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Cotorsion Theories Involving Auslander and BassClassesZhen Zhang a , Xiaosheng Zhu a & Xiaoguang Yan aa Department of Mathematics, Nanjing University, Nanjing, ChinaPublished online: 11 Sep 2012.

To cite this article: Zhen Zhang , Xiaosheng Zhu & Xiaoguang Yan (2012): Cotorsion Theories Involving Auslander and BassClasses, Communications in Algebra, 40:10, 3771-3782

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Communications in Algebra®, 40: 3771–3782, 2012Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927872.2011.594971

COTORSION THEORIES INVOLVING AUSLANDERAND BASS CLASSES

Zhen Zhang, Xiaosheng Zhu, and Xiaoguang YanDepartment of Mathematics, Nanjing University, Nanjing, China

Let R be a commutative ring, � be a semidualizing R-module. We show that theAuslander class ���R� with respect to � is the first left orthogonal class of somepure injective module M , that is, ���R� = ⊥1M� and the Bass class ���R� is thefirst right orthogonal class of some G�-projective module N , that is, ���R� = N⊥1 .As applications, we can see that ����R�����R�

⊥� is a cotorsion theory generated bya set. Especially, we show that �⊥���R�����R�� is a complete hereditary cotorsiontheory cogenerated by a set.

Key Words: Auslander class; Bass class; Cotorsion theory; (Pre)covering; (Pre)enveloping; Puresubmodule; Semidualizing module.

2010 Mathematics Subject Classification: 13D02; 13D07; 13E15; 16D10; 18A20.

INTRODUCTION

In 1994, Foxby [6] introduced two important classes of modules over a localCohen–Macaulay ring admitting a dualizing module, which we call Foxby classesnow. Enochs, Jenda, and Xu showed that these two classes are precisely the classesof modules with finite Gorenstein projective dimensions and finite Gorensteininjective dimensions, respectively, see [5, Corollaries 2.4 and 2.6]. We know thatFoxby classes are defined via dualizing modules, and the notion of semidualizingmodules (see Definition 1.4), which was also introduced by Foxby in [7], can beviewed as a generalization of dualizing modules and the Foxby classes associatedwith a semidualizing module � are the Auslander class ���R� and the Bass class���R� (see Definition 1.5).

Relative algebra with respect to a semidualizing module has caught manyauthors’ attention. For this topic, we refer the reader to see Holm and White’swork [13], and also [10, 17–19]. The Auslander class and the Bass class play veryimportant role in these studies. In [13], Holm and White characterized these twoclasses with the so-called �-projective and �-injective modules [13, Definition 5.1],where � is a semidualizing module. In this article, we will show the followingtheorem (Theorems 2.5 and 2.10).

Received February 22, 2011; Revised May 15, 2011. Communicated by I. Swanson.Address correspondence to Dr. Zhen Zhang, Department of Mathematics, Nanjing University,

Nanjing 210093, China; E-mail: zhangzhenbiye@gmail.com

3771

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3772 ZHANG ET AL.

Theorem. Let � be a semidualizing module over a commutative ring R. Then:

(1) The Auslander class ���R� = ⊥1M , where M is a pure injective R-module;(2) The Bass class ���R� = N⊥1 , where N is a G�-projective R-module.

By [12, Theorem 3.1] we know that both ���R� and ���R� are closed underpure submodules and pure quotient modules, and with our main theorem wecan also get this result. Therefore, we can get the existence of (pre)covers and(pre)envelopes by ���R� and ���R�, which was also given in [12]. Note thatHolm and Jørgensen showed that ����R�����R�

⊥� is a perfect cotorsion theory,see [12, Theorem 3.2]. Thus by our theorem, we can say that this cotorisontheory is generated by a set. Moreover, we prove the following proposition, seeProposition 2.11.

Proposition. �⊥���R�����R�� is a hereditary complete cotorsion theorycogenerated by a set with kernel ��.

Throughout this article, R is always a commutative ring and � is always asemidualizing R-module. A subcategory or a class of R-modules is a full subcategoryof the category of R-modules, which is closed under isomorphisms. For unexplainedconcepts and notations, we refer the reader to [4, 8].

1. PRELIMINARIES

In this section, we introduce a number of notions and results which willbe used throughout this work. First, we employ some notions used by Avramovand Martsinkovsky in [1], Holm in [9], also Sather-Wagstaff, Sharif, and Whitein [17–19].

