Foxby equivalences of Cartan-Eilenberg complexes

ZHANG Chunxia,QUE Chunyue

Abstract:LetRbe a commutative noetherian ring with a semi-dualizing moduleC.We introduce CE(abbreviation for Cartan-Eilenberg)Auslander classCE-AC(R)and CE Bass classCE-ℬC(R),and extend the Foxby equivalence to the setting of CE complexes.

Key Words:CE complex；semi-dualizing module；Auslander class；Bass class

0 Introduction

Foxby duality has proven to be an important tool in studying the category of modules over a local Cohen-Macaulay ring admitting a dualizing module.Later as a generalization of a dualizing module and a free module of rank one,the notion of a semidualizing module has been given in[1].LetRbe a commutative noetherian ring.A semi-dualizing module overRis a finite generated moduleCsuch that the homothety morphismR→RHomR(C,C)is invertible in the derived categoryD(R).If furthermoreChas finite injective dimension thenCis said to be a dualizingR-module.

Given a semi-dualizingR-moduleC,the relative Foxby classes which are called the Auslander classAC(R)and Bass classℬC(R)can be defined[1-2]and there is stillan associated Foxby dualitywhich is restricted an adjoint pairoffunctorsonD(R)to triangulated subcategories.The Auslander classAC(R)and the Bass classℬC(R)are defined as follows:

is an isomorphism inD(R),andYis an isomorphism inD(R),andRHomR(C,Y)∈Db(R)}.

Foxby equivalence has been extensively studied,see for instance[3-7].Foxby[8]studied modules in Auslander classA0C(R)and Bass classℬ0C(R),where

Exti≥1R(C,C⊗RM),and the map

M→HomR(C,C⊗RM)is an isomorphism},ℬ0C(R)={N∈R-Mod|Exti≥1R(C,N)=0=

TorR i≥1(C,HomR(C,N)),and the map

C⊗RHomR(C,N)→Nis an isomorphism}.

We always takeCas a complex concentrated in degree zero,and then it is a semi-dualizing complex in the sense of[1]definition 2.1.By[1]observation 4.10,A0C(R)andℬ0C(R)coincide respectively,with the subcategories ofAC(R)andℬC(R)consisting ofR-complexes concentrated in degree zero.

Recall that anR-module isC-projective if it has the formC⊗RPfor some projectiveR-moduleP.AnR-module isC-injective if it has the form HomR(C,E)for some injectiveR-moduleE.LetPCandICdenote the class ofC-projective andC-injective modules,respectively.By[9]lemmas 4.1 and 5.1,the Auslander classA0C(R)contains every projectiveR-module and everyC-injectiveR-module,and the Bass classℬ0C(R)contains every injectiveR-module and everyC-projectiveR-module.HOLM et al[2]introduced theC-Gorenstein projective andC-Gorenstein injective modulesusing semi-dualizing modulesand their associated projective,injective classes and they connected the study of semi-dualizing modules to associated Auslander and Bass classes.Denote byGP(R),GI(R),GPC(R)andGIC(R)the classes of Gorenstein projective,Gorenstein injective,C-Gorenstein projective andC-Gorenstein injective modules,respectively.From[10]theorem 4.6,we have the following Foxby equivalent diagram:

On the other hand,in his thesis,VERDIER[11]introduced the notion of Cartan-Eilenberg injective complexes.A complexIis said to be Cartan-Eilenberg(CE for short)injective provided thatI,the cycle complexZ(I),the boundary complexB(I)and the homology complexH(I)are all complexes of injective modules.Similarly,CE projective complexesaredefined.Itwasshown thatCE projectiveand CE injectivecomplexesform the relative projective and injective objects for the chain homotopy category with respect to a proper class of triangles[12].ENOCHS[13]showed that Cartan-Eilenberg resolutions can be defined in terms of precovers and preenvelopes by CE projective and injective complexes,and he further introduced CE Gorenstein projective and injective complexes.Denote byCE-P(R),CE-I(R),CE-GP(R)and CE-GI(R)the classesofCE projective,CE injective,CE Gorenstein projective and CE Gorenstein injective complexes,respectively.

Thus,a natural question arises:what are the counterparts to the CE projective,CE injective and CE Gorenstein projective,CE Gorenstein injective complexes under Foxby equivalence?

To this end,we introduce CE Auslander class CE-AC(R)and CE Bass classCE-ℬC(R),and extend the Foxby equivalence in some existed literature such as[3,6-7],to the setting of CE complexes.

Throughout,Ris a commutative noetherian ring,andCis a semi-dualizing module overR.

1 Foxby equivalences of CE complexes

Definition 1TheCEAuslander class andCEBass class with respect toC,denoted byCE-AC(R)and CE-ℬC(R),are defined as follows:

CE-AC(R)={X∈Db(R)|X≃A,whereA,Z(A),B(A)andH(A)are complexes consisting of modules in A0C(R)},

CE-ℬC(R)={X∈Db(R)|X≃D,whereD,Z(D),B(D)andH(D)are complexes consisting of modules in ℬ0C(R)}.

