Stability of Cartan-Eilenberg Gorenstein categories

Stability of Cartan-Eilenberg Gorenstein categories

LILIANG(*) - GANGYANG(**)

ABSTRACT- We study Cartan-Eilenberg Gorenstein categories by introducingCE- projectiveCE-generators andCE-injectiveCE-cogenerators in the paper. We give a relationship between injective cogenerators (resp., projective generators) introduced by Sather-Wagstaff, Sharif and White andCE-injectiveCE-cogen- erators (resp.,CE-projectiveCE-generators). As applications, we prove some stability results of Cartan-Eilenberg Gorenstein categories.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 18G10, 18G25, 18G35.

KEYWORDS. Gorenstein categories, Cartan-Eilenberg Gorenstein categories, pro- jective generators, injective cogenerators.

1. Introduction

Throughout this work,Rdenotes an associative ring with identity, and by anR-module we mean a leftR-module.

Auslander and Bridger [1] introduced the Gorenstein dimension for finitely generated modules over a commutative Noetherian ring. Enochs and Jenda [4] defined Gorenstien projective modules over arbitrary rings whether the modules are finitely generated or not. It is well known that, for finitely generated modules over commutative Noetherian rings, Gor- enstein projective modules are actually the modules of Gorenstein di- mension zero. Dually, Gorenstein injective modules were defined in [4].

Sather-Wagstaff, Sharif and White [7] introduced the concept ofW-Gor- enstein modules that unifies the concepts of Gorenstein projective and (*) Indirizzo dell'A.: School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China.

E-mail: lliangnju@gmail.com

(**) Indirizzo dell'A.: School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China.

E-mail: yanggang@mail.lzjtu.cn

injective modules, and they studied the stability of Gorenstein categories by using so called projective generators and injective cogenerators (see (2.3) for definitions) that are very useful tools for studying Gorenstein categories.

In his thesis Verdier [8] considered Cartan-Eilenberg projective and injective resolutions of a complex using the ideas of Cartan and Eilenberg [2]. Furthermore, Enochs [3] introduced Cartan-Eilenberg Gorenstein projective and injective complexes in terms of so called Cartan-Eilenberg complete projective and injective resolutions, respectively. Using the ideas of Sather-Wagstaff, Sharif and White [7], we introduce CE-projective CE- generators and CE-injective CE-cogenerators in this paper (see Definition 3.2), which are useful for studying Cartan-Eilenberg Gorenstein cate- gories. In particular, we give a relationship between injective cogenerators (resp., projective generators) and CE-injective CE-cogenerators (resp., CE-projective CE-generators) as follows (see Theorem 3.3):

THEOREMA. Assume thatWandXare classes ofR-modules such that W X and W is closed under finite direct sums. Let CE(W) (resp., CE(X)) be the class of CE-W(resp., CE-X) complexes ofR-modules. IfX is closed under extensions, then the following statements hold.

(1) W is an injective cogenerator forX if and only if CE(W) is a CE- injective CE-cogenerator for CE(X).

(2) W is a projective generator forX if and only if CE(W) is a CE- projective CE-generator for CE(X).

As an application of Theorem A, we get the next result that compares to [3, Theorem 8.5] (see Proposition 3.7).

THEOREM B. If W is a class of R-modules closed under finite direct sums satisfying W ? W, then CE(GM(W))ˆ G_CE(CE(W)) GC(CE(W)).

Furthermore, if W contains R or a faithfully injective R-module, then G_CE(CE(W))ˆ G_C(CE(W)).

Sather-Wagstaff, Sharif and White gave a stability result of Gorenstein categories as follows.

THEOREM. ([7, Theorem 4.9]) Assume thatW is a class ofR-modules closed under finite direct sums satisfying W ? W. Let ! X₁! X0!X 1 ! be an exact complex ofR-modules inGM(W) such that it is Hom(GM(W); ) and Hom( ;GM(W))-exact, then each K_iˆ Ker(X_i!X_{i 1}) is in GM(W) for i2Z.

The next result shows that the corresponding stability result of Cartan- Eilenberg Gorenstien categories is also true (see Theorem 3.12).

THEOREMC. Assume thatVis a class of complexes ofR-modules closed under finite direct sums. Let !G ¹!G⁰!G¹! be a CE-exact complex of complexes inG_CE(V) such that it is Hom(V; ) and Hom( ;V)- exact. Then eachKⁱˆKer(Gⁱ!G^i‡1) is inGCE(V) fori2Z. In particular, G_CE(G_CE(V))ˆ G_CE(V).

