δ-Koszulity of Finitely Generated Graded Modules

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δ-Koszulity of Finitely Generated Graded ModulesJia-Feng Lü a & Xiao-Yan Lu aa Institute of Mathematics , Zhejiang Normal University , Jinhua , Zhejiang , ChinaPublished online: 09 May 2013.

To cite this article: Jia-Feng Lü & Xiao-Yan Lu (2013): δ-Koszulity of Finitely Generated Graded Modules, Communications inAlgebra, 41:5, 1727-1750

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Communications in Algebra®, 41: 1727–1750, 2013Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927872.2011.649507

�-KOSZULITY OF FINITELY GENERATED GRADEDMODULES

Jia-Feng Lü and Xiao-Yan LuInstitute of Mathematics, Zhejiang Normal University,Jinhua, Zhejiang, China

The �-Koszulity of finitely generated graded modules is discussed and the notionof weakly �-Koszul module is introduced. Let M ∈ gr�A� and �Sd1

� Sd2� � � � � Sdm

�

denote the set of minimal homogeneous generating spaces of M where Sdiconsists

of homogeneous elements of M of degree di. Put �1 = �Sd1�, �2 = �Sd1

� Sd2�, � � � ,

�m = �Sd1� Sd2

� � � � � Sdm�. Then M admits a chain of graded submodules: 0 = �0 ⊂

�1 ⊂ �2 ⊂ · · · ⊂ �m = M . Moreover, it is proved that M is a weakly �-Koszulmodule if and only if all �i/�i−1�−di are �-Koszul modules, if and only if theassociated graded module G�M� is a �-Koszul module. Further, as applications, therelationships of minimal graded projective resolutions among M , G�M� and thesequotients �i/�i−1 are established. The Ext module

⊕i≥0 Ext

iA�M�A0� of a weakly

�-Koszul module M is proved to be finitely generated in degree zero.

Key Words: �-Koszul algebras; �-Koszul modules; Weakly �-Koszul modules.

2000 Mathematics Subject Classification: Primary 16S37, 16W50; Secondary 16E30, 16E40.

1. INTRODUCTION

The noncommutative graded algebras play an important role in algebra,topology, and mathematical physics. Probably the most interesting class of suchalgebras is the class of Koszul algebras (Priddy [15]), which give a nice connectionto algebraic objects (dual algebras) and homological objects (Yoneda algebras).In the last decade, several extensions of this theory to some more general caseshave been developed. Berger introduced the notion of nonquadratic Koszul algebrain [1]; Brenner, Butler, and King introduced the notion of almost Koszul algebrain [2]; Lü, He, and Lu introduced the notion of piecewise-Koszul algebra in [8];Green and Snashall introduced the notion of �D�A� B�-stacked monomial algebrain [7]. Note that all the algebras mentioned above are “pure” algebras, i.e., eachterm in the minimal graded projective resolution of its trivial module is generatedin a single degree. Taken in this sense, Green and Marcos generalized such classof algebras to the most extensive level and introduced the notion of �-Koszulalgebra in [3]. Recall that Koszul modules, D-Koszul modules, and piecewise-Koszul

Received August 23, 2011; Revised December 6, 2011. Communicated by D. Zacharia.Address correspondence to Dr. Jia-Feng Lü, Institute of Mathematics, Zhejiang Normal

University, Jinhua, Zhejiang 321004, China; E-mail: [email protected]

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1728 LÜ AND LU

modules are graded pure modules. In order to study Koszulity, D-Koszulity andpiecewise-Koszulity of finitely generated graded modules, the notion of weaklyKoszul modules, weakly D-Koszul modules and weakly piecewise-Koszul moduleswere introduced in [9–11, 14], respectively.

Note that �-Koszul modules are also a class of graded pure modules, inspiredby [9–11, 14], we mainly discuss the �-Koszulity of arbitrary finitely generatedgraded modules over a �-Koszul algebra in this article (see also [12]). Similarly, thenotion of weakly �-Koszul module is introduced and the formal definition can befound in the next section.

The whole article is arranged as follows.In Section 2, we give some notations and definitions which will be used

throughout the article.In Section 3, let A be a �-Koszul algebra. For any given finitely generated

graded A-module M , we construct a special filtration for M :

0 = �0 ⊂ �1 ⊂ �2 ⊂ · · · ⊂ �m = M�

where �i = �Sd1� Sd2� � � � � Sdi�� i = 1� 2� � � � � m�, and �Sd1� Sd2� � � � � Sdm� denotes theset of minimal homogeneous generating spaces of M and Sdi consists ofhomogeneous elements of M of degree di. In particular, we prove that M is a weakly�-Koszul module if and only if all the quotients �i/�i−1�−di of the fixed filtrationare �-Koszul modules.

In Section 4, we give some applications of the results obtained in Section 3.More precisely, the relationship of the minimal graded projective resolutionsbetween a weakly �-Koszul module and all the quotients of its graded submodulefiltration is discussed. As an easy consequence, we show that the finitistic dimensionconjecture is true in the category of weakly �-Koszul modules over a finitedimensional �-Koszul algebra.

In Section 5, the associated graded module of a weakly �-Koszul module isstudied. We prove that a finitely generated graded module is weakly �-Koszul if andonly if its associated graded module is �-Koszul. Further, the relationship of theminimal graded projective resolutions between a weakly �-Koszul module and itsassociated graded module is discussed.

In Section 6, the Ext module of a weakly �-Koszul module is investigated and weprove that the Ext module of a weakly �-Koszul module is finitely generated in degreezero. We also provide an example to suggest that the converse is not true in general.

In Section 7, the notion of block �-Koszul module is introduced. We generalizesome results of weakly �-Koszul modules to the case of block �-Koszul modules.

2. NOTATIONS AND DEFINITIONS

Throughout the whole article, k denotes a fixed field, and � and � denotethe sets of natural numbers and integers, respectively. All the positively graded k -algebra A = ⊕

i≥0 Ai in this article are assumed with the following properties:

i) A0 = k × · · · × k , a finite product of k ;ii) Ai · Aj = Ai+j for all 0 ≤ i� j < �;iii) dim

kAi < � for all i ≥ 0.

