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On Generalized Cohen–Macaulay Canonical ModulesNguyen Thi Hong Loan a & Le Thanh Nhan ba Vinh University , Nghe An , Vietnamb Thai Nguyen College of Sciences , Thai Nguyen , VietnamPublished online: 23 Sep 2013.

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Communications in Algebra®, 41: 4453–4462, 2013Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927872.2012.703737

ON GENERALIZED COHEN–MACAULAYCANONICAL MODULES

Nguyen Thi Hong Loan1 and Le Thanh Nhan21Vinh University, Nghe An, Vietnam2Thai Nguyen College of Sciences, Thai Nguyen, Vietnam

Let �R��� be a Noetherian local ring which is a quotient of a Gorenstein local ring. Afinitely generated R-module M is called generalized Cohen–Macaulay canonical if thecanonical module K�M� of M is generalized Cohen–Macaulay. In this paper, we givecharacterizations of generalized Cohen–Macaulay canonical modules in term of certainsystems of parameters.

Key Words: Generalized Cohen–Macaulay canonical modules; Residual length; Strict f-sequences.

2000 Mathematics Subject Classification: 13D45; 13H10; 13E05.

1. INTRODUCTION

Throughout this paper, let �R��� be a Noetherian local ring which is aquotient of a n-dimensional Gorenstein local ring �R′��′�. Let M be a finitelygenerated R-module with dimM = d. Denote by Ki�M� the R-module Extn−i

R′ �M�R′�.Then Ki�M� is a finitely generated R-module. Following Schenzel [9], Ki�M� is calledthe ith deficiency module of M for i = 0� � � � � d − 1, and K�M� = Kd�M� is called thecanonical module of M .

Schenzel [9] introduced the notion of Cohen–Macaulay canonical modules: Mis called Cohen Macaulay canonical if the canonical module K�M� of M is CohenMacaulay. The Cohen–Macaulay canonical property is related to some importantquestions. For example, if M is Cohen–Macaulay canonical, then the monomialconjecture raised by Hochster [5] is valid for the ring R/AnnRM . Furthermore, ifR is a domain, then R is Cohen–Macaulay canonical if and only if R posseses abirational Macaulayfication R1, i.e., an extension ring R ⊆ R1 ⊆ Q (where Q is thefield of fractions) such that R1 is finitely generated as an R-module and R1 is aCohen–Macaulay ring, cf. [9, Theorem 1.1].

For a system of parameters (s.o.p. for short) x = �x1� � � � � xd� of M , set

I�x�M� = �R�M/�x1� � � � � xd�M�− e�x�M��

Received April 14, 2012. Communicated by I. Swanson.Address correspondence to Prof. Nhan Le, Thai Nguyen College of Sciences, Thai Nguyen,

Vietnam; E-mail: trtrnhan@yahoo.com

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4454 NGUYEN AND LE

where e�x�M� is the multiplicity of M with respect to x� It is well known that Mis Cohen–Macaulay if and only if I�x�M� = 0 for all s.o.p x� This is the motivationof the following characterization of Cohen–Macaulay canonical modules in term ofsystems of parameters given in Theorem 4.1 of [7]: M is Cohen–Macaulay canonical

if and only ifd−3∑k=0

Rl(Hd−k−1

� �M/�x1� � � � � xk�M�) = 0 for all s.o.p. �x1� � � � � xd� of M

which are strict f-sequences in sense of Cuong-Morales-Nhan [3]. Here, Rl�−�denotes for the residual length defined by Sharp–Hamieh [11].