We notice that Theorem C does not need the conditionV ? V, and one can find a partial converse in Proposition 3.15.

2. Preliminaries

We begin with some notation and terminology for use throughout this paper.

2.1. LetAbe an abelian category. A complex !X1!^dX1 X0!^dX0 X 1! of objects ofAwillbedenoted by(X;d^X) or simplyX. Thenthboundary(resp., cycle, homology) ofXis defined as Imd^X_n‡1(resp., Kerd^X_n, Kerd^X_n=Imd^X_n‡1) and it is denoted by Bn(X) (resp., Zn(X), Hn(X)). We let Z(X);B(X) and H(X) denote the complexes of cycles, boundaries and homologies ofX, respectively.

For any m2Z, S^mX denotes the complex with the degree-n term (S^mX)_n ˆX_{n m} and whose boundary operators are ( 1)^md^X_{n m}. We set SMˆS¹M.

IfX and Y are both complexes of objects of A, then by a morphism a:X!Y we mean a sequencean:Xn!Yn such thatan 1d^X_n ˆd^Y_nanfor eachn2Z. We let Hom(X;Y) denote the set of morphisms of complexes from XtoY, and Extⁱ(X;Y) (i1) denote the right derived functors of Hom. The mapping cone Cone(a) ofais defined as Cone(a)_nˆY_nX_{n 1} withnth boundary operatord^Cone(a)_n ˆd^N_n an 1

0 d^M_{n 1}

. For an objectM of A, we letHom(M;X) denote the complex

!HomA(M;X1)!HomA(M;X0)!HomA(M;X 1)! : The complexHom(X;M) can be given dually. Given a classDof objects ofA, we say that a complex X is Hom(D; )-exact if Hom_A(M; ) exacts the complex Xfor any M2 D, that is, the complexHom(M;X) is exact. The term Hom( ;D)-exact is defined dually.

LetMbe an object ofA.Mis the complex !0!M!^1M M!0!

with all terms 0 except M in the degrees 1 and 0, and M denotes the complex

!0!M!0! with all terms 0 exceptMin the degree 0.

2.2. LetAbe an abelian category and Da class of objects of A. Fol- lowing from [7], an exact complex of objects inDis calledtotallyD-acyclic if it is Hom_A(D; )-exact and Hom_A( ;D)-exact. Let G_A(D) denote the subcategory ofAwith objects of the formMKer(d^X₁) for some totally D-acyclic complexX; in this case we say thatMisD-Gorenstein.

2.3. Throughout the paper, M always denotes the category of R- modules, and C denotes the category of complexes of R-modules. We usually useWandXto denote two classes ofR-modules such thatW X and W is closed under finite direct sums. We write X ? W if Extⁱ_R(X;W)ˆ0 for anyX2 X,W2 W andi1. Following from [7],W is called a cogenerator for X if, for each X2 X, there exists an exact sequence 0!X!W!X⁰!0 withW 2 WandX⁰2 X.Wis called an injective cogenerator for X if W is a cogenerator for X and X ? W.

Generatorsandprojective generatorscan be defined dually.

The next definition can be found in [3].

DEFINITION2.4. A sequence !X ¹!X⁰!X¹! of complexes ofR-modules is said to be CE-exact if

(1) !X ¹ !X⁰!X¹! ,

(2) !Z(X ¹)!Z(X⁰)!Z(X¹)! , (3) !B(X ¹)!B(X⁰)!B(X¹)! ,

(4) !X ¹=B(X ¹)!X⁰=B(X⁰)!X¹=B(X¹)! , (5) !X ¹=Z(X ¹)!X⁰=Z(X⁰)!X¹=Z(X¹)! , and (6) !H(X ¹)!H(X⁰)!H(X¹)!

are exact.

By [3, Lemma 5.2], if (1) and (2), or (1) and (4) of Definition 2.4 are exact then all of (1) (6) are exact.

2.5. LetWbe a class ofR-modules. Following from [3], a complexXof R-modules is called a CE-W complex if X, Z(X), B(X) and H(X) are complexes ofR-modules inW. We denote the class of CE-Wcomplexes by CE(W). We sometimes name the CE-Wcomplex by the name of the class W. For example, the CE-projective (resp. CE-injective) complexes are actually the CE-P(resp. CE-I)-complexes, whereP(resp.I) is the class of projective (resp. injective)R-modules.

