TC-Gorenstein Projective,LC-Gorenstein Injective and HC-Gorenstein Flat Modules?

Zhen ZHANGXiaosheng ZHUXiaoguang YAN

1 Introduction

Semidualizing modules are the common generalizations of dualizing modules and free modules of rank one.Foxby[4],Vasconcelos[12]and Golod[5]initiated the study of semidualizing modules under different names.A semidualizing moduleCinduces some interesting classes of modules,such as the Auslander classAC(R),the Bass classBC(R),theC-projective modulesPC(R),theC-injective modulesIC(R)and theC-flat modulesFC(R),etc.These classes of modules were investigated by many authors and the Foxby equivalence between the Auslander classAD(R)and the Bass classBD(R)with respect to a dualizing moduleDwas also extended to the semidualizing case(more details can be found in[7,10]).

Recall that Enochs and Jenda introduced and studied Gorenstein projectiveR-modules as a generalization of Auslander’s G-modules to the non-finitely modules.AnR-moduleMis called Gorenstein projective if there exists an exact sequence

such that the complex Hom(,Q)is exact for each projective moduleQandThe class of all Gorenstein projectiveR-modules,denoted byGP(R)and the class of all Goren-stein injectiveR-modules,denoted byGI(R),are defined dually,while anR-moduleMis called Gorenstein flat if there is an exact sequence offlatR-modules,

such that F?Eis exact for any injectiveR-moduleEandThe class of all Gorenstein flatR-modules is denoted byGF(R).

Recently,Holm,J?rgensen,Sather-Wagstaff,and White extended the Gorenstein projective(injective,flat)modules toC-Gorenstein projective(injective,flat)modules via the completePPC-resolution(FFC-resolution,ICI-resolution).Recall that a completePPC-resolution is an exact sequence ofR-modules

wherePiandPifori∈are projective,and the complex Hom(,C?RQ)is exact for each projectiveR-moduleQ.And the completeICI-resolution is defined similarly.Note that the completeFFC-resolution is an exact sequence ofR-modules

whereFiandFifori∈Z are flat,and the complex Hom(C,E)?F is exact for all injective modulesE.The class of allC-Gorenstein projective(injective or flat)modules is denoted byNote that if the semidualizing moduleCis the regular moduleR,thenC-Gorenstein projective(injective or flat)modules are just Gorenstein projective(injective or flat).

Note that the two functors Hom(C,?)andC??provide equivalences between the class of projective modules andC-projective modules,injective modules andC-injective modules,and flat modules andC- flat modules.A natural question arises:Do the functors HomandC??provide the equivalence between the classesandandor

The authors noticed that Sather-Wagstaff,Sharif and White[8]defined the classG(PC(R)),whichis consists of the modules that are built by a complete resolution ofPC(R)-modules and they[10]also proved that the functorsC??and Hom(C,?)provide natural equivalence between the classesGP(R)∩AC(R)andG(PC(R)),which helps us to answer the above question significantly.

In this paper,in order to study the relations between the classes of Gorenstein projective modules andC-Gorenstein projective modules,we define theTC-Gorenstein projective modules.Similarly,we define theLC-Gorenstein injective andHC-Gorenstein flat modules.And we get the following Foxby equivalences,in which the first two can be deduced from Theorem 3.1 and the work of[10](see Corollary 3.1 and Theorem 5.2).

Corollary ALetting C be a semidualizing R-module,we have the following equivalent classes provided by the functors C??andHom(C,?):

where(3)holds when R is coherent.

In Section 4,we study theTC(R)-projective andLC(R)-injective dimensions,and theTC(R)-precovers orLC(R)-preenvelopes,which extends the results of Holm[6].Particularly,we have(see Theorem 4.1)the following result.

Theorem ALet M be an R-module and n a nonegative integer.Denote P<∞by the class of R-modules with finite projective dimensions.The following are equivalent:

(1)TC(R)-pd(M)=n<∞.

(2)M admits a special TC(R)-precover:0→K→T→M→0with T∈TC(R)and pd(K)=n?1.

(3)M admits a special P<∞-preenvelope:0→M→L→T′→0with pd(L)=n and T′∈TC(R).

As an application,we prove that the classical finitistic projective dimension and the injective dimension are equal to the finitisticTC(R)-projective dimension and theLC(R)-injective dimension,respectively(see Proposition 4.1).Moreover,we get the following result(see Theorem 4.3).