Proposition 1LetXbe anR-complex.IfX∈CEAC(R),then the complexesX,Z(X),B(X)andH(X)are all inAC(R).Dually,ifX∈CE-ℬC(R),then the complexesX,Z(X),B(X)andH(X)are all inℬC(R).

ProofLetX∈CE-AC(R).Then there exists an isomorphismX≃AinD(R),whereAis a complex such thatA,Z(A),B(A)andH(A)are complexes consisting of modules inA0C(R).SetsupH(X)=s,infH(X)=i.Consider a truncated complex

By noting that in the exact sequence 0→Bs(A)→As→As/Bs(A)→0 the firsttwo entries are inA0C(R),we haveAs/Bs(A)is also in A0C(R),and thenA≃A()s,i∈CE-AC(R).Thus,wemaychooseAto beabounded complex consisting of modules inA0C(R).

Letα:P·→Cbe a projective resolution of the semi-dualizing moduleC,whereP·=…→P1→P0→0.Thenαis a quasi-isomorphism,and we representC⊗LRX≃C⊗LRAby the complexP·⊗RA.For anyn∈ Z,it follows from ToriR(C,An)=0thatα⊗RAn:P·⊗RAn→C⊗RAnis a quasi-isomorphism.Then,by[14]proposition 2.14α⊗RA:P·⊗RA→C⊗RAis a quasiisomorphism,andhenceC⊗LRA≃P·⊗RA≃C⊗RA∈Db(R).

Since ExtiR(C,C⊗An)=0,itfollows that HomR(C,C⊗RAn)→ HomR(P·,C⊗RAn)is a quasi-isomorphism.Then,we have a quasiisomorphism

HomR(α,C⊗RA):HomR(C,C⊗RA)→HomR(P·,C⊗RA)

by [14]proposition 2.7.HenceRHomR(C,C⊗LRA)≃ HomR(P·,C⊗RA)≃HomR(C,C⊗RA).

Moreover,A→HomR(C,C⊗RA)is a canonical isomorphism.ThenA→RHomR(C,C⊗LRA)is an isomorphism inD(R),and this impliesX≃A∈AC(R).Similarly,the bounded complexH(A)is also in AC(R)and soH(X)≃H(A)∈ AC(R).In the next,we need to prove that the complexesZ(X)andB(X)are in AC(R).

Conversely,we supposeZ(X)is not inAC(R).Note thatAC(R)is a triangulated subcategory ofDb(R),which satisfies 2-out-of-3 property,that is,if any two items in an exact sequence are inAC(R)then so is the third.Then it follows from the exact sequence 0 →B(X)→Z(X)→H(X)→ 0 thatB(X)∉AC(R)(otherwise,contradict to Z(X)∉AC(R)).Moreover,there is an exact sequence0→Z(X)→X→B(X)[1]→0,whereB(X)[1]stands for shiftingB(X)one-degree to the left.HenceX∉AC(R),and a contradiction occurs.This implies thatZ(X)∈AC(R),and furthermore,B(X)∈AC(R)as well.

ForX∈CE-ℬC(R),the assertions can be proved dually,and then is omitted.

Proposition 2There is an equivalence of categories

ProofLetX∈CE-AC(R).Then there exists an isomorphismX≃AinD(R),whereAis a bounded complex such thatA,Z(A),B(A)andH(A)are complexes consisting of modules inA0C(R).

Itfollows from the arguments above thatC⊗LRX≃C⊗RA.Obviously,C⊗RAis a complex of modules inℬ0C(R).It remains to prove thatZ(C⊗RA),B(C⊗RA)andH(C⊗RA)are complexes of modules inℬ0C(R).

For anyn∈Z,consider the exact sequence

SinceBn-1(A)∈A0C(R),there is an exact sequence

SoZn(C⊗RA)=C⊗RZn(A)∈ℬ0C(R)and

Bn-1(C⊗RA)=C⊗RBn-1(A)∈ℬ0C(R).

Similarly,consider the exact sequence

0 →Bn(A)→Zn(A)→Hn(A)→ 0,and we have,from the exact sequence

0→C⊗RBn(A)→C⊗RZn(A)→C⊗RHn(A)→0,thatHn(C⊗RA)=C⊗RHn(A)∈ℬ0C(R).

TheproofthatRHomR(C,-)takesCE-ℬC(R)into CE-AC(R)is similar.Finally,it follows from proposition 1 that there are inclusions of categories CE-AC(R)⊆AC(R)and CE-ℬC(R)⊆ ℬC(R),and hence the equivalence of categories is immediate by[1]theorem 4.6.This completes the proof.

Following [3]it is denoted by(R)the category composed by complexesXsuch thatX≃UinDb(R),for a bounded complexUof projective modules.We denote the subcategorywithPa bounded CE projectiveofDb(R)byCE-(R).Similarly,(R)andCE-(R)are defined.