Given two complexes X and Y of R-modules, it follows from [3, Theorems 5.5 and 5.7] that there exist two CE-exact sequences !P ¹ !P⁰!X!0 and 0!Y !I⁰!I¹ ! , wherePⁱis a CE- projective complex of R-modules for eachi0 andI^k is a CE-injective complex of R-modules for each k0. By [3, Proposition 6.3], we can compute derived functors of Hom using either of the two sequences. We denote these derived functors by Extⁱ(X;Y). Note that CE-exact se- quence of complexes ofR-modules is exact. Thus we have Ext¹(X;Y) Ext¹(X;Y). One can check easily that for any CE-exact sequence 0!A!B!C!0 of complexes ofR-modules, there exist exact se- quences

0!Hom(X;A)!Hom(X;B)!Hom(X;C)!Ext¹(X;A)! and

0!Hom(C;Y)!Hom(B;Y)!Hom(A;Y)!Ext¹(C;Y)! : The next lemma was given in [3, Proposition 2.1 and Lemmas 9.1, 9.2 and 9.3].

LEMMA2.6. IfX is a complex ofR-modules and M is an R-module, then for any k2Zwe have the following statements.

(1) Hom(X;S^kM)Hom(X_k=B_k(X);M).

(2) Hom(S^kM;X)Hom(M;Zk(X)).

(3) Hom(X;S^kM)Hom(X_k;M).

(4) Hom(S^kM;X)Hom(M;X_k‡1).

(5) Ext¹(X;S^kM)Ext¹(X_k=B_k(X);M).

(6) Ext¹(S^kM;X)Ext¹(M;Z_k(X)).

(7) Ext¹(X;S^kM)Ext¹(Xk;M).

(8) Ext¹(S^kM;X)Ext¹(M;X_k‡1).

3. CE-Gorenstein categories

The next lemma that will be used frequently throughout the paper is akin to [3, Proposition 3.3].

LEMMA 3.1. If W?W, then a complex X ofR-modules is a CE-W complex ifand only ifX can be written as the form of (_i2ZSⁱX_i)(_i2ZSⁱX_i⁰) or (Q

i2ZSⁱX_i)(Q

i2ZSⁱX_i⁰)

with X_i and X_i⁰in Wfor each i2Z.

PROOF. LetXˆ(i2ZSⁱXi)(i2ZSⁱX_i⁰), whereXiandX_i⁰are inW for eachi2Z. Since_i2ZSⁱX_iand_i2ZSⁱX⁰_iare CE-Wcomplexes, we get thatXis a CE-Wcomplex.

Conversely, assume thatXis a CE-Wcomplex. Then the sequences 0!Zn(X)!Xn!Bn 1(X)!0

and

0!B_n(X)!Z_n(X)!H_n(X)!0

are split for each n2Z since W?W, and so Xnˆ(Bn(X)Bn 1(X)) H_n(X). This implies that Xˆ(_i2ZSⁱB_i(X))(_i2ZSⁱH_i(X)). By as- sumption B_i(X) and H_i(X) are inWfor eachi2Z.

DEFINITION3.2. Let V and Y be classes of complexes of R-modules such that V Y. V is said to be a CE-cogenerator for Y if, for any Y2 Y, there is a CE-exact sequence 0!Y!V!Y⁰!0 such that V 2 V and Y⁰2 Y. We say that V is a CE-injective CE-cogenerator for Y if V is a CE-cogenerator for Y and Extⁱ(Y;V)ˆ0 for any Y2 Y, V 2 V and i1. CE-generators and CE-projective CE-generators can be defined dually.

The next result gives a relationship between injective cogenerators (resp., projective generators) and CE-injective CE-cogenerators (resp., CE-projective CE-generators).

THEOREM 3.3. If X is closed under extensions, then the following statements hold.

(1) W is an injective cogenerator for X if and only if CE(W) is a CE-injective CE-cogenerator forCE(X).

(2) W is a projective generator for X if and only if CE(W) is a CE-projective CE-generator for CE(X).

PROOF. We prove part (1); the proof of part (2) is dual. Assume that Wis an injective cogenerator forX, and letX2CE(X). Then B_i(X) and H_i(X) are in X for all i2Z, and so there are exact sequences 0!Bi(X)!W_i⁰!T_i⁰!0 and 0!Hi(X)!W_i⁰⁰!T⁰⁰_i !0 withW_i⁰and W_i⁰⁰ in W, andT_i⁰ and T⁰⁰_i in X. Note that the sequence

0!B_i(X)!Z_i(X)!H_i(X)!0

is exact and Hom( ;W)-exact sinceX ? W. Then by [5, Remark 8.2.2], we get an exact sequence