The notion of generalized Cohen–Macaulay modules was introduced byCuong–Schenzel–Trung [4]: M is generalized Cohen–Macaulay if there exists anumber I�M� such that I�x�M� < I�M� for all s.o.p x of M . Note that M isgeneralized Cohen–Macaulay if and only if ��Hi

��M�� < � for all i < d� By anobvious way, we say that M is generalized Cohen–Macaulay canonical if the canonicalmodule K�M� of M is generalized Cohen–Macaulay. It is clear that if d � 3 thenM is generalized Cohen–Macaulay canonical. The aim of this paper is to givecharacterizations of generalized Cohen–Macaulay canonical modules in term ofsystems of parameters for the case where d ≥ 4�

For a s.o.p. x = �x1� � � � � xd� of M , set Mx�k = M/�x1� � � � � xk�M for k =1� � � � � d − 3�

Main Theorem. Let d ≥ 4� The following statements are equivalent:

(i) M is generalized Cohen–Macaulay canonical;

(ii) There exists a number c�M� such thatd−3∑k=1

Rl(Hd−k−1

� �Mx�k�)� c�M� for all s.o.p. x

of M which are strict f-sequences;(iii) There exists a s.o.p. x = �x1� � � � � xd� of M which is a strict f-sequence and a

number c�x�M� such thatd−3∑k=1

Rl(Hd−k−1

� �M/�xn11 � � � � � x

nkk �M�

)� c�x�M� for all

n1� � � � � nd−3 ∈ ��

Furthermore, if (i), (ii), (iii) are satisfied, then

d−3∑k=1

Rl�Hd−k−1� �Mx�k�� �

d−3∑k=1

k∑i=0

(ki

)��Hi+2

� �K�M���

for all s.o.p. x of M which are strict f-sequences. The equality holds when x ⊆ �2d−4r �where

r = min�t ∈ � � �tHi��K�M�� = 0 for all i < d��

In Section 2, we outline some properties of strict f-sequences which will beused later. The proof of the main theorem is given in the last section.

2. STRICT f-SEQUENCES

The theory of secondary representation was introduced by Macdonald [6]which is in some sense dual to that of primary decomposition for Noetherianmodules. Note that every Artinian R-module A has a minimal secondary

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ON GENERALIZED COHEN–MACAULAY CANONICAL MODULES 4455

representation A = A1 + · · · + An� where Ai is �i-secondary. The set ��1� � � � � �n� isindependent of the choice of the minimal secondary representation of A. This set iscalled the set of attached primes of A� and denoted by AttRA.

Definition 2.1 ([3]). An element x ∈ � is called a strict filter regular element (strictf-element for short) with respect to M if x � � for all � ∈ ⋃d

i=1 AttRHi��M�\���. A

sequence �x1� � � � � xt� of elements in � is called a strict filter regular sequence (strictf-sequence for short) with respect to M if xi+1 is a strict f-element with respect toM/�x1� � � � � xi�M for every i = 0� � � � � t − 1.

It should be mentioned that Cuong–Morales–Nhan [3] used strict f-sequencesas a useful tool to study the finiteness for attached primes of local cohomologymodules, to study the polynomial property of the length of generalized fractionsdefined in [11], and to characterize the pseudo generalized Cohen–Macaulaymodules defined in [2]. In [7], strict f-sequences were used to study the monomialconjecture raised by Hochster [5] and to characterize Cohen–Macaulay canonicalmodules.

Lemma 2.2 ([3, Lemma 3.4]). If �x1� � � � � xt� is a strict f-sequence w.r.t. M , then sois �xn11 � � � � � x

ntt � for all positive integers n1� � � � � nt.

Following Cuong–Schenzel–Trung [4], an element x ∈ � is called a filterregular element with respect to M if x � � for all � ∈ AssRM\���� A sequence�x1� � � � � xt� of elements in � is called a filter regular sequence with respect toM if xi+1 is a filter regular element with respect to M/�x1� � � � � xi�M for alli = 0� � � � � t − 1.

Lemma 2.3. Each strict f-sequence w.r.t. M is a filter regular sequence w.r.t. M . Inparticular, if x ∈ � is a strict f-element w.r.t. M then ��0 �M x� < ��

Proof. By [1, 11.3.3], AssRM ⊆ ⋃di=0 AttRH

i��M�� Therefore, if x is a strict f-element

w.r.t. M , then x is a filter regular element w.r.t. M . So, the result follows. �

It is clear by Prime Avoidance that for each integer t > 0, there exists a strictf-sequence w.r.t. M of length t. Also, by Lemma 2.3, if �x1� � � � � xt� is a strict f-sequence w.r.t. M and t � d, then it is a part of a system of parameters of M .

