On Gorenstein flat preenvelopes of complexes

On Gorenstein Flat Preenvelopes of Complexes

GANGYANG(*) - ZHONGKUILIU(**) - LILIANG(***)

ABSTRACT- In this paper we show that if the classBofR-modules is closed under well ordered direct limits, then the classBis preenveloping in the category ofR- modules if and only if the classdwBis preenveloping in the category of R- complexes, wheredwBdenotes the class of all complexes with all components in B. As an immediate consequence, we get that over commutative and Noetherian rings with dualizing complexes every complex admits a Gorenstein flat preen- velope.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 16E05, 18G35.

KEYWORDS. (pre)envelopes, Gorenstein injective modules (complexes), FP-injective modules, Gorenstein flat modules (complexes).

1. Introduction

The study of the existence of envelopes and covers started with the introduction of the concepts of injective envelopes ([6]), projective covers ([4]) as well as pure-injective envelopes ([18]). An independent research of Auslander, Reiten and Smal ([2] and [3]) in the finite dimensional case, and Enochs ([7]) for arbitrary modules, has led to a general theory of left and right approximations, or preenvelopes and precovers of modules (see [10]

and [15]).

(*) Indirizzo dell'A.: Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, P. R. China.

E-mail: [email protected]

(**) Indirizzo dell'A.: Department of Mathematics, Northwest Normal Uni- versity, Lanzhou 730070, P. R. China.

E-mail: [email protected]

(***) Indirizzo dell'A.: Department of Mathematics, Lanzhou Jiaotong Uni- versity, Lanzhou 730070, P. R. China.

E-mail: [email protected]

In a natural way, (pre)envelopes and (pre)covers have been studied in more general settings than that of modules. A particular and important example is the study of (pre)envelopes and (pre)covers of complexes of modules (see [1], [13] and [14]). It was shown [14, Theorem 5.2.2] that a ring R is right coherent if and only if every complex of R-modules has a flat preenvelope. It was proved [14, Theorem 3.2.9] that over Gorenstein rings every complex has a Gorenstein injective (pre)envelope. In this paper we are inspired to consider the existence of Gorenstein flat preenvelopes. We show that over commutative and Noetherian rings with a dualizing complex every complex admits a Gorenstein flat preenvelope. This complements the work of Enochs, Estrada and Iacob who proved in [8] that over such rings every complex admits a Gorenstein flat (pre)cover. The method we use to show that the class of Gorenstein flat complexes is preenveloping in the category of complexes works in a more general setting, and so the existence of preenvelopes by many important classes of complexes is obtained.

2. Preliminaries

Throughout the paper, we assume that all rings have an identity and all modules are unitary. Unless stated otherwise, an R-module will be un- derstood to be a leftR-module. We useR-Mod (respectively, Mod-R) to denote the category of left (respectively, right)R-modules.

DEFINITION 2.1. A complex C of R-modules is a sequence ! C ^{2 d}!^C2 C ^{1 d}!^C1 C^{0 d}!^0C C^{1 d}!^1C C²! of R-modules and R-homo- morphisms such thatd^m_Cd^m_C ¹ˆ0 for allm2Z. Themth cycle module of the complex C is defined as Ker(d^m_C) and is denoted Z^m(C), and the mth boundary module is defined as Im(d^m_C ¹) and denotedB^m(C). A complexCis said to be exact ifZ^m(C)ˆB^m(C) for allm2Z.

DEFINITION2.2 ([10]). A module M is called Gorenstein injective, if there exists an exact complex !I ²!I ¹!I⁰!I¹!I²! with allIⁱinjective such thatMˆKer(I⁰!I¹) and the sequence remains exact whenever HomR(J; ) is applied for any injective moduleJ.

DEFINITION2.3 ([10]). A moduleN is called Gorenstein flat, if there exists an exact complex !F ²!F ¹!F⁰!F¹!F²! with all Fⁱ flat such that NˆKer(F⁰!F¹) and the sequence remains exact wheneverJR is applied for any injective rightR-moduleJ.

The Gorenstein injective and flat complexes are defined in a similar manner, but using resolutions of complexes.

