). Then it implies from the basic facts of local cohomology the

that dimR/adj = dj , as required. ✷To give a characterization of sequentially generalized CM modules, we need

facts of the theory of secondary representation of Artinian modules from [M,SH]:Artinian R-moduleA has aminimal secondary representationA = A1 + · · · + An ofpi -secondary submodulesAi . The set{p1,p2, . . . ,pn} is independent of the choice ominimal representation ofA and is denoted by AttR A. From now on, for any positiveintegerj we set(Att A)j = {p ∈ Att A: dimR/p = j }. Note that�(A/mnA) is finite and


d

r

rii).

atimi-

f

we

independent ofn whenn large. Therefore we denote this length byRl(A) for n large. It isclear that ifx ∈ m andx /∈ p for all p ∈ Att A\{m} then�(A/xnA) = Rl(A) for n large.

Theorem 5.3.Let 0 = M0 ⊂ M1 ⊂ · · · ⊂ Mt = M be the dimension filtration ofM anddi = dimMi for i = 1, . . . , t . Suppose thatR possesses a dualizing complex. Then

(a) The following statements are equivalent:(i) M is sequentially generalized CM.(ii) Mi is pseudo generalized CM for alli = 1, . . . , t .(iii) For all j = 1, . . . , d , the modulesKj(M) are either of finite length or generalize

Cohen–Macaulay of dimensionj .(iv) For all j = 1, . . . , d − 1, the modulesKj(M) are either of finite length o

generalized Cohen–Macaulay of dimensionj .(b) Suppose thatM satisfies the equivalent conditions above. Then

di−1 = dimMi−1 = p(Mi),

wherep(Mi) is the polynomial type ofMi for all i = 1, . . . , t .

Proof. (a): (i) ⇔ (ii). Assume thatM is sequentially generalized CM. ThenMi/Mi−1 isgeneralized Cohen–Macaulay by Lemma 4.4(iii). Thus,Mi is pseudo generalized CM foall i = 1, . . . , t by Lemma 2.4(ii). The converse follows immediately by Theorem 3.1(

(i) ⇒ (iii ). Suppose thatM is sequentially generalized CM. ThenMi/Mi−1 is gen-eralized Cohen–Macaulay for alli = 1, . . . , t . We claim thatKdi (Mi/Mi−1) is gener-alized Cohen–Macaulay for alli = 1, . . . , t . In fact, let p ∈ SuppKdi (Mi/Mi−1)\{m}.Then p ∈ Supp(Mi/Mi−1)\{m}. Therefore we have by [Sch1, 2.2.3] and [CK] th(Kdi (Mi/Mi−1))p is Cohen–Macaulay and therefore the claim follows by [CST]. Slarly to the proof of Theorem 5.1,(i) ⇒ (iii ), and by the claim, it follows thatKj(M) isgeneralized Cohen–Macaulay of dimensionj for all j = d1, . . . , dt , and�(Kj (M)) < ∞for all j /∈ {d1, . . . , dt }.

(iii ) ⇒ (iv) is trivial.(iv) ⇒ (i). Let a(M) = a0(M)a1(M) . . .ad−1(M), whereai (M) = AnnHi

m(M), i =0, . . . , d − 1. Then there exists by [C1] a system of parametersx = (x1, . . . , xd) of M suchthatxd ∈ a(M) andxi ∈ a(M/(xi+1, . . . , xd)M), for all i = 1, . . . , d −1. A such system oparameters is called ap-standard system of parameters.First of all we show the followingclaim.

Claim. Let x = (x1, . . . , xd) be ap-standard system of parameters ofM. ThenJM,x(n) isbounded above by a constant for alln = (n1, . . . , nd).

Proof of the claim. Let n = (n1, . . . , xd) be a set of positive integers. For shortput Mj = M/(x

n11 , . . . , x

njj )M, xj = (xj+1, . . . , xd), and nj = (nj+1, . . . , nd) for all

j = 1, . . . , d − 1. It follows by [C1, 3.4] and [CM, 2.5] that�(0 :Mj−1 xnj) < ∞ for all j .
j


ct

lay

at

Thereforexj /∈ p for all p ∈ AssMj−1 \ {m}. Hencexj /∈ p for all p ∈ (Att(H im(Mj−1)))i .

So by the Matlis duality (see [BS, 10.2.20]),

(Ass

(0 :Ki(Mj−1)

xnjj

))i= (

Att(Hi

m(Mj−1))/

xnjj H i

m(Mj−1))i= ∅.

