arXiv:1208.5934v1 [] 29 Aug 2012

ON THE LOCAL COHOMOLOGY MODULES DEFINED BY A
PAIR OF IDEALS AND SERRE SUBCATEGORIES
KH. AHMADI-AMOLI AND M.Y. SADEGHI

Abstract. This paper is concerned about the relation between local cohomology modules defined by a pair of ideals and Serre classes of R-modules, as
a generalization of results of J. Azami, R. Naghipour and B. Vakili (2009)
and M. Asgharzadeh and M.Tousi (2010). Let R be a commutative Noetherian ring, I , J be two ideals of R and M be an R-module. Let a ∈ W̃ (I, J)
i (M )) ∈ S
and t ∈ N0 be such that ExttR (R/a, M ) ∈ S and ExtjR (R/a, HI,J
t (M ) such that
for all i < t and all j ≥ 0. Then for any submodule N of HI,J
t (M )/N ) ∈ S.
Ext1R (R/a, N ) ∈ S, we obtain HomR (R/a, HI,J

1. Introduction
Throughout this paper, R is denoted a commutative Noetherian ring, I , J are
denoted two ideals of R, and M is denoted an arbitrary R-module. By N0 , we shall
mean the set of non-negative integers. For basic results, notations and terminologies
not given in this paper, the reader is referred to [7] and [22], if necessary.
As a generalization of the usual local cohomology modules, Takahashi, Yoshino and
Yoshizawa [22], introduce the local cohomology modules with respect to a pair of
ideals (I, J). To be more precise, let W (I, J) = { p ∈ Spec(R) | I n ⊆ p + J for
some positive integer n} and W̃ (I, J) denote the set of ideals a of R such that
I n ⊆ a + J for some integer n. In general, W (I, J) is closed under specialization,
but not necessarily a closed subset of Spec(R). For an R-module M , we consider
the (I, J)-torsion submodule ΓI,J (M ) of M which consists of all elements x of M
with Supp(Rx) in W (I, J). Furthermore, for an integer i, the local cohomology
i
functor HI,J
with respect to (I, J) is defined to be the i-th right derived functor of
i
ΓI,J . Also HI,J (M ) is called the i-th local cohomology module of M with respect
i
to (I, J). If J = 0, then HI,J
coincides with the ordinary local cohomology functor
HIi with the support in the closed subset V (I).
Recently, some authors approached the study of properties of these extended
modules, see for example [9], [10], [19] and [23].
It is well known that an important problem in commutative algebra is to determine when the R-module HomR (R/I, HIi (M )) is finite. Grothendieck in [14]
conjectured the following:
If R is a Noetherian ring, then for any ideal I of R and any finite R-module M ,
the modules HomR (R/I, HIi (M )) are finite for all i ≥ 0.
2010 Mathematics Subject Classification. Primary 13D45, 13E05.
Key words and phrases. Local cohomology modules defined by a pair of ideals, local cohomology, Goldie dimension, (I, J)-minimax modules, Serre subcategory, (S, I, J)-cominimax modules,
associated primes.
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KH. AHMADI-AMOLI AND M.Y. SADEGHI

