WANG Zhan-ping,GUO Shou-tao,MA Hai-yu

(School of Mathematics and Statistics,Northwest Normal University,Lanzhou 730070,China)

STABILITY OF GORENSTEIN MODULES WITH RESPECT TO A SEMIDUALIZING MODULE

WANG Zhan-ping,GUO Shou-tao,MA Hai-yu

(School of Mathematics and Statistics,Northwest Normal University,Lanzhou 730070,China)

This paper mainly study the stability of Gorenstein modules with respect to a semidualizing moduleC(i.e.,GorensteinC-projective,GorensteinC-injective and GorensteinC- fl at modules).With the method of homology,we gain the result that GorensteinC-projective(resp.,C-injective,C- fl at)modules have a nice stability,which generalizes the result that Gorenstein projective(resp.,injective, fl at)modules have a nice stability.

GorensteinC-projective(injective, fl at)modules;stability

1 Introduction

Gorenstein homological theory originated from the works of Auslander and Bridger[1],where they introduced modules ofG-dimension 0.Enochs et al.[2,3]extended these ideas and introduced Gorenstein projective,Gorenstein injective and Gorenstein fl at modules and corresponding dimensions over arbitrary rings.From then,Gorenstein homological theory was extensively studied and developed,see for example,[4,5]and references therein.

Recall from White[6]that anR-moduleCis said to be semidualizing ifCadmits a degreewise finite projective resolution,the natural homothety mapR→HomR(C,C)is an isomorphism andExamples include the rank one free modules and a dualizing(Canonical)modules when one exists.With this notion,Gorenstein homological theory with respect to a semidualizing module was extensively studied,see for example,[7]and[8].

LetRbe a ring.White[6]initiated the study ofGC-projectiveR-modules as the modules of the form Im(P0→C⊗RP0)for some exact sequence ofR-modules

where eachPiandPiis projective,and such that the complex HomR(P,C⊗RQ)is exact for each projectiveR-moduleQ.The complex P is called a completePC-resolution.WhenC=R,the de finition corresponds to the de finition of Gorenstein projective modules.In[6],Proposition 2.6 showed that ifPisR-projective,thenPandC⊗RPareGC-projective,and Proposition 2.9 showed that every cokernel in a completePC-resolution isGC-projective,so every image and kernel in it is alsoGC-projective.

ForGC-projectiveR-modules,we give another generalization in this paper,which is different from the de finition above and that in[9].AnR-moduleMis called GorensteinC-projective if there exists an exact sequence ofR-modules

where eachPiandPiis projective withM≌Im(C⊗RP0→C⊗RP0),such that the complex HomR(X,C⊗RQ)is exact for each projectiveR-moduleQ.It is easy to see that,for every projectiveR-moduleP,C⊗RPis GorensteinC-projective from de finition,while the projective modulePis not GorensteinC-projective,so GorensteinC-projective modules are a generalization ofPC-projective modules not projective modules,which distinguishesGC-projective modules.WhenC=R,GorensteinC-projective modules are exactly Gorenstein projective modules.From this point of view,the notion of GorensteinC-projective modules is a generalization of Gorenstein projective modules.

LetAbe an abelian category andXan additive full-subcategory ofA.Sather-Wagsta ff,Sharif and White[9]introduced the de finition of Gorenstein categoryG(X)onX.G(X)={A∈A|there exists a both HomA(X,−)-exact and HomA(−,X)-exact exact sequence X:···→X1→X0→X0→X1→ ···inAwith allXiandXiinXandA≌Im(X0→X0)}.Especially,ifA=R-Mod,X=P(R),thenG(P(R))is nothing but the subcategory of Gorenstein projective modules.Dually,G(I(R))is nothing but the subcategory of Gorenstein injective modules.LetG0(X)=X,G1(X)=G(X),for arbitraryn≥1,Gn+1(X)=Gn(G(X)).They proved that whenXis self-orthogonal,Gn(X)=G(X)for arbitraryn≥1;and they proposed the question whetherGn(X)=G(X)holds for an arbitrary subcategoryX.See[9,4.10 and 5.8].Huang[10]proved that the answer to this question is affirmative.Kong and Zhang[11]gave a slight generalization of this stability by a different method.Recently,stability of other Gorenstein classes of modules was extensively studied by many authors,see for example,[12–15].Inspired by these results,we consider the following questions:

Question 1Given an exact sequence of GorensteinC-projectiveR-modules G:···→G1→G0→G0→G1→ ···such that the complex HomR(G,C⊗RP)is exact for eachP∈P(R),is the module Im(G0→G0)GorensteinC-projective?