0!Z_i(X)!W_i⁰W_i⁰⁰!T_i!0 such that the diagram

with exact rows and columns is commutative. SinceXis closed under ex- tensions,T_i2 X. Note that the sequence

0!Zi(X)!Xi!Bi 1(X)!0

is exact and Hom( ;W)-exact. Then by [5, Remark 8.2.2], we get an exact sequence

0!X_i!W_i⁰W_i⁰⁰W_i⁰ ₁!X_i⁰!0 such that the diagram

with exact rows and columns is commutative. SinceXis closed under ex- tensions,X_i⁰2 X. Now consider the following commutative diagram

Then we get a CE-exact sequence

0!X!W!X⁰!0;

whereWˆ(_i2ZSⁱW_i⁰)(_i2ZSⁱW_i⁰⁰). From the above constructions on W andX⁰, we getW2CE(W) andX⁰2CE(X). Thus CE(W) is a CE-co- generator for CE(X).

Let X2CE(X) and M2 W, and let 0!M!I₀!I ₁! be an injective resolution ofM. Then 0!S^kM!S^kI₀!S^kI ₁! is a CE- exact sequence withS^kI_iCE-injective for eachi0, and so

Extⁱ(X;S^kM)Ext¹(X;S^kC2 i)

for each i1, where Cj ˆCoker(Ij‡1!Ij) for j 1 with C0 ˆ Coker(M!I0) andC1ˆM. By Lemma 2.6,

Ext¹(X;S^kC_{2 i})Ext¹(X_k=B_k(X);C_{2 i})Extⁱ(X_k=B_k(X);M)ˆ0 sinceX_k=B_k(X)2 Xby the exactness of the sequence

0!H_k(X)!X_k=B_k(X)!B_{k 1}(X)!0

and the fact thatXis closed under extensions. Thus Extⁱ(X;S^kM)ˆ0 for eachi1. Similarly, Extⁱ(X;S^kM)ˆ0. Note thatW ? W, then a CE-W complexWcan be written as the form of (Q

i2ZSⁱWi)(Q

i2ZSⁱW_i⁰) with W_iandW_i⁰inWby Lemma 3.1. Thus Extⁱ(X;W)ˆ0 for eachW2CE(W).

This implies that CE(W) is a CE-injective CE-cogenerator for CE(X).

Conversely, assume that CE(W) is a CE-injective CE-cogenerator for CE(X). LetM2 X. ThenM2CE(X), and so there is a CE-exact sequence

0!M!W !X⁰!0

withW2CE(W) andX⁰2CE(X). Thus we get an exact sequence 0!M!W0!X₀⁰ !0

withW02 WandW02 X. This implies thatWis a cogenerator forX. On the other hand, for any M2 X and N2 W, by Lemma 2.6, we have Extⁱ(M;N)Ext¹(M;C)Ext¹(M;C)Extⁱ(M;N)ˆ0 since M2CE(X) andN2CE(W), whereCis anR-module. This implies thatWis an injective

cogenerator forX. p

DEFINITION3.4. LetVbe a class of complexes ofR-modules. We call that a complexXofR-modules has a CE-completeV-resolution if there is a CE- exact sequence !X ¹!X⁰!X¹! of complexes inV such that the sequence is Hom(V; ) and Hom( ;V)-exact andXKer(X⁰!X¹).

The class of complexesXthat has a CE-completeV-resolution is denoted by GCE(V).

COROLLARY 3.5. Let X be closed under extensions and W both an injective cogenerator and a projective generator for X. Then CE(X) G_CE(CE(W)).

PROOF. Suppose thatX2CE(X), then there is a CE-exact sequence 0!X!W⁰!X⁰!0

()

withW⁰2CE(W) andX⁰2CE(X) by Theorem 3.3. ForW 2CE(W), we get an exact sequence

0!Hom(X⁰;W)!Hom(W⁰;W)!Hom(X;W)!Ext¹(X⁰;W):

Since Ext¹(X⁰;W)ˆ0 by Theorem 3.3, the sequence () is Hom( ;CE(W))-exact. Similarly, one can check that it is also Hom(CE(W); )-exact. Using the same procedure we can construct a CE-exact sequence

0!X!W⁰!W¹! (y)

with Wⁱ2CE(W) such that it is Hom(CE(W); ) and Hom( ;CE(W))- exact.

Using the similar method we can construct another CE-exact sequence !W ²!W ¹!X!0

(z)

with Wⁱ2CE(W) such that it is Hom(CE(W); ) and Hom( ;CE(W))- exact.