In [15], Hartshorne gave a counterexample to Grothendieck, s conjecture and
he defined the concept of I-cofinite modules to generalize the conjecture. In [6],
Brodmann and Lashgari showed that if, for a finite R-module M and an integer
t, the local cohomology modules H 0 (M ) , H 1 (M ) , · · · , H t−1 (M ) are finite, then
R-module HomR (R/I, HIt (M )) is finite and so Ass(HIt (M )/N ) is a finite set for any
finite submodule N of HIt (M ). A refinement of this result for I-minimax R-modules
is as follows, see [4].
Theorem 1.1. Let M be an I-minimax R-module and t be a non-negative integer
such that HIi (M ) is I-minimax for all i < t. Then for any I-minimax submodule
N of HIt (M ), the R-module HomR (R/I, HIt (M )) is I-minimax.
Also authors in [1] and [3] studied local cohomology modules by means of Serre
subcategories. As a consequence, for an arbitrary Serre subcategory S, authors in
[3] showed the following result.
Theorem 1.2. Let s ∈ N0 be such that ExtsR (R/I, M ) ∈ S and ExtjR (R/I, HIi (M ))
∈ S for all i < s and all j ≥ 0. Let N be a submodule of HIs (M ) such that
Ext1R (R/I, N ) ∈ S. Then HomR (R/I, HIs (M )/N ) ∈ S.
The aim of the present paper is to generalize the concept of I-cominimax Rmodule, introduced in [4], to an arbitrary Serre subcategory S, to verify situations
i
in which the R-module HomR (R/I, HI,J
(M )) belongs to S. To approach it, we use
the methods of [3] and [4]. Our paper consists of four sections as follows.
In Section 2, by using the concept of (I, J)-relative Goldie dimension, we introduce the (I, J)-minimax R-modules and we study some properties of them, see
Proposition 2.7.
In Section 3, for an arbitrary Serre subcategory S, we defined (S, I, J)-cominimax
R-modules. This concept of R-modules can be considered as a generalization of Icofinite R-modules [14], I-cominimax R-modules [4], and (I, J)-cofinite R-modules
[23]. Also, as a main result of our paper, we prove the following. (See Theorem
3.4).
Theorem 1.3. Let a ∈ W̃ (I, J). Let t ∈ N0 be such that ExttR (R/a, M ) ∈ S and
i
ExtjR (R/a, HI,J
(M )) ∈ S for all i < t and all j ≥ 0. Then for any submodule N of
t
t
HI,J (M ) such that Ext1R (R/a, N ) ∈ S, we have HomR (R/a, HI,J
(M )/N ) ∈ S.
One can see, by replacing various Serre classes with S and using Theorem 1.3,
the main results of [2, Theorem 1.2] , [3, Theorem 2.2] , [4, Theorem 4.2] , [5,
Lemma 2.2] , [6] , [12, Corollary 2.7] , [16] , [17, Corollary 2.3], and [23, Theorem
3.2] are obtained.(See Theorem 3.14 and Proposition 3.15).
At last, in Section 4, as an application of results of the previous sections, we give
the following consequence about finiteness of associated primes of local cohomology
modules.(See Proposition 4.1 and Corollary 4.2)
i
Proposition 1.4. Let t ∈ N0 be such that ExttR (R/I, M ) ∈ SI,J and HI,J
(M ) ∈
t
C(SI,J , I, J) for all i < t. Let N be a submodule of HI,J
(M ) such that Ext1R (R/I, N )
t
t
(M )/N < ∞ and so
(M )/N ) ⊆ V (I) then GdimHI,J
belongs to SI,J . If Supp(HI,J
t
t
(M ).
HI,J (M )/N has finitely many associated primes; in particular, for N = JHI,J