From now on, let R be a quotient of a n-dimensional Gorenstein local ring�R′��′�� Denote by Ki�M� the R-module Extn−i

R′ �M�R′�� Then Ki�M� is a finitelygenerated R-module. Following Schenzel [8, Def. 2.1], Ki�M� is the ith deficiencymodule of M for i < d, and K�M� = Kd�M� is the canonical module of M�

Lemma 2.4. An element x ∈ � is a strict f-element w.r.t. M if and only if x is a filterregular element w.r.t. Ki�M� for all i. In particular, if d > 0 and x is a strict f-elementw.r.t. M , then x is K�M�-regular and ��0 �Ki�M� x� < � for all i < d.

Proof. By the local duality (cf. [1, 11.2.6]), we have an isomorphism

Hi��M� � HomR�K

i�M�� E�R/����

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4456 NGUYEN AND LE

where E�R/�� is the injective hull of R/�� Therefore we get by [10, Theorem 2.3]that AssRK

i�M� = AttRHi��M� for all i. Now the result follows. �

Lemma 2.5. Let x ∈ � be a strict f-element w.r.t. M . Then we have an exact sequence

0 −→ Ki+1�M�/xKi+1�M� −→ Ki�M/xM� −→ �0 �Ki�M� x� −→ 0

for each integer i ≥ 1. In particular, Hi�

(K�M�/xK�M�

) � Hi�

(K�M/xM�

)for any

i ≥ 2.

Proof. Since ��0 �M x� < � by Lemma 2.3, we have Hi��M/�0 �M x�� � Hi

��M�

for all i ≥ 1� Therefore, from the exact sequence 0 −→ M/�0 �M x�x−→ M −→

M/xM −→ 0, we have the exact sequence for all i ≥ 1

0 −→ Hi��M�/xHi

��M� −→ Hi��M/xM� −→ �0 �Hi+1

� �M� x� −→ 0�

By local duality (cf. [1, 11.2.6]), we have the exact sequence for all i ≥ 1

0 −→ Ki+1�M�/xKi+1�M� −→ Ki�M/xM� −→ �0 �Ki�M� x� −→ 0�

In particular, for i = d − 1, we have the exact sequence

0 −→ K�M�/xK�M� −→ K�M/xM� −→ �0 �Kd−1�M� x� −→ 0�

Since �0 �Kd−1�M� x� is of finite length by Lemma 2.4, we have for any i ≥ 2 that

Hi�

(K�M�/xK�M��

) � Hi�

(K�M/xM�

)�

�

Remark 2.6. Let d ≥ 1 and let x ∈ �, be a strict f-element w.r.t. M . Then x isK�M�-regular by Lemma 2.4. Therefore, we have the exact sequence

0 −→ K�M�x−→ K�M� −→ K�M�/xK�M� −→ 0�

This induces the exact sequence

0 −→ Hi��K�M��/xHi

��K�M�� −→ Hi��K�M�/xK�M��� −→ �0 �Hi+1

� �K�M�� x� −→ 0

for all integers i ≥ 0.