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�-KOSZULITY OF FINITELY GENERATED GRADED MODULES 1729

Under the above assumptions, it is easy to see that the graded Jacobson radicalof A, which we denote by J , is obvious

⊕i≥1 Ai. For any finitely generated graded

A-module M , M possesses a graded projective resolution

· · · −→ Qn

dn−→ · · · −→ Q1

d1−→ Q0

d0−→ M −→ 0

such that ker di ⊆ JQi for all i ≥ 0, i.e., the resolution is “minimal.” Let Gr�A�denote the category of graded A-modules and gr�A� the category of finitelygenerated graded A-modules, which is a full subcategory of Gr�A�. We denoteGrs�A� and grs�A� the full subcategory of Gr�A� and gr�A� whose objects aregenerated in degree s. An object in Grs�A� or grs�A� is called a graded pureA-module. Endowed with the Yoneda product,

⊕i≥0 Ext

iA�A0� A0� is a bigraded

algebra. Let M ∈ gr�A�. Then⊕

i≥0 ExtiA�M�A0� is a bigraded

⊕i≥0 Ext

iA�A0� A0�-

module. For simplicity, we write

E�A� = ⊕i≥0

ExtiA�A0� A0�� M� = ⊕i≥0

ExtiA�M�A0�

and call E�A� the Yoneda algebra of A, ��M� the Ext module of M , respectively.

Definition 2.1 ([3]). Let A be a positively graded algebra. A is called �-Koszulprovided the following two conditions:

(1) The trivial A-module A0 admits a minimal graded projective resolution

· · · −→ Pn −→ · · · −→ P1 −→ P0 −→ A0 −→ 0

such that each Pn is generated in a single degree, say ��n� for all n ≥ 0, where �is a strictly increasing set function from � to �;

(2) The Yoneda algebra E�A� is finitely generated as a graded algebra.

Let A be a �-Koszul algebra and M ∈ gr�A�. We call M a �-Koszul moduleif there is a graded projective resolution · · · −→ Qn −→ · · · −→ Q1 −→ Q0 −→M −→ 0 such that each Qn is generated in degree ��n�.

Let ��A� denote the category of �-Koszul modules.

It is easy to see that Koszul algebras (modules), D-Koszul algebras(modules) and piecewise-Koszul algebras (modules) are all special �-Koszul algebras(modules).

The following observation motivates the notion of weakly �-Koszul module.

Proposition 2.2. Let A be a �-Koszul algebra, M ∈ gr0�A�, and

· · · −→ Qn

fn−→ · · · −→ Q1

f1−→ Q0

f0−→ M −→ 0

be a minimal graded projective resolution of M . Then M is a �-Koszul module if andonly if ker fn ⊆ J��n+1�−��n�Qn and J ker fn = ker fn ∩ J��n+1�−��n�+1Qn for all n ≥ 0.

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Proof. Note that Qn+1 −→fn+1 ker fn −→ 0 is a graded projective cover. Thusker fn is generated in degree ��n+ 1�, which forces that ker fn ⊆ J��n+1�−��n�Qn sinceQn is generated in degree ��n�. Let x ∈ ker fn ∩ J��n+1�−��n�+1Qn be a homogeneouselement. Then the degree of x is larger than ��n+ 1�+ 1, which implies thatx ∈ J ker fn. Hence we finish the proof of necessity since J ker fn ⊆ ker fn ∩J��n+1�−��n�+1Qn is obvious. Conversely, the conditions ker fn ⊆ J��n+1�−��n�Qn andJ ker fn = ker fn ∩ J��n+1�−��n�+1Qn guarantee that Qn+1 is generated in degree ��n+1� for all n ≥ 0 since Q0 and M0 are generated in degree ��0� = 0. �

Definition 2.3. Let A be a �-Koszul algebra and M ∈ gr�A�. Let

· · · −→ Qn

fn−→ · · · −→ Q1

f1−→ Q0

f0−→ M −→ 0

be a minimal graded projective resolution of M . Then M is called a weakly �-Koszulmodule if for all n� k ≥ 0, we have ker fn ⊆ J��n+1�−��n�Qn and Jk ker fn = ker fn ∩J��n+1�−��n�+kQn. Let ��A� denote the category of weakly �-Koszul modules.

Example 2.4.

(1) Weakly Koszul modules, introduced by Martínez-Villa and Zacharia in [14] is aspecial class of weakly �-Koszul modules in the sense of ��i� = i for all i ≥ 0.

(2) Weakly d-Koszul modules, introduced in [9], is a special class of weakly �-Koszul modules in the sense of

��n� =

nd

2� if n is even�

�n− 1�d2

+ 1� if n is odd�

where d ≥ 2 is a given integer.(3) Weakly piecewise-Koszul modules, introduced in [10], is a special class of weakly

�-Koszul modules in the sense of

��n� =

nd

p� if n ≡ 0 �mod p��

�n− 1�dp

+ 1� if n ≡ 1 �modp��

� � � � � ��n−p+1�d

p+ p− 1� if n ≡ p− 1 �modp��

where d ≥ p ≥ 2 are given integers.(4) �-Koszul modules are weakly �-Koszul modules. In particular, we have ��A� =

gr0�A� ∩��A� for a �-Koszul algebra A.(5) Let A = A0 ⊕ A1 ⊕ A2 ⊕ � � � be a �-Koszul algebra. Then for all l ∈ �− �0�,

Ml = A0 ⊕ A0�1⊕ A0�2⊕ · · · ⊕ A0�l is a weakly �-Koszul module.

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3. WEAKLY �-KOSZUL MODULES

In this section, we will discuss the �-Koszulity of finitely generated gradedmodules over a �-Koszul algebra.

We begin with the following lemma.

Lemma 3.1. Let A be a positively graded algebra and 0 −→ K −→ M −→ N −→ 0be an exact sequence in gr�A�. Then JK = K ∩ JM if and only if we have the followingcommutative diagram with exact rows and columns:

where P0, Q0, and L0 are graded projective covers of K, M , and N , respectively.

Proof. �⇒� We get the exact sequence 0 −→ K/JK −→ M/JM −→ N/JN −→ 0since JK = K ∩ JM . Set P0 = A⊗A0

K/JK, Q0 = A⊗A0M/JM , and L0 = A⊗A0

N/JN . We have the following exact sequence since A0 is semisimple:

0 −→ P0 −→ Q0 −→ L0 −→ 0�

Now by “3× 3” Lemma, we get the following exact sequence

0 −→ �1�K� −→ �1�M� −→ �1�N� −→ 0�

Therefore, we get the desired diagram.