ON THE LOCAL COHOMOLOGY MODULES DEFINED BY A PAIR OF IDEALS

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2. Serre Classes And (I, J)-Minimax Modules
Recall that for an R-module H, the Goldie dimension of H is defined as the
cardinal of the set of indecomposable submodules of E(H), which appear in a
decomposition of E(H) in to direct sum of indecomposable submodules. Therefore,
H is said to have finite Goldie dimension if H does not contain an infinite direct sum
of non-zero submodules, or equivalently the injective hull E(H) of H decomposes as
a finite direct sum of indecomposable (injective) submodules. We shall use Gdim H
to denote the Goldie dimension of H. For a prime ideal p, let µ0 (p, H) denotes the
0-th Bass number of H with respect to prime ideal p. It is known that µ0 (p, H) > 0
iff pP∈ Ass(H). It is clear by
of the Goldie dimension that Gdim H
P the definition
0
0
µ
(p,
H). Also, the (I, J)-relative Goldie
µ
(p,
H)
=
=
p∈Spec(R)
p∈Ass(H)
P
dimension of H is defined as GdimI,J H := p∈W (I,J) µ0 (p, H).(See [19, Definition
3.1]). If J = 0, then GdimI,J H = GdimI H (see[11, Definition 2.5]), moreover if I
= 0, we obtain GdimI,J H = Gdim H. It is known that when R is a Noetherian
ring, an R-module H is minimax if and only if any homomorphic image of H has
finite Goldie dimension (see [13], [25], or [26]). This motivates the definition of
(I, J)-minimax modules.
Definition 2.1. An R-module M is said to be minimax with respect to (I, J) or
(I, J)-minimax if the (I, J)-relative Goldie dimension of any quotient module of M
is finite.
Remarks 2.2. By Definition 2.1, it is clear that
(i) GdimI M ≤ GdimI,J M ≤ Gdim M.
This inequalities maybe strict (see [19, Definition 3.1]).
(ii) For Noetherian ring R, an R-module M is minimax iff for any R-module N
of M , Gdim M/N < ∞. Therefore, by (i), in Noetherian case, the class of
I-minimax R-modules contains the class of (I, J)-minimax R-modules and
it contains the class of minimax R-modules.
Example and Remarks 2.3. It is easy to see that
(i) Every quotient of finite modules, Artinian modules, Matlis refelexive modules and linearly compact modules have finite (I, J)-relative Goldie dimension, and so all of them are (I, J)-minimax modules.
(ii) If I = 0, then W (I, J) = Spec(R) = V (I) and so an R-module M is minimax
iff is (I, J)-minimax iff is I-minimax.
(iii) If J = 0, then W (I, J) = V (I), so that an R-module M is (I, J)-minimax
iff is I-minimax.
(iv) let M be an I-torsion module. Then, by [11, Lemma 2.6] and [19, Lemma
3.3], M is minimax iff is (I, J)-minimax iff is I-minimax.
(v) If M is (I, J)-torsion module, then M is minimax iff is (I, J)-minimax. (By
the definition and [19, Lemma 3.3]). Specially, when (0) ∈ W̃(I, J) or (0)
∈ W(I, J).
(vi) By [22, Corollary (1.8)(2)], the class of (I, J)-torsion is a Serre subcategory of R-modules. Therefore, by part(v), in this category, the concept of
minimax modules coincides with the concept of (I, J)-minimax modules;
i
(M ) (i ≥ 0).
specially, for the (I, J)-torsion module of HI,J
(vii) If M in(M ) ⊆ W (I, J) and GdimI,J M < ∞, then by [19, Lemma 3.3] and
definition, Gdim M < ∞, and so |Ass(M )| < ∞.

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KH. AHMADI-AMOLI AND M.Y. SADEGHI

The following proposition shows that the class of (I, J)-minimax R-modules is
a Serre subcategory.
Proposition 2.4. Let 0 → M ′ → M → M ′′ → 0 be an exact sequence of Rmodules. Then M is (I, J)-minimax if and only if M′ and M′′ are both (I, J)minimax.
Proof. One can obtain the result, by replacing W (I, J) with V (I) and a modification of the proof of proposition 2.3 of [4].

Remark 2.5. Recall that a class S of R-modules is a ′′ Serre subcategory′′ or ′′ Serre
class′′ of the category of R-modules, when it is closed under taking submodules,
quotients and extensions. For example, the following class of R-modules are Serre
subcategory.
(a) The class of Zero modules.
(b) The class of Noeterian modules.
(c) The class of Artinian modules.
(d) The class of R-modules with finite support.
(e) The class of all R-modules M with dimR M ≤ n, where n is a non-negative
integer.
(f) The class of minimax modules and the class of I-cofinite minimax Rmodules. (see [18, Corollary 4.4] )
(g) The class of I-minimax R-modules. (see [4, Proposition 2.3 ] )
(h) The class of I-torsion R-modules and the class of (I, J)-torsion R-modules.
(see [22, Corollary 1.8])
(i) The class of (I, J)-minimax R-modules. (Proposition 2.4)
Notations 2.6. In this paper, the following notations are used for the following
Serre subcategories:
′′
′′
′′
′′

S ′′ for an arbitrary Serre class of R-modules;
S 0 ′′ for the class of minimax R-modules;
S I ′′ for the class of I-minimax R-modules;
S I,J ′′ for the class of (I, J)-miniax R-modules.

