On Rings with Finite Global Gorenstein Dimensions

(Department of Mathematics,Northwest Normal University,Lanzhou 730070,China)

§1.Introduction

A new branch in homological algebra which is known as Gorenstein homological algebra has developed rapidly during the past several years.An interesting problem in this branch is to get Gorenstein counterparts of classical homological algebra.For instance,the Auslander-Bridger Formula for Gorenstein projective dimension of modules parallels to the classical Auslander-Buchsbaum Formula for the projective dimension of modules,see[9].There is also a Gorenstein counterpart for the global dimension.Bennis and Mahdou proved an equality in[3]

The common value of this equality is called the global Gorenstein dimension ofRand is denoted by Ggldim(R).Recently,Emmanouil[6]also obtained this equality.

It is well known that the equality for global dimension holds since both projective and injective dimensions of modules are characterized by vanishing of derived functors Ext.However,Bennis and Mahdou established the global Gorenstein dimension by applying strongly Gorenstein projective and injective modules they introduced in[2]and Emmanouil got the equality by comparing Gorenstein projective and injective dimensions with two invariants introduced by Gedrich and Gruenberg[8],the projective lengths of injectiveR-modules spli(R)and the injective lengths of projectiveR-modules silp(R).As far as the methods used to determine the global Gorenstein dimension are concerned,they do not parallel to the one used to determine the global dimension in classical homological algebra.

In this paper,we intend to give a new proof of global Gorenstein dimension by vanishing of Gorenstein derived functors,serving as a case to support the metatheorem[12]:every result in classical homological algebra has a counterpart in Gorenstein homological algebra.To this end,the ring having finite global Gorenstein projective and injective dimensions,called generalized Gorenstein ring,is considered.Some examples are given to show that generalized Gorenstein rings are non-trivial generalizations of Gorenstein rings and rings with finite global dimension.WhenRis a generalized Gorenstein ring,Gorenstein projective and injective dimensions for anyR-module are characterized by vanishing of Holm’s Gorenstein Ext-functors.Then,the equality for global Gorenstein dimension follows immediately.

§2.Rings with Finite Global Gorenstein Dimensions

Throughout,Rdenotes a ring with unity,and modules are leftR-modules.Recall that anR-moduleMis Gorenstein projective provided that there exists a totally acyclic complex of projective modules(i.e.an exact sequence P=···→P1→P0→P−1→···of projective modules with HomR(P,Q)exact for any projective moduleQ),such thatM=Z0(P)=Ker(P0→P−1).The Gorenstein projective dimension,GpdRM,of anR-moduleMis defined by declaring that GpdRM≤nif and only if,Mhas a Gorenstein projective resolution of lengthn.Dually,Gorenstein injective modules and Gorenstein injective dimension of modules are defined.

Lemma 2.1LetRbe a ring.The following statements are equivalent.

(1)sup{GpdR(M)|Mis anR-module}is finite.

(2)sup{GidR(M)|Mis anR-module}is finite.

(3)Both spli(R)and silp(R)are finite.

Proof(1)(3)For any injectiveR-moduleE,we have pdR(E)=GpdR(E)by[13,Theorem 2.2]and hence spli(R)＜∞.It is easy to see from[11,Theorem 2.20]that every projective module has finite injective dimension,so silp(R)＜∞.

Similarly,the equivalence between(2)and(3)holds.

Definition 2.2If a ringRsatisfies one of the above equivalent conditions,thenRis called a generalized Gorenstein ring.

A ring with finite global dimension is obviously a generalized Gorenstein ring.It follows from[7]that ifRis Gorenstein(i.e.,a left and right noetherian ring with finite left and right self-injective dimensions),thenRis a generalized Gorenstein ring.The following examples show that generalized Gorenstein rings are not necessarily be Gorenstein or not be of finite global dimension.We remark that for finitely generated modules over two-sided noetherian rings,the conditions in Lemma 2.1 are equivalent to Gorenstein rings and are symmetrical,that is,which hold for right modules[14,Theorem 1.4].

Example 2.3

(1)For a domainAof field of quotientsK,letK[[t]]be the power series ring with coefficients inKandA[[t))be the subring ofK[[t]]consisting of the series with constant term inA.Now takeAto be a noetherian local ring of global dimension two in whichKhas projective dimension one.ThenA[[t))is a local ring of global dimension two which is neither a valuation domain nor a noetherian ring([15,p383]),i.e.,A[[t))is a generalized Gorenstein ring but is not Gorenstein.

(2)By[4,Example 2.8],letSn=S[X1,X2,···,Xn]be the polynomial ring innindeterminates over a non-noetherian hereditary ringS.LetRi=Si−1⊗Si−1be the trivial extension ofSi−1bySi−1fori≥1(such thatS0=S).ThenRihas global Gorenstein dimensionifor everyi≥1,i.e.,Riis a generalized Gorenstein ring.WhereasRiis a non-noetherian coherent ring and hence is not Gorenstein.