Now assembling the sequences (y) and (z) we get a CE-complete CE(W)-resolution ofX, and soX2 GCE(CE(W)). p LEMMA3.6. LetW ? W. If X2CE(W), then X=B(X)is a complex ofR- modules inW.

PROOF. IfW ? W, thenWis closed under extensions sinceWis closed under finite direct sums. Note that Hi(X) and Bi(X) are inWfor alli2Z, thenX_i=B_i(X)2 Wsince the sequence

0!H_i(X)!X_i=B_i(X)!B_{i 1}(X)!0

is exact. This implies thatX=B(X) is a complex ofR-modules inW. p The next result augments Corollary 3.5 in the special caseX ˆ GM(W).

PROPOSITION 3.7. If W ? W, then CE(G_M(W))ˆ G_CE(CE(W)) G_C(CE(W)). Furthermore, if W contains R or a faithfully injective R- module, thenGCE(CE(W))ˆ GC(CE(W)).

PROOF. By [7, Theorem B and Corollary 4.7],GM(W) is closed under extensions andW is both a projective generator and an injective cogen- erator forG_M(W). Thus CE(G_M(W)) G_CE(CE(W)) by Corollary 3.5. For the inverse containment, we letX2 GCE(CE(W)). Then there is a CE-exact

sequence

!W ¹!W⁰!W¹! ()

of complexes in CE(W) such that it is Hom(CE(W); ) and Hom( ;CE(W))-exact and XˆKer(W⁰!W¹). Let M2 W. Then S^kM2CE(W), and so Hom(S^kM; ) and Hom( ;S^kM) exact the se- quence (). Thus Hom(M; ) and Hom( ;M) exact the sequence

!W_k¹!W_k⁰!W_k¹!

for any k2Z by Lemma 2.6. Note that X_k ˆKer(W_k⁰!W_k¹), then X_k2 G_M(W) for eachk2Z.

SinceS^kM2CE(W), we get an exact sequence

!Hom(W¹;S^kM)!Hom(W⁰;S^kM)!Hom(W ¹;S^kM)! ; and so the sequence

!Hom(W_k¹=B_k(W¹);M)!Hom(W_k⁰=B_k(W⁰);M)!

!Hom(W_k¹=B_k(W ¹);M)! is exact by Lemma 2.6. Consider the following commutative diagram

Note that the first and second rows of the above diagram are exact, and all columns are exact since the sequence

0!B_k(Wⁱ)!W_kⁱ !W_kⁱ=B_k(Wⁱ)!0

is split exact for anyi2Zby Lemma 3.6. Thus the last row of the above diagram is exact. This implies that the sequence

!Bk(W ¹)!Bk(W⁰)!Bk(W¹)! (y)

is Hom( ;W)-exact for anyk2Z.

On the other hand, since the sequence

!Hom(S^kM;W ¹)!Hom(S^kM;W⁰)!Hom(S^kM;W¹)! is exact, we get an exact sequence

!Hom(M;Z_k(W ¹))!Hom(M;Z_k(W⁰))!Hom(M;Z_k(W¹))! by Lemma 2.6. Consider the following commutative diagram

Note that the first and second rows of the above diagram are exact, and all columns are exact since the sequence

0!Z_k(Wⁱ)!W_kⁱ!B_{k 1}(Wⁱ)!0

is split exact for anyi2Z. Thus the last row of the above diagram is exact.

This implies that the sequence (y) is Hom(W; )-exact for anyk2Z. Thus B_k(X)ˆKer(B_k(W⁰)!B_k(W¹))2 G_M(W) for any k2Z. Similarly, one can check that Z_k(X) and H_k(X) are in G_M(W) for any k2Z. Thus X2CE(GM(W)). This implies that CE(GM(W))ˆ GCE(CE(W)).

By definition, G_CE(CE(W)) GC(CE(W)). Let X2 GC(CE(W)). Then there is an exact sequence

!X ¹!X⁰!X¹ ! (z)

with Xⁱ2CE(W), such that the sequence is Hom(CE(W); ) and Hom( ;CE(W))-exact and XˆKer(X⁰!X¹). If R2 W, then S^kR2 CE(W) for anyk2Z, and so the sequence

!Hom(S^kR;X ¹)!Hom(S^kR;X⁰)!Hom(S^kR;X¹)! is exact. Thus the sequence

!Z_k(X ¹)!Z_k(X⁰)!Z_k(X¹)!

is exact by Lemma 2.6. This implies that the sequence (z) is CE-exact, and soX2 GCE(CE(W)). IfWcontains a faithfully injectiveR-moduleE, then S^kE2CE(W) for anyk2Z, and so the sequence

!Hom(X¹;S^kE)!Hom(X⁰;S^kE)!Hom(X ¹;S^kE)! is exact. Thus the sequence

!X_k¹=Bk(X ¹)!X_k⁰=Bk(X⁰)!X¹_k=Bk(X¹)!

is exact by Lemma 2.6. This implies that the sequence (y) is CE-exact, and

soX2 GCE(CE(W)). p

The next result extends [6, Theorem 3.6]. We prove it using a different method for our convenience.