Hence

dim(0 :Ki(Mj−1)

xnjj

)� i − 1 (1)

for all i = 1, . . . , d − j + 1. First, we prove by induction onj thatKi(Mj ) is either offinite length or generalized Cohen–Macaulay of dimensioni, for all i = 1, . . . , d − j . Thecase wherej = 0 follows by the hypothesis. Letj > 0. From the exact sequences

0 → 0 :Mj−1 xnjj → Mj−1 → Mj−1/0 :Mj−1 x

njj → 0,

0 → Mj−1/0 :Mj−1 xnjj

xnjj−→Mj−1 → Mj → 0

with notice that�(0 :Mj−1 xnjj ) < ∞, we get by the local duality the following exa

sequence:

0 → Ki+1(Mj−1)/xnjj Ki+1(Mj−1) → Ki(Mj ) → 0 :Ki(Mj−1)

xnjj → 0 (2)

for i = 1, . . . , d − j . By induction hypothesis, either�(Ki(Mj−1)) < ∞ or Ki(Mj−1) isgeneralized Cohen–Macaulay of dimensioni. Therefore any submodule ofKi(Mj−1) iseither of finite length or is of dimensioni. It follows by (1) that�(0 :Ki(Mj−1)

xnjj ) < ∞ and

therefore by the inductive hypothesis thatKi+1(Mj−1)/xnjj Ki+1(Mj−1) is generalized

Cohen–Macaulay. HenceKi(Mj) is either of finite length or generalized Cohen–Macauof dimensioni for all i = 1, . . . , d − j by (2). On the other hand, sinceJMj ,xj (nj ) =JMj/H

0m(Mj ),xj

(nj ), we can assume thatxj+1 is non-zero-divisor ofMj . Therefore it canbe derived by [CM, 2.1] that

JMj ,xj (nj ) � JMj+1,xj+1(nj+1)+ Rl(H

d−j−1m (Mj )

)for all j = 0, . . . , d − 1. Note thatJMd−1,(xd)(nd) = 0. Therefore we have

JM,x(n) �d−1∑j=1

Rl(H

jm(Md−j−1)

).

Next, sinceRl(H im(Mj )) = �(H 0

m(Ki(Mj ))), the claim is proved if we can show th�(Hk

m(Ki(Mj ))) is bounded above by a constant for allj = 1, . . . , d − 1, i = 1, . . . ,


ing
d − j − 1, andk = 0, . . . , i − 1. Indeed, by the exact sequence (2), we get the followexact sequence:
Hkm

(Ki+1(Mj−1)

/xnjj Kj+1(Mj−1)

) → Hkm

(Ki(Mj )

) → Hkm

(0 :Ki(Mj−1)

xnjj

),

for all k = 0, . . . , i − 1. Since�(0 :Ki(Mj−1)xnjj ) < ∞, H k

m(0 :Ki(Mj−1)xnjj ) = 0 for all

k > 0 and

H 0m

(0 :Ki(Mj−1)

xnjj

) = (0 :Ki(Mj−1)

xnjj

) ⊆ H 0m

(Ki(Mj−1)

).

Therefore we obtain

�(Hk

m

(Ki(Mj)

))� �

(Hk

m

(Ki+1(Mj−1)

/xnjj Ki+1(Mj−1)

))(3)

for k > 0 and

�(H 0

m

(Ki(Mj)

))� �

(H 0

m

(Ki+1(Mj−1)

/xnjj Ki+1(Mj−1)

))+ �

(H 0

m

(Ki(Mj−1)

)). (4)

From the short exact sequence

0 → Ki+1(Mj−1)/

0 :Ki(Mj−1)xnjj → Ki+1(Mj−1)

→ Ki+1(Mj−1)/xnjj Ki+1(Mj−1) → 0

we get the following exact sequence:

Hkm

(Ki+1(Mj−1)

) → Hkm

(Ki+1(Mj−1)

/xnjj Ki+1(Mj−1)

) → Hk+1m

(Ki+1(Mj−1)

)for all k = 0, . . . , i − 1. Thus, with the observation that�(0 :Ki(Mj−1)

xnjj ) < ∞ and

Ki+1(Mj−1) is generalized Cohen–Macaulay, we have by (3), (4) that

�(Hk

m

(Ki(Mj)

))� �

(Hk

m

(Ki+1(Mj−1)

)) + �(Hk+1

m

(Ki+1(Mj−1)

))+ �

(H 0

m

(Ki(Mj−1)

))for all j = 1, . . . , d − 1, i = 1, . . . , d − j − 1, andk = 0, . . . , i − 1. Then the result followseasily by induction onj .