The same question of GorensteinC-injectiveR-modules can be considered dually.

Question 2Given an exact sequence of GorensteinC- fl atR-modules G:···→G1→G0→G0→G1→ ···such that the complex HomR(C,E)⊗RG is exact for eachE∈I(R),is the module Im(G0→G0)GorensteinC- fl at?

In this paper,we give an affirmative answer to Question 1(see Theorem 3.5 and Theorem 3.9),and a partial positive answer to Question 2(see Theorem 3.12).It is shown that GorensteinC-projective(injective, fl at)modules have a nice stability.

2 Preliminaries

Throughout this paper,Ris a commutative ring with identity and all modules are unitary.R-Mod denotes the category ofR-modules.P(R)(resp.,I(R),F(R))denotes projective(resp.,injective, fl at)R-modules.

In this section,we mainly recall some necessary notions and de finitions.

De finition 2.1[14]LetX,Ybe the class ofR-modules,Mbe anR-module.

(1)AnX-resolution ofR-moduleMis an exact sequence

withXi∈X(i≥0).

(2)AY-coresolution ofR-moduleMis an exact sequence

withYi∈Y(i≥0).

De finition 2.2[5]LetXbe a class ofR-modules.

(1)Xis calledPC(R)-resolving ifPC(R)⊆X,and for every short exact sequence 0→X′→X→X′′→0 withX′′∈Xthe conditionsX′∈XandX∈Xare equivalent.

(2)Xis calledIC(R)-resolving ifIC(R)⊆X,and for every short exact sequence 0→X′→X→X′′→0 withX′∈Xthe conditionsX∈XandX′′∈Xare equivalent.

Recall from[6]that a moduleCis semidualizing ifCadmits a degreewise finite projective resolution,the natural homothety mapR→HomR(C,C)is an isomorphism and ExtThe rank one free modules and a dualizing module are semidualizing modules over a noetherian ringR.We refer the reader to[16–19].

In the remainder of the paper,letCbe a fixed semidualizingR-module.

Recall from[18]that theR-modules in the following classes

are calledC-projective,C- fl at,andC-injective,respectively.

De finition 2.3AnR-moduleMis said to be GorensteinC-projective,if there exists an exact sequence ofC-projectiveR-modules

withM≌Im(C⊗RP0→C⊗RP0),such that the complex HomR(X,C⊗RQ)is exact for each projectiveR-moduleQ.The complex X is called completePC-resolution,then by symmetry,all the images,and hence also all the kernels,and cokernels of X are GorensteinC-projective modules.

Dually,GorensteinC-injective modules are de fined.

AnR-moduleLis said to be GorensteinC- fl at,if there exists an exact sequence ofC- fl at modules

withL≌Im(C⊗RF0→C⊗RF0)such that the complex HomR(C,I)⊗RX is exact for each injectiveR-moduleI.

We use the symbolGPC(R)(resp.,GIC(R),GFC(R))to stand for the class of GorensteinC-projective(resp.,GorensteinC-injective,GorensteinC- fl at)R-modules.It is obvious thatPC(R)⊆GPC(R),IC(R)⊆GIC(R),FC(R)⊆GFC(R).

1.4 治疗方案 所有患者入院后立即参考《中国心力衰竭诊断和治疗指南2014》及《室性心律失常中国专家共识》中方案接受常规药物治疗，包括积极控制原发病、对症治疗等，同时应用胺碘酮（杭州赛诺菲制药有限公司生产，200毫克／片）抗心律失常。在此基础上，观察组联合应用人参果总皂苷（吉林省集安益盛药业股份有限公司生产，0.25克／粒），口服，2粒／次，3次／天。所有患者均治疗4周为1个疗程，联用3个疗程后观察疗效。

3 Stability of Gorenstein C-Projective(resp.,C-Injective,C-Flat)Modules

We begin with the following:

Lemma 3.1If{Xλ}λ∈Λis a collection of completePC-resolution,thenЦΛXλis a completePC-resolution.Thus,the class of GorensteinC-projectiveR-modules is closed under direct sums.