PROPOSITION3.8. Assume that Ais an abelian category andDis a class ofobjects ofA. Let

0!M⁰!^f M!^g M⁰⁰!0 ()

be a sequence inA, and let

W^0‡ !W₂^{0 d}!^W

0 2 W₁⁰ !^dW

0

1 W₀⁰ !^a0 M⁰!0 be a sequence with W_i⁰2 D, and

W^‡ !W2!^dW2 W1!^dW1 W0!^a M!0 aHom(D; )-exact sequence. Then there is a sequence

W^00‡ !W2W₁⁰ !W1W₀⁰ !W0!^a00 M⁰⁰!0 satisfying the following statements.

(1) If () is exact and H_i(W^‡)ˆ0ˆH_{i 1}(W^0‡), then H_i(W^00‡)ˆ0, where i2Z.

(2) If Hom(A; ) exacts () and Hi(Hom(A;W^‡))ˆ0ˆ H_{i 1}(Hom(A;W^0‡)), then H_i(Hom(A;W^00‡))ˆ0, where A is an object ofAand i2Z.

(3) If Hom( ;A) exacts () and H_i(Hom(W^‡;A))ˆ0ˆ Hi 1(Hom(W^0‡;A)), then Hi(Hom(W^00‡;A))ˆ0, where A is an object ofAand i2Z.

Furthermore, ifAis the category ofcomplexes ofR-modules, then we have the next statement.

(4) If();W^‡and W^0‡areCE-exact, then so is W^00‡.

PROOF. By hypothesis, there is a mapl^‡:W^0‡!W^‡of complexes of R-modules withl^‡₁ˆf. Consider the following commutative diagram

wherea⁰⁰ˆga. In the following, we show thatW^00‡satisfies the conditions desired.

(1). If Hi(W^‡)ˆ0ˆHi 1(W^0‡), then Hi(Cone(l^‡))ˆ0 since the se- quence

0!W^‡!Cone(l^‡)!SW^0‡!0 (y)

is exact. Note that the sequence

0!S ¹M⁰!Cone(l^‡)!W^00‡!0 is exact, then H_i(W^00‡)ˆ0 since H_j(S ¹M⁰)ˆ0 for anyj2Z.

(2). Note that the exact sequence (y) is degree-wise split, then the se- quence

0! Hom(A;W^‡)! Hom(A;Cone(l^‡))! Hom(A;SW^0‡)!0 of complexes is exact, and so Hi(Hom(A;Cone(l^‡)))ˆ0 since

H_i(Hom(A;W^‡))ˆ0ˆH_{i 1}(Hom(A;W^0‡)):

On the other hand, since the sequence

0! Hom(A;S ¹M⁰)! Hom(A;Cone(l^‡))! Hom(A;W^00‡)!0 of complexes is exact clearly under the hypotheses, and H_j(Hom(A;S ¹M⁰))ˆ0 for anyj2Z, we get

H_i(Hom(A;W^00‡))ˆ0:

(3) can be proved in the same way as in the proof of (2).

(4). Assume that A is the category of complexes of R-modules, and ();W^‡andW^0‡are CE-exact. Then the sequences

!Hom(S^kR;W₁⁰)!Hom(S^kR;W₀⁰)!Hom(S^kR;M⁰)!0 and

!Hom(S^kR;W₁)!Hom(S^kR;W₀)!Hom(S^kR;M)!0 are exact by Lemma 2.6. Consider the following commutative diagram

Since all columns of the above diagram are exact, we get that the sequence !Hom(S^kR;W₁W₀⁰)!Hom(S^kR;W₀M⁰)!Hom(S^kR;M)!0 is exact. Consider the following commutative diagram

Since the sequence () is CE-exact, we get that all columns of the above

diagram are exact. On the other hand, note that the first and second rows of the above diagram are exact, then so is the third row. This implies that the sequence

!Z_k(W1W₀⁰)!Z_k(W0)!Z_k(M⁰⁰)!0

is exact for anyk2Zby Lemma 2.6. The complexW^00‡is exact by (1), and

so it is CE-exact. p

The next corollary is immediate by Proposition 3.8.