Now we continue to prove Theorem 5.3. We prove by induction ond that M issequentially generalized CM. Ifd = 1, it is trivial. Supposed > 1. It follows by the claimand Theorem 2.1(ii) thatM is pseudo generalized CM. ThereforeM/Mt−1 is generalizedCohen–Macaulay by Theorem 3.1(ii). From the exact sequence

0 → Mt−1 → M → M/Mt−1 → 0


s.

gay(b)

local

t

by

rs

we get the exact sequences

Kj(M/Mt−1) → Kj(M) → Kj(Mt−1) → Kj−1m (M/Mt−1)

for all j = 1, . . . , d − 1. Let Nj be the kernel of the mapKj(M/Mt−1) → Ki(M) andPj be the image of the mapKj(Mt−1) → Kj−1(M/Mt−1) in the above exact sequenceThenNj andPj are of finite length. Therefore from the exact sequence

0→ Kj(M)/Nj → Kj(Mt−1) → Pj → 0

for all i = 1, . . . , d−1, we can check thatMt−1 satisfies the hypothesis of (iv). By applyinthe induction assumption toMt−1, the modulesMi/Mi−1 is generalized Cohen–Macaulfor all i = 1, . . . , t − 1. ThereforeM is sequentially generalized CM. The statementfollows similarly to the proof of Theorem 5.1(b).✷

Analogous to Corollary 5.2, we get the following consequence about thecohomology modules of a sequentially generalized CM module.

Corollary 5.4. Suppose thatM is a sequentially generalized CM module and0 =M0 ⊂ M1 ⊂ · · · ⊂ Mt = M the dimension filtration ofM. Setdi = dimMi and aj =Ann(H j

m(M)) for j = 0,1, . . . , d . Then�(Hjm(M)) < ∞ if and only if j /∈ {d1, . . . , dt }

anddimR/aj = j for all j ∈ {d1, . . . , dt }.

The following example show that Theorem 5.3 is not true in general ifR does not admia dualizing complex.

Example 5.5.There exists a finitely generatedR-moduleM such thatM is not sequentiallygeneralized CM, but the dimension filtration 0= M0 ⊂ M1 ⊂ · · · ⊂ Mt = M has propertiesMi is pseudo CM for alli = 1, . . . , t .

Proof. Denote by(A,m) the Noetherian local domain of dimension 2 constructedD. Ferrand and M. Raynaud in [FR] for which them-adic completionA of A hasan associated primeq of dimension 1 (see also [N, Appendix, Example 2]). LetR =a❏x1, . . . , xd❑ be the ring of formal series of variablesx1, . . . , xd over A. Let Mi =R/(xi, . . . , xd)R, i = 1, . . . , d . Let M = M1 ⊕ M2 ⊕ · · · ⊕ Md ⊕ R andN0 = 0, Ni =M1 ⊕M2 ⊕ · · · ⊕ Mi for i = 1, . . . , d , andNd+1 = M. Then dimM = d + 2 and

0= N0 ⊂ N1 ⊂ N2 ⊂ · · · ⊂ Nd ⊂ Nd+1 = M

is the dimension filtration ofM. We haveNi/Ni−1 ∼= Mi∼= A❏x1, . . . , xi−1❑. Let (f, g)

be a system of parameters ofA. Thenz = (f, g, x1, . . . , xi−1) is a system of parameteof Mi . Sincex1, . . . , xi−1, f is a regular sequence ofMi , it follows that JMi,z(n) = 0for all n = (nf ,ng, n1, . . . , ni−1). ThereforeMi is pseudo CM for alli = 1, . . . , d + 1.However,Ni/Ni−1 is never generalized Cohen–Macaulay. ThereforeM is not sequentiallygeneralized CM. ✷


d

l

.at

]

by

l

e

ions,

lizing

Now we study the unmixedness ofp(M)-th module of deficiency of pseudo CM anpseudo generalized CM modules.

Proposition 5.6.Suppose thatR has a dualizing complex. Letp = p(M) be the polynomiatype ofM. Then we have

(i) If M is pseudo CM thenKp(M) is unmixed.(ii) If M is pseudo generalized CM thenKp(M) is unmixed up to anm-primary

component.

Proof. Let 0= M0 ⊂ M1 ⊂ · · · ⊂ Mt = M be the dimension filtration ofM. We prove (i).SinceM is pseudo CM, it follows by Theorem 3.1(i) thatM/Mt−1 is Cohen–MacaulayThereforeKi(M) ∼= Ki(Mt−1) for all i < d . Therefore we get by Lemma 2.6(i) thKi(Mt−1) = 0 for all i = p+1, . . . , d −1. Hence dimMt−1 � p. Fori = 0, . . . , d −1, weget by [Sch1, 2.2.4] that dimKi(M) � i for all i = 0, . . . , d − 1. Therefore, by [C1, 1.1and Lemma 2.6(i), we have

p = maxi=0,...,d−1

dimKi(M) = maxi=0,...,p

dimK1(Mt−1) � p.