Given complexesXandY, we letHom(X;Y) denote the usual complex associated complexesQ X and Y, i.e., the complex with Hom(X;Y)ⁿ ˆ

i2ZHom(Xⁱ;Y^n‡i) and with differential given by dⁿ((fⁱ)_i2Z)ˆ(d^n‡i_Y fⁱ ( 1)ⁿf^i‡1dⁱ_X)_i2Zforf ˆ(fⁱ)_i2ZinHom(X;Y)ⁿ. Amorphism of complexes f :X!Yis defined to be a family (fⁱ)_i2Zof homomorphisms ofR-modules fⁱ:Xⁱ!Yⁱsuch thatdⁱ_Yfⁱˆf^i‡1dⁱ_X for eachi2Z. We use Hom(X;Y) to present the group of all morphisms from X toY. We let C(R-Mod) (re- spectively, C(Mod-R)) be the category of complexes of left (respectively, right) R-modules. Unless stated otherwise, an R-complex will be under- stood to be a complex of leftR-modules.

We also recall that a complexIisinjectiveif the functor Hom( ;I) is exact. Equivalently, a complex Iinjective is if and only if it is exact and Zⁱ(I) is an injective module for eachi2Z. For example, ifNis an injective module then !0!N ^Id!N!0! is injective. In fact, any in- jective complex is uniquely up to isomorphism the direct sum of such complexes. The dual notion is that of projective complex. A complexPis projectiveif the functor Hom(P; ) is exact. Equivalently,Pis projective if and only ifPis exact andZⁱ(P) is a projective module for eachi2Z. For example, ifMis a projective module then !0!M ^Id!M!0! is projective. In fact, any projective complex is uniquely up to isomorphism the direct sum of such complexes. Thus the category of complexes of modules has enough projectives.

DEFINITION2.4 ([14]). A complex His called Gorenstein injective, if there exists an exact sequence of complexes !I ²!I ¹!I⁰! I¹!I²! with allIⁱinjective such thatHˆKer(I⁰!I¹) and the se- quence remains exact whenever Hom(J; ) is applied for any injective complexJ.

REMARK2.5. It was shown by Liu and Zhang [17, Theorem 8] that a complexHis Gorenstein injective if and only if each termHⁱis a Goren- stein injective module for alli2ZwhenRis left Noetherian.

The definition of Gorenstein flat complexes is introduced in terms of tensor product of complexes.

Recall from [14] that ifX is a complex of right R-modules andY is a complex of leftR-modules, then the usual tensor productXYis defined

by (XY)ⁿ ˆ L

i‡jˆnXⁱRY^jin degreen, and the differentialdⁿis defined on the generators by dⁱ_X(x)y‡( 1)ⁱxd_Y^j(y) for x2Xⁱ,y2Y^j. Let XYdenote the complex (XY)

B(XY)with the maps (XY)ⁿ

Bⁿ(XY)! (XY)^n‡1

B^n‡1(XY);xy7!dX(x)y;

wherexyis used to denote the coset in (XY)ⁿ

Bⁿ(XY). Then we note that the functor is right exact in the category of complexes and the category of complexes has enough projectives, thus we can construct left derived functors which we denote by Tori.

Recall from [14] that a complexFisflatif the functor Fis exact. By [14, Theorem 4.1.3], a complexFis flat if and only ifFis exact andZⁱ(F) is a flat module for eachi2Z. It is clear that any projective complex is flat.

DEFINITION 2.6 ([14]). A complex G is called Gorenstein flat if there exists an exact sequence of complexes !F ²!F ¹!F⁰

!F¹!F²! with allFⁱ flat such that GˆKer(F⁰!F¹) and the sequence remains exact whenever E is applied for any injective complex E of right R-modules.

We also recall the definitions of precovers and preenvelopes as well as right and left resolutions in an abelian categoryA.

DEFINITION2.7. LetVbe a class of objects ofA, andMis an object of A. We say that a morphismf :M!Qis anV-preenvelope ifQ2Vand the sequence Hom(Q;Q⁰)!Hom(M;Q⁰)!0 is exact for any Q⁰2V. If moreover, gf ˆf implies that g is an automorphism whenever g2End(Q), thenf is called anV-envelope.V-precovers andV-covers are defined dually. We say that the class V is preenveloping (respectively, enveloping) if every object of A has an V-preenvelope (respectively, V- envelope).