Definition 1.1. Let � be a class of R-modules and M be an R-module. An �-resolution of M is an exact sequence � = · · · → X1 → X0 → M → 0 with each Xi ∈� . If for any X ∈ � , Hom�X��� is exact, then � = · · · → X1 → X0 → 0 is calleda proper �-resolution of M . The �-coresolution and the proper �-coresolution of Mare defined dually.

Let res � denote the class of modules admitting proper �-resolutions, andcores � denote the class of modules admitting proper �-coresolutions.

Definition 1.2. Let � be a class of R-modules. We call � projectively resolving if� contains all projective R-modules, and for every short exact sequence 0 → X′ →X → X′′ → 0 with X′′ ∈ � , the conditions X′ ∈ � and X ∈ � are equivalent. Dually,we can define injectively resolving class.

We denote �⊥ by the subcategory of R-modules M such that ExtiR�X�M� =0 for all i ≥ 1 and all X ∈ � . Similarly, ⊥� denotes the subcategory of modulesM such that ExtiR�M�X� = 0 for all i ≥ 1 and all X ∈ � . Especially, �⊥1 =�M �Ext1R�X�M� = 0 for any X ∈ �� and ⊥1� = �M �Ext1R�M�X� = 0 for any X ∈��. It is common to know that the ith kernel of a projective resolution of a module

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AUSLANDER CLASS AND BASS CLASS 3773

M is called the ith syzygy of M for i ≥ 0, denoted by �i�M�. Dually, we denote theith cosyzygy of M by �i�M�. Note that if the class � is projectively resolving, then� is syzygy closed, and �⊥ = �⊥1 . If � is injectively resolving, then � is cosyzygyclosed, and ⊥� = ⊥1� . The following remark is easy to be shown by dimensionshifting.

Remark 1.3. Let M be an R-module. Then:

(1) M⊥ = �N �Ext1�∐i≥0 �i�M�⊕M�N� = 0� = �∐

i≥0 �i�M�⊕M�⊥1 ;(2) ⊥M = �N �Ext1�N�∏i≥0 �

i�M�⊕M� = 0� = ⊥1�∏

i≥0 �i�M�⊕M�.

Semidualizing modules have already been defined over arbitrary associativerings [13, Definition 2.1]. Note here the ring is commutative.

Definition 1.4 ([23, 1.8]). An R-module � is called semidualizing if:

(1) � admits a degreewise finitely generated projective resolution;(2) The natural homothety map R −→ HomR����� is an isomorphism; and(3) Ext≥1

R ����� = 0.

It is well known that dualizing R-modules are just those semidualizing moduleswith finite injective dimensions. Avramov and Foxby defined the Auslander classand the Bass class via dualizing modules, while Foxby introduced two classes ofmodules with respect to a PG-module [7, Section 1]. In fact, semidualizing modulesare always PG-modules, thus the Auslander class and the Bass class with respect toa semidualizing module can be defined similarly.

Definition 1.5. Let � be a semidualizing R-module. The Auslander class withrespect to �, denoted by ���R�, consists of all R-modules M satisfying:

(1) TorR≥1���M� = Ext≥1R ���� ⊗M� = 0; and

(2) The natural map M → Hom���� ⊗M� is an isomorphism.

The Bass class with respect to �, denoted by ���R�, consists of all R-modules Msatisfying:

(1) Ext≥1R ���M� = TorR≥1���Hom���M�� = 0; and

(2) The natural evaluation map � ⊗Hom���M� → M is an isomorphism.

These two classes were characterized by Holm and White in [13] with the so-called �-projective (�-flat) and �-injective modules, which are of the form � ⊗P (� ⊗ F ) and Hom��� I� with P projective (F flat) and E injective, respectively.We denote the class of �-projective (flat) modules by �� (��) and the class of�-injective modules by ��. They also defined faithfully semidualizing bimodulesover noncommutative rings, see [13, Definition 3.1]. It was showed that if R iscommutative, then a semidualizing module is always faithfully semidualizing, see[13, Proposition 3.1].

Remark 1.6. Note that � is finitely presented and admits a degreewise finitelygenerated projective resolution, then it is not difficult to see that both ���R� and

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3774 ZHANG ET AL.

