On Injective and Gorenstein Injective Dimensions of Local Cohomology Modules

Abstract

Let $(R,\fm)$ be a commutative Noetherian local ring and let $M$ be an
$R$-module which is a relative Cohen-Macaulay with respect to a proper ideal
$\fa$ of $R$ and set $n:=\h_{M}\fa$. We prove that $\ind M<\infty$ if and only
if $\ind\H^{n}_\fa(M)<\infty$ and that $\ind\H^{n}_\fa(M)=\ind M-n$. We also
prove that if $R$ has a dualizing complex and $\Gid_{R} M<\infty$, then
$\Gid_{R}\H^{n}_\fa(M)<\infty$ and $\Gid_{R}\H^{n}_\fa(M)=\Gid_{R} M-n$.
Moreover if $R$ and $M$ are Cohen-Macaulay, then it is proved that $\Gid_{R}
M<\infty$ whenever $\Gid_{R}\H^{n}_\fa(M)<\infty$. Next, for a finitely
generated $R$-module $M$ of dimension $d$, it is proved that if $K_{\hat M}$ is
Cohen-Macaulay and $\Gid_{R}\H_{\fm}^{d}(M)<\infty$,
then$\Gid_{R}\H_{\fm}^{d}(M)=\depth R- d.$ The above results have consequences
which improve some known results and provide characterizations of Gorenstein
rings.

Full text

arXiv:1204.2394v5 [math.AC] 26 Feb 2013
ON INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS OF
LOCAL COHOMOLOGY MODULES
MAJID RAHRO ZARGAR AND HOSSEIN ZAKERI
Abstract. Let (R, m) be a commutative Noetherian local ring and let Mbe an R-
module which is a relative Cohen-Macaulay with respect to a proper ideal aof Rand
set n:= htMa. We prove that injdimM < ∞if and only if injdimHn
a(M)<∞and
that injdimHn
a(M) = injdimM−n. We also prove that if Rhas a dualizing complex
and Gi d RM < ∞, then Gid RHn
a(M)<∞and Gid RHn
a(M) = Gid RM−n. More-
over if Rand Mare Cohen-Macaulay, then it is proved that Gid RM < ∞whenever
Gid RHn
a(M)<∞. Next, for a finitely generated R-module Mof dimension d, it is proved
that if Kc
Mis Cohen-Macaulay and Gid RHd
m(M)<∞, then Gid RHd
m(M) = depth R−d.
The above results have consequences which improve some known results and provide
characterizations of Gorenstein rings.
1. introduction
Throughout this paper, Ris a commutative Noetherian ring, ais a proper ideal of R
and Mis an R-module. For a prime ideal pof R, the residue class field Rp/pRpis denoted
by k(p). For each non-negative integer i, let Hi
a(M) denotes the i-th local cohomology
module of Mwith respect to a; see [1] for its definition and basic results. Also, we use
injdimRMto denote the usual injective dimension of M. The notion of Gorenstein injective
module was introduced by E.E. Enochs and O.M.G. Jenda in [4]. The class of Gorenstein
injective modules is greater than the class of injective modules; but they are same classes
whenever Ris a regular local ring. The Gorenstein injective dimension of M, which is
denoted by Gid RM, is defined in terms of resolutions of Gorenstein injective modules. This
notion has been used in [3, 15, 21] and has led to some interesting results. Notice that
Gid RM≤injdimRMand the equality holds if injdimRM < ∞.
The principal aim of this paper is to study the injective (resp. Gorenstein injective)
dimension of certain R-modules in terms of injective (resp. Gorenstein injective) dimension
of its local cohomology modules at support a.
In this paper we will use the concept of relative Cohen-Macaulay modules which has been
studied in [7] under the title of cohomologically complete intersections. The organization
of this paper is as follows. In section 2, among other things, we prove, in 2.5, that if
Mis relative Cohen-Macaulay with respect to a, then injdimMand injdimHht Ma
a(M) are
simultaneously finite and there is an equality injdimHht Ma
a(M) = injdimM−ht Ma.Then,
2000 Mathematics Subject Classification. 13D05, 13D45.
Key words and phrases. Injective dimension, Gorenstein injective dimension, Local cohomology, Goren-
stein ring, Relative Cohen-Macaulay module.
1