DEFINITION 2.8. A right V-resolution of M is a complex 0!M! Q⁰!Q¹! with each Qⁱ2V and which remains exact whenever Hom( ;Q⁰) is applied for any Q⁰2V. We note that 0!M! Q⁰!Q¹! is a rightV-resolution ofMif and only ifM!Q⁰and each Coker(Q^{i 2}!Q^{i 1})!Qⁱfori1 areV-preenvelopes (withQ ¹ˆM).

The left V-resolutions are defined dually. Note that ifV contains all injective (respectively, projective) objects then any right (respectively, left)V-resolution is an exact complex.

We note that there is no loss of generality in what follows in assuming that ifBis a class ofR-modules andM,NareR-modules such thatMN andM2 B, thenN2 B. Hence we will always assume that the classBis closed under isomorphisms and contains 0. LetBbe a class ofR-modules, we say a complexDis adwBcomplex if each termDⁱis inBfori2Z(the

``dw'' is meant to be thought of as ``degreewise'').

3. Gorenstein flat complexes over coherent rings

The aim of this section is to give some characterizations of Gorenstein flat complexes over coherent rings. So throughout this section Rwill de- note a right coherent ring.

Since over a right coherent ring every module has a flat preenvelope by [10, Proposition 3.3], then every module has a right flat resolution. Thus one can define right derived functors of _R by using (right) injective resolutions and right flat resolutions in the first and second variables re- spectively (see [10]). These new derived functors are denoted by Torⁿ_R, and such functors can be used to characterize Gorenstein flat modules (see [11, Proposition 2.1]). Similarly since over a right coherent ring every complex has a flat preenvelope by [14, Theorem 5.2.2], then every complex has a right flat resolution. Thus we can define right derived functors of in the category of complexes by using (right) injective resolutions and right flat resolutions in the first and second variables respectively. These new derived functors are denoted by Torⁿ. We also note that there exists a natural morphismXY!Tor⁰(X;Y).

Given an R-module M, we denote by M the complex !0! M ^Id!M!0! with all components 0 exceptMin the degrees 1 and 0. Given a complex Cand an integerm, we let C[m] denote the complex such thatC[m]ⁿ ˆC^m‡nand whose differentials are ( 1)^md^m‡n. We first give the following useful lemma.

LEMMA3.1. Let C be acomplex, M aright R-module and let m be an integer. Then the following statements hold.

(1) [Torⁿ(M[m];C)]ⁱTorⁿ_R(M;C^i‡m)for any i2Zand any n0.

(2) [Torn(M[m];C)]ⁱTor^R_n(M;C^i‡m)for any i2Zand any n0.

PROOF. (1). Let 0!M!I⁰!I¹!I²! be an injective resolu- tion of the moduleM. Then 0!M[m]!I⁰[m]!I¹[m]!I²[m]! is an injective resolution of the complexM[m], and so we have the following commutative diagram

where the vertical isomorphisms follow from [14, Proposition 4.2.1(4)]. Now it is easy to see that [Torⁿ(M[m];C)]ⁱTorⁿ_R(M;C^i‡m) for anyi2Zand anyn0.

(2). The proof is similar to that of (1). p

Note that Hom(X;Y)ˆZ⁰(Hom(X;Y)). If we let Hom(X;Y)ˆ Z(Hom(X;Y)) then we see that Hom(X;Y) can be made into a complex with Hom(X;Y)^mthe abelian group of morphisms fromXtoY[m] and with a differential given by:f 2Hom(X;Y)^m, thend^m(f):X!Y[m‡1] with d^m(f)ⁿ ˆ( 1)^md^m‡n_Y fⁿ;8n2Z. For a complex C, we letC^‡ denote the complex Hom(C;Q=Z), that is,C^‡ˆHomR(C;Q=Z).

In the following we give a characterization of Gorenstein flat complexes over right coherent rings by using the new derived functors Torⁿ.

PROPOSITION3.2. Let G be acomplex. Then the following conditions are equivalent.

(1) G is Gorenstein flat.