Theorem BLet R be a Gorenstein ring and C be a semidualizing module.Then theAuslander classand the Bass class

In Section 5,we define and study theHC-Gorenstein flat modules over a commutative coherent ringR.We have the following results(see Theorem 5.1 and Proposition 5.1).

Proposition ALet R be coherent and M an R-module.Then M∈HC(R)?M+∈LC(R).

Hence,many properties of theHC-Gorenstein flat modules can be obtained from theLCGorenstein injective modules.Particularly,we extend the Foxby equivalence

Notation AThroughout this paper,Ris a commutative ring with an identity,all the modules are unitary,andCis a semidualizingR-module.The class of all the projective,injective or flatR-modules is denoted byP(R),I(R)orF(R),respectively.For anR-moduleM,letpd(M),id(M),Gpd(M)andGid(M)denote the projective,injective,Gorenstein projective and Gorenstein injective dimensions ofM,respectively.For unexplained concepts and notations,we refer the readers to[8–10].

2 Preliminaries

In this section,we introduce a number of definitions,notions and facts which will be used throughout this paper.

Definition 2.1(cf.[13,1.8])An R-module C is called semidualizing if

(1)C admits a degreewise finite generated projective resolution,

(2)the natural homothety map R→HomR(C,C)is an isomorphism,and

(3)

Definition 2.2(cf.[11])Let C be a semidualizing R-module.The Auslander class with respect to C,denoted by AC(R),consists of all the R-modules M satisfying

(1)for any i≥1,

(2)the natural map M→Hom(C,C?M)is an isomorphism.

Dually,the Bass class with respect to C,denoted by BC(R),consists of all the R-modulesM satisfying

(1)for any i≥1,

(2)the natural evaluation map C?Hom(C,M)→M is an isomorphism.

Fact 2.1LetCbe a semidualizingR-module.The classesAC(R)andBC(R)are closed under extensions,kernels of epimorphisms and Cokernels of monomorphisms(cf.[7,Corollary 3.6]).The classAC(R)contains all theR-modules offinite flat dimensions and those offiniteIC-injective dimensions,and the categoryBC(R)contains all theR-modules offinite injective dimensions by[7,Corollaries 6.1–6.2].

LetXbe a class ofR-modules.We denote bythe subcategory ofR-modulesMsuch thatSimilarly,denotes the subcategory of modulesMsuch thatfor all.

Definition 2.3(cf.[2])Let X be a class of R-modules and M be any R-module.An X-precover of M is called special if there is an exact sequence0→L→X→M→0withandThe special preenvelope is defined dually.

3 TC-Gorenstein Projective and LC-Gorenstein Injective Modules

In this section,we give the definitions and some properties of theTC-Gorenstein projective andLC-Gorenstein injective modules.

Definition 3.1Let C be a semidualizing R-module.An R-module M is called TCGorenstein projective if there exists an exact complex of projective R-modules

such that the following conditions hold:

(1)The complex C?is exact.

(2)The complexHom(,Q)is exact for all the projective R-modules Q.

(3)There is an isomorphism M≌ Coker(P1→P0).

Denote the class of all TC-Gorenstein projective modules by TC(R).

An R-module M is called-Gorenstein injective if there exists an exact complex of injectiveR-modules

such that the following conditions hold:

(1)The complexHom(C,I)is exact.

(2)The complexHom(E,I)is exact for all the injective R-modules E.

(3)There exists an isomorphism

Denote the class of all-Gorenstein injective modules by(R).

Remark 3.1LetCbe a semidualizingR-module.

(1)WhenC=R,we have that

(2)By symmetry,every kernel or cokernel of the morphisms in the complexisTCGorenstein projective and every kernel or cokernel of the morphisms in the complex I is-Gorenstein injective.

(3)By definition,we have

The following theorem implies that the class ofTC-Gorenstein projective modules or the class ofLC-Gorenstein injective modules shares many common properties with the class of Gorenstein projective or injective modules.