2 M.R. ZARGAR AND H. ZAKERI
as a corollary, we obtain a characterization of Gorenstein rings. Next, in 2.10, for all n≥0
and any p∈Supp (M), we establish a comparison between the Bass numbers of Hn
pRp(Mp)
and Hn+dim (R/p)
m(M) whenever (R, m) is a homomorphic image of a Gorenstein ring and
Mis finitely generated.
In section 3, we first prove some basic properties about Gorenstein injective dimension
of a module. In particular, Proposition 3.6 indicates that Gorenstein injective dimension is
a refinement of the injective dimension. As a main result, in Theorem 3.8 we establish a
Gorenstein injective version of 2.5. Indeed, it is proved that if, in addition to the hypothesis
of 2.5, Rhas a dualizing complex, then Gid RM < ∞implies Gid RHn
a(M)<∞and the
converse holds whenever Rand Mare Cohen-Macaulay. This theorem has consequences
which recover some interesting results that have currently been appeared in the literature.
As a first corollary of 3.8, we deduce that Gid RHn
m(M) = Gid RM−n, wherever Mis a
Cohen-Macaulay module over the Cohen-Macaulay local ring (R, m) and dim M=n. This
corollary improves the main result [15, Theorem 3.10](see the explanation which is offered
before 3.9). As a second corollary, we obtain a characterization of Gorenstein local rings
which recovers [21, Theorem 2.6]. As a main result, it has been shown in [15, Theorem
3.10] that if Rand Mare Cohen-Macaulay with dim M=dand Gid RHd
m(M)<∞, then
Gid RHd
m(M) = dim R−d. In 3.12, we will use the canonical module of a module to improve
the above result without Cohen-Macaulay assumption on Rand M. This result provides
some characterizations of Gorenstein local rings.
2. local cohomology and injective dimension
The starting point of this section is the next proposition, which plays essential role in the
proof of Theorems 2.5 and 3.8.
Proposition 2.1. Let nbe a non-negative integer and let Nbe an a-torsion R-module.
Suppose that Hi
a(M) = 0 for all i6=n. Then
Ext i
R(N, Hn
a(M)) ∼
=Ext i+n
R(N, M )
for all i≥0.
Proof. First we notice that HomR(N, M ) = Hom R(N, Γa(M)). Hence, in view of [14,
Theorem 10.47], we have the Grothendieck third quadrant spectral sequence with
Ep,q
2= Ext p
R(N, Hq
a(M)) =⇒
pExt p+q
R(N, M ).
Now, since Hq
a(M) = 0 for all q6=n,Ep,q
2= 0 for all q6=n. Therefore, this spectral sequence
collapses in the column q=n; and hence one gets, for all i≥0, the isomorphism
Ext i
R(N, Hn
a(M)) ∼
=Ext i+n
R(N, M ),
as required. 
The following corollary, which is an immediate consequence of 2.1, determines the Bass
numbers µi(p,Hn
a(M)) := vdim k(p)Ext i
Rp(k(p),Hn
aRp(Mp)) of the local cohomology module
Hn
a(M).

INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS 3
Corollary 2.2. Let nand Mbe as in 2.1. Then, for all p∈V(a),µi(p,Hn
a(M)) =
µi+n(p, M )for each i≥0.
Definition 2.3. We say that a finitely generated R-module Mis relative Cohen Macaulay
with respect to aif there is precisely one non-vanishing local cohomology module of Mwith
respect to a. Clearly this is the case if and only if grade (a, M ) = cd (a, M ), where cd (a, M )
is the largest integer ifor which Hi
a(M)6= 0 and grade (a, M ) is the least integer isuch that
Hi
a(M)6= 0.
Observe that the above definition provides a generalization of the concept of Cohen-
Macaulay modules. Also, notice that the notion of relative Cohen-Macaulay modules is
connected with the notion of cohomologically complete intersection ideals which has been
studied in [7] and has led to some interesting results. Furthermore, such modules have been
studied in [8] over certain rings.
Remark 2.4. Let Mbe a relative Cohen-Macaulay module with respect to aand let
cd (a, M ) = n. Then, in view of [1, Theorems 6.1.4, 4.2.1, 4.3.2], it is easy to see that
Supp Hn
a(M) = Supp (M/aM) and ht Ma= grade (a, M ), where ht Ma= inf{dim RpMp|p∈
Supp (M/aM)}.
The following theorem, which is one of the main results of this section, provides a compar-
ison between the injective dimensions of a relative Cohen-Macaulay module and its non-zero
local cohomology module. Here we adopt the convention that the injective dimension of the
zero module is to be taken as −∞.
Theorem 2.5. Let (R, m)be local and let nbe a non-negative integer such that Hi
a(M) = 0
for all i6=n.
(i) If injdimM < ∞, then injdimHn
a(M)<∞.
(ii) The converse holds whenever Mis finitely generated.
Furthermore, if Mis non-zero finitely generated, then injdimHn
a(M) = injdimM−n.
Proof. (i). Let s:= injdimM < ∞. We may assume that Hn
a(M)6= 0; and hence s−n≥0.
Therefore, in view of 2.2, µi+(s−n)(p,Hn
a(M)) = 0 for all p∈Spec (R) and for all i > 0; so
that injdimHn
a(M)≤s−n.
(ii). Suppose that Mis finitely generated. We first notice that Hn
a(M) = 0 if and only if
M=aM; and this is the case if and only if M= 0. Therefore we may assume that Hn
a(M)6=
0. Suppose that t:= injdimHn
a(M)<∞. Then there exists a prime ideal qof Rsuch that
µt(q,Hn
a(M)) 6= 0. Hence, by 2.2, µt+n(q, M )6= 0. Next we show that µt+n+i(p, M ) = 0
for all p∈Spec (R) and for all i > 0. Assume the contrary. Then there exists a prime ideal
pof Rsuch that µt+n+j(p, M )6= 0 for some j > 0. Let r:= dim R/p. Then, by [11, §18,
Lemma 4], we have µt+n+j+r(m, M )6= 0. Hence, by 2.2, µj+t+r(m,Hn
a(M)) 6= 0 which is a
contradiction in view of the choice of t. Therefore, injdimM≤t+n. The final assertion is
a consequence of (i) and (ii). 