�⇐� Suppose that we have the above given commutative diagram with exactrows and columns. Note that the projective cover of a module is unique up toisomorphisms. We may assume that P0 = A⊗A0

K/JK, Q0 = A⊗A0M/JM , and

L0 = A⊗A0N/JN . From the middle row of the diagram, we have the following

exact sequence:

0 −→ A⊗A0K/JK −→ A⊗A0

M/JM −→ A⊗A0N/JN −→ 0�

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1732 LÜ AND LU

That is, we have the following short exact sequence as A0-modules

0 −→ K/JK −→ M/JM −→ N/JN −→ 0

since A0 is semisimple. Thus, JK = K ∩ JM . �

Lemma 3.2. Let A be a �-Koszul algebra and 0 −→ K −→ M −→ N −→ 0 anexact sequence in gr�A�.

(1) If K�M ∈ ��A� and JkK = K ∩ JkM for all k ≥ 0, then N ∈ ��A�.(2) If K�N ∈ ��A� and JK = K ∩ JM , then M ∈ ��A�.(3) If Jk�i�K� = �i�K� ∩ Jk�i�M� for all k� i ≥ 0, then K ∈ ��A� provided that

M ∈ ��A�.

Proof. We only prove (1) since the proofs of (2) and (3) are similar.By the hypothesis JkK = K ∩ JkM for all k ≥ 0, we have in particular that

JK = K ∩ JM , which implies the exact sequence 0 −→ JK −→ JM −→ JN −→ 0.By Lemma 3.1, we have the commutative diagram with exact rows and columns

Apply the functor A/Jk ⊗A − �k ≥ 0� to the above diagram, we obtain the followingcommutative diagram with exact rows and columns

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Note that

JK ∩ Jk+1M = JK ∩ K ∩ Jk+1M = JK ∩ Jk+1K = Jk+1K = Jk�JK��

By Lemma 2.1 of [14], we have �1 is a monomorphism. Recall that K and M areweakly �-Koszul modules, we have Jk�1�K� = �1�K� ∩ Jk+1P0 = �1�K� ∩ Jk�JP0�and Jk�1�M� = �1�M� ∩ Jk+1Q0 = �1�M� ∩ Jk�JQ0�, by Lemma 2.1 of [14] again,we get that �1 and 1 are monomorphisms. Thus �1 is a monomorphism. By Lemma2.1 of [14], we have that Jk�1�K� = �1�K� ∩ Jk�1�M� for all k ≥ 0. Now by “3× 3”Lemma, we have that �1 is a monomorphism, which implies that Jk�1�N� = �1�N� ∩Jk+1L0 for all k ≥ 0.

Now consider the exact sequence

0 −→ �1�K� −→ �1�M� −→ �1�N� −→ 0�

we have proved that Jk�1�K� = �1�K� ∩ Jk�1�M� for all k ≥ 0, which implies theexact sequence

0 −→ J��2�−��1��1�K� −→ J��2�−��1��1�M� −→ J��2�−��1��1�N� −→ 0�

By Lemma 3.1, we have the commutative diagram with exact rows and columns

Apply the functor A/Jk ⊗A − �k ≥ 0� to the above diagram, we obtain thecommutative diagram with exact rows and columns

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1734 LÜ AND LU

Note that

J��2�−��1��1�K� ∩ Jk+��2�−��1�M = J��2�−��1�K ∩ K ∩ Jk+��2�−��1�M

= J��2�−��1�K ∩ Jk+��2�−��1�K

= Jk+��2�−��1�K�

By Lemma 2.1 of [14], we have �2 is a monomorphism. Recall that K and Mare weakly �-Koszul modules, we have Jk�2�K� = �2�K� ∩ Jk+��2�−��1�P1 = �1�K� ∩Jk�J��2�−��1�P1� and Jk�2�M� = �2�M� ∩ Jk+��2�−��1�Q1 = �2�M� ∩ Jk�J��2�−��1�Q1�, byLemma 2.1 of [14] again, we get that �2 and 2 are monomorphisms. Thus �2 is amonomorphism. By Lemma 2.1 of [14], we have that Jk�2�K� = �2�K� ∩ Jk�2�M�for all k ≥ 0. Now by “3× 3′′ lemma, we have that �2 is a monomorphism, whichimplies that Jk�1�N� = �1�N� ∩ Jk+��2�−��1�L1 for all k ≥ 0.

Now repeat the above procedure and by induction, we finish the proof. �

Lemma 3.3 ([8]). Let 0 −→ K −→ M −→ N −→ 0 be an exact sequence in gr�A�with JkK = K ∩ JkM for all k ≥ 0. Assume that K, M and N are supported in �i � i ≥d�. Then �Kd� = K ∩ �Md�.

Lemma 3.4. Let A be a positively graded algebra, M = ⊕i≥0 Mi ∈ gr�A� and

�Sd1� Sd2� Sd3 , � � � � Sdm� be the set of its minimal homogeneous generating spaces. Thenfor all m ≥ i ≥ 0, we have Jk�Sdi� = �Sdi� ∩ JkM for all k ≥ 0.

Proof. Jk�Sdi� ⊆ �Sdi� ∩ JkM is obvious since Jk�Sdi� ⊆ �Sdi� and �Sdi� ⊆ M . Forthe inverse inclusion, let x ∈ �Sdi� ∩ JkM be a homogeneous element of degree j.Now it is clear that j ≥ di + k since �Sdi� is generated in degree di as a gradedA-module. Therefore, x ∈ Jk�Sdi� since �Sd1� Sd2� Sd3� � � � � Sdm� is the set of minimalhomogeneous generating spaces of M . �

Lemma 3.5. Let A be a �-Koszul algebra and M = ⊕i≥0 Mi be a weakly �-Koszul

module with M0 �= 0. Let �M0� denote the graded submodule of M generated by M0.Then:

(1) �M0� ∈ ��A�;(2) M/�M0� ∈ ��A�.

Proof. Let · · · → Pn → · · · → P2 → P1 → P0 → �M0� → 0 be a minimal gradedprojective resolution of �M0�. We want to show that each Pi is generatedin degree ��i�. Note that M is weakly �-Koszul, by definition, M admits a

minimal graded projective resolution · · · → Qn

fn→ · · · → Q1

f1→ Q0

f0→ M → 0 withJk ker fn = ker fn ∩ J��n+1�−��n�+kQn for all n� k ≥ 0. Up to an isomorphism, Qi =A⊗A0

S�i�M�, where S�i�M� = S��i� ⊕ S��i�+1 ⊕ · · · ⊆ �i�M� is a minimal graded sub-A0-module of �i�M� such that A · S�i�M� = �i�M� and i = 0� 1� 2� � � � . Set Pi =�Qi�

��i� = A⊗A0S��i� for all i ≥ 0.