Using the above notations and Remark 2.2, we have S 0 ⊆ S I,J ⊆ S I .
Now, we exhibit some of the properties of SI,J .
Proposition 2.7. Let I,J,I ′ ,J ′ be ideals of R and M be an R-module. Then
(i) SI,J = S√I,J = S√I,√J = SI,√J .
√
(ii) If I n ⊆ J, for some n ∈ N (or equally, if R/J is an I-torsion R-module),
then S0 √
= SI,J .
(iii) If I n ⊆ √I ′ , for some n ∈ N, then SI,J ⊆ SI ′ ,J .
(iv) If J n ⊆ √ J ′ , for some n ∈ N, then SI,J ′ ⊆ SI,J .
(v) If I n ⊆ I ′ , for some n ∈ N and M is (I ′ , J)-torsion, then M ∈ SI,J iff
M ∈ S0 √iff M ∈ SI ′ ,J .
(vi) If J n ⊆ J ′ , for some n ∈ N and M is (I, J)-torsion, then M ∈ SI,J iff
M ∈ S0 iff M ∈ SI,J ′ .
Proof. All these statements follow easily from [22, Proposition 1.4 and 1.6] and
Remark 2.3. As an illustration, we just prove statement (iii).

ON THE LOCAL COHOMOLOGY MODULES DEFINED BY A PAIR OF IDEALS

5

√
√
Let H ∈ SI,J . Since I n ⊆ I ′ , we have W ( I ′ , J) ⊆ W (I, J), by [22, Proposition
1.6]. Now, since H is (I, J)-minimax, the assertion follows from definition.

Lemma 2.8. (i) If N ∈ S and M is a finite R-module, then for any submodi
ule H of ExtiR (M, N ) and T of T orR
(M, N ), we have ExtiR (M, N )/H ∈ S and
i
T orR (M, N )/T ∈ S, for all i ≥ 0 .
(ii) For all i ≥ 0, we have ExtiR (R/I, M ) ∈ SI,J iff ExtiR (R/I, M ) ∈ S0 iff
ExtiR (R/I, M ) ∈ SI .
Proof. (i) The result follows from [3, Lemma 2.1].
(ii) Since, for all i, ExtiR (R/I, M ) and T oriR (R/I, M ) are (I, J)-torsion R-modules,
the assertion holds by Remark 2.3 (iv).

The following proposition can be thought of as a generalization of Proposition
2.6 of [4], in case of J = 0 and S = SI .
i
Proposition 2.9. Let M in(M ) ⊆ W (I, J). If M ∈ S, then HI,J
(M ) ∈ S for all
i ≥ 0.

Proof. By hypothesis and [21, Corollary 1.7], M is (I, J)-torsion R-module and so
0
i
HI,J
(M ) = ΓI,J (M ) = M . Therefore, HI,J
(M ) = 0 for all i ≥ 1, by [22, Corollary
1.13]. Thus the assertion holds.

Now, we are in position to prove the main results of this section, which is a
generalization of Theorem 2.7 of [4], for S = SI . Some applications of these results
are appeared in Section 3.
Theorem 2.10. Let M be a finite R-module and N an arbitrary R-module. Let
t ∈ N0 . Then the following conditions are equivalent:
(i) ExtiR (M, N ) ∈ S for all i ≤ t .
(ii) For any finite R-module H with Supp(H) ⊆ Supp(M ), ExtiR (H, N ) ∈ S
for all i ≤ t.
Proof. (i)⇒(ii) Since Supp(H) ⊆ Supp(M ), according to Gruson’s Theorem [24,
Theorem 4.1], there exists a chain of submodules of M ,
0 = H0 ⊂ H1 ⊂ · · · ⊂ Hk = H,
such that the factors Hj /Hj−1 are homomorphic images of a direct sum of finitely
many of M . Now, consider the exact sequences
0 → K → M n → H1 → 0
0 → H1 → H2 → H2 /H1 → 0
..
.
0 → Hk−1 → Hk → Hk /Hk−1 → 0,

for some positive integer n. Considering the long exact sequence
i
i
i
· · · → Exti−1
R (Hj−1 , N ) → ExtR (Hj /Hj−1 , N ) → ExtR (Hj , N ) → ExtR (Hj−1 , N ) → . . .