ProofFor each projectiveR-moduleQ,there is an isomorphism

So if the complex HomR(Xλ,C⊗RQ)is exact for everyλ,then the complex HomR(ЦΛXλ,C⊗R Q)is also exact.It follows that a direct sum of GorensteinC-projectiveR-modules is GorensteinC-projective.

Lemma 3.2Let 0→M′→M→M′′→0 be the exact sequence ofR-modules andM′′be a GorensteinC-projectiveR-module.ThenM′is GorensteinC-projective if and only ifMis GorensteinC-projective.Namely,the class of GorensteinC-projective modules is closed under extensions and direct summands.

ProofFirstly,assume thatM′andM′′are GorensteinC-projective modules with completePC-resolution X′and X′′,respectively.By Horseshoe lemma[20,Lemma 6.20]and[6,Lemma 1.13],we can construct a complex

where eachPiandPiis projective,and a degreewise split exact sequence of complexes

such thatM≌Im(C⊗RP0→C⊗RP0).To show thatMis GorensteinC-projective,it suffices to show complex HomR(X,C⊗RQ)is exact for every projectiveR-moduleQ.The sequence

Next,assume thatMandM′′are GorensteinC-projective modules with completePC-resolution X and X′′,respectively.Comparison lemma provides a morphism of chain complexesϕ:X→X′′induced byπ:M→M′′on the degree 0.By adding complexes of the formto X,we assumeϕis surjective.SincePC(R)is closed under kernels of epimorphisms,thus the complex X′=Ker(ϕ)has the form

Next is a key lemma,which plays an important part in solving Question 1 from the introduction.

Lemma 3.3LetMbe anR-module.ThenMhas aPC(R)-resolution which is HomR(−,PC(R))-exact if and only ifMhas aGPC(R)-resolution which is HomR(−,PC(R))-exact.

ProofIt is enough to show the“if”part.LetMhas a HomR(−,PC(R))-exactGPC(R)-resolution···−→G2−→G1−→G0−→M−→0 withGi∈GPC(R),i≥0.SetN=Im(G1→G0),then 0→N→G0→M→0 be an exact sequence which is HomR(−,PC(R))-exact,whereG0∈GPC(R)andNhas aGPC(R)-resolution which is HomR(−,PC(R))-exact.Then we have the following pullback diagram

withP0∈P(R),G′∈GPC(R)and all rows and columns HomR(−,PC(R))-exact.Note that there exists an exact sequence 0→K→G1→N→0,which is HomR(−,PC(R))-exact,whereG1∈GPC(R)andKhas aGPC(R)-resolution which is HomR(−,PC(R))-exact.

Consider the following pullback diagram

The exactness of the middle row impliesL∈GPC(R)by Lemma 3.2.Since the short exact sequence

is HomR(−,PC(R))-exact by Snake lemma,andKhas a HomR(−,PC(R))-exactGPC(R)-resolution,thusHhas aGPC(R)-resolution which is HomR(−,PC(R))-exact.Note that

is HomR(−,PC(R))-exact,HandMhave the same properties,so by repeating the preceding process,we have thatMhas aPC(R)-resolution which is HomR(−,PC(R))-exact,as required.

Dually,we have the following lemma.

Lemma 3.4LetMbe anR-module.ThenMhas aPC(R)-coresolution which is HomR(−,PC(R))-exact if and only ifMhas aGPC(R)-coresolution which is HomR(−,PC(R))-exact.

ProofIt is enough to show the“if”part.LetMhas a HomR(−,PC(R))-exactGPC(R)-coresolution

withGi∈GPC(R),i≥0.SetN=Im(G0→G1),then 0→M→G0→N→0 be an exact sequence which is HomR(−,PC(R))-exact,whereG0∈GPC(R)andNhas aGPC(R)-coresolution which is HomR(−,PC(R))-exact.Then we have the following pushout diagram

withP0∈P(R),G(1)∈GPC(R)and all rows and columns HomR(−,PC(R))-exact.Note that there exists an exact sequence 0→N→G1→K→0,which is HomR(−,PC(R))-exact,whereG1∈GPC(R)andKhas aGPC(R)-coresolution which is HomR(−,PC(R))-exact.Consider the following pushout diagram.