Theorem 3.1Let C be a semidualizing R-module.Then

ProofWe only prove(1).By Fact 2.1,the classesAC(R)andBC(R)are closed under extensions,kernels of epimorphisms and cokernels of monomorphisms,soGP(R)∩AC(R)?TC(R)is straightforward to prove.On the other hand,by Remark 3.1(3),TC(R)?GP(R).We only need to showTC(R)?AC(R).In fact,for anyR-moduleM,ifM∈TC(R),then there exists an exact sequence of projective modules

such thatC?is exact andSofori≥1 andC?is an exact complex ofPC(R)andBy Fact 2.1,Pi∈AC(R)fori∈Z,so HomThus HomClearly,Exti(C,C?M)=0 by[9,Lemma 1.9(b)].HenceM∈AC(R)and the result follows.

Following from the well-known properties of the classesGP(R),GI(R),AC(R)andBC(R)(cf.[6–7]),by Theorem 3.1,we have the following proposition.

Proposition 3.1Let C be a semidualizing R-module.Then

(1)the class TC(R)is closed under direct sums and the class(R)is closed under direct products;

(2)the class TC(R)is projectively resolving and(R)is injectively resolving;

(3)both the classes TC(R)and(R)are closed under direct summands.

Sather-Wagstaff,Sharif and White proved that

And by Theorem 3.1,we can prove the following equivalence provided by the functorsC??and Hom(C,?),which answers partially the question put forward in the introduction.As the conclusion was also showed by Sather-Wagstaff,Sharif and White[10,Remark 2.11],we omit the proof.

Corollary 3.1

4 TC(R)-Precovers and LC(R)-Preenvelopes

In this section we want to study the existence ofTC(R)-precovers and(R)-preenvelopes.Moreover,we also study theTC(R)projective dimensions and(R)injective dimensions and we get some good results which extend the results of Holm[6].

Letbe a class ofRmodules.We denote bythe class ofR-modules with finite-projective dimensions and bythe class ofR-modules with finite-injective dimensions.

Firstly,we prove the following lemma.

Lemma 4.1Let M be an R-module.Denote by TC(R)-pd(M)and(R)-id(M),the TC-Gorenstein projective and-Gorenstein injective dimensions of M,respectively.

(1)If TC(R)-pd(M)<∞,then TC(R)-pd(M)=Gpd(M).

(2)If(R)-id(M)<∞,then(R)-id(M)=Gid(M).

In particular,and

ProofWe only prove(1)and the proof of(2)is similar.By Remark 3.1(3),we have an inequalityGpd(M)≤TC(R)-pd(M).Next letGpd(M)=n<∞.Then there exists an exact sequence

with eachPi∈P(R)andG∈GP(R).By assumption,TC(R)-pd(M)<∞,soM∈AC(R)by Theorem 3.1 and Fact 2.1.SoG∈AC(R)also by Fact 2.1.ThusG∈GP(R)∩AC(R).SoG∈TC(R)by Theorem 3.1.HenceTC(R)-pd(M)≤n=Gpd(M)and(1)follows.

Theorem 4.1Let M be an R-module and n a nonegative integer.Denote by P<∞the class of R-modules with finite projective dimensions.The following are equivalent.

(1)TC(R)-pd(M)=n<∞.

(2)M admits a special TC(R)-precover:0→K→T→M→0with T∈TC(R)and pd(K)=n?1.

(3)M admits a special P<∞-preenvelope:0→M→L→T′→0with pd(L)=n and T′∈TC(R).

Proof(1)?(2).By Lemma 4.1,GpdR(M)=TC(R)-pd(M)=n.SoMadmits a surjective Gorenstein projective precover:withGbeing Gorenstein projective andpd(K)=n?1 by[6,Theorem 2.1].Sincepd(K)=n?1<∞,K∈GP(R)⊥.So the Gorenstein projective precover Φ is special by Definition 2.3.We claim thatG∈TC(R).In fact,asTC(R)-pd(M)=n,M∈AC(R)by Theorem 3.1 and Fact 2.1.Clearly,K∈AC(R),and thusG∈AC(R)also by Fact 2.1.SoG∈TC(R)by Theorem 3.1.Hence letT=G,and then Φ:T→M→0 is the desired specialTC(R)-precover ofM.

(2)?(3).Consider the exact sequence 0→K→T→M→0 withT∈TC(R)andpd(K)=n?1.SinceT∈TC(R),there is an exact sequence 0→T→P→T′→0 withP∈P(R)andT′∈TC(R).Thus we have the following pushout diagram:

Sincepd(K)=n?1,pd(L)=n<∞by the exact sequence in the middle row of the above pushout diagram.By Definition 3.1,so the exact sequencewithis a specialP<∞-preenvelope ofMby Definition 2.3 and[6,Proposition 2.3].