It implies that dimMt−1 = p and henceKp(Mt−1) is unmixed. ThusKp(M) is unmixed.We prove (ii). The case wherep(M) � 0 is trivial. Assume thatp(M) > 0. Since

M is pseudo generalized CM, the moduleM/Mt−1 is generalized Cohen–MacaulayTheorem 3.1(ii). Therefore from the exact sequence 0→ Mt−1 → M → M/Mt−1 → 0,we get the exact sequences

0 → Ki(M)/Qi → Ki(Mt−1) → Pi → 0

for all i < d , where�(Qi) < ∞ and �(Pi) < ∞. So, dimKi(M) = dimKi(Mt−1) forall i < d . Therefore we have by Lemma 2.6(ii) thatKi(Mt−1) has finite length for ali = p + 1, . . . , d − 1. It follows that dimMt−1 � p. Sincep(M) > 0, it follows by [Sch1,2.2.4] and [C1, 1.1] that dimMt−1 = p. ThereforeKp(Mt−1) is unmixed, and hencKp(M) is unmixed up to anm-primary component. ✷

The following result follows immediately by the proof of Proposition 5.6.

Corollary 5.7. Let M be pseudo generalized CM andN the largest submodule ofM ofdimension at mostd − 1. Then we havep(M) = dimN .

References

[AB] M. Auslander, D.A. Buchsbaum, Codimension and multiplicity, Ann. of Math. 68 (1958) 625–657.[BS] M. Brodmann, R.Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applicat

Cambridge Univ. Press, 1998.[C1] N.T. Cuong, On the dimension of the non-Cohen–Macaulay locus of local ring admitting dua

complexes, Math. Proc. Cambridge Philos. Soc. (2) 109 (1991) 479–488.


ths and4.atlis

y, and

type,

urier

hr. 85

Norm.

tem,

(1973)

(1973)

Notes

Ferrara.1985)

6.

(1986)

[C2] N.T. Cuong, On the least degree of polynomials bounding above the differences between lengmultiplicities of certain systems of parameters in local rings, Nagoya Math. J. 125 (1992) 105–11

[CK] N.T. Cuong, V.T. Khoi, Modules whose local cohomology modules have Cohen–Macaulay Mduals, in: D. Eisenbud (Ed.), Proc. of Hanoi Conf. on Commutative Algebra, Algebraic GeometrComputational Methods, Springer-Verlag, 1999, pp. 223–231.

[CM] N.T. Cuong, N.D. Minh, Lengths of generalized fractions of modules having small polynomialMath. Proc. Cambridge Philos. Soc. (2) 128 (2000) 269–282.

[CMN] N.T. Cuong, M. Morales, L.T. Nhan, Lengths of generalized fractions, Prépublication de l’Institut Fon◦ 539, 2001.

[CST] N.T. Cuong, P. Schenzel, N.V. Trung, Verallgemeinerte Cohen–Macaulay Moduln, Math. Nac(1978) 57–75.

[FR] D. Ferrand, M. Raynaund, Fibres formelles d’un anneau local noetherian, Ann. Sci. ÉcoleSup. 3 (4) (1970) 295–311.

[G] S. Goto, Approximately Cohen–Macaulay rings, J. Algebra 76 (1981) 214–225.[GK] J.-L. Garcia Roig, D. Kirby, On the Koszul homology modules for the powers of a multiplicity sys

Mathematika 33 (1986) 96–101.[H] M. Hochster, Contracted ideals from integral extensions of regular rings, Nagoya Math. J. 51

25–43.[HS] J. Herzog, E. Sbarra, Sequentially Cohen–Macaulay modules and local cohomology, Preprint.[M] I.G. Macdonald, Secondary representation of modules over a commutative ring, Sympos. Math. 11

23–43.[N] M. Nagata, Local Rings, Interscience, New York, 1962.[Sch1] P. Schenzel, Dualisierende Komplexe in der lokalen Algebra und Buchsbaum Ringe, in: Lecture

in Math., Vol. 907, Springer-Verlag, Berlin, 1982.[Sch2] P. Schenzel, On the dimension filtration and Cohen–Macaulay filtered modules, in: Proc. of the

Meeting in honour of Mario Fiorentini, University of Antwerp, Wilrijk, Belgium, 1998, pp. 245–264[SH] R.Y. Sharp, M.A. Hamieh, Lengths of certain generalized fractions, J. Pure Appl. Algebra 38 (

323–336.[St] R.P. Stanley, Combinatorics and Commutative Algebra, Second edition, Birkhäuser Boston, 199[SV] J. Stückrad, W. Vogel, Buchsbaum Rings and Applications, Springer-Verlag, Berlin, 1986.[T] N.V. Trung, Toward a theory of generalized Cohen–Macaulay modules, Nagoya Math. J. 102

1–49.

Pseudo Cohen–Macaulay and pseudo generalized Cohen–Macaulay modules

Documents

Transcript of Pseudo Cohen–Macaulay and pseudo generalized Cohen–Macaulay modules