The exactness of the middle column impliesL∈GPC(R). Since the short exact sequence 0→H→L→K→0 is HomR(−,PC(R))-exact by Snake lemma,andKhas a HomR(−,PC(R))-exactGPC(R)-coresolution,thusHhas aGPC(R)-coresolution which is HomR(−,PC(R))-exact.Note that 0→M→C⊗P0→H→0 is HomR(−,PC(R))-exact,HandMhave the same properties,so by repeating the preceding process,we have thatMhas aPC(R)-coresolution which is HomR(−,PC(R))-exact.

LetG2PC(R)be the class ofR-modulesMfor which there exists an exact sequence of GorensteinC-projectiveR-modules

such that the complex HomR(G,C⊗RP)is exact for eachP∈P(R)andM≌Im(G0→G0).It is obvious thatGPC(R)⊆G2PC(R),as a consequence of Lemmas 3.3 and 3.4,we have the following result.

Theorem 3.5GPC(R)=G2PC(R).

LetG(GPC(R))={M∈R-Mod|there exists a HomR(−,H)-exact exact sequence

inR-Mod with allGiandGiinGPC(R),H∈GPC(R)andM≌Im(G0→G0)}.ThenGPC(R)⊆G(GPC(R))⊆G2PC(R),by Theorem 3.5,we haveGPC(R)=G(GPC(R))=G2PC(R).

The stability of GorensteinC-injectiveR-modules can be given dually.

Lemma 3.6Let 0→M′→M→M′′→0 be the exact sequence ofR-modules andM′′be a GorensteinC-injectiveR-module.ThenM′is GorensteinC-injective if and only ifMis GorensteinC-injective.

Lemma 3.7LetMbe anR-module.ThenMhas aIC(R)-resolution which is HomR(IC(R),−)-exact if and only ifMhas aGIC(R)-resolution which is HomR(IC(R),−)-exact.

Lemma 3.8LetMbe anR-module.ThenMhas aIC(R)-coresolution which is HomR(IC(R),−)-exact if and only ifMhas aGIC(R)-coresolution which is HomR(IC(R),−)-exact.

LetG2IC(R)be the class ofR-modulesMfor which there exists an exact sequence of GorensteinC-injectiveR-modules

such that the complex HomR(HomR(C,E),G)is exact for eachE∈I(R)andM≌Im(G0→G0).It is obvious thatGIC(R)⊆G2IC(R),as a consequence of Lemmas 3.7 and 3.8,we have the following result.

Theorem 3.9GIC(R)=G2IC(R).

LetG(GIC(R))={M∈R-Mod|there exists a HomR(H,−)-exact exact sequence

inR-Mod with allGiandGiinGIC(R),H∈GIC(R)andM≌Im(G0→G0)}.ThenGIC(R)⊆G(GIC(R))⊆G2IC(R),by Theorem 3.9,we haveGIC(R)=G(GIC(R))=G2IC(R).

Theorem 3.5 and Theorem 3.9 give an affirmative answer to Question 1.

Finally,we discuss the stability of GorensteinC- fl at modules.

Lemma 3.10LetMbe anR-module.IfGFC(R)is closed under extensions,thenMhas anFC(R)-resolution which isIC(R)⊗R-exact if and only ifMhas aGFC(R)-resolution which isIC(R)⊗R-exact.

ProofIt is enough to show the“if”part.Let 0→N→G0→M→0 be anIC(R)⊗R-exact exact sequence withG0∈GFC(R)andNhaving aGFC(R)-resolution,which isIC(R)⊗R-exact.Then we have the following pullback diagram

withF0∈F(R),G′∈GFC(R).Since the bottom row isIC(R)⊗R-exact,so is the middle row.Note that there is anIC(R)⊗R-exact exact sequence 0→K→G1→N→0 withG1∈GFC(R)andKhaving aGFC(R)-resolution,which isIC(R)⊗R-exact.Consider the following pullback diagram.

SinceGFC(R)is closed under extensions,L∈GFC(R).Thus,Hhas aGFC(R)-resolution,which isIC(R)⊗R-exact.Note that 0→H→C⊗F0→M→0 isIC(R)⊗R-exact.By repeating the the preceding process,we have thatMhas anFC(R)-resolution which isIC(R)⊗R-exact.

Dually,we can prove the following lemma.

Lemma 3.11LetMbe anR-module.IfGFC(R)is closed under extensions,thenMhas anFC(R)-coresolution which isIC(R)⊗R-exact if and only ifMhas aGFC(R)-coresolution which isIC(R)⊗R-exact.