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Now consider the following commutative diagram with exact rows:

We claim that ��1�M��1�� = �1��M0��. Since M is a weakly �-Koszul module, wehave �1�M� ⊆ JQ0. Thus, we have the following exact sequence 0 → �1�M� →JQ0 → JM → 0� Note that �1�M� ∩ Jk�JQ0� = �1�M� ∩ Jk+1Q0 = Jk�1�M�, hence

��1�M��1�� = �1�M� ∩ ��JQ0��1�� = �1�M� ∩ ��J�Q0��0��1��

Recall that �Q0��0� is generated in degree ��0�, we have

�1�M� ∩ ��J�Q0��0��1�� = �1�M� ∩ �Q0�

��0� = �1��M0��

Hence �1��M0�� = ��1�M��1��, which is generated in degree ��1�.For the general cases, consider the following commutative diagram with exact

rows:

Now we claim that ��i+1�M��i+1�� = �i+1��M0��. Since M is a weakly �-Koszul, wehave �i+1�M� ⊆ J��i+1�−��i�Qi. Thus, we have the following exact sequence

0 → �i+1�M� → J��i+1�−��i�Qi → J��i+1�−��i��i�M� → 0�

Note that�i+1�M�∩ Jk�J��i+1�−��i�Qi�=�i+1�M�∩ Jk+��i+1�−��i�Qi = Jk�i+1�M�, hence

��i+1�M��i+1�� = �i+1�M� ∩ ��J��i+1�−��i�Qi��i+1��= �i+1�M� ∩ ��J��i+1�−��i��Qi�

��i��i+1��

Recall that �Qi��i� is generated in degree ��i�, we have

�i+1�M� ∩ ��J��i+1�−��i��Qi��i��i+1�� = �i+1�M� ∩ �Qi�

��i� = �i+1��M0��

Hence �i+1��M0�� = ��i+1�M��i+1��, which is generated in degree ��i+ 1� for alli ≥ 0. Thus assertion (1) holds.

The assertion (2) is immediate from the exact sequence 0 → �M0� → M →M/�M0� → 0 and Lemmas 3.2 and 3.4. �

Theorem 3.6. Let A be a �-Koszul algebra and M ∈ gr�A�. Let �Sd1� Sd2� � � � � Sdm�denote the set of minimal homogeneous generating spaces of M and Sdi consists of

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1736 LÜ AND LU

homogeneous elements of degree di. Consider the following natural filtration of M:0 = �0 ⊂ �1 ⊂ �2 ⊂ · · · ⊂ �m = M� where �1 = �Sd1�, �2 = �Sd1� Sd2�, � � � , �m =�Sd1� Sd2� � � � � Sdm�. Then M ∈ ��A� if and only if all �i/�i−1�−di ∈ ��A� for all1 ≤ i ≤ m.

Proof. �⇒� Let M ∈ ��A�. If m = 1, then M = �Sd1� is a graded pure module.By Lemma 3.5, we are done. If m ≥ 2, then by Lemmas 3.5 and 3.4, we have�Md1

��−d1 ∈ ��A� and �Md1� ∩ JkM = Jk�Md1

� for all k ≥ 0. Now setW = M/Md1.

By Lemma 3.5, W ∈ ��A�. Consider the exact sequence 0 → �Wd2� → W →

W/�Wd2� → 0� where �Wd2

� = �2/�1. By Lemmas 3.5 and 3.4 again, we get that�Wd2

��−d2 = �2/�1�−d2 ∈ ��A� and �Wd2� ∩ JkM = Jk�Wd2

� for all k ≥ 0. Repeatthe above argument, we obtain�i/�i−1�−di ∈ ��A� for all 1 ≤ i ≤ m.

�⇐� Consider the exact sequence 0 → �1 → �2 → �2/�1 → 0� Obviously,J�1 = �1 ∩ J�2. Since �1 and �2/�1 are �-Koszul modules, by Lemma 3.2, we getthat �2 is a weakly �-Koszul module. Now we claim that for all 1 ≤ i ≤ m− 1, wehave J�i = �i ∩ J�i+1, which will be proved in Corollary 4.3. Now repeat the aboveprocedures and apply Lemma 3.2 several times, we get that M is a weakly �-Koszulmodule. �

Corollary 3.7. LetA be a �-Koszul algebra. Use the notations of Theorem 3.6. ThenM isa weakly �-Koszul module if and only if all�i �1 ≤ i ≤ m� are weakly �-Koszul modules.

Proof. �⇒� Clearly, �i possesses a following natural filtration 0 = �0 ⊆ �1 ⊆�2 ⊆ · · · ⊆ �i such that all�j/�j−1�−dj are �-Koszul modules, where 1 ≤ j ≤ i. ByTheorem 3.6, all�i �1 ≤ i ≤ m� are weakly �-Koszul modules.

�⇐� It is immediate from the fact thatM = �m. �

4. APPLICATIONS OF THE CHAIN THEOREM

Let A be a �-Koszul algebra, M ∈ gr�A� a weakly �-Koszul module and�Sd1� Sd2� � � � � Sdm� be the set of its minimal homogeneous generating spaces. Set

�i = �Sd1� Sd2� � � � � Sdi�� Ki = �i/�i−1� �1 ≤ i ≤ m��

By Theorem 3.6, M is a weakly �-Koszul module if and only if, for all 1 ≤ i ≤ m,all Ki�−di are �-Koszul modules, where �0 = 0. Now one can ask the followingquestion: Do there exist some relationships of minimal resolutions between M andKi’s? The main aim of this section is to discuss this question. In particular, weprove �n �

⊕mi=1 �

in for all n ≥ 0, where �∗ → M → 0 and �i

∗ → Ki → 0 denote theminimal graded projective resolutions ofM and Ki’s, respectively.

We begin with the following lemma.

Lemma 4.1. LetM = ⊕i≥0 Mi be a weakly �-Koszul module withM0 �= 0.

(1) Denote K = �M0� and N = M/K. Then the “Minimal Horseshoe Lemma” holds forthe natural exact sequence 0 −→ K −→ M −→ N −→ 0.