and an easy induction on k, the assertion follows. So, it suffices to prove the case
k = 1. From the exact sequence
0 → K → M n → H → 0,

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KH. AHMADI-AMOLI AND M.Y. SADEGHI

where n ∈ N and K is a finite R-module, and the induced long exact sequence, by
using induction on i, we show that ExtiR (H, N ) ∈ S for all i. For i = 0, we have
the exact sequence
0 → HomR (H, N ) → HomR (M n , N ) → HomR (K, N ).
n
L
HomR (M, N ), hence in view of the assumption and
Since HomR (M n , N ) ∼
=
Lemma 2.8, Ext0R (H, N ) ∈ S. Now, let i > 0. We have, for any R-module H with
Supp(H) ⊆ Supp(M ), the R-module Exti−1
R (H, N ) ∈ S, in particular for K. Now,
from the long exact sequence
i
i
n
· · · → Exti−1
R (K, N ) → ExtR (H, N ) → ExtR (M , N ) → . . .

and by Lemma 2.8, we can conclude that ExtiR (H, N ) ∈ S.
(ii)⇒(i) It is trivial.



Corollary 2.11. Let r ∈ N0 . Then, for any R-module M , the following conditions
are equivalent:
(i) ExtiR (R/I, M ) ∈ S for all i ≤ r.
(ii) For any ideal a of R with a ⊇ I, ExtiR (R/a, M ) ∈ S for all i ≤ r.
(iii) For any finite R-module N with Supp(N ) ⊆ V (I), ExtiR (N, M ) ∈ S for all
i ≤ r.
(iv) For any p ∈ M in(I), ExtiR (R/p, M ) ∈ S for all i ≤ r.
Proof. In view of Theorem 2.10, it is enough to show that (iv) implies (i). To
do this, let p1 ,p2 ,· · · ,pn be the minimal elements of V (I). Then, by assumption,
the R-modules ExtiR (R/pj, M ) ∈ S for all j = 1, 2, · · · , n. Hence, by Lemma
2.8, ExtiR (⊕nj=i R/pj, M ) ∼
= ⊕nj=1 ExtiR (R/pj, M ) ∈ S. Since Supp(⊕nj=1 R/pj) =
Supp(R/I), it follows from Theorem 2.10 that ExtiR (R/I, M ) ∈ S, as required. 
i
3. (S, I, J)-Cominimax Modules And HI,J
(M )

Recall that M is said to be (I, J)-cofinite if M has support in W (I, J) and
ExtiR (R/I, M ) is a finite R-module for each i ≥ 0 (see [23, Definition 2.1]). In
fact this definition is a generalization of I-cofinite modules, which is introduced
by Hartshorne in [15]. Considering an arbitrary Serre subcategory of R-modules
instead of finitely generated one, we can give a generalization of (I, J)-cofinite
modules as follows.
Definition 3.1. Let R be a Noetherian ring and I, J be two ideals of R. For the
Serre subcategory S of the category of R-modules, an R-module M is called an
(S, I, J)-cominimax precisely when Supp(M ) ⊆ W (I, J) and ExtiR (R/I, M ) ∈ S
for all i ≥ 0.
Remark 3.2. By applying various Serre classes of R-modules in 3.1, we may
obtain different concepts. But in view of [22, Proposition 1.7], the class of (S, I, J)cominimax R-modules is contained in the class of (I, J)-torsion R-modules. Moreover, for every R-module M and all i ≥ 0, ExtiR (R/I, M ) is I-torsion, so by
Lemma 2.8 (ii), we have ExtiR (R/I, M ) ∈ SI,J iff ExtiR (R/I, M ) ∈ S0 . In other
words, the class of (S0 , I, J)-cominimax R-modules and the class of (SI,J , I, J)cominimax R-modules are the same. Also, since Supp(M ) ⊆ V (I) implies that
Supp(M ) ⊆ W (I, J), hence the class of (SI,J , I, J)-cominimax R-modules contains
the class of (SI , I, J)-cominimax R-modules.