���R� are closed under direct products and direct sums. By [13, Proposition 4.2], weknow that these two classes are also closed under direct summands and direct limits.Moreover, if two modules in a short exact sequence are in ���R� (resp., ���R�),so is the third, see [13, Corollary 6.3]. Moreover, ���R� contains modules of finiteflat dimensions and those of finite ��-injective dimensions, and ���R� containsmodules of finite injective dimensions and those of finite ��-projective dimensions[13, Corollaries 6.2 and 6.1].

The following lemma is used frequently in this article.

Lemma 1.7 ([20, Theorem 2.8]). Let � be a semidualizing R-module and M be anR-module. Then the following hold:

(1) M ∈ ���R� if and only if Hom���M� ∈ ���R��

(2) M ∈ ���R� if and only if � ⊗M ∈ ���R��

Now we turn our attention to Wakamatsu tilting modules, which are essentialelements in the proof of our main results. The Wakamatsu tilting module isa kind of generalization of the tilting module [8, Definition 5.1.1], and it wasfirst introduced by Wakamatsu in [22]. Mantese and Reiten [14] characterizedWakamatsu tilting modules in terms of suitable subcategories of finitely generatedmodules and in terms of cotorsion theories over an Artin algebra. In fact,our inspiration to prove our main results is just from the similarity betweensemidualizing modules and Wakamatsu tilting modules and Mantese and Reiten’swork in [14]. Note that Wakamatsu tilting modules can be defined over commutativerings, see [22, p. 8], and some results in [14] also hold true in this setting. Recallthat for an R-module M , AddM�addM� denotes the full subcategory of R-moduleswhose objects are the direct summands of (finite) direct sums of copies of M .

Definition 1.8. An R-module T is called Wakamatsu tilting module if:

(1) T admits a degreewise finitely generated projective resolution;(2) Exti�T� T� = 0 for all i ≥ 1;

(3) There exists an exact sequence: 0 → Rf0→ T0

f1→ T1

f2→ · · · , where Ti ∈ addT andcokerfi ∈ ⊥T for any i ≥ 0.

Note that tilting modules in [21] are known as Wakamatsu tilting modulesnow. By [21, Corollary 3.2], we can deduce that over a commutative ringR, a semidualizing R-module T is always a Wakamatsu tilting R-module withEndR T = R. However, a Wakamatsu tilting R-module T is a left R-, right EndR T -semidualizing bimodule, but it is not a semidualizing R-bimodule in general.Here we give a different approach to show that over a commutative ring R, asemidualizing module is a Wakamatsu tilting module.

Lemma 1.9. Let � be a semidualizing R-module. Then � is a Wakamatsu tiltingmodule.

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AUSLANDER CLASS AND BASS CLASS 3775

Proof. We only need to prove that Definition 1.8(3) also holds for a semidualizingmodule �. Since � is semidualizing, there exists a HomR�−���-exact exactsequence, where ni are positive integers:

· · · → Rn1g1→ Rn0

g0→ � → 0�

Since HomR����� R, so it is not hard to see that when applying HomR�−��� tothe above finitely generated free resolution, we can get an exact and HomR�−���-exact sequence

0 → Rf0→ �n0

f1→ �n1 → · · · �

where fi = Hom�gi���. Let Ki = coker fi for i ≥ 0, then by dimension shifting, it iseasy to show that Ki = coker fi ∈ ⊥�. Thus � is a Wakamatsu tilting module. �

Now we know that for a semidualizing module �, there exists a long exact

sequence 0 → Rf0→ �n0

f1→ �n1 → · · · where ni are positive integers and coker fi ∈⊥�. In the rest of this article, we always denote the modules coker fi by Ki for i ≥ 0,then Ki ∈ ⊥� by Lemma 1.9. Moreover, by [23, Lemma 1.7], we conclude that Ki

admits a degreewise finitely generated projective resolution. Hence by [23, Definition2.1], we know that Ki is in fact a G�-projective module for each i ≥ 0.

We conclude this section by recalling the notion of cotorsion theory, whichwas first introduced by Salce in [16] for abelian groups, and which is a useful toolfor showing the existence of (pre)covers and (pre)envelopes, see [4, Definitions 5.1.1and 6.1.1].

Definition 1.10 ([4, Definition 7.1.2]). A pair of classes of modules �� �� is acotorsion theory if �⊥1 = and ⊥1 = � .