(2) Use the notions of Theorem 3.6. Then for all integers j ≥ 1, the “Minimal HorseshoeLemma” holds for 0 −→ �j −→ �j+1 −→ �j+1/�j −→ 0�

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Proof. (1) By Lemma 3.4, we get JK = K ∩ JM . By Lemma 3.1, we have thecommutative diagram as Lemma 3.1 and the following commutative diagram withexact rows:

Here P0, Q0, and L0 are graded projective covers. Of course, L0 = P0 ⊕Q0 since theexact sequence 0 −→ P0 −→ L0 −→ Q0 −→ 0 and Q0 is a graded projective module.Note thatM and N are weakly �-Koszul modules. Apply the functor A/J ⊗A − to theabove diagram, we get the following commutative diagram:

By Lemma 3.5, K is a �-Koszul module, which implies J�1�K� = �1�K� ∩J��1�−��0�+1P0. Thus, � is a monomorphism, which implies that is also amonomorphism. Hence we have J�1�K� = �1�K� ∩ J�1�M�. Now replace

0 −→ K −→ M −→ N −→ 0

by

0 −→ �1�K� −→ �1�M� −→ �1�N� −→ 0�

repeat the above argument, and we are done.

(2) By (1), we have the following commutative diagram with exact rows andcolumns

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1738 LÜ AND LU

Here �1∗ , �

2∗ , and �2

∗ are the minimal graded projective resolutions. Clearly, for eachi, the terms P1

i , P2i , and Q2

i in the complexes �1∗ , �

2∗ , and �2

∗, respectively, satisfy P2i =

P1i ⊕Q2

i .Similarly, we also have the following commutative diagram with exact rows and

columns:

Here �2∗, �, and �3

∗ are the minimal graded projective resolutions.Now consider the following commutative diagram:

Here�1∗ ,�

3∗ , and � are the minimal graded projective resolutions. If we further denote

the terms in complexes �3∗ and �3

∗ by P3i and Q3

i . Then it is clear that P3i = P1

i ⊕Q2

i ⊕Q3i .

For exact sequence 0 −→ �2 −→ �3 −→ �3/�2 −→ 0, by the HorseshoeLemma, we have the following commutative diagram:

with exact rows and columns, where �2∗ and �3

∗ are the minimal graded projectiveresolutions. For each term Pi in �∗, it is clear that Pi = P2

i ⊕Q3i = P1

i ⊕Q2i ⊕Q3

i ,which shows that �∗ is the minimal graded projective resolution of �3. Then we canget the desired result by induction. �

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The following example shows that the minimal Horseshoe Lemma is not true ingeneral.

Example 4.2. Let A = k �x, a graded polynomial algebra, M = A/�x2�, N =A/�x��−1 and K = k . Then we have the following short exact sequence:

0 −→ N −→ M −→ K −→ 0�

Under a routine computation, we get the following diagram:

where the vertical columns are minimal graded projective resolutions. Now it is easyto see that the minimal Horseshoe Lemma is not true for such an exact sequence.

Corollary 4.3. Let M = ⊕i≥0 Mi be a weakly �-Koszul module. Use the notations of

Theorem 3.6. Then J�j−1 = �j−1 ∩ J�j for all 1 ≤ j ≤ m, whereM0 = 0.

Proof. It is immediate from Lemmas 3.1 and 4.1. �

Theorem 4.4. Let M ∈ ��A� and 0 = �1 ⊂ �2 ⊂ · · · ⊂ �m−1 ⊂ �m = M itsnatural submodule filtration. Set Ki = �i/�i−1 for i = 1� 2� � � � � m. Let �∗ → M →0 and �i

∗ → Ki → 0 be the minimal graded projective resolutions of M and Ki’s,respectively. Then for all n ≥ 0, we have

�n �m⊕i=1

�in�

Proof. Consider the following exact sequence

0 −→ �1 −→ M −→ M/�1 −→ 0�

By Lemma 4.1 (1), we have the following commutative diagram with exact rowsand columns:

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1740 LÜ AND LU

Here �1∗ , �∗, and 1

∗ are the minimal graded projective resolutions of �1, M ,and M/�1, respectively. Clearly, �∗ = �1

∗ ⊕1∗ . Setting W = M/�1. Then �Wd2

� =�2/�1 = K2. Consider the exact sequence

0 −→ K2 −→ W −→ W/K2 −→ 0�

By Lemma 4.1 (1) again, we have the following commutative diagram with exact rowsand columns:

Here�2∗ ,

1∗ , and2

∗ are the minimal graded projective resolution ofK2,W andW/K2,respectively. Clearly, 1

∗ = �2∗ ⊕2

∗ . Repeat the above argument and by induction,we finally get the following commutative diagram with exact rows and columns:

Therefore, we have �n �⊕m

i=1 �in for all n ≥ 0. �

Corollary 4.5. Use the notations of Theorems 3.6 and 4.4. LetM ∈ gr�A� and

· · · −→ Pi −→ · · · −→ P1 −→ P0 −→ M −→ 0

a minimal projective resolution ofM . Set

�∗ = · · · −→ Pi −→ · · · −→ P1 −→ P0 −→ 0�

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ThenM ∈ ��A� if and only if the complex �∗ has a filtration

�∗ 0 = �0∗ ⊂ �1

∗ ⊂ · · · ⊂ �m−1∗ ⊂ �m

∗ = �∗�

such that for all 1 ≤ j ≤ m,

· · · −→ Pji /P

j−1i −→ · · · −→ P

j1/P

j−11 −→ P

j0/P

j−10 −→ 0

has only one nonzero homology Kj at Pj0/P

j−10 , is a �-Koszul module. In fact, Pi =⊕m

j=1 Pji /P

j−1i . Moreover, M has a filtration 0 = �0 ⊂ �1 ⊂ �2 ⊂ · · · ⊂ �m−1 ⊂ �l =

M such that�j/�j−1 � Kj and all Kj are �-Koszul modules.

Proof. (⇒) Let M = ⊕i≥0 Mi be a weakly �-Koszul module with its natural

filtration:

0 = �0 ⊂ �1 ⊂ �2 ⊂ · · · ⊂ �m−1 ⊂ �l = M�

By Theorem 3.6, all �i/�i−1�−di� �1 ≤ i ≤ m� are �-Koszul modules. By Lemma 4.1(2), we can get the following commutative diagram with exact columns:

which induces minimal graded projective resolutions of �j/�j−1�−dj = Kj for each1 ≤ j ≤ m,

· · · −→ Pji /P

j−1i −→ · · · −→ P

j1/P

j−11 −→ P

j0/P

j−10 −→ Kj −→ 0�

(⇐) IfM ∈ gr�A� has the minimal projective resolution

· · · −→ Pi −→ · · · −→ P1 −→ P0 −→ M −→ 0

and the complex �∗ · · · −→ Pi −→ · · · −→ P1 −→ P0 −→ 0 has a filtration asstated. Then it is not hard to check that for each 1 ≤ j ≤ m,

�j · · · −→ Pji −→ · · · −→ P

j1 −→ P

j0 −→ 0

has only one nonzero homology, sayKj , at Pj0. Therefore, the filtration of the complex

�∗ induces a filtration of the moduleM :

0 = �0 ⊂ �1 ⊂ �2 ⊂ · · · ⊂ �m−1 ⊂ �l = M�

Moreover, for each 1 ≤ j ≤ m, we have Kj = �j/�j−1�−dj is a �-Koszul module.Then of course,M is a weakly �-Koszul module, as desired. �

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1742 LÜ AND LU

Corollary 4.6. Let M ∈ ��A�. Use the notations of Theorem 4.4. Then pd�M� =max�pd�Ki� i = 1� 2� � � � � m�, where “pd” means projective dimension.

Proof. It is immediate from Theorem 4.4. �

Now we will give an example to explain Theorem 5.2 and Corollary 4.6.

Example 4.7. Let � be the quiver

Let A = k�/�u1u2u3�. Then A is a �-Koszul algebra, where

��n� =

3n2� if n is even�

3�n− 1�2

+ 1� if n is odd�

Let e1� � � � � e5 be the idempotents of A corresponding to the vertices. Let V = kv0 ⊕kv1 be a graded vector space with basis v0 and v1. Assume that the degree of v0 is 0and that of v1 is 1. Define a left A0-module action on V as follows: e4 · v0 = v0 andei · v0 = 0 for i �= 4; e5 · v1 = v1 and ei · v1 = 0 for i �= 5. Let

M = A⊗A0V

�u2 ⊗A0u3 ⊗A0

v0 − u4 ⊗A0v1�

�

Now it is not hard to check thatM is a weakly �-Koszul module.Under a routine computation, M admits the following minimal projective

resolution:

0 −→ P2�2 −→ P4 ⊕ P5�1 −→ M −→ 0�

where Pi is the indecomposable projective module corresponding to vertex i.Obviously, M has the following natural filtration: 0 = �0 ⊆ �1 ⊆ �2 = M�

where �1 = P4. Now using the notations of Theorem 4.4, K1 = P4 and K2 = M/P4 �S5�1. It is not difficult to work out that K1 and K2 admit the following minimalprojective resolutions, respectively:

0 −→ P4 −→ K1 −→ 0

and

0 −→ P2�2 −→ P5�1 −→ K2 −→ 0�

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Therefore, the projective resolution ofM is the direct sum of the resolutions of K1 andK2 and pd�M� = max�pd�K1�� pd�K2�� = 2.

The finitistic dimension conjecture, is one of the important and interestingconjectures in the representation theory of Artin algebras. Now we will recall thecontents of the original conjecture.

Let� be an arbitrary Artin R-algebra, where R is a commutative Artin ring withidentity. Let mod�� be the category of finitely generated �-modules and

Bound�� = �M ∈ mod��pd��M� < ��

The following is the well-known finitistic dimension conjecture. Let � be as above.Then

sup�pd��M� �M ∈ Bound�� < ��

It is too far to solve the conjecture completely and it still remains open now. Therefore,it is also interesting to find certain subcategories in which the finitistic dimensionconjecture holds.

Lemma 4.8. Let A be a finite dimensional �-Koszul algebra. Then the finitisticdimension conjecture holds in��A�.

Proof. It is immediate from ([6], Theorem 4.5). �

Corollary 4.9. Let ��A� denote the category of weakly �-Koszul modules over afinite dimensional �-Koszul algebra A. Then the finitistic dimension conjecture is true in��d�A�.

Proof. Let M ∈ ��A� and 0 = �1 ⊂ �2 ⊂ · · · ⊂ �m−1 ⊂ �m = M its specialfixed graded submodule filtration, the same as in Theorem 3.6. Set Ki = �i/�i−1

for i = 1� 2� � � � � m. First note that for each M ∈ ��A� with finite projectivedimension. Then by Corollary 4.6, we have pd�M� = max�pd�Ki� i = 1� 2� � � � � m�,which implies that pd�Ki� < � for all i = 1� 2� � � � � m. Therefore, we have

sup�pd�M� < ��M ∈ ��A�� = sup�max�pd�K1�� pd�K2�� pd�Km�� < ��

That is, the finitistic dimension conjecture is true in ��d�A�. �

5. THE ASSOCIATED GRADEDMODULE OF AWEAKLY �-KOSZULMODULE

LetM ∈ gr�A�. We can define another graded module, denoted by G�M�, calledthe associated graded module ofM as follows:

G�M� = M/JM ⊕ JM/J 2M ⊕ J 2M/J 3M ⊕ � � � �

Similarly, we can define G�A�.

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1744 LÜ AND LU

Lemma 5.1 ([14]). LetM = M0 ⊕M1 ⊕M2 ⊕ · · · ∈ gr�A� withM0 �= 0. Then we havea split exact sequence in gr�G�A�� = gr�A�

0 −→ G��M0�� −→ G�M� −→ G�M/�M0�� −→ 0�

Theorem 5.2. Let A be a �-Koszul algebra and M ∈ gr�A�. Then M ∈ ��A� if andonly if G�M� ∈ ��A�.

Proof. Assume thatM is generated by a minimal set of homogeneous elements lyingin degrees k0 < k1 < · · · < kp since M ∈ gr�A�. By Lemma 5.1, we get a split exactsequence,

0 −→ G��Mk0�� −→ G�M� −→ G�M/�Mk0

�� −→ 0�

�⇒� Recall that M is a weakly �-Koszul module, to prove that G�M� is a �-Koszul module, by induction on p. If p = 0,M is a weakly �-Koszul module generatedin a single degree, clearly M is a �-Koszul module and M � G�M� as a graded A-module. Hence G�M� is a �-Koszul module. Now we assume that the statement holdsfor less than p. By Lemma 5.1, �Mk0

� is a �-Koszul module, of course is a weakly �-Koszul module. Consider the exact sequence

0 −→ �Mk0� −→ M −→ M/�Mk0

� −→ 0�

By Lemmas 3.2 and 3.4, we get that M/�Mk0� is a weakly �-Koszul module. Observe

that the number of generators ofM/�Mk0� is less than p, by the induction assumption,

G�M/�Mk0�� ia a �-Koszul module. Note thatG��Mk0

�� is obvious a �-Koszul module,we get that G�M� is a �-Koszul module induced from the split exact sequence

0 −→ G��Mk0�� −→ G�M� −→ G�M/�Mk0

�� −→ 0�

�⇐� Suppose that G�M� is a �-Koszul module, we also do it by induction. Ifp = 0, thenG�M� � M as a gradedA-module. ClearlyM is a weakly �-Koszul module.Now we assume that the statement holds for less than p. We get that G��Mk0

�� andG�M/�Mk0

�� are �-Koszul modules induced from the split exact sequence

0 −→ G��Mk0�� −→ G�M� −→ G�M/�Mk0

�� −→ 0�

By the induction assumption, �Mk0� and M/�Mk0

� are weakly �-Koszul modules.Consider the exact sequence 0 −→ �Mk0

� −→ M −→ M/�Mk0� −→ 0 and by Lemmas

3.2 and 3.4, we get thatM is a weakly �-Koszul module. �

Lemma 5.3. Let 0 −→ K −→ M −→ N −→ 0 be an exact sequence in gr�A� such thateach term has the same highest degree. Then

0 −→ G�K� −→ G�M� −→ G�N� −→ 0

is exact if and only if, for all k ≥ 0, JkK = K ∩ JkM .

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Proof. The necessity is proved by the exact sequence

0 −→ JkK/Jk+1K −→ JkM/Jk+1M −→ JkN/Jk+1N −→ 0�

which is induced from 0 −→ G�K� −→ G�M� −→ G�N� −→ 0�The sufficiency is proved by the exact sequence

0 −→ JkK/Jk+1K −→ JkM/Jk+1M −→ JkN/Jk+1N −→ 0�

which is induced from the following commutative diagram with exact rows

�

Motivated by Theorem 5.2, one can ask the following question: Do there existany relationships between the resolutions of M and G�M�, where M is a weakly�-Koszul module? We have the following result.

Theorem 5.4. Let M be a weakly �-Koszul module and G�M� be its associated gradedmodule. Let

· · · −→ P2

d2−→ P1

d1−→ P0

d0−→ M −→ 0

and

· · · −→ Q2 −→ Q1 −→ Q0 −→ G�M� −→ 0

be the minimal graded projective resolutions. Then for all i ≥ 0, we have

Qi � G�Pi��i��

Proof. Let M be a weakly �-Koszul module. Then for all i ≥ 0, we have the exactsequences

0 −→ ker di −→ J��i+1�−��i�Pi −→ J��i+1�−��i� ker di−1 −→ 0�

where ker d−1 = M . By the definition of weakly �-Koszul modules, we have

Jk ker di = ker di ∩ J��i+1�−��i�+kPi = ker di ∩ Jk�J��i+1�−��i�Pi��

By Lemma 5.3, for all i ≥ 0, we have the exact sequences

0 −→ G�ker di� −→ G�J��i+1�−��i�Pi� −→ G�J��i+1�−��i� ker di−1� −→ 0�

which imply the following exact sequences

0 −→ G�ker di��i+ 1�− ��i� −→ G�Pi� −→ G�ker di−1� −→ 0�

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1746 LÜ AND LU

Putting all the above exact sequences together, we have the following minimal gradedprojective resolution of G�M�,

· · · −→ G�P1��1� −→ G�P0��0� −→ G�M� −→ 0�

Thus, we complete the proof since all the minimal projective resolution of a moduleare isomorphic. �

6. THE EXT MODULE OF AWEAKLY �-KOSZUL MODULE

Lemma 6.1. Let A be a �-Koszul algebra and M a �-Koszul module over A. Then as agraded E�A�-module, we have ��M� = �Ext0A�M�A0��.

Proof. It is immediate from (Proposition 3.5, [5]). �

Corollary 6.2. LetM ∈ gr�A�. Using the notations of Theorem 3.6. ThenM is a weakly�-Koszul module if and only if ��i/�i−1� is generated in degree 0 as a graded E�A�-module for all 1 ≤ i ≤ m.

Proof. By Theorem 3.6 and Lemma 6.1, we are done. �

Theorem 6.3. LetM be a weakly �-Koszul module. Using the notations of Theorem 3.6.Then ��M� is finitely generated in degree 0 as a graded E�A�-module.

Proof. First we claim that ��M� is generated in degree 0 as a graded E�A�-module.Indeed, consider the exact sequence

0 −→ �1 −→ �2 −→ �2/�1 −→ 0�

For all i ≥ 1, we have the exact sequences

0 −→ �i��1� −→ �i��2� −→ �i��2/�1� −→ 0�

which imply the following exact sequences for all i ≥ 1:

0 −→ ��2/�1� −→ ��2� −→ ��1� −→ 0�

ByTheorem 3.6 and Lemma 6.1, we have��1� and��2/�1� are generated in degree0 as a graded E�A�-module, which forces ��2� is generated in degree 0 as a gradedE�A�-module.

Next, we prove that ��M� is finitely generated as a graded E�A�-module. From[3], it is obvious that��M� is finitely generated as a gradedE�A�-module for a �-Koszulalgebra A and a �-Koszul module M over A. Thus, ��2� is finitely generated as agraded E�A�-module since the exact sequence

0 −→ ��2/�1� −→ ��2� −→ ��1� −→ 0

and the fact that the category of finitely generated modules is closed under extensions.Thus, we finish the proof by induction. �

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Remark 6.4. Different to �-Koszul modules, the converse of Theorem 6.3 is not true.The following is a counterexample.

The following example also appeared in Example 4.2 of [13].

Example 6.5. Let � be the quiver of Example 4.7 and A = k�/�u1u4�. Then A is a�-Koszul algebra, where ��i� = i for all i ≥ 0. Let e1� � � � � e5 be the idempotents of Acorresponding to the vertices. Let V = kv0 ⊕ kv1 be a graded vector space with basisv0 and v1. Assume that the degree of v0 is 0 and that of v1 is 1. Define a left A0-moduleaction on V as follows: e4 · v0 = v0 and ei · v0 = 0 for i �= 4; e5 · v1 = v1 and ei · v1 = 0for i �= 5. Let

M = A⊗A0V

�u2 ⊗A0u3 ⊗A0

v0 − u4 ⊗A0v1�

�

It is not hard to check ��M� = ��0�M�� as a graded E�A�-module. Let V0 = �v0� andV1 = �v1�. Then V = V0 ⊕ V1 as a left A0-module. But

�M0� �A⊗A0

V0

�u1u2u3 ⊗A0v0�

�

Clearly, �M0� is not a �-Koszul module. By Lemma 3.4,M is impossible to be a weakly�-Koszul module.

7. BLOCK �-KOSZUL MODULES

Definition 7.1. LetM ∈ gr�A�. Then we can find a set SM = �Sd0� Sd1� � � � � Sdm� ofA0-submodules ofM such that:

(1) d0 < d1 < · · · < dm;(2) Each Sdi is concentrated in degree di;(3) M/JM = Sd0 ⊕ Sd1 ⊕ · · · ⊕ Sdm as graded A0-modules.

Then SM is called a minimal homogeneous generating space of M . M is called a block�-Koszul module in case M = �Sd0� ⊕ �Sd1� ⊕ · · · ⊕ �Sdm� and M is a weakly �-Koszulmodule.

Example 7.2.

(1) Koszul modules, D-Koszul modules, piecewise-Koszul modules and �-Koszulmodules are special block �-Koszul modules.

(2) LetA be a �-Koszul algebra. ThenM = A0�i1⊕ A0�i2⊕ · · · ⊕ A0�iswith integers0 ≤ i1 < i2 < · · · < is is a block �-Koszul module.

The following provides a general method to construct block �-Koszul modulesfrom a given weakly �-Koszul module.

Proposition 7.3. Let M ∈ gr�A� be generated in degrees d0 < d1 < · · · < dm. Thenthere exist graded pure A-modules K0� K1� � � � � Km, such that:

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1748 LÜ AND LU

(1) As A0-modules,M � ⊕mi=0 Ki;

(2) G�M� � G�⊕m

i=0 Ki�;(3) G�M� � ⊕m

i=0 Ki�−di.

Moreover, if M is a weakly �-Koszul module, then for all 0 ≤ j ≤ l,⊕j

i=0 Ki areblock �-Koszul modules.

Proof. In fact, let Sd0� Sd1� � � � � Sdm be the minimal generating spaces of M and eachSdi is an A0-submodule of M consisting of homogeneous elements of degree di, (0 ≤i ≤ m). Let K0 = �Sd0�, where �Sd0� denotes the graded A-submodule of M generatedby Sd0 . Let

K1 = ��M/�Sd0��d1�� K2 = ��M/�Sd0��/K1�d2��

Now it is easy to see that each Ki is a graded pure module generated in degree di.From the construction of each Ki, the statements are clear. If M is a weakly �-Koszulmodule, by Theorem 3.6, one can get that each Ki is a �-Koszul module. Therefore,for all 0 ≤ j ≤ l,

⊕ji=0 Ki are block �-Koszul modules. �

Theorem 7.4. Let A be a �-Koszul algebra, M ∈ gr�A�. Using the notations ofTheorem 3.6. Then M is a block �-Koszul module if and only if M = ⊕m

i=0�Sdi� and all�Sdi� are �-Koszul modules.

Proof. �⇒� By the definition of block �-Koszul module,M = ⊕li=0�Sdi� is obvious.

Note that block �-Koszul modules are special weakly �-Koszul modules, by Theorem3.6, all�i/�i−1 are �-Koszul modules. Note also that

�i

�i−1

= �Sd0� Sd1� � � � � Sdi��Sd0� Sd1� � � � � Sdi−1

�

= �Sd0� ⊕ �Sd1� ⊕ · · · ⊕ �Sdi��Sd0� ⊕ �Sd1� ⊕ · · · ⊕ �Sdi−1

�= �Sdi��

Thus, all �Sdi� are �-Koszul modules.�⇐�Note that

�Sdi� =�Sd0� ⊕ �Sd1� ⊕ · · · ⊕ �Sdi��Sd0� ⊕ �Sd1� ⊕ · · · ⊕ �Sdi−1

�

= �Sd0� Sd1� � � � � Sdi��Sd0� Sd1� � � � � Sdi−1

�

= �i

�i−1

�

By assumption, all �Sdi� are �-Koszul modules, which implies that all �i/�i−1 are �-Koszul modules. By Theorem 3.6 again, we get that M is a weakly �-Koszul module.

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Now combining the condition M = ⊕li=0�Sdi�, we obtain that M is a block �-Koszul

module. �

Corollary 7.5. Let M = ⊕i≥k0

Mi be a block �-Koszul module and use the notations ofTheorem 3.6. Then

(1) All �Sdi� are �-Koszul modules, where 0 ≤ i ≤ m;(2) �Sdi� ∩ JkM = Jk�Sdi� for all k ≥ 0;

(3) All the A-modules�Sdi1 �Sdi2 ��Sdip ��Sdj1 �Sdj2 ��Sdjn �

are block �-Koszul modules, where �Sdj1� Sdj2

, � � � ,

Sdjn� ⊂ �Sdi1

� Sdi2� � � � , Sdip � ⊆ �Sd0� Sd1� � � � � Sdm�.

Proof. (1) is immediate from Theorem 7.4. (2) is a routine check. For (3), note

that�Sdi1 �Sdi2 ��Sdip ��Sdj1 �Sdj2 ��Sdjn �

=⊕p

a=1�Sdia �⊕nb=1�Sdjb �

= ⊕c�Sdic � for some 0 ≤ ic ≤ m. By Theorem 7.4, each

�Sdic � is a �-Koszul module. Therefore,⊕

c�Sdic � is a block �-Koszul module. �

Theorem 7.6. Using the notations of Theorem 3.6. Let M = ⊕mi=0�Sdi�. Then M is a

block �-Koszul module if and only if the Ext module of M , ��M� = ⊕n≥0 Ext

nA�M�A0� is

generated in degree 0 as a graded E�A�-module.

Proof. �⇒� It is immediate from Theorem 6.3.�⇐� Note that M = ⊕m

i=0�Sdi� and ��M� = ⊕n≥0 Ext

nA�M�A0� is generated

in degree 0 as a graded E�A�-module. Then each⊕

n≥0 ExtnA��Sdi�� A0� is a direct

summand of ��M�. Hence all ��Sdi�� =⊕

n≥0 ExtnA��Sdi�� A0� are generated in degree

0 as a graded E�A�-module. Thus each �Sdi� is a �-Koszul module. By Theorem 7.4,we finish the proof. �

ACKNOWLEDGMENTS

The authors would like to give their sincere thanks to the anonymous refereefor the careful reading and improved suggestions, which improve the quality of themanuscript a lot.

This work was supported by National Natural Science Foundation of China(11001245, 11271335 and 11101288), Natural Science Foundation of ZhejiangProvince (Y6110323).

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δ-Koszulity of Finitely Generated Graded Modules