Let A be an Artinian R-module. Following Sharp–Hamieh [11], the stabilityindex of A, denoted by s�A�� is the least positive integer s such that �sA = �nAfor all n ≥ s� Denote by Rl�A� the length of A/�s�A�A� Then Rl�A� is finite, and iscalled the residual length of A� It is clear that Rl�A� = 0 if and only if � � AttRA�Moreover, if x � � for all � ∈ AttRA\���, then ��A/xA� � Rl�A� and in this case,��A/xnA� = Rl�A� for all n ≥ s�A��

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ON GENERALIZED COHEN–MACAULAY CANONICAL MODULES 4457

Lemma 2.7. Let i ≥ 0 be an integer, and let x be a strict f-regular element w.r.t. M .Then for all n 0, we have

Rl�Hi��M�� = ��H0

��Ki�M��� = ��0 �Ki�M� x

n��

Proof. Note that xn is a strict f-regular element w.r.t. M for any integer n >0. Therefore ��0 �Ki�M� x

n� < � by Lemma 2.4. Since H0��K

i�M�� is the largestsubmodule of Ki�M� of finite length, we have for n 0 that

H0��K

i�M�� = �0 �Ki�M� �n� ⊆ �0 �Ki�M� x

n� ⊆ H0��K

i�M���

So, ��H0��K

i�M��� = ��0 �Ki�M� xn� for n 0� Note that Hi

��M� � D�Ki�M�� by [1,11.2.6], where D�−� is the Matlis dual functor. Since ��0 �Ki�M� x

n� < �� we have

��0 �Ki�M� xn� = ��D�0 �Ki�M� x

n�� = �(D�Ki�M��/xnD�Ki�M��

)= �

(Hi

��M�/xnHi��M�

)�

Because x � � for all � ∈ AttRHi��M�\���� we have �

(Hi

��M�/xnHi��M�

) =Rl�Hi

��M�� for n 0. Therefore, the lemma follows. �

3. PROOF OF MAIN THEOREM

Firstly, we introduce the notion of generalized Cohen–Macaulay canonicalmodules.

Definition 3.1. M is called a generalized Cohen–Macaulay canonical module if thecanonical module K�M� of M is generalized Cohen–Macaulay.

Note that if dimM � 3 or M is generalized Cohen–Macaulay then Mis generalized Cohen–Macaulay canonical. If M/UM�0� is generalized Cohen–Macaulay, where UM�0� is the largest submodule of M of dimension less thand, then M is generalized Cohen–Macaulay canonical. In particular, sequentiallygeneralized Cohen–Macaulay modules and pseudo generalized Cohen–Macaulaymodules defined in [2] are generalized Cohen–Macaulay canonical.

Lemma 3.2. If M is a generalized Cohen–Macaulay canonical module then so isM/xM , for any strict f-regular element x w.r.t. M .

Proof. If d � 2, then the result is clear. Let d � 3� Then dim�M/xM� ≥ 2. Hencedepth�K�M/xM�� ≥ 2 and hence Hi

��K�M/xM�� = 0 for i = 0� 1� By Lemma 2.5,

Hi��K�M/xM�� � Hi

��K�M�/xK�M��

for all i ≥ 2� Note that x is K�M�-regular. Because K�M� is generalized Cohen–Macaulay, so is K�M�/xK�M�. Hence �

(Hi

��K�M�/xK�M��)< � for all i < d − 1�

So, Hi��K�M/xM�� is of finite length for all i < d − 1� Thus, M/xM is generalized

Cohen–Macaulay canonical. �

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4458 NGUYEN AND LE

Lemma 3.3. Let d ≥ 4� If M is generalized Cohen–Macaulay canonical, then

Rl�Hd−2� �M/xM�� � ��H2

��K�M���+ ��H3��K�M���

for any strict f-element x w.r.t. M . The equality holds when x ∈ �r , where

r = min�t ∈ � � �tHi��K�M�� = 0 for all i < d��

Proof. Let x be a strict f-element w.r.t. M . Set N = M/xM� Let n ∈ � and let ybe a strict f-element w.r.t. N . Then yn is a strict f-element w.r.t. N . Therefore, byapplying Lemma 2.5 to the module N with the strict f-element yn and the integeri = d − 2, we have the following exact sequence:

0 −→ K�N�/ynK�N� −→ K�N/ynN� −→ �0 �Kd−2�N� yn� −→ 0�

Note that dim�N/ynN� = d − 2 ≥ 2. Therefore, depth(K�N/ynN�

) ≥ 2� It followsthat Hi

��K�N/ynN�� = 0 for i = 0� 1. Note that �

(0 �Kd−2�N� y

n)< � by Lemma 2.4.