(2) Tor_n(E;G)ˆ0 8n1 for every injective complex E of right R-modules, Torⁿ(R;G)ˆ0 8n1 and G!Tor⁰(R;G) is an isomorphism.

(3) Gⁱare Gorenstein flat modules for all i2Z.

(4) Tor_n(E;G)ˆ0 8n1 for every injective complex E of right R- modules, Torⁿ(P;G)ˆ0 8n1 and PG!Tor⁰(P;G) is an isomorphism for every projective complex P.

PROOF. (1))(2). It is clear that Tor_n(E;G)ˆ0 8n1 for every injective complex E of right R-modules. Furthermore since R is right coherent then there exists an exact sequence 0!G!F⁰!F¹ ! that can be used to compute Torⁿ(R;G)ˆ0, 8 n1. Finally Gˆ Ker(F⁰!F¹)Ker(RF⁰!RF¹)ˆTor⁰(R;G).

(2))(3). It follows from Lemma 3.1 and [11, Proposition 2.1].

(3))(4). Since the derived functors Tornand Torⁿof the tensor product preserve arbitrary direct sums, every projective module is a direct sum- mand of a freeR-module and every projective complexPis a direct sum of some projective complexesK[i], then the result follows from Lemma 3.1 and [11, Proposition 2.1].

(4))(1). Let 0!G!F⁰!F¹! be a right flat resolution. Then by the fact that Torn(R;G)ˆ08n1 we get thatF⁰!F¹! is exact.

But since GRG!Tor⁰(R;G)ˆKer(RF⁰!RF¹) is an iso- morphism, we get that 0!G!F⁰!F¹! is exact. Let E be an injective complex of right R-modules. In order to show that 0!EG!EF⁰!EF¹! is exact, it is enough to show that !(EF¹)^‡!(EF⁰)^‡!(EG)^‡!0 is exact. But this follows from isomorphisms (EC)^‡Hom(C;E^‡) for any complex C, and the exactness of the sequence !Hom(F¹;E^‡)!Hom(F⁰;E^‡)! Hom(G;E^‡)!0 because E^‡ is flat. On the other hand Tor_n(E;G)ˆ0 8n1 implies that any (left) flat resolution !F1!F0!G!0 remains exact whenever the functor E is applied for any injective

complexEof rightR-modules. p

REMARK3.3. The equivalence of statements (1) and (3) of Proposition 3.2 was shown over more general rings by Yang and Liu [21, Theorem 3.11].

Since a leftR-moduleM is Gorenstein flat if and only if the character moduleM^‡is Gorenstein injective as a rightR-module by [16, Theorem 3.6], the following result follows directly from Remark 2.5 and Proposition 3.2.

COROLLARY3.4. Let R be aright Noetherian ring. Then acomplex G is Gorenstein flat in C(R-Mod)if and only if G^‡is Gorenstein injective in C(Mod-R).

4. The existence of Gorenstein flat preenvelopes

In this section we will prove that over commutative and Noetherian rings with a dualizing complex every complex has a Gorenstein flat preenvelope. To obtain this result, we develop a general approach. More precisely, for a given classBofR-modules we will show that if the classBof R-modules is closed under direct limits, then the classBis preenveloping in R-Mod if and only if the class dwB of complexes is preenveloping in

C(R-Mod), wheredwBdenotes the class of complexes of all components in B. We first start with the following lemma.

LEMMA4.1. If the class of R-modulesBis enveloping in R-Mod then every left bounded complex has a dwB-envelope. Furthermore, if the class of R-modulesBis preenveloping in R-Mod then every left bounded complex has a dwB-preenvelope which is also left bounded.

PROOF. We assume without loss of generality that Cˆ:0! C^{0 d}!^0C C^{1 d}!^1C C²! . Using an idea dual to the one used in [14, Theorem 3.3.10], we are going to construct a complexD2dwBand a morphism of complexesf :C!Dwhich is adwB-envelope ofCin the following way.

Firstly, for n50, let Dⁿˆ0 andfⁿˆ0. Letf⁰:C⁰!D⁰ be a B-en- velope ofC⁰. Now forn0, we proceed inductively. Suppose that we have constructed:

We consider the push-out diagram of dⁿ_C :Cⁿ!C^n‡1 and pⁿfⁿ:Cⁿ! Coker(d_D^{n 1}),

…1†

wherepⁿ:Dⁿ!Coker(d_D^{n 1}) is the natural epimorphism.