4 M.R. ZARGAR AND H. ZAKERI
Next, we provide an example to show that if Ris non-local, then Theorem 2.5(ii) is no
longer true. Also, in 3.11, we present two examples which show that 2.5(ii) and 2.5(i),
respectively, are no longer true without the finiteness and the relative Cohen-Macaulayness
assumptions on M.
Example 2.6. Suppose that Ris a non-local Artinian ring with injdimR=∞. Let
Max (R) = {m1, ..., mn}. Then, in view of [17, Exercise 8.49], we have R=Lm∈Max(R)Γm(R).
Now, since the injective dimension of Ris infinite, there exists a maximal ideal mtof R
such that the injective dimension of Γmt(R) is infinite. Set M:= ER(R/ms)LΓmt(R),
where ms∈Max (R) with ms6=mt. Then Mis a finitely generated R-module with infinite
injective dimension and Hi
ms(M) = 0 for all i6= 0; but Γms(M) is injective.
It is well-known that if (R, m, k) is a d-dimensional local ring, then Ris Gorenstein if and
only if Ris Cohen-Macaulay and Hd
m(R)∼
=ER(k). The following corollary, which recovers
this fact, is an immediate consequence of 2.5.
Corollary 2.7. Let (R, m)be local and let Rbe relative Cohen-Macaulay with respect to a.
Then Ris Gorenstein if and only if injdimHht Ra
a(R)is finite.
In particular, if x=x1, ..., xnis an R-sequence for some non-negative integer n, then R
is Gorenstein if and only if injdimHn
(x)(R)is finite.
The following proposition, which is needed in the proof of 3.8, provides an explicit minimal
injective resolution for the non-zero local cohomology module of a relative Cohen-Macaulay
module.
Proposition 2.8. Suppose that Mis relative Cohen-Macaulay with respect to aand that
n= cd (a, M ). Then
0−→ Hn
a(M)−→ M
p∈V(a)
µn(p, M )E(R/p)−→ M
p∈V(a)
µn+1(p, M )E(R/p)−→ · · ·
is a minimal injective resolution for Hn
a(M). Furthermore, Ass RHn
a(M) = {p∈V(a)|µn(p, M )6=
0}.
Proof. Let
0−→ Md−1
−→ E0(M)d0
−→ · · · −→ En−1(M)dn−1
−→ En(M)dn
−→ En+1(M)dn+1
−→ · · ·
be a minimal injective resolution for M. If there exists a prime ideal pin V(a) with
µn−1(p, M )6= 0, then depth RpMp≤n−1. On the other hand, since p∈Supp (M/aM), 2.4
implies that Hn
aRp(Mp)6= 0. Therefore n= grade Rp(aRp, Mp)≤depth RpMp≤n−1 which
is a contradiction. It follows that Γa(En−1(M)) = 0; and hence we obtain the minimal
injective resolution
0−→ Hn
a(M)−→ Γa(En(M)) −→ Γa(En+1(M)) −→ · · ·
for Hn
a(M). Now, we may use this resolution to complete the proof. 

INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS 5
The following elementary lemma, which is needed in the proof of the next theorem, can
be proved by using a minimal free resolution for Mand the concept of localization.
Lemma 2.9. Let (R, m, k)be local and let Mbe finitely generated. Then, for any prime
ideal pof R,vdim k(p)Tor Rp
i(k(p), Mp)≤vdim kTor R
i(k, M )for all i≥0.
The next theorem provides a comparison of Bass numbers of certain local cohomology
modules.
Theorem 2.10. Suppose that (R, m, k)is a local ring which is a homomorphic image of a
Gorenstein local ring and that Mis finitely generated. Let n, m be non-negative integers.
Then µm(p,Hn
p(M)) ≤µm(m,Hn+dim R/p
m(M)) for all p∈Spec (R).
Proof. Let (R′,m′) be a Gorenstein local ring of dimension n′for which there exists a sur-
jective ring homomorphism f:R′→R. Let pbe a prime ideal of Rand set p′=f−1(p).
Now R′
p′is a Gorenstein local ring and dim R′/p′= dim R/p. Since R′is Gorenstein, we
have dim R′
p′= dim R′−dim R′/p′. In view of [1, Exercise 11.3.1] there is, for each j∈Z,
an Rp-isomorphism Ext j
R′
p′(Mp, R′
p′)∼
=(Ext j
R′(M, R′))p. Also, by the Local Duality The-
orem [1, Theorem 11.2.6], we have Hn
pRp(Mp)∼
=Hom RpExt n′−n−t
R′
p′(Mp, R′
p′),ERp(k(p))
as Rp-modules, where t:= dim R/p, and Hn+t
m(M)∼
=Hom R(Ext n′−n−t
R′(M, R′),ER(k)). It
therefore follows that
Ext m
Rp(k(p),Hn
pRp(Mp)) ∼
=Ext m
Rp(k(p),Hom Rp(Ext n′−n−t
R′
p′(Mp, R′
p′),ERp(k(p))))
∼
=Hom Rp(Tor Rp
m(k(p),Ext n′−n−t
R′
p′(Mp, R′
p′)),ERp(k(p))).
and
Ext m
R(k, Hn+t
m(M)) ∼
=Ext m
R(k, Hom R(Ext n′−n−t
R′(M, R′),ER(k)))
∼
=Hom R(Tor R
m(k, Ext n′−n−t
R′(M, R′)),ER(k)).
Since by 2.9
vdim k(p)(Tor Rp
m(k(p),Ext n′−n−t
R′
p′(Mp, R′
p′))) ≤vdim k(Tor R
m(k, Ext n′−n−t
R′(M, R′))), one
may use the above isomorphisms to complete the proof. 
It is known as Bass’s conjecture that if a local ring admits a finitely generated module
of finite injective dimension, then it is a Cohen-Macaulay ring. For the proof of this fact
the reader is referred to [12] and [13]. In the next corollary we shall use this fact and the
concept of a generalized Cohen-Macaulay module. Recall that, over a local ring (R, m), a
finitely generated module of positive dimension is a generalized Cohen-Macaulay module if
Hi
m(M) is finitely generated for all 0 ≤i < dim M.
Corollary 2.11. Let the situation be as in 2.10. Then the following statements hold.
(i) injdimRpHn
pRp(Mp)≤injdimRHn+dim R/p
m(M)for all prime ideals pof Rand for
any n≥0.
(ii) If Mis generalized Cohen-Macaulay with dimension dsuch that Hd
m(M)is injective,
then Mpis Gorenstein, in the sense of [18], for all p∈Supp (M)\ {m}.

6 M.R. ZARGAR AND H. ZAKERI
Proof. (i) is clear by 2.10.
(ii) Let p∈Supp (M)\ {m}. By [1, Exercise 9.5.7], Mpis Cohen-Macaulay and dim Mp+
dim R/p= dim M. Hence, in view of (i) and 2.5, we have injdimMp= dim Mp. Therefore,
by [18, Theorem 3.11], [2, Theorem 3.1.17] and Bass’s conjecture, Mpis Gorenstein. 
3. local cohomology and gorenstein injective dimension
We first recall some definitions that we will use in this section.
Definition 3.1. Following [4], an R-module Mis said to be Gorenstein injective if there
exists a Hom (Inj , −) exact exact sequence
· · · → E1→E0→E0→E1→ · · ·
of injective R-modules such that M= Ker (E0→E1). We say that an exact sequence
0→M→G0→G1→G2→ · · ·
of R-modules and R-homomorphisms is a Gorenstein injective resolution for M, if each Gi
is Gorenstein injective. We say that Gid RM≤nif and only if Mhas a Gorenstein injective
resolution of length n. If there is no shorter resolution, we set Gid RM=n. Dually, an
R-module Mis said to be Gorenstein projective if there is a Hom (−, P roj ) exact exact
sequence
· · · → P1→P0→P0→P1→ · · ·
of projective R-modules such that M= Ker (P0→P1). Similarly, one can define the
Gorenstein pro jective dimension, Gpd RM, of M.
Definition 3.2. For a local ring Radmitting the dualizing complex DR, we denote by KM
the canonical module of an R-module M, which is defined to be
KM= Hd−n(RHom R(M, DR)),
where d= dim Rand n= dim M. Note that if Ris Cohen-Macaulay, then KRcoincides
with the classical definition of the canonical module of Rwhich is denoted by ωR.
Definition 3.3. Let Rbe a Cohen-Macaulay local ring of Krull dimension dwhich admits
a canonical module ωR. Following [4], let J0(R) be the class of R-modules Mwhich satisfies
the following conditions.
(i) Ext i
R(ωR, M ) = 0 , for all i > 0.
(ii) Tor R
i(ωR,Hom R(ωR, M )) = 0, for all i > 0.
(iii) The natural map ωR⊗RHom R(ωR, M )→Mis an isomorphism.
This class of R-modules is called the Bass class.
Definition 3.4. Following [19], let aand bbe ideals of R. We set
W(a,b) = {p∈Spec (R)|an⊆p+bfor some integer n > 0}.
For an R-module M, Γa,b(M) denotes a submodule of Mconsisting of all elements of M
with support in W(a,b), that is, Γa,b(M) = {x∈M|Supp (Rx)⊆W(a,b)}.

INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS 7
The following lemma has been proved in [10, Lemma 4.2] and is of assistance in the proof
of Proposition 3.6.
Lemma 3.5. Let (R, m, k)be local. Then
Ext i
R(ER(k), M )∼
=Ext i
R(E(k), M ⊗Rˆ
R)∼
=Ext i
ˆ
R(E ˆ
R(k), M ⊗Rˆ
R)
for all i≥0.
Let (R, m, k) be local and let Mbe a non-zero non-injective R-module of finite Goren-
stein injective dimension. It was shown in [5, Corollary 4.4] that if Ext i
R(E, M ) = 0
for all i≥0 and all indecomposable injective R-modules E 6= ER(k), then Gid RM=
sup{i|Ext i
R(ER(k), M )6= 0}.Our next proposition, which is concerned with this result,
indicates that Gorenstein injective dimension is a refinement of the injective dimension.
However we will use 3.6 and 3.7 to prove the main theorem 3.8.
Proposition 3.6. Let (R, m, k)be local and let Mbe non-zero with Gid RM < ∞. If either
Mis finitely generated or Artinian, then
Gid RM= sup{i|Ext i
R(ER(k), M )6= 0}.
Proof. First assume that Mis finitely generated. Then, by [3, Theorem 3.24], Gid RM=
Gid b
Rc
M. Now, since b
Ris complete and c
Mis finitely generated as an b
R-module, the proof
of [6, Proposition 2.2] in conjunction with [5, Corollary 4.4 ] implies that
Gid b
Rc
M= sup{i|Ext i
b
R(E b
R(b
R/ b
m),c
M)6= 0}.
Therefore we can use 3.5 to complete the proof.
Next, we consider the case where Mis Artinian. By [15, Lemma 3.6 ] and [1, Exercise
8.2.4], Gid R(M) = Gid b
R(M) and, by [3, Theorem 4.25], Gfd b
R(Hom b
R(M, Eb
R(b
R/ b
m))) =
Gid b
R(M), where, for an R-module X, Gfd R(X), denotes the Gorenstein flat dimension of
X. Therefore, since Hom b
R(M, Eb
R(b
R/ b
m)) is finitely generated as an b
R-module, in view of
[3, Theorem 4.24] and [3. Theorem 1.10] we have the first equality in the next display
Gfd b
R(Hom b
R(M, Eb
R(b
R/ b
m))) = sup{i|Ext i
b
R(Hom b
R(M, Eb
R(b
R/ b
m)),b
R)6= 0}
= sup{i|Ext i
R(ER(k), M )6= 0}.
The last equality follows from [10, Theorem 4.3], because ER(k) and Mare Artinian. 
Lemma 3.7. Let (R, m)be a Cohen-Macaulay local ring and let Mbe finitely generated.
Suppose that x∈mis both R-regular and M-regular. Then the following statements are
equivalent.
(i) Gid RM < ∞.
(ii) Gid R/xRM/xM < ∞.
Furthermore, Gid R/xR M/xM = Gid RM−1.
Proof. In view of [3, Theorem 3.24], we can assume that Ris complete; and hence it admits
a canonical module ωR.