Proposition 3(1)There are inclusions of categories:

(2)There is an equivalence of categories:

ProofThe inclusions of categories are obvious.We only need to prove(2).

LetX∈CEP(R).Then the reexistsan isomorphismX≃PinD(R),wherePis a bounded CE projective complex.By[13]proposition 3.4,we may choosePto be a graded module of projective.Note thatC⊗LRX≃C⊗RPandC⊗RPis a graded module with entries beingC-projective modules.ThenC⊗LRXis in CEP(R).

Similarly,for anyX∈CEPC(R),one can prove thatRHomR(C,X)∈CEP(R).Moreover,by proposition 2，the equivalence of categories holds.

Letwbe a class ofR-modules.wis called selforthogonal ifExtiR(W,W′)=0for allW,W′∈wand alli≥1.AnR-moduleMis said to bew-Gorenstein(refe.[6],definition 2.2）if there exists an exact sequence

W·=…→W1→W0→W-1→W-2→…of modules inwsuch thatM=Ker(W-1→W-2)andW·isHomR(w,-)andHomR(-,w)exact.In this case,W·is called a completew-resolution ofM.This covers a various of examples by different choices ofw,for instance,Gorenstein projective and Gorenstein injective modules by takingwto be the class of projective and injective modules respectively.

By[6]theorem 3.1,PC=AddCandIC=ProdC+,where AddCstandsforthecategory consisting of allmodules isomorphic to direct summandsof direct sumsof copiesofC,and ProdC+the category consisting ofallmodules isomorphic to direct summands of direct products of copies ofC+=HomR(C,Q)withQan injective cogenerator.ThusPCandICare self-orthogonal.In particular,PC-Gorenstein andIC-Gorenstein modules,by settingw=PCandw=IC，respectively,are namedC-Gorenstein projective andC-Gorenstein injective modules in[6].We denote byG(P),G(I),G(PC)andG(IC)the class of Gorenstein projective,Gorenstein injective,C-Gorenstein projective andC-Gorenstein injective modules respectively.

Recallfrom [2]and [7]that a completePPC-resolution isacomplexXofR-modules satisfying: (1)The complexXis exact and HomR(-,PC)-exact；(2)TheR-moduleXiis projective ifi≥ 0andXiisC-projective ifi＜ 0.AnR-moduleMisGC-projective if there exists a completePPC-resolutionXsuch thatM≅Z-1(X).Dually,a completeICI-coresolution and aGC-injective module are defined.The classes ofGC-projective andGC-injective modules are denoted byGPCandGICrespectively.

In the following lemma,(1)is from [15]proposition 5.2 or[6]proposition 3.6,and(2)is from[6]theorem 3.11.

Lemma 1(1)G(PC)=GPC∩ ℬ0C(R)，G(IC)=GIC∩A0C(R).

(2)LetMbe anR-module.IfM∈G(P)∩A0C(R),thenC⊗RM∈G(PC)；ifM∈G(PC),thenHomR(C,M)∈G(P)∩A0C(R).Dually,ifM∈G(IC),thenC⊗RM∈G(I)∩ℬ0C(R)；ifM∈G(I)∩ ℬ0C(R),thenHomR(C，M)∈G(IC).

Inspired by[13,6],we introduced the notion of CEw-Gorensteincomplex,andprovedin[16]theorem 3.10 that two potential choices for defining these complexes are equivalent,that is,Gis a complex such thatG,B(G),Z(G),andH(G)are complexes ofw-Gorenstein modules,if and only ifGadmits a completeCEw-resolution in the sense of[16]definition 2.4.Obviously,we recover ENOCHS's CE Gorenstein projective complexes by settingw=P.Thenotionof CEC-Gorenstein projective complexes follows by lettingw=PC.

In the next,we consider the subcategory ofDb(R)withrespecttoCE GorensteinandCEC-Gorenstein projective complexes.The subcategory {X∈Db(R)|X≃G} ofDb(R)is denoted by(R)ifGis a bounded CE Gorenstein projective complex,and ifGis a bounded CEC-Gorenstein projective complex,then we use the notationfor it.

Proposition 4(1)There are inclusions of categories:

(2)There is an equivalence of categories:

ProofLetXbe a complex inDb(R).IfX∈CE-thenX≃G,whereGis a bounded CE Gorenstein projective complex and is inCE-AC(R).By the argument aforementioned in proof of proposition 1,we haveC⊗LRX≃C⊗RG.Analogous to the proof in proposition 3,one gets thatC⊗RGisa CEC-Gorenstein projective complex by combining with lemma 1.Thus

for a bounded CEC-Gorenstein projective complexG.Similarly,we haveRHomR(C，X)≃HomR(C,G)withHomR(C,G)beingbothCE Gorenstein projective and in CE-AC(R).Consequently,we have the equivalence ofthe categories by proposition 2.

By assembling the information above,we have the following extension of the Foxby equivalence for the CE version.

Theorem 1(Foxby equivalence)There are equivalences of categories：