Then we take aB-envelopetⁿ:Xⁿ!D^n‡1of the moduleXⁿand define d_Dⁿ:Dⁿ!D^n‡1as the compositiond_Dⁿˆtⁿsⁿpⁿandf^n‡1:C^n‡1!D^n‡1as the compositionf^n‡1ˆtⁿcⁿ. It is easy to see thatd_Dⁿd_D^{n 1}ˆ0 (d_D^{n 1}ˆ0 if nˆ0) and thus this construction produces a complex D2dwB and a morphism of complexesf :C!D.

Secondly, we will show that the above morphism of complexes f :C!D is a dwB-preenvelope of the complex C. Let A2dwB and W:C!A be a morphism of complexes. We are going to construct a morphism of complexes g:D!A satisfying Wˆgf. For n50, we take gⁿˆ0. For nˆ0, since f⁰ :C⁰!D⁰ is a B-envelope ofC⁰, there exists

g⁰ :D⁰!A⁰ such that g⁰f⁰ˆW⁰. By induction, suppose gⁿ:Dⁿ!Aⁿ is defined such that gⁿfⁿˆWⁿ and gⁿd_D^{n 1}ˆd_A^{n 1}g^{n 1}. We notice that Ker(pⁿ)Ker(dⁿ_Agⁿ). Sincepⁿ:Dⁿ!Coker(d_D^{n 1}) is epimorphic, by the factor lemma, there exists a morphismgⁿ:Coker(d_D^{n 1})!A^n‡1such that the following diagram is commutative:

i.e., we have gⁿpⁿˆd_Aⁿgⁿ. Hencegⁿpⁿfⁿˆd_Aⁿgⁿfⁿ ˆdⁿ_AWⁿˆW^n‡1dⁿ_C. This yields the induced commutative diagram by the following push-out diagram:

…2†

Sincetⁿ:Xⁿ!D^n‡1is aB-envelope ofXⁿ andA^n‡1 is inB, there exists g^n‡1:D^n‡1!A^n‡1 such that g^n‡1tⁿˆrⁿ. Therefore, we get from the commutative diagram (2) that g^n‡1f^n‡1ˆg^n‡1tⁿcⁿˆrⁿcⁿˆW^n‡1. By above construction, it is not hard to check thatg^n‡1d_Dⁿˆd_Aⁿgⁿ. So it is easy to see that in this way we can get a morphism of complexesg:D!Asa- tisfyinggf ˆW, thus we proved.

In the last, letg:D!Dbe a morphism of complexes such thatgf ˆf. Forn50, we have thatgⁿis the zero morphism. Fornˆ0, we have thatg⁰ is an automorphism sincef⁰:C⁰!D⁰is aB-envelope ofC⁰. Forn0, we proceed inductively. Suppose thatg⁰;g¹; ;gⁿare automorphisms, we will show that g^n‡1:D^n‡1!D^n‡1 is also an automorphism. By the factor lemma, there exists a morphismgⁿ:Coker(d_D^{n 1})!Coker(d_D^{n 1}) such that the following diagram commutes:

…3†

By the five lemma, we get that gⁿ:Coker(d_D^{n 1})!Coker(d_D^{n 1}) is an iso- morphism. Combining the commutative diagrams (1) and (3), we havecⁿdⁿ_Cˆ sⁿpⁿfⁿˆsⁿpⁿ(gⁿfⁿ)ˆsⁿ(pⁿgⁿ)fⁿˆsⁿ(gⁿpⁿ)fⁿ. Hence, the push-out diagram

maps into itself by an automorphism. This implies that there exists an au- tomorphismgbⁿ:Xⁿ!Xⁿ such thatgbⁿcⁿˆcⁿand the following diagram commutes on each surface