8 M.R. ZARGAR AND H. ZAKERI
(i)⇒(ii). It follows from [3, Prop osition 3.9] that Gid RM/xM < ∞. Therefore we
can use [4, Proposition 10.4.22], [11, p.140,lemma 2] and [2, Theorem 3.3.5], to see that
Gid R/xRM/xM < ∞.
(ii)⇒(i). By [4, Proposition 10.4.23], M/xM ∈ J0(R/xR). Since ωR/xR ∼
=ωR/xωR, in
view of [11, p.140, lemma 2] we have Ext i
R(ωR, M/xM ) = Tor R
i(ωR,Hom R(ωR, M/xM )) =
0 for all i > 0 and ωR/xωR⊗R/xR Hom R(ωR, M /xM)∼
=M/xM . Now, using the exact
sequence Ext i
R(ωR, M )x
−→ Ext i
R(ωR, M )−→ Ext i
R(ωR, M/xM ) and Nakayama’s lemma,
we deduce that Ext i
R(ωR, M ) = 0 for all i > 0. Thus we have the exact sequence
(3.1) 0 −→ Hom R(ωR, M )x
−→ Hom R(ωR, M )−→ Hom R(ωR, M/xM )−→ 0.
Now, we may use (3.1) and similar arguments as above to see that Tor i
R(ωR,Hom R(ωR, M )) =
0 for all i > 0. Also, in view of [2, Lemma 3.3.2], we can see that ωR⊗RHom R(ωR, M )∼
=M.
Therefore by [4, Proposition 10.4.23], Gid RM < ∞. The final assertion is an immediate
consequence of [3, Theorem 3.24]. 
Theorem 3.8, which is a Gorenstein injective version of 2.5, is one of the main results of
this section. As we will see, this theorem has consequences which recover some interesting
results that have currently been appeared in the literature. Here we adopt the convention
that the Gorenstein injective dimension of the zero module is to be taken as −∞.
Theorem 3.8. Suppose that the local ring (R, m)has a dualizing complex and let nbe a
non-negative integer such that Hi
a(M) = 0 for all i6=n.
(i) If Gid RM < ∞, then Gid RHn
a(M)<∞.
(ii) The converse holds whenever Rand Mare Cohen-Macaulay.
Furthermore, if Mis non-zero finitely generated with finite Gorenstein injective dimension,
then Gid RHn
a(M) = Gid RM−n.
Proof. (i) Notice that if Hn
a(M) = 0, then there is nothing to prove. So, we may assume
that Hn
a(M)6= 0. Hence, by [20, Lemma 1.1], we have n≤d, where d= Gid RM. Let
0−→ Md−1
−→ G0d0
−→ G1d1
−→ · · · −→ Gn−1dn−1
−→ Gndn
−→ Gn+1 dn+1
−→ · · · −→ Gd−1dd−1
−→ Gd−→ 0
be a Gorenstein injective resolution for M. By applying the functor Γa(−) on this exact
sequence, we obtain the complex
0−→ Γa(M)Γa(d−1)
−→ Γa(G0)Γa(d0)
−→ Γa(G1)Γa(d1)
−→ · · · −→ Γa(Gn−1)
Γa(dn−1)
−→ Γa(Gn)Γa(dn)
−→ Γa(Gn+1)Γa(dn+1 )
−→ · · · −→ Γa(Gd−1)Γa(dd−1)
−→ Γa(Gd)−→ 0
in which, by [15, Theorem 3.2], Γa(Gi) is Gorenstein injective for all 0 ≤i≤d. If n= 0
the result is clear. So suppose that n > 0. Now, since by [20, Lemma 1.1] each Giis
Γa-acyclic for all i, we may use [1, Exercise 4.1.2 ] in conjunction with our assumption on
local cohomology modules of Mto obtain the following two exact sequences
0−→ Γa(M)Γa(d−1)
−→ Γa(G0)Γa(d0)
−→ · · · −→ Γa(Gn−1)Γa(dn−1)
−→ Γa(Gn)−→ Γa(Gn)
Im Γa(dn−1)−→ 0
and

INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS 9
0−→ Im Γa(dn) = Ker Γa(dn+1)֒→Γa(Gn+1 )−→ · · · −→ Γa(Gd−1)−→ Γa(Gd)−→ 0.
But, by assumption, Γa(M) = 0. Therefore, by using the first above exact sequence and
[4, Theorem 10.1.4], we see that Γa(Gn)
Im Γa(dn−1)is Gorenstein injective. Notice that Hn
a(M) =
ker Γa(dn)
Im Γa(dn−1). Therefore, patching the second above long exact sequence together with the
exact sequence
0−→ Hn
a(M)−→ Γa(Gn)
Im Γa(dn−1)−→ Γa(Gn)
ker Γa(dn)−→ 0,
yields the following long exact sequence
0−→ Hn
a(M)−→ Γa(Gn)
Im Γa(dn−1)−→ Γa(Gn+1)−→ · · · −→ Γa(Gd−1)−→ Γa(Gd)−→ 0.
Hence, Gid RHn
a(M)≤Gid RM−n.
(ii) Suppose that Rand Mare Cohen-Macaulay. Since H0
m(E(R/m)) = E(R/m) and,
for any non-maximal prime ideal pof R, H0
m(E(R/p)) = 0, we may apply 2.8 to see that
Hi
m(Hn
a(M)) = Hn+i
m(M) for all i≥0. Therefore, we can use the Cohen-Macaulayness of M
to deduce that
Hi
m(Hn
a(M)) = 