(3)?(1).Sincepd(L)=n,there exists an exact sequencewithP0projective andpd(L′)=n?1.Consider the following commutative diagram with exact rows:

SinceTC(R)is projectively resolving,T′′∈TC(R).Moreover,by the Snake lemma,we get an exact sequence

Aspd(L′)=n?1,TC(R)-pd(M)=n.And the theorem follows.

Similarly we have the following result.

Theorem 4.2Let M be an R-module and n a nonegative integer.Denote by I<∞the class of R-modules with finite injective dimensions.The following are equivalent.

(1)(R)-id(M)=n<∞.

(2)M admits a special LC(R)-preenvelope:0→M→L→K→0with L∈(R)and id(K)=n?1.

(3)M admits a special I<∞-precover:0→L→K→M→0with id(K)=n and L∈LC(R).

The next proposition is an application of Theorems 4.1–4.2.

Recall that the finitistic projective dimension FPD(R)is defined as FPD(R)=sup{pd(M)|pd(M)<∞}and the finitistic injective dimension FID(R)=sup{id(M)|id(M)<∞}.Holm[6]defined the finitistic Gorenstein projective dimension FGPD(R)and the finitistic Gorenstein injective dimension FGID(R),and he proved the equalities FGPD(R)=FPD(R)and FGID(R)=FID(R)(cf.[6,Theorems 2.28 and 2.29]).Similarly,we prove the following equalities and note that we use a different way from Holm’s.

Proposition 4.1LetFTPD(R)=sup{TC(R)-pd(M)|TC(R)-pd(M)<∞}andFLID(R)=sup{(R)-id(M)|-id(M)<∞}denote the finitistic TC(R)-Gorenstein projective and(R)-Gorenstein injective dimensions of the base ring R,respectively.ThenFPD(R)=FTPD(R)andFID(R)=FLID(R).

ProofWe only prove FPD(R)=FTPD(R).Clearly FPD(R)≤FTPD(R).On the other hand,ifMis a module with 0≤TC(R)-pd(M)≤n,wherenis a nonegative integer,then there exists a moduleLwithpd(L)=nby Theorem 4.1.Hence,if we assume that 0≤FTPD(R)=n,then we can find anR-moduleLwithpd(L)=n,so FPD(R)≥n,and FPD(R)=FTPD(R).

Enochs,Jenda and Xu[3,Corollaries 2.4 and 2.6]showed that whenRis a local Cohen-Macaulay ring with a dualizing moduleD,the Auslander classes with respect toDare exactly theR-modules with finite Gorenstein projective(flat)dimensions and the Bass classes with respect toDare exactly theR-modules with finite Gorenstein injective dimensions.While the Gorenstein ring is always a local Cohen-Macaulay ring,in this case,Ris the only dualizing module(cf.[2,Remark 9.5.15]).Enochs and Jenda[2,Theorem 12.3.1]proved that everyR-module has finite Gorenstein projective dimensions,if and only if everyR-module has finite Gorenstein flat dimensions,if and only if everyR-module has finite the Gorenstein injective dimensions over Gorenstein ringR.Hence we have the following result,noting that whenC=R,the result is exactly the[3,Corollaries 2.4 and 2.6]:

Theorem 4.3Let R be a Gorenstein ring and C be a semidualizing module.Then theAuslander classand the Bass class

ProofBy Lemma 4.1,we know thatandBC(R).Moreover,Ris Gorenstein,so everyR-module has a finite Gorenstein projective and Gorenstein injective dimension.Hence we have thatandSo

5 HC(R)-Gorenstein Flat Modules

In this section,we will give the definition ofHC(R)-Gorenstein flat modules which share the common properties with the Gorenstein flatR-modules.

Definition 5.1Let C be a semidualizing R-module.An R-module M is called HCGorenstein flat if there is an exact complex offlat R-modules

such that the following conditions hold:

(1)The complex C?Fis exact.

(2)The complex I?Fis exact for any injective R-module I.

(3)There exists an isomorphism

Denote the class of allHC-Gorenstein flat modules byHC(R).

Clearly,any flat module isHC-Gorenstein flat,and anyHC-Gorenstein flat module is Gorenstein flat.Moreover,whenC=R,HC(R)is exactly the class of Gorenstein flat modules.