Since

f^n‡1dⁿ_Cˆd_Dⁿfⁿ ˆtⁿsⁿpⁿfⁿˆtⁿsⁿpⁿ(gⁿfⁿ)ˆtⁿsⁿ(pⁿgⁿ)fⁿ

ˆtⁿsⁿ(gⁿpⁿ)fⁿˆtⁿ(sⁿgⁿ)pⁿfⁿˆtⁿ(gbⁿsⁿ)pⁿfⁿ

ˆ(tⁿgbⁿsⁿ)(pⁿfⁿ);

there exists an unique morphismv:Xⁿ!D^n‡1 such that the following diagram is commutative

By standard computation, we can substitutevwith eithertⁿgbⁿorg^n‡1tⁿ, so they are identical, i.e.,tⁿgbⁿˆg^n‡1tⁿ, wheretⁿ:Xⁿ!D^n‡1is aB-envelope ofXⁿ. We get thatg^n‡1:D^n‡1!D^n‡1 is an automorphism. By induction, we get that g:D!D is an automorphism. This shows that every left bounded complex has adwB-envelope.

To complete the proof it only remains to argue that if the class ofR- modulesBis preenveloping inR-Mod then every left bounded complex has adwB-preenvelope which is also left bounded. But this is obvious from the

proof above. p

The following theorem gives our main result in this section, which presents a nice relation of preenvelopes between R-modules and un- bounded complexes ofR-modules.

THEOREM4.2. LetBbe aclass of R-modules which is closed under well ordered direct limits. ThenBis preenveloping in R-Mod if and only if the class dwBis preenveloping in C(R-Mod).

PROOF. For the necessity, we need only to show that every complex of R-modules has adwB-preenvelope. Let

Aˆ: !A2 ^@!^2A A1 ^@!^A1 A0 ^@!^A0 A 1! and we write

A(n)ˆ: 0!A_n ^@!^nA A_{n 1} ^@An!¹ A_{n 2} ^@An!²

Then we get that ((A(n));(r_mn))_n0 is a well ordered direct system in C(R-Mod) and lim_!A(n)ˆA, wherer_mn:A(m)!A(n) is a natural injec- tion for anymn.

By Lemma 4.1, there exists adwB-preenvelopeh₀:A(0)!D(0) ofA(0) with D(0) left bounded. Then we consider the push-out diagram of morphismsh₀:A(0)!D(0) andr₀₁:A(0)!A(1)

Clearly,Uis left bounded Since the others of above diagram are so. Again using Lemma 4.1, we have adwB-preenvelopen:U!D(1) ofUwithD(1)

left bounded. In the following we will show that the composition nm₀:A(1)!D(1) is adwB-preenvelope ofA(1). Suppose thata:A(1)!B is any morphism with Ba dwB complex. Then there exists a morphism e

a:D(0)!Bsuch that ar₀₁ ˆeah₀ sinceh₀:A(0)!D(0) is a dwB-preen- velope ofA(0), and so we have a morphismba:U!Bsuch that the com- pleted diagram

commutes. Sincen:U!D(1) is adwB-preenvelope of U, there exists a morphism b:D(1)!B such that baˆbn, and so aˆbam₀ˆbnm₀. This proves thatnm₀:A(1)!D(1) is adwB-preenvelope ofA(1). Therefore we get, by the construction above, a commutative diagram

wherel01 ˆnl0andh₁ˆnm₀, and it has the property that if

is commutative withBindwBthen there exists a morphism of complexes D(1)!Bsuch that the completed diagram

commutes. If we continue this process, we get a commutative diagram

whereh_n :A(n)!D(n) is adwB-preenvelope ofA(n) for eachn0 and each diagram

has the preceding property. Clearly, ((D(n));(lmn))_n0forms a well ordered direct system inC(R-Mod).

Note thatdwBis closed under well ordered direct limits and we have lim_!D(n) in dwB. We will see in the following that the morphism h:lim_! A(n)!lim_!D(n),hˆlim_!h_n, is adwB-preenvelope of lim_!A(n)ˆA.

In fact, if f :A!X is a morphism with X2dwB, then there exists x₀:D(0)!Xsuch that the following diagram

commutes since h₀:A(0)!D(0) is a dwB-preenvelope of A(0), where A(1)!X is the composition of morphisms of the natural injection A(1)!Aandf :A!X. Thus, by construction above, there is a morphism x₁:D(1)!Xsuch that the completed diagram

commutes. By using this procedure continuously, we get that the following diagram

is commutative for any n0, now it is easy to see that there exists a morphismx:lim_!D(n)!Xsuch thatf ˆxh. Thush:lim_!A(n)!lim_!D(n) is adwB-preenvelope of lim_!A(n)ˆA.