(3)LetRbe a non-semisimple quasi-Frobenius ring(e.g.k(x)/(x2)for a fieldk).Consider the polynomial ringR[x]in one indeterminate.

ThenR[x]is a generalized Gorenstein ring,which is noetherian and is not of finite global dimension.Indeed,sinceRis quasi-Frobenius,everyR-module is Gorenstein projective(injective)and it follows from[5,Theorem 2.1]thatR[x]is a generalized Gorenstein ring(precisely,R[x]is a ring of global Gorenstein dimension one).Moreover,Ris noetherian and henceR[x]is also noetherian by the well-known Hilbert basis theorem.Note that the projective dimension of a Gorenstein projective module is either zero or infinite[7,Theorem 10.2.3]and the global dimension ofRcan not be zero(Ris non-semisimple).This implies that the global dimension ofRis infinite.It follows immediately from an equality gldim(R[x])=gldim(R)+1(see[16,Theorem 4.3.7])thatR[x]has infinite global dimension.

LetM,NbeR-modules.It follows from[11,Theorem 2.10]that if GpdR(M)＜∞thenMhas a proper Gorenstein projective resolution,i.e.,a HomR(GP,−)-exact exact sequence G=···→G1→G0→M→0withGiGorenstein projective and then it is defined inis the deleted complex.Dually,if GidR(N)＜∞,then one defines(M,N):=Hn(HomR(M,E·)),where E·is a deleted complex of a coproper Gorenstein injective coresolution ofN.Since both proper Gorenstein projective resolution and coproper Gorenstein injective coresolution are unique up to homotype by a version of comparison theorem,the functors are well defined.

The following is due to Holm[10,Theorem 3.6],which implies that the Gorenstein Extfunctor is balanced.This is crucial in Definition 2.4.

LemmaFor allR-modulesMandNwith GpdR(M)＜∞and GidR(N)＜∞,there are isomorphisms

It seems to be reasonable to describe Gorenstein projective and injective dimensions ofR-modules by vanishing of(−,−)under the assumption thatRis a generalized Gorenstein ring.We remark that similar notions of Gorenstein Ext-functors have been introduced by Avramov and Martsinkovsky[1]for finitely generated modules of finiteG-dimension over a noetherian ring,and by Enochs and Jenda[1,12.1]when the ring is Gorenstein.By analogy with[7,Proposition 4.6]and its dual,there exist long exact sequences for the functorsGExt∗(−,−).

Lemma 2.5

(1)Let 0→M′→M→M′′→0 be a HomR(GP,−)-exact sequence ofR-modules with finite Gorenstein projective dimension.Then for anyR-moduleNwith GidR(N)＜∞,there is an exact sequence

(2)Let 0→N′→N→N′′→0 be a HomR(−,GI)-exact sequence ofR-modules with finite Gorenstein injective dimension.Then for anyR-moduleMwithGpdR(M)＜∞,there is an exact sequence

The following are analogous to the ones in classical homological algebra.

Theorem 2.6LetRbe a generalized Gorenstein ring.For anR-moduleMand a nonnegative integern,the following statements are equivalent.

thatKnis a direct summand ofGn.HenceKn∈GP.

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(3)=⇒(4).In the following,the lower row is a proper Gorenstein projective resolution and then there exist vertical arrows making the diagram commutative

By the associated exact sequence

we haveif and only ifKn∈GP.This completes the proof.

Dually,we have the following.

Theorem 2.7LetRbe a generalized Gorenstein ring.For anR-moduleNand a nonnegative integern,the following statements are equivalent.

(3)For any coproper Gorenstein injective coresolution 0→N→E·,theR-moduleLn=Coker(En−2−→En−1)is Gorenstein injective.

(4)For any Gorenstein injective coresolution 0−→N−→E′·,theR-moduleL′n=Coker(E′n−2−→E′n−1)is Gorenstein injective.

Then,we have a new proof for Bennis and Mahdou’s equality by combining Lemma 2.1 with theorems 2.6～2.7.The argument here is a Gorenstein counterpart of the one which is used to determine the global dimension.

Corollary 2.8LetRbe a ring.The following equality holds:

RemarkIfRis a generalized Gorenstein ring,Emmanouil established the equality spli(R)=silp(R)=Ggldim(R)in[6,Theorem 4.1]by comparing these invariants.Moreover,ifRis a ring with finite global dimension,then it is immediately from[11,Proposition 2.27]that spli(R)=silp(R)=Ggldim(R)=gldim(R).

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