So, we have by the above exact sequence that

�0 �Kd−2�N� yn� = H0

��0 �Kd−2�N� yn� � H1

�

(K�N�/ynK�N�

)�

By Lemma 2.7, Rl�Hd−2� �N�� = �

(H0

��Kd−2�N��

) = ��0 �Kd−2�N� yn� for n 0. Hence

Rl�Hd−2� �N�� = �

(H1

��K�N�/ynK�N��

)� (1)

Since dimN = d − 1 ≥ 3, we have depthK�N� ≥ 2. Hence H1��K�N�� = 0. Also, we

can apply Remark 2.6 to the module N with the strict f-element yn and the integeri = 1 in order to obtain the isomorphism

H1�

(K�N�/ynK�N�

) � �0 �H2��K�N�� y

n��

Note that K�N� is generalized Cohen–Macaulay of dimension d − 1 ≥ 3 by Lemma3.2. Therefore, ��H2

��K�N��� < �. Hence �0 �H2��K�N�� y

n� = H2��K�N�� for n 0, and

hence

�(H1

��K�N�/ynK�N��

) = ��H2��K�N���� (2)

It follows by Lemma 2.5 that

H2��K�N�� � H2

��K�M�/xK�M���� (3)

By Remark 2.6, we have the following exact sequence:

0 −→ H2��K�M��/xH2

��K�M�� −→ H2��K�M�/xK�M��� −→ �0 �H3

��K�M�� x� −→ 0�

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ON GENERALIZED COHEN–MACAULAY CANONICAL MODULES 4459

Since d ≥ 4, we have by the choice of r that �0 �H3��K�M�� x� = H3

��K�M�� andxH2

��K�M�� = 0 whenever x ∈ �r . Therefore, we have by the above exact sequencethat

�(H2

��K�M�/xK�M��)� �

(H2

��K�M��)+ �

(H3

��K�M��)

(4)

and the equality holds if x ∈ �r . Now, combining (1)–(4) we have

Rl�Hd−2� �N�� = �

(H1

�

(K�N�/ynK�N�

)) = ��H2�

(K�N���

= �(H2

�

(K�M�/xK�M�

))� �

(H2

��K�M��)+ �

(H3

��K�M��)�

and the equality holds when x ∈ �r . �

For a s.o.p. x = �x1� � � � � xd� of M , set Mx�k = M/�x1� � � � � xk�M for k =1� � � � � d − 3.

Lemma 3.4. Suppose that M is generalized Cohen–Macaulay canonical. Let k be aninteger such that 1 � k � d − 3. Then

Rl�Hd−k−1� �Mx�k�� �

k∑i=0

(ki

)��Hi+2

� �K�M���

for any s.o.p. x = �x1� � � � � xd� of M which is a strict f-sequence. The equality holdswhen �x1� � � � � xd� ⊆ �2k−1r , where r = min�t ∈ � � �tHi

��K�M�� = 0 for all i < d�.

Proof. If d � 3, then there is nothing to do. So we assume that d ≥ 4. Let x =�x1� � � � � xd� be a s.o.p of M which is a strict f-sequence. We will show by inductionon d that

Rl�Hd−k−1� �Mx�k�� �

k∑i=0

(ki

)��Hi+2

� �K�M���

for any k = 1� � � � � d − 3, and the equality holds when x1� � � � � xk ∈ �2k−1r . The caseof d = 4 follows immmediately by Lemma 3.3. Let d > 4 and assume that the resultis true for d − 1� Set N = M/x1M . Note that N is generalized Cohen–Macaulaycanonical of dimension d − 1 by Lemma 3.2. Since �x2� � � � � xd� is a strict f-sequences.o.p of N , we have by induction that