A set is said to generate the cotorsion theory if ⊥1 = � and a set ′ is saidto cogenerate �� �� if

′⊥1 = .

The existence of a set that generates or cogenerates a cotorsion theoryis closely related to the question of the existence of enough injectives orenough projectives for the cotorsion theory. In Section 2, we will show that����R�����R�

⊥� is a cotorsion theory generated by a set and �⊥���R�����R�� isa cotorsion theory cogenerated by a set.

For a cotorsion theory �� ��, we usually call � ∩ the kernel of �� ��.A cotorsion theory �� �� is called perfect if � is covering and is enveloping,and �� �� is complete if � is special precovering, equivalently, is specialpreenveloping [8, Lemma 2.2.6]. Note that a perfect cotorsion theory is alwayscomplete by [8, Corollary 2.2.5]. A cotorsion theory �� �� is hereditary if � isprojectively resolving, equivalently, is injectively resolving, or Ext≥1

R �F�G� = 0 forany F ∈ � and G ∈ [8, Lemma 2.2.10].

2. MAIN RESULTS

In this section, we prove our main results given as a theorem in theintroduction. First, we will show that the Auslander class ���R� is the left

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3776 ZHANG ET AL.

orthogonal class of a pure injective module. Let’s begin with two well-knownproperties of pure injective modules.

Proposition 2.1. The class of pure injective R-modules is closed under directproducts and cosyzygies.

Proof. It is easy to be seen by the definition of pure injective modules and [8,Lemma 3.2.10]. �

Proposition 2.2. For any R-module M , HomR�M�E� with E injective is a pureinjective module.

Proof. It is straightforward to prove. �

For an R-module M , we set M∗ = HomR�M�E� with E an injectivecogenerator. From the above proposition, M∗ is pure injective. We claim that⊥M∗ is independent of the choice of the injective cogenerator. In fact, N ∈ ⊥M∗ ⇔Exti≥1�N�M∗� = 0 ⇔ �Tori≥1�M�N��∗ = 0 ⇔ Tori≥1�M�N� = 0, where the secondequivalence is from [4, Theorem 3.2.1].

Proposition 2.3. Let � be a semidualizing R-module. Then the Auslander class���R� = ⊥�∗ ∩ cores ��.

Proof. By the above argument, A ∈ ⊥�∗ if and only if Tori≥1��� A�=0. On theother hand, for an R-module M , M ∈ cores �� if and only if Hom���� ⊗M� Mand Exti≥1���� ⊗M�=0 by Definition 1.1 and [20, Proposition 2.2]. Therefore, wehave ���R� = ⊥�∗ ∩ cores ��. �

Proposition 2.4. If � is a semidualizing module, then ⊥�∗ ∩ cores �� =⊥�∏

i≥0 K∗i ⊕�∗�, where Ki is the coker fi in the exact sequence 0 → R

f0→ �n0f1→

�n1 → · · · .

Proof. Let M ∈ ⊥�∏

i≥0 K∗i ⊕�∗�. Then M ∈⊥ �∗ and M ∈⊥ K∗

i for each i. Nextwe show that M ∈ cores ��. Note that �� is enveloping by [13, Proposition 5.3(c)].Then we first prove that M can be embedded into its ��-envelope.

Applying �−�∗ to the exact sequence 0 → R → �n0 → K0 → 0, we get anotherexact sequence 0 → K∗

0 → ��n0�∗ → E → 0. Since E is an injective cogenerator, wehave an exact sequence: 0 → M → ∏

E for an index set . Hence we have thefollowing commutative diagram with exact rows, where G is a pullback:

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AUSLANDER CLASS AND BASS CLASS 3777

Since Ext1�M�∏

K∗0�

∏Ext1�M�K∗

0� = 0, the first row is split, therefore we havea monomorphism 0 → M → ∏

��n0�∗. Note that

∏��

n0�∗ ∈ ��, so there existsan exact sequence 0 → M

→ Hom��� I0� → M ′ → 0 with I0 injective and the��-envelope of M . By [13, Theorem 6.4(b)] we have that Extk�Hom��� I0���

∗� Extk�I0� E� = 0 for all k ≥ 1. Thus Hom��� I0� ∈ ⊥�∗. As M ∈ ⊥�∗ and is an ��-envelope, so M ′ ∈ ⊥�∗. Moreover, we claim that M ′ ∈ ⊥�

∏i≥0 K

∗i �. In fact, consider

the following two exact sequences:

0 → K∗i+1 → ��ni+1�∗ → K∗

i → 0 and (†)

0 → M → Hom��� I0� → M ′ → 0� (‡)

Applying Hom�M ′�−� to �†�, we get Extk�M ′� K∗i � Extk+1�M ′� K∗

i+1� for all k ≥ 1.On the other hand, applying Hom�−� K∗

i+1� to �‡�, we get an exact sequence

Extk�Hom��� I0�� K∗i+1� → Extk�M�K∗

i+1� → Extk+1�M ′� K∗i+1�

→ Extk+1�Hom��� I0�� K∗i+1�

for all k ≥ 1. Since Ki+1 admits a degreewise finitely generated projective resolution,we have the following sequence:

Extk�Hom��� I0�� K∗i+1� �Tork�Ki+1�Hom��� I0���

∗

�Hom�Extk�Ki+1���� I0��∗�

where the first and the second isomorphisms follow from [4, Theorems 3.2.1] and[8, Lemma 1.2.11(d)], respectively. Note that � is a Wakamatsu tilting module andKi+1 ∈ ⊥�, so we can get that

Extk�Hom��� I0�� K∗i+1� = 0 = Extk+1�Hom��� I0�� K

∗i+1��

Thus Extk�M�K∗i+1� Extk+1�M ′� K∗

i+1� and Extk�M ′� K∗i � Extk�M�K∗

i+1� for i≥ 0and all k ≥ 1. So M ′ ∈ ⊥�

∏i≥0 K

∗i �. Hence M ′ ∈ ⊥�

∏i≥0 K

∗i ⊕�∗� and M ′ has an

injective ��-envelope. Repeating the same argument for M ′ and so on, we obtainthat M ∈ cores ��. Thus M ∈ ⊥�∗∩ cores ��.

Conversely, if M ∈ ⊥�∗ ∩ cores �� = ��, then there exists an exact sequence0 → M

→ Hom��� I0� → M ′ → 0 with M ′ ∈ �� = ⊥�∗∩ cores ��. By the aboveargument, Extk�M�K∗

i+1� Extk�M ′� K∗i � for all i ≥ 0 and k ≥ 1. So we only need to

show Ext1�M�K∗0�=0. Consider the following exact sequence,

0 → K∗0 → ��n0�∗

h→ E → 0 (�)

which is obtained by applying �−�∗ to the sequence 0 → Rf0→ �n0 → K0 → 0 and

h = f ∗0 . Applying Hom�M�−� to ���, we get the exact sequence:

Hom�M� ��n0�∗� → Hom�M�E� → Ext1�M�K∗0� → Ext1�M� ��n0�∗� = 0�

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3778 ZHANG ET AL.

Therefore, if we prove that Hom�M��∗��0�� → Hom�M�E� is surjective, i.e., for anyelement g ∈ Hom�M�E�, there exists an f ∈ Hom�M��∗��0�� such that hf = g, thenwe are done. Now consider the following diagram:

As E is an injective module, so there exists a morphism g′ ∈Hom�Hom��� I0�� E� such that g = g′. Applying Hom�Hom��� I0��−� to thebottom row, we get an exact sequence

Hom�Hom��� I0�� ��n0�∗� → Hom�Hom��� I0�� E� → Ext1�Hom��� I0�� K

∗0��

By [4, Theorems 3.2.1] and [8, Lemma 1.2.11(d)], we have the following sequence:

Extk≥1�Hom��� I0�� K∗0� �Tork≥1�K0�Hom��� I0���

∗

�Hom�Ext≥1�K0���� I0��∗�

But by Lemma 1.9 we know that K0 ∈ ⊥�, and hence Extk≥1�K0��� = 0, whichimplies that Extk≥1�Hom��� I0�� K

∗0� = 0. So

Hom�Hom��� I0�� ��n0�∗� → Hom�Hom��� I0�� E� → 0

is exact. Thus for the morphism g′ ∈ Hom�Hom��� I0�� E�, there exists a morphismg′′ ∈ Hom�Hom��� I�� ��n0�∗� such that hg′′ = g′. Let f = g′′. Then hf = hg′′ =g′ = g. Therefore, Hom�M� ��n0�∗� → Hom�M�E� → 0 is exact. �

Theorem 2.5. Let � be a semidualizing R-module. Then ���R� = ⊥�∏

i≥0 K∗i ⊕�∗�,

where∏

i≥0 K∗i ⊕�∗ is pure injective. In particular, ���R� = ⊥1M for some pure

injective R-module M .