For the sufficiency, letM be an R-module and considerf :M!Da dwB-preenvelope of M in C(R-Mod). Then it is easy to see that f⁰:M!D⁰is aB-preenvelope ofMinR-Mod, and so we have completed

our proof. p

By Proposition 3.2, a complex C over any right coherent ring R is Gorenstein flat if and only if eachC^mis a Gorenstein flat module form2Z.

COROLLARY4.3. Let R be commutative and Noetherian with a dualiz- ing complex. Then every complex of R-modules has a Gorenstein flat preenvelope.

PROOF. By [5, Theorem 5.7], over such a ring the class of Gorenstein flat modules is closed under direct products. Note also that the class is a Kaplansky class closed under direct limits by [12, Proposition 2.10].

Thus the class of Gorenstein flat modules is preenveloping inR-Mod by [12, Theorem 2.5]. By Theorem 4.2, the class of Gorenstein flat com- plexes is preenveloping inC(R-Mod), as desired. p

REMARK4.4. The result above can also be obtained from Corollary 2.7, Theorem 4.2 and the Theorem on page 2, in [20].

Recall that anR-moduleMissaidto be FP-injective (also absolutely pure), if Ext¹_R(A;M)ˆ0 for all finitely presentedR-modulesA. A ringRis left co- herent if and only if the class of all FP-injective leftR-modules is closed under arbitrary direct limits [19, Theorem 3.2]. Thus we have the following result.

COROLLARY4.5. Let R be aleft coherent ring andPbe the class of all FP-injective left R-modules. Then every complex of R-modules has a dwP-preenvelope.

PROOF. Note that the class of FP-injective R-modules is preen- veloping for any ringRby [10, Proposition 6.2.4]. Thus the result holds by

Theorem 4.2. p

In the following, we denote byF the class of flat leftR-modules. It is showed in [1] and [13] that every complex has a flat cover and adwF-cover.

As mentioned in the introduction that a ringRis right coherent if and only if every complex ofR-modules has a flat preenvelope, we extend this result to the following.

COROLLARY 4.6. A ring R is right coherent if and only if every complex of R-modules has a dwF-preenvelope.

PROOF. A ringRis right coherent if and only if everyR-module has a flat preenvelope [10, Proposition 6.5.1], and by Theorem 4.2, if and only if every complex ofR-modules has adwF-preenvelope since the classF is

closed under direct limits. p

Recall from [14] that a short exact sequence 0!P!X!X=P!0 of complexes ispureif for any complexY of rightR-modules, the sequence 0!YP!YX!YX=P!0 is exact, and a complex is said to be pure injectiveif it is injective relative to any pure sequence of complexes.

According to [14, Proposition 5.1.4], the complexC^‡ is pure injective, and the evaluationC!C^‡‡is a pure monomorphism for any complexC.

COROLLARY4.7. Let R be commutative and Noetherian with a dualiz- ing complex. Then every pure injective complex C has a Gorenstein injective precover.

PROOF. Since every complex over such a ring has a Gorenstein flat preenvelope by Theorem 4.2, so hasC^‡. Letf :C^‡!Gbe a Gorenstein flat preenvelope ofC^‡. ThenG^‡is Gorenstein injective by Corollary 3.4. Now it is easily seen thatf^‡:G^‡!C^‡‡is a Gorenstein injective precover ofC^‡‡

by [9, Corollary 3.2]. On the other hand,Cis a direct summand ofC^‡‡by [14, Proposition 5.1.4] sinceCis pure injective, and then it is easy to see that pf^‡:G^‡!Cis a Gorenstein injective precover ofC, wherep:C^‡‡!Cis

the canonical projection. p

Acknowledgements. The work was supported by NSF of China (No.

11101197, 11261050, 11226059) and the Program of Science and Technique of Gansu Province (No. 1107RJZA233). The authors like to thank the referee for valuable suggestions and comments, which improved the present article.

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Manoscritto pervenuto in redazione l'8 Novembre 2011.