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7

Notation 3.3. For a Serre classes S of R-modules and two ideals I, J of R, we use
C(S, I, J) to denote the class of all (S, I, J)-cominimax R-modules.
Example and Remark 3.4. (i) Let N ∈ S be such that Supp(N ) ⊆ W (I, J).
Then it follows from Lemma 2.8 (i) that N ∈ C(S, I, J).
(ii) Let N be a pure submodule of R-module M . By using the following exact
sequence 0 → ExtiR (R/I, N ) → ExtiR (R/I, M ) → ExtiR (R/I, M/N ) → 0, for all
i ≥ 0, [20, Theorem 3.65], M ∈ C(S, I, J) iff N, M/N ∈ C(S, I, J); in particular,
when S = SI,J .
Proposition 3.5. let 0 → M ′ → M → M ′′ → 0 be an exact sequence of R-modules
such that two of the modules belong to S. Then the third one is (S, I, J)-cominimax
if its support is in W (I, J).
Proof. The assertion follows from the induced long exact sequence
i+1
′
· · · → ExtiR (R/I, M ) → ExtiR (R/I, M ′′ ) → Exti+1
R (R/I, M ) → ExtR (R/I, M ) → . . .

and Lemma 2.8 (i).



An immediate consequence of Proposition 3.5 and Lemma 2.8 is as follows.
Corollary 3.6. Let f : M → N be a homomorphism of R-modules such that
M, N ∈ S. Let one of the three modules Kerf , Imf and Cokerf be in S. Then
two others belong to C(S, I, J) if their supports are in W (I, J).
Proposition 3.7. Let I, J, I ′ , J ′ are ideals of R. Then
√
√ √
√
(i) M ∈ C(S, I, J) iff M ∈ C(S, I, J) iff M ∈ C(S, I, J) iff M ∈ C(S, I, J).
(ii) If M is I-cominimax, then M ∈ C(SI,J , I, J).
√
(iii) If M in(M ) ⊆ W (I ′ , J) , ExtiR (R/I, M ) ∈ SI,J , and I n ⊆ I ′ for some
n ∈ N and all i ≥ 0, then M ∈ C(SI,J , I, J) and M ∈ C(SI ′ ,J , I ′ , J). In
′
′
particular, if M ∈ C(SI,J , I, J), then
√ we have M ∈ C(SI ,J , I , J).
n
′
(iv) If M in(M ) ⊆ W (I, J) and J ⊆ J for some n ∈ N, then M ∈ C(SI,J , I, J)
iff M ∈ C(SI,J ′ , I, J ′ ).
√
Proof. (i) Since V (I) = V ( I), the assertions follow from [22, Proposition 1.6],
Corollary 2.11, and Definition 3.1.
(ii) By assumption and Lemma 2.8 (ii), Supp(M ) ⊆ V (I) ⊆ W (I, J) and ExtiR (R/I,
M ) ∈ SI,J for all i ≥ 0.
(iii), (iv) Apply [22, Proposition 1.6 and 1.7], Corollary 2.11 and Proposition 2.7(iii),
(iv).