Theorem 5.1Particularly,an R-module M is in GF(R)∩AC(R),if and only if there exists an exact sequence

such that both C?Fand I?Fare exact for any injective R-module I and

ProofClearlyAssumeso there exists an exact complex offlat modules F such thatThus ToriAsfor every flat moduleF,HomHence the exact complexC?is Hom(C,?)-exact and HomMoreover,by[9,Lemma 1.9],we get that Extifori≥1.ThusM∈AC(R)by Definition 2.2 andThe converse containment follows from Fact 2.1 and Definition 5.1.

Hence,following the properties of the classesGF(R)andAC(R)(cf.[6–7]),we know that the class ofHC(R)is projective resolving.Furthermore,HC(R)is closed under direct sums and direct summands.

WhenRis coherent,Holm[6,Theorem 3.6]showed thatMis a Gorenstein flat module if and only if the Pontryagin dualM+=HomZ(M,Q/)is Gorenstein injective,and Sather-Wagstaff,Sharif and White[9,Lemma 4.2]proved thatMisGC-flat,if and only if the Pontryagin dualM+isGC-injective.

By Theorem 5.1,we have the extension result.

Proposition 5.1Let R be coherent and M an R-module.Then M∈HC(R)?M+∈C(R).

ProofOn one hand,M∈AC(R)?M+∈BC(R)by[1,(3.2.9)].On the other hand,M∈GF(R)?M+∈GI(R)by[6,Theorem 3.6].Hence the result follows from Theorem 5.1.

Based on Proposition 5.1,we can easily get the following result.

Theorem 5.2The functorsC??andHom(C,?)provide the equivalence between theclasses HC(R)and

ProofOn one hand,by Proposition 5.1,M∈HC(R)?M+∈LC(R).Moreover,by Corollary 3.1,

But Hom(C,M+)～=(C?M)+,so

by[9,Lemma 4.2]and[1,(3.2.9)].On the other hand,by[9,Lemma 4.2]and[1,(3.2.9)],By Corollary 3.1,by Proposition 5.1,we have that

As any projective module isHC-Gorenstein flat,everyR-module has anHC(R)-projective dimension.By Fact 2.1 and Theorem 5.1,we can easily get that

Hence we have the extended Foxby equivalence

AcknowledgementsThe authors would like to express their sincere thanks to the referees for their careful reading of the manuscript and helpful suggestions.

[1]Christensen,L.W.,Gorenstein Dimensions,Lecture Notes in Mathematics,Springer-Verlag,Berlin,2000.

[2]Enochs,E.E.and Jenda,O.M.G.,Relative homological algebra,De Gruyter,Berlin,New York,2000.

[3]Enochs,E.E.,Jenda,O.M.G.and Xu,Jinzhong,Foxby duality and Gorenstein injective and projective modules,Trans.Amer.Math.Soc.,348(8),1996,3223–3234.

[4]Foxby,H.B.,Gorenstein modules and related modules,Math.Scand,31,1972,267–284.

[5]Golod,E.S.,G-dimension and generalized perfect ideals,Algebraic geometry and its applications,Trudy Mat.Inst.Steklov,165,1984,62–66(in Russian).

[6]Holm,H.,Gorenstein homological dimensions,J.Pure Appl.Algebra,189,2004,167–193.

[7]Holm,H.and White,D.,Foxby equivalence over associative rings,J.Math.Kyoto Univ.,47,2008,781–808.

[8]Sather-Wagstaff,S.,Sharif,T.and White,D.,Stability of Gorenstein categories,J.London Math.Soc.,77(2),2008,481–502.

[9]Sather-Wagstaff,S.,Sharif,T.and White,D.,AB-context and stabiity for Gorenstein flat modules with respect to semidualizing modules,Algebra and Representation Theory,54,2010,430–435.

[10]Sather-Wagstaff,S.,Sharif,T.and White,D.,Tate cohomology with respect to semidualizing modules,J.Algebra,324,2010,2336–2368.

[11]Takahashi,R.and White,D.,Homological aspects of semidualizing modules,Math.Scand,106(1),2010,5–22.

[12]Vasconcelos,W.V.,Divisor theory in module categories,North-Holland Publishing Co.,Amsterdam,1974.

[13]White,D.,Gorenstein projective dimension with respect to a semidualizing module,J.Commut.Algebra,2(1),2010,111–137.