LetG2FC(R)be the class ofR-modulesMfor which there exists an exact sequence of GorensteinC- fl atR-modules

such that the complex HomR(C,E)⊗RG is exact for eachE∈I(R)andM≌Im(G0→G0).It is obvious thatGFC(R)⊆G2FC(R),as a consequence of Lemmas 3.10 and 3.11,we have the following result.

Theorem 3.12IfGFC(R)is closed under extensions,thenGFC(R)=G2FC(R).

LetG(GFC(R))={M∈R-Mod|there exists aH⊗R-exact exact sequence

inR-Mod with allGiandGiinGFC(R),H∈GIC(R)andM≌Im(G0→G0)}.It is obvious thatGFC(R)⊆G(GFC(R))⊆G2FC(R),by Theorem 3.12,we have

Theorem 3.12 gives a partial positive answer to Question 2.

[1]Auslander M,Bridger M.Stable module theory[M].Providence RI:Mem.Amer.Math.Soc.,1969.

[2]Enochs E E,Jenda O M G.Gorenstein injective and projective modules[J].Math.Z.,1995,220:611–633.

[3]Enochs E E,Jenda O M G,Torrecillas B.Gorenstein fl at modules[J].J.Nanjing Univ.Math.Biq.,1993,10:1–9.

[4]Enochs E E,Jenda O M G.Relative homological algebra[M].Berlin:Walter de Gruyter,2000.

[5]Holm H.Gorenstein homological dimensions[J].J.Pure Appl.Alg.,2004,189:167–193.

[6]White D.Gorenstein projective dimension with respect to a semidualizing module[J].J.Commut.Alg.,2010,2:111–137.

[7]Sather-Wagsta ffS,Sharif T,White D.Tate cohomology with respect to a semidualizing modules[J].J.Alg.,2010,324:2336–2368.

[8]Takahashi R,White D.Homological aspects of semidualizing modules[J].Math.Scand.,2010,106:5–22.

[9]Sather-Wagsta ffS,Sharif T,White D.Stability of Gorenstein categories[J].J.London Math.Soc.,2008,77:481–502.

[10]Huang Zhaoyong.Proper resolutions and Gorenstein categories[J].J.Alg.,2013,393:142–169.

[11]Kong Fan,Zhang Pu.From CM- finite to CM-free[J].J.Pure Appl.Alg.,2016,220:782–801.

[12]Bouchiba S.Stability of Gorenstein classes of modules[J].Alg.Colloq.,2013,20:623–636.

[13]Wang Zhanping,Liu Zhongkui.Stability of strongly Gorenstein fl at modules[J].Vietnam J.Math.,2014,42:171–178.

[14]Xu Aiming,Ding Nanqing.On stability of Gorenstein categories[J].Comm.Alg.,2013,41:3793–3804.

[15]Yang Gang,Liu Zhongkui.Stability of Gorenstein fl at categories[J].Glasg.Math.J.,2012,54:177–191.

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

[17]Golod E S.G-dimension and generalized perfect ideals[J].Trudy Mat.Inst.Steklov,1984,165:62–66.

[18]Holm H,White D.Foxby equivalence over associative rings[J].J.Math.Kyoto Univ.,2007,47:781–808.

[19]Vasconcelos,Vercosa W.Divisor theory in module categories[M].Amsterdam:North-Holland Math.Stud.,1974,Vol.14.

[20]Rotman J J.An introduction to homological algebra[M].New York:Academic Press,1979.

关于半对偶化模的Gorenstein模的稳定性

王占平,郭寿桃,马海玉
(西北师范大学数学与统计学院,甘肃兰州 730070)

本文研究了相对于半对偶化模C的Gorenstein模(即GorensteinC-投射模,GorensteinC-内射模和GorensteinC-平坦模)的稳定性的问题.利用同调的方法,获得了GorensteinC-投射(C-内射,C-平坦)模具有很好的稳定性的结果,推广了Gorenstein投射(内射,平坦)模具有很好的稳定性的结果.

GorensteinC-投射(内射,平坦)模;稳定性

O154.2

16D40;16D50;16E05;16E30

A

0255-7797(2017)06-1143-11

date:2016-09-26Accepted date:2016-11-08

Supported by National Natural Science Foundation of China(11561061;11261050).

Biography:Wang Zhanping(1978–),female,born at Lanzhou,Gansu,associated professor,major in homological algebra.