The following Remark plays an important role in the proof of our main theorems
in this section.
Remark 3.8. In view of proof [22, Theorem 3.2], Γa (M ) ⊆ ΓI,J (M ), for any
a ∈ W̃ (I, J). Thus ΓI,J (M )= 0 implies that Γa (M ) = 0, for all a ∈ W̃ (I, J). Now,
let M̄ = M/ΓI,J (M ) and E = ER (M̄ ) be the injective hull of R-module M̄ . Put
L = E/M̄ . Since ΓI,J (M̄ ) = 0, then ΓI,J (E) = 0 and also for any a ∈ W̃ (I, J),
we have Γa (M̄ ) = 0 = Γa (E). In particular, the R-module HomR (R/a, E) is zero.
Now, from the exact sequence 0 → M̄ → E → L → 0, and applying HomR (R/a, −)
and ΓI,J (−), we have the following isomorphisms
Exti (R/a, L) ∼
= Exti+1 (R/a, M̄ ) and H i (L) ∼
= H i+1 (M ),
R

R

I,J

I,J

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KH. AHMADI-AMOLI AND M.Y. SADEGHI

for any a ∈ W̃ (I, J) and all i ≥ 0. In particular, ExtiR (R/I, L) ∼
= Exti+1
R (R/I, M̄ ).
i
Proposition 3.9. Let t ∈ N0 be such that HI,J
(M ) ∈ C(S, I, J) for all i < t. Then
ExtiR (R/I, M ) ∈ S for all i < t.

Proof. We use induction on t. When t = 0, there is nothing to prove. For t = 1, since
HomR (R/I, ΓI,J (M )) = HomR (R/I, M ), and ΓI,J (M ) is (S, I, J)-cominimax, the
result is true. Now, suppose that t ≥ 2 and the case t − 1 is settled. The exact
sequence 0 → ΓI,J (M ) → M → M̄ → 0 induced the long exact sequence
· · · → ExtiR (R/I, ΓI,J (M )) → ExtiR (R/I, M ) → ExtiR (R/I, M̄ ) → . . . .

Since ΓI,J (M ) ∈ C(S, I, J), we have ExtiR (R/I, ΓI,J (M )) ∈ S for all i ≥ 0. Therefore, it is enough to show that ExtiR (R/I, M̄ ) ∈ S for all i < t. For this purpose,
let E = ER (M̄ ) and L = E/M̄ . Now, by Remark 3.8, for all i ≥ 0, we get the
i+1
i
isomorphisms HI,J
(L) ∼
(M ) and ExtiR (R/I, L) ∼
= HI,J
= Exti+1
R (R/I, M̄ ). Now,
i+1
i
(L) ∈ C(S, I, J)
by assumption, HI,J (M ) ∈ C(S, I, J) for all i < t − 1, and so HI,J
i
for all i < t − 1. Thus, by the inductive hypothesis, ExtR (R/I, L) ∈ S and so
Exti+1

R (R/I, M̄ ) ∈ S.
The next corollary generalizes Proposition 3.7 of [4].
i
Corollary 3.10. Let HI,J
(M ) ∈ C(S, I, J) for all i ≥ 0. Then ExtiR (R/I, M ) ∈ S
for all i ≥0; particularly, when S is the class of I-minimax modules or the class of
(I, J)-minimax modules.

The Proposition 3.8 of [4] can be obtained from the following theorem when
J = 0 and S = SI .

Theorem 3.11. Let ExtiR (R/I, M ) ∈ S for all i ≥ 0. Let t ∈ N0 be such that
i
t
HI,J
(M ) ∈ C(S, I, J), for all i 6= t, then HI,J
(M ) ∈ C(S, I, J).
Proof. We use induction on t. If t = 0, we must prove that ExtiR (R/I, ΓI,J (M )) ∈ S
for all i ≥ 0. By the exact sequence
i
i
· · · → Exti−1
R (R/I, M̄ ) → ExtR (R/I, ΓI,J (M )) → ExtR (R/I, M ) → . . .

and the hypothesis, it is enough to show that ExtiR (R/I, M̄ ) ∈ S for all i ≥
i
0. Now, by Remark 3.8 and our assumption, we obtain HI,J
(L) ∈ C(S, I, J).
i
Therefore Corollary 3.10 implies that ExtR (R/I, M̄ ) ∈ S for all i ≥ 0 (note that