Rl�H�d−1�−�k−1�−1� �N/�x2� � � � � xk�N�� �

k−1∑i=0

(k− 1i

)�(Hi+2

� �K�N��)

(5)

for any k = 2� � � � � d − 3, and the equality holds when x2� � � � � xk ∈ �2k−2s, where

s = min�t ∈ � � �t�Hi��K�N�� = 0 for all i < d − 1��

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4460 NGUYEN AND LE

Note that N/�x2� � � � � xk�N � Mx�k for all k = 2� � � � � d − 3. On the other hand, byLemma 2.5, we have an isomorphism

Hi��K�N�� � Hi

��K�M�/x1K�M��

for all i ≥ 2. Therefore, from the exact sequence as in Remark 2.6

0 −→ Hi��K�M��/x1H

i��K�M�� −→ Hi

��K�M�/x1K�M�� −→ �0 �Hi+1� �K�M�� x1� −→ 0�

we get that

��Hi��K�N��� = ��Hi

��K�M�/x1K�M��� � ��Hi��K�M���+ ��Hi+1

� �K�M���

for all i = 2� � � � � d − 2. Note that � k−1i �+ � k−1

i−1 � = � ki �. So, we have by (5) that

Rl�Hd−k−1� �Mx�k�� �

k−1∑i=0

(k− 1i

)��Hi+2

� �K�N���

�

k−1∑i=0

(k− 1i

)(��Hi+2

� �K�M���+ ��Hi+3� �K�M���

)

=k∑

i=0

(ki

)��Hi+2

� �K�M���

for all k = 2� � � � � d − 3. The case of k = 1 follows by Lemma 3.3. From the exactsequence as in Remark 2.6

0 −→ Hi��K�M��/xHi

��K�M�� −→ Hi��K�M�/xK�M��� −→ �0 �Hi+1

� �K�M�� x� −→ 0�

we have by the choice of r that �2rHi��K�M�/xK�M��� = 0 for all i < d −

1� Note that depthK�N� ≥ 2 since dimN ≥ 4� Therefore, Hi��K�N�� = 0 for i =

0� 1� Since Hi��K�N�� � Hi

��K�M�/x1K�M�� for all i ≥ 2 by Lemma 2.5, we have�2rHi

��K�N�� = 0 for all i = 2� � � � � d − 2� Hence s � 2r� Therefore, if x1� � � � � xk ∈�2k−1r , then

Rl�Hd−k−1� �Mx�k�� =

k∑i=0

(ki

)��Hi+2

� �K�M����

�

Lemma 3.5. Let x = �x1� � � � � xd� be a s.o.p. of M which is a strict f-sequence. Letc� k be integers such that 1 � k � d − 3 and ��Hk+2

� �K�M��� > c� Then there existpositive integers n1� � � � � nk such that Rl

(Hd−k−1

� �M/�xn11 � � � � � x

nkk �M�

)> c.