COROLLARY 3.9. Assume that A is an abelian category andD is a class ofobjects ofA. Let

0!M⁰!M!M⁰⁰!0 be an exact sequence inA, and let

W_n⁰ ₁! !W₁⁰ !W₀⁰ !M⁰!0 be an exact sequence with W_i⁰2 D, and

W_n!W_{n 1}! !W₁!W₀!M!0

an exact andHom(D; )-exact sequence. Then there is an exact sequence W_nW_n⁰ ₁! !W₁W₀⁰ !W₀!M⁰⁰!0

satisfying the statements(2),(3)and(4)ofProposition3.8.

The following corollary can be checked easily using induction ont.

COROLLARY3.10. Assume thatAis an abelian category and Dis a class ofobjects ofA. Let

M ^t! !M ¹ !M⁰!M!0 be an exact sequence inA, and let

W_t‡iⁱ ! !W₁ⁱ!W₀ⁱ!Mⁱ!0

be exact and Hom(D; )-exact with each W_jⁱ2 D. Then there is an exact sequence

W₀^tW₁^t‡1 W_t⁰! !W₀¹W₁⁰!W₀⁰!M!0 satisfying the statements(2),(3)and(4)ofProposition3.8.

Furthermore, we have the next result that is a tool for the proof of Theorem 3.12.

COROLLARY3.11. Assume thatAis an abelian category and Dis a class ofobjects ofA. Let

!M ²!M ¹!M⁰!M!0 be an exact sequence inA, and let

!W₂ⁱ!W₁ⁱ!W₀ⁱ!Mⁱ!0

be exact and Hom(D; )-exact with each W_jⁱ2 D. Then there is an exact sequence

!W₀²W₁¹W₂⁰!W₀¹W₁⁰!W₀⁰!M!0 satisfying the statements(2),(3)and(4)ofProposition3.8.

The dual versions of Proposition 3.8 and Corollaries 3.9, 3.10 and 3.11 can be given easily.

The next result compares to [6, Theorem 4.1] and [7, Corollary 4.10]. We notice that this result does not need the conditionV ? V.

THEOREM3.12. Assume thatV is a class ofcomplexes ofR-modules closed under finite direct sums. Let G !G ¹!G⁰!G¹! be a CE-exact sequence ofcomplexes ofR-modules. IfGⁱ2 G_CE(V)for each i2Z and G is Hom(V; ) and Hom( ;V)-exact, then KⁱˆKer(Gⁱ!G^i‡1)2 GCE(V)for each i2Z. In particular, we haveGCE(GCE(V))ˆ GCE(V).

PROOF. We only need to prove that K¹ˆKer(G¹!G²) is inG_CE(V).

SinceGⁱ2 G_CE(V) for eachi2Z, there is a CE-exact sequence !W₁ⁱ!W₀ⁱ!Gⁱ!0

of complexes of R-modules for each i0 with W_jⁱ2 V such that it is Hom(V; ) and Hom( ;V)-exact. Thus there is a CE-exact sequence

!W₀²W₁¹W₂⁰ !W₀¹W₁⁰!W₀⁰!K¹!0 (y)

of complexes ofR-modules such that it is Hom(V; ) and Hom( ;V)-exact by Corollary 3.11.

On the other hand, note that there is a CE-exact sequence 0!Gⁱ!Wⁱ₁!Wⁱ₂!

of complexes of R-modules for each i1 with W_jⁱ2 V, such that it is Hom(V; ) and Hom( ;V)-exact, then there is a CE-exact sequence

0!K¹!W¹₁!W¹₂W²₁!W¹₃W²₂W³₁! (z)

of complexes ofR-modules such that it is Hom(V; ) and Hom( ;V)-exact by the dual version of Corollary 3.11.

Now assembling the sequences (y) and (z) we getK¹2 G_CE(V) sinceVis

closed under finite direct sums. p

LEMMA3.13. The following statements hold.

(1) W ? X ifand only ifExtⁱ(C;D)ˆ0 for any C 2CE(W), D2CE(X) and i1.

(2) X ? W if and only if Extⁱ(D;C)ˆ0 for any C2CE(W), D2CE(X)and i1.

PROOF. We prove part (1); the proof of part (2) is dual. LetW2 Wand k2Z. Then Extⁱ(S^kW;D)Ext¹(S^kT;D)Ext¹(T;D_k)Extⁱ(W;D_k)ˆ0 by Lemma 2.6, whereTis anR-module. On the other hand, Extⁱ(S^kW;D) Ext¹(S^kT;D)Ext¹(T;Zk(D))Extⁱ(W;Zk(D))ˆ0 by Lemma 2.6. Note that W ? W, then C can be written as the form of (i2ZSⁱWi) (i2ZSⁱW_i⁰) withWiandW_i⁰inWby Lemma 3.1, and so Extⁱ(C;D)ˆ0 for eachi1.