0 if i6= dim M/aM
Hd
m(M) if i= dim M/aM,
where d= dim M. Hence, by using part(i) for Hn
a(M), we have Gid RHd
m(M)<∞. Now,
we proceed by induction on dto show that Gid RMis finite. The case d= 0 is clear. Let
d > 0 and assume that the result has been proved for d−1. Suppose that x∈mis both
R-regular and M-regular. Then one can use the induced exact sequence
0−→ Hd−1
m(M/xM )−→ Hd
m(M)−→ Hd
m(M)−→ 0
and [3, Proposition 3.9] to see that Gid R(Hd−1
m(M/xM )) is finite. Hence, by inductive
hypothesis, Gid RM/xM is finite. Therefore, since, in view of [9, Corollary 6.2], Radmits a
canonical module, one can use the same argument as in the proof of 3.7(i)⇒(ii) to deduce
that Gid R/xRM/xM < ∞. Therefore Gid RMis finite by 3.7. Now the result follows by
induction.
For the final assertion, let Mbe non-zero finitely generated with Gid RM=s < ∞.
Then, by part(i), we have Gid RHn
a(M)≤s−n. If Gid RHn
a(M)< s −n, Then, in view
of [3, Theorem 3.6], we deduce that Ext s−n
R(E(k),Hn
a(M)) = 0. Hence, by Proposition 2.1,
Ext s
R(E(k), M ) = 0 which is a contradiction by 3.6. Therefore, Gid RHn
a(M) = Gid RM−
n. 
Let (R, m) be a local ring. As a main theorem, it was proved in [15, Theorem 3.10]
that if Rand Mare Cohen-Macaulay with dim M=nand Gid RHn
m(M)<∞, then
Gid RHn
m(M) = dim R−n. Notice that if Gid RHn
m(M)<∞, then, in view of [15, Lemma
3.6], 3.8(ii) and [3, Theorem 3.24], we have depth R= Gid RM. Therefore, the next corollary,
which is established without the assumption that Gid RHn
m(M)<∞, recovers [15, Theorem
3.10]. Another improvement of the above result will be given in 3.12.

10 M.R. ZARGAR AND H. ZAKERI
Corollary 3.9. Let (R, m)be a Cohen-Macaulay local ring and let Mbe Cohen-Macaulay
of dimension n. Then Gid RHn
m(M) = Gid RM−n.
Proof. First we notice that M⊗Rb
Ris a Cohen-Macaulay b
R-module of dimension n. By
using [15, Lemma 3.6] and [1, Exercise 8.2.4], we have GidRHn
m(M) = Gid b
RHn
b
m(M⊗Rb
R).
Also, in view of [3, Theorem 3.24], Gid RMand Gid b
Rc
Mare simultaneously finite. Therefore
we may assume that Ris complete; and hence it has a dualizing complex. Now, one can use
3.8 to obtain the assertion. 
In [21, Theorem 2.6] a characterization of a complete Gorenstein local ring R, in terms of
Gorenstein injectivity of the top local cohomology module of R, is given. The next corollary
together with 2.7 recover that characterization.
Corollary 3.10. Let (R, m)be a Cohen-Macaulay local ring which has a dualizing complex.
Then the following conditions are equivalent.
(i) Ris Gorenstein.
(ii) Gid RHn
a(R)<∞for any ideal aof Rsuch that Ris relative Cohen-Macaulay with
respect to aand that ht Ra=n.
(iii) Gid RHn
a(R)<∞for some ideal aof Rsuch that Ris relative Cohen-Macaulay with
respect to aand that ht Ra=n.
Proof. The implication (i)⇒(ii) follows from 2.7 and the implication (ii)⇒(iii) is clear. The
implication (iii)⇒(i) follows from 3.8(ii) and [3, Proposition 3.11]. 
Concerning the above corollary, we notice that if Hn
a(R) is Artinian, then it is not needed
to impose the hypothesis that Rhas a dualizing complex. Therefore [21, Theorem 2.6]
follows from 3.10 without the completeness assumption on R.
Next, as promised before, we provide examples to show that if Mis not finitely generated
or Mis not relative Cohen-Macaulay, then 2.5(ii) and 2.5(i), respectively, are no longer true.
Examples 3.11. (i). Let (R, m) be a Gorenstein lo cal ring with dim R≥2 such that
Rpis not regular for some non-maximal prime ideal pof R(for example, one can take
R=K[[X,Y,Z ]]
(X2)and p= (x, y)R, where Kis a field). Then one can use [3, Theorem 3.14]
to see that there exists a non-zero Gorenstein injective Rp-module Mpwhich is neither
injective nor finitely generated. Hence, by [4, Proposition 10.1.2 ], injdimRpMp=∞; so
that injdimRMp=∞. Now, we notice that, for all x∈m−p, injdimRMx=∞because
(Mx)pRx∼
=Mp. It is easy to check that Hi
m(Mx) = 0 for all iand that Mxis not finitely
generated as an R-module. Set N=Mx⊕ER(k). Then injdimN=∞, but Γm(N) is
injective. This example shows that, in 2.5(ii), the finiteness assumption on Mis required.
(ii). Let R=k[[x, y]]/(xy), where kis a field. Then Ris a 1-dimensional complete
Gorenstein local ring. Let mbe the maximal ideal of Rand let J= (y)R. In view of [1,
Theorem 8.2.1] and [21, Corollary 2.10], H1
J(R) is a non-zero Gorenstein injective R-module.
Note that ΓJ(R)6= 0. Now, we show that H1
J(R) is not injective. If H1
J(R) were injective,
then Hom R(H1
J(R),ER(R/m)) = Rnfor some positive integer n. Therefore, by using [19,

INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS 11
Theorem 5.11] and [21, Lemma 3.1], we get an R-isomorphism
ψ:Rn= Hom R(H1
J(R),ER(R/m)) →Γm,J (R) = J.
Now, ψ(xRn) = xJ = 0 which is a contradiction. Hence, by [4, Proposition 10.1.2], we have
injdimH1
J(R) = ∞. Therefore, by using the exact sequences
0−→ ΓJ(R)−→ R−→ R/ΓJ(R)−→ 0
and
0−→ R/ΓJ(R)−→ Ry+(xy)−→ H1
J(R)−→ 0,
we achieve injdimΓJ(R) = ∞.
As a mentioned just above 3.9, the next theorem is an improvement of [15, Theorem
3.10]. Indeed, we will use the canonical module of a module to prove the above result
without assuming that Rand Mare Cohen-Macaulay. Notice that if Mis Cohen-Macaulay,
then KMis Cohen-Macaulay. But the converse does not hold in general; see for example
[16, Lemma 1.9] and [16, Theorem 1.14].
Theorem 3.12. Assume that (R, m)is local, and Mis non-zero finitely generated of di-
mension d. Then the following statements hold.
(i) Gid RHd
a(M) = Gpd b
RΓm
b
R,a
b
R(Kc
M).
(ii) If Kc
Mis Cohen-Macaulay and Gid RHd
m(M)<∞, then Gid RHd
m(M) = depth R−d.
Proof. (i) By [1, Theorem 7.1.6], Hd
a(M) is Artinian. Therefore, by use of [1, Theorem
4.3.2] and [19, Theorem 5.11], we have Hd
a(M) = 0 if and only if Γm
b
R,a
b
R(Kc
M) = 0. Hence,
we may assume that Hd
a(M)6= 0. Now, by [15, Lemma 3.6], Gid RHd
a(M) = Gid b
RHd
a(M)
and, by [3, Theorem 4.25], Gid b
RHd
a(M) = Gfd b
RHom b
R(Hd
a(M),ER(k)). Therefore, since
Hom b
R(Hd
a(M),ER(k)) is finitely generated as an b
R-module, one can use [3, Theorem 4.24]
and [19, Theorem 5.11] to establish the result.
(ii) First notice that Γm
b
R,m
b
R(Kc
M) = Kc
M.Hence, by part(i), Gid RHd
m(M) = Gpd b
R(Kc
M).
Therefore, in view of [3, Proposition 2.16] and [3, Theorem 1.25], Gid RHd
m(M) = depth b
R−
depth Kc
M. Now, one can use [16, Lemma 1.9(c)] to complete the proof. 
The following corollary is a generalization of the main result [21, Theorem 2.6].
Corollary 3.13. Assume that (R, m)is local with dim R=dand that Kb
Ris Cohen-
Macaulay. Then the following statements are equivalent.
(i) Ris Gorenstein.
(ii) injdimRHd
m(R)<∞.
(iii) Gid RHd
m(R)<∞.
Proof. The implication (i)⇒(ii) follows from 2.5 while the implication (ii)⇒(iii) is clear.
(iii)⇒(i). By 3.12(ii), we have Gid RHd
m(R) = depth R−dim R; and hence Ris Cohen-
Macaulay. Now one can use 3.9 to obtain the assertion. 

12 M.R. ZARGAR AND H. ZAKERI
Corollary 3.14. Let (R, m)be local with dim R=d≤2. Then the following statements
are equivalent.
(i) Ris Gorenstein.
(ii) Gid RHd
m(R)<∞.
(iii) Hd
a(M)is Gorenstein injective for all finitely generated R–modules Mand for all
ideals aof R.
Proof. Let Mbe a non-zero finitely generated R-module. Then, by [1, Theorem 7.1.6],
Hd
a(M) and Hd
m(R) are Artinian. Therefore, in view of [15, Lemma 3.5], we may assume
that Ris complete. Since, by [16, Lemma 1.9], KRis Cohen-Macaulay, (i)⇔(ii) follows
immediately from 3.13. The implication (iii)⇒(i) is clear and the implication (i)⇒(iii)
follows from [21, Corollary 2.10].

Acknowledgements.The authors would like to thank Alberto Fernandez Boix and the
referee for careful reading of manuscript and helpful comments.
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M.R. Zargar and H. Zakeri, Faculty of mathematical sciences and computer, Kharazmi Uni-
versity, 599 Taleghani Avenue, Tehran 15618, Iran
E-mail address:zargar9077@gmail.com
E-mail address:zakeri@tmu.ac.ir
M.R. Zargar, School of Mathematics, Institute for Research in Fundamental Sciences (IPM),
P.O. Box 19395–5746, Tehran, Iran.