Proof. The proof is by induction on k = 1� � � � � d − 3. Let k = 1� Then��H3

��K�M��� > c� Note that H3��K�M�� is �-torsion, i.e., H3

��K�M�� =

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ON GENERALIZED COHEN–MACAULAY CANONICAL MODULES 4461

⋃n≥0

�0 �H3��K�M�� �

n�� Therefore, we can choose an integer n1 such that ��0 �H3��K�M��

xn11 � > c� By Remark 2.6, we have an exact sequence

0 −→ H2��K�M��/x

n11 H2

��K�M�� −→ H2��K�M�/x

n11 K�M��� −→ �0 �H3

��K�M�� xn11 �−→ 0�

It follows that �(H2

��K�M�/xn11 K�M���

)> c� Set M1 = M/x

n11 M� Then by Lemma 2.5,

we have an isomorphism

H2��K�M�/x

n11 K�M��� � H2

��K�M1���

Therefore, ��H2��K�M1��� > c� So, there exists n2 ∈ � such that ��0 �H2

��K�M1��xn22 � >

c� Since xn22 is a strict f-element with respect to M1 by Lemma 2.2, we have by

Remark 2.6 the exact sequence

0 → H1��K�M1��/x

n22 H1

��K�M1�� → H1��K�M1�/x

n22 K�M1��� → �0 �H2

��K�M1��xn22 � → 0�

It follows that �(H1

��K�M1�/xn22 K�M1���

)> c� Also, we have by Lemma 2.5 an exact

sequence

0 −→ K�M1�/xn22 K�M1� −→ K�M1/x

n22 M1� −→ �0 �Kd−2�M1�

xn22 � −→ 0�

Since d ≥ 4� we have dim�M1/xn22 M1� ≥ 2� Therefore, depth�K�M1/x

n22 M1� ≥ 2� Since

0 �Kd−2�M1�xn22 is of finite length, we get by the above exact sequence that

H1�

(K�M1�/x

n22 K�M1�

) � H0�

(0 �Kd−2�M1�

xn22

) = (0 �Kd−2�M1�

xn22

)�

Hence �(0 �Kd−2�M1�

xn22

)> c and hence Rl

(Hd−2

� �M/xn11 M�

)> c by Lemma 2.7. Thus,

the claim is true for k = 1�Let 2 � k � d − 3� Since �

(Hk+2

� �K�M��)> c, we can choose a positive integer

n1 such that ��0 �Hk+2� �K�M�� x

n11 � > c� By Lemma 2.5, we have an exact sequence

0 → Hk+1� �K�M��/x

n11 Hk+1

� �K�M�� → Hk+1� �K�M�/x

n11 K�M��� → �0 �Hk+2

� �K�M�� xn11 � → 0�

It follows that �(Hk+1

� �K�M�/xn11 K�M���

)> c� Since k ≥ 2� we get by Lemma 2.5 an

isomorphism

Hk+1� �K�M�/x

n11 K�M��� � Hk+1

� �K�M/xn11 M���

Set N = M/xn11 M . Then �

(H

�k−1�+2� �K�N��

)> c, where 1 � k− 1 � dimN − 3. So, we

have by induction for the the integer k− 1 with the module N that

Rl(HdimN−�k−1�−1

� �N/�xn22 � � � � � x

nkk �N�

)> c

for some positive integers n2� � � � � nk� It means that Rl(Hd−k−1

� �M/�xn11 � � � � �

xnkk �M�

)> c, and the lemma is proved. �

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4462 NGUYEN AND LE

Proof of Main Theorem. The assertion (i)⇒(ii) follows by Lemma 3.4. The claim(ii)⇒(iii) is trivial. Now we prove (iii)⇒(i). Let �x1� � � � � xd� be a s.o.p. of M whichis a strict f-sequence and c�x�M� a number such that

d−3∑k=1

Rl�Hd−k−1� �M/�x

n11 � � � � � x

nkk �M�� � c�x�M�

for all n1� � � � � nd ∈ �. Suppose that M is not generalized Cohen–Macaulaycanonical. Since K�M� satisfies the condition Serre �S2� and K�M� isequidimensional, it follows by [8, Lemma 3.2.1] that ��H1

��K�M�� < � for alli � 2� Therefore, there exists an integer k such that 1 � k � d − 3 and Hk+2

� �K�M��is of infinite length. It follows that ��Hk+2

� �K�M��� > c�x�M� for some integer1 � k � d − 3. So, there exist by Lemma 3.5 integers n1� � � � � nk such thatRl

(Hd−k−1

� �M/�xn11 � � � � � x

nkk �M�

)> c�x�M�. This gives a contradiction. �

ACKNOWLEDGMENT

The authors are supported by the Vietnam National Foundation for Scienceand Technology Development (Nafosted) under grant numbers 101.01-2011.20 and101.01-2011.49.

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[8] Schenzel, P. (1982). Dualisierende Komplexe in der lokalen Algebra und BuchsbaumRinge. Lecture Notes in Math., Vol. 907. Berlin- Heidelberg-New York: Springer-Verlag.

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