Conversely, fix W2 W and X2 X, then Extⁱ(W;X)Ext¹(T;X) Ext¹(T;X)Extⁱ(W;X) by Lemma 2.6, whereTis anR-module. Note that W2CE(W) andX2CE(X), then Extⁱ(W;X)ˆ0, and so Extⁱ(W;X)ˆ0

for eachi1. p

The next result can be checked easily.

LEMMA3.14. LetWbe closed under extensions, and let 0!X⁰!X!X⁰⁰!0

be anCE-exact sequence ofcomplexes ofR-modules. IfX⁰and X⁰⁰are in CE(W), then so is X.

We close the paper with the following result that contains a partial converse of Theorem 3.12.

PROPOSITION 3.15. Let W ? W, and let G !G ¹!G⁰! G¹! be a CE-exact sequence ofcomplexes ofR-modules. Then Gⁱ2 G_CE(CE(W)) for each i2Z and G is Hom(CE(W); ) and Hom( ;CE(W))-exact ifand only ifKⁱˆKer(Gⁱ!G^i‡1)2 G_CE(CE(W)) for each i2Z.

PROOF. By Theorem 3.12, we only need to prove the ``if'' part. Let Kⁱ2 G_CE(CE(W)) for each i2Z. Since CE(G(W))ˆ G_CE(CE(W)) by Proposition 3.7, we get thatGⁱ2CE(G_M(W)) for eachi2Zby Lemma 3.14 sinceG_M(W) is closed under extensions by [7, Theorem B], and so Gⁱ2 G_CE(CE(W)). Now we only need to prove that Ext¹(T;K)ˆ0 and Ext¹(K;T)ˆ0 for any T2CE(W) and K2 GCE(CE(W)). Let !C ¹!C⁰!C¹! be a CE-exact sequence of complexes of R-modules with Cⁱ2CE(W) such that it is Hom(CE(W); ) and Hom( ;CE(W))-exact and KˆKer(C⁰!C¹). Since Ext¹(T;C⁰)ˆ0 by Lemma 3.13, we get an exact sequence

0!Hom(T;K)!Hom(T;C⁰)!Hom(T;L¹)!Ext¹(T;K)!Ext¹(T;C⁰)ˆ0;

whereL¹ˆKer(C¹!C²). On the other hand, the sequence 0!Hom(T;K)!Hom(T;C⁰)!Hom(T;L¹)!0

is exact, and so Ext¹(T;K)ˆ0. Similarly, one can check that Ext¹(K;T)ˆ0. ThusGis Hom(CE(W); ) and Hom( ;CE(W))-exact. p Acknowledgments. This work was partly supported by NSF of China (Grant Nos. 11301240, 11226059, 11101197), the Young Scholars Science Foundation of Lanzhou Jiaotong University (Grant No. 2012020), and NSF of Gansu Province of China (No. 145RJZA079). The authors would like to thank the referee for a careful reading of the paper and for many useful comments and suggestions.

REFERENCES

[1] M. AUSLANDER- M. BRIDGER,Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969.

[2] H. CARTAN- S. EILENBERG,Homological algebra, Princeton University Press, Princeton, N.J., 1956.

[3] E. E. ENOCHS,Cartan-Eilenberg complexes and resolutions, J. Algebra342 (2011), pp. 16-39.

[4] E. E. ENOCHS- O. M. G. JENDA,Gorenstein injective and projective modules, Math. Z.220(1995), pp. 611-633.

[5] E. E. ENOCHS- O. M. G. JENDA,Relative homological algebra, de Gruyter Expositions in Mathematics, vol. 30, Walter de Gruyter & Co., Berlin, 2000.

[6] Z. HUANG, Proper resolutions and Gorenstein categories, J. Algebra 393 (2013), pp. 142-169.

[7] S. SATHER-WAGSTAFF - T. SHARIF - D. WHITE, Stability ofGorenstein categories, J. London Math. Soc.77(2008), pp. 481-502.

[8] J.-L. VERDIER,Des cateÂgories deÂriveÂes des cateÂgories abeÂliennes, AsteÂrisque 239, Soc. Math. France, Paris, 1996.

Manoscritto pervenuto in redazione il 2 Ottobre 2012.