Proof. The first assertion follows from Lemma 2.1 and Propositions 2.2, 2.3, and2.4. On the other hand, let H = ∏

i≥0 K∗i ⊕�∗ and M = ∏

j≥0 �j�H�⊕H . Then M is

also pure injective by Proposition 2.1 and so ���R� = ⊥1M by Remark 1.3. �

Remark 2.6.

(1) By Theorem 2.5, we can prove that ����R�����R�⊥� is a cotorsion theory

generated by a pure injective module. Thus it is a hereditary perfect cotorsiontheory by [8, Theorem 3.2.9], which was also shown by Holm and Jørgensen[12, Theorem 3.2]. Moreover, we claim that �� is the kernel of the cotorsiontheory. On the one hand, �� ⊆ ���R� ∩ ���R�

⊥ by Remark 1.6 and [19, Lemma4.7]. On the other hand, let M ∈ ���R� ∩ ���R�

⊥, then M ∈ ���R�, so thereexists an exact sequence 0 → M → Hom��� I� → M ′ → 0 with I injective byProposition 2.3. Thus M ′ ∈ ���R� by Remark 1.6. Since M ∈ ���R�

⊥, the

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AUSLANDER CLASS AND BASS CLASS 3779

exact sequence is split, so M ∈ ��. Hence, the kernel of the cotorsion theory����R�����R�

⊥� is ��.(2) By Theorem 2.5, the Auslander class ���R� = ⊥1M for some pure injective R-

module M , hence it is easy to show that both ���R� and ���R� are closedunder pure submodules and pure quotients by the definition of pure injectivemodules, Remark 1.6, Lemma 1.7, and [4, Proposition 5.3.8]. Therefore, by [15,Proposition 2.8 and Corollary 3.5(c)] and [11, Theorem 2.5] we conclude that���R� is preenveloping and ���R� is covering, which was also proved by Holmand Jørgensen [12, Theorem 3.1], note that the proof in it is valid even if R isnot Noetherian.

Next, we will show that the Bass class ���R� is the right orthogonal classof some module. We employ the notions of Mantese and Reiten in [14]. For aWakamatsu tilting module, Gen∗�T� denotes the subcategory of all modules M suchthat there exists an exact sequence · · · → T 1

g1→ T 0g0→ M → 0 where T i ∈ AddT and

Ext1�T� kergi� = 0 for i ≥ 0. Mantese and Reiten [14, Proposition 3.6] showed that,if is an Artin algebra, we have

T⊥ ∩Gen∗�T� =(⊕

i≥0

Ki ⊕ T

)⊥�

On the other hand, by Lemma 1.9 we know that a semidualizing module � is infact a wakamatsu tilting module, and Ki = coker fi, which is defined in Lemma1.9, admits a degreewise finitely generated projective resolution, so Extj�Ki��

�I�� �Extj�Ki�����I� = 0 for all j ≥ 1, i ≥ 0 and some index set I by [8, Lemma 3.1.6].Hence it is not hard to see from the proof of [14, Proposition 3.6] that over acommutative ring R, we also have a similar equality for a semidualizing module�, that is, �⊥ ∩Gen∗��� = �

⊕i≥0 Ki ⊕��⊥. Now we show that ���R� = �⊥ ∩

Gen∗���. To this end, we need the following two lemmas.

Lemma 2.7. Let � be semidualizing. Then Add� = ���

Proof. It is trivial by the definition of Add � and Remark 1.6. �

Lemma 2.8. Let � be a semidualizing R-module. Then res �� = Gen∗���.

Proof. Let N ∈ Gen∗���, then there exists an exact sequence

· · · → �1 f 1→ �0 f 0→ N → 0�

where �i ∈ Add� = �� by Lemma 2.7 and Ext1��� ker f i� = 0 for any i ≥ 0, whichimplies that the sequence is Hom���−� exact. Therefore, it is also Hom�� ⊗ P�−�exact with P projective, since Hom�� ⊗ P�−� Hom�P�Hom���−��. Hence N ∈res ���

Conversely, let M ∈ res ��. Then M admits a proper ��-resolution:

· · · → � ⊗ P1

g1→ � ⊗ P0

g0→ M → 0�

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3780 ZHANG ET AL.

Note that � ⊗ Pi ∈ Add� by Lemma 2.7 and the sequence is Hom���−� exact by

Definition 1.1. Applying Hom���−� to the exact sequence 0 → ker gi → � ⊗ Pi

gi→ker gi−1 → 0, where i ≥ 0 and ker g−1 = M , we get a long exact sequence:

Hom���� ⊗ Pi� → Hom��� ker gi−1� → Ext1��� ker gi� → Ext1���� ⊗ Pi��

By [13, Theorem 6.4], Ext1���� ⊗ Pi� = 0. So Ext1��� ker gi� = 0 for i ≥ 0.Therefore, M ∈ Gen∗���. �

Proposition 2.9. Let � be a semidualizing R-module. Then the Bass class ���R� =�⊥ ∩ res �� = �⊥ ∩Gen∗���.

Proof. By Definition 1.5, for an R-module M , M ∈ ���R� if and only if M ∈ �⊥,� ⊗Hom���M� M and Tori≥1���Hom���M�� = 0� But � ⊗Hom���M� M

and Tori≥1���Hom���M�� = 0 if and only if M ∈ res �� by Definition 1.1 and [20,Proposition 2.2]. Hence ���R� = �⊥ ∩ res ��. The second equality follows fromLemma 2.8. �

Theorem 2.10. Let � be a semidualizing R-module. Then ���R� = �⊕

i≥0 Ki ⊕��⊥.In particular, �� = N⊥1 , where N is a G�-projective R-module.

Proof. The first assertion is the immediate consequence of Lemma 1.9,Proposition 2.9 and remarks above Lemma 2.7. Moreover, let W = ⊕

i≥0 Ki ⊕� andN = ∐

j∈� �j�W�⊕W , then ���R� = N⊥1 by Remark 1.3 and N is G�-projectivefollows from the augments below Lemma 1.9 and [23, Propositions 2.4 and 2.6]. �

Holm and Jørgensen in [12, Theorem 3.2] showed that ���R� is covering andpreenveloping. On the other hand, by Remark 2.6(2), we also conclude that ���R�is covering. Moreover, with Theorem 2.10, we have the following Proposition.

Proposition 2.11. �⊥���R�����R�� is a hereditary complete cotorsion theorycogenerated by a set with kernel ��. Therefore, ���R� is a preenveloping class.

Proof. By Theorem 2.10 and [8, Theorem 3.2.1], we know that �⊥1���R�����R��is a complete cotorsion theory cogenerated by a set. Since ���R� is aninjectively resolving class, ⊥1���R� = ⊥���R�. Therefore, �

⊥���R�����R�� is just�⊥1���R�����R��.

Clearly, �� ⊆ ���R� ∩ ⊥���R� by Remark 1.6 and [19, Lemma 4.7].Conversely, let M ∈ ���R� ∩ ⊥���R�, then M ∈ ���R�. So there exists a proper��-resolution 0 → M ′ → � ⊗ P → M → 0 by Proposition 2.9. Thus M ′ ∈ ���R�and the exact sequence is split. So M ∈ ��� Thus, the kernel of the cotorsion theory�⊥���R�����R�� is ��. �

Remark 2.12. When �R�m� k� is a local Cohen–Macaulay ring with a dualizingmodule D, the existence of covers and preenvelopes by modules of finite �-Gorenstein projective (flat) dimensions and modules of finite �-Gorenstein injectivedimensions is easy to be shown by Remark 2.6, Proposition 2.11 and [2, 10], which

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AUSLANDER CLASS AND BASS CLASS 3781

was a special case of [12, Theorem 3.3]. Particularly, when � = R, we can deducethe existence of covers and preenvelopes by modules of finite Gorenstein projective(flat) dimensions and modules of finite Gorenstein injective dimensions, which wereshown by Enochs and Holm [3, Corollary 3.13].

ACKNOWLEDGMENTS

The authors would like to express their sincere thanks to the referee for his orher careful reading of the manuscript and helpful suggestions.

This research was partially supported by the National Natural ScienceFoundation of China (No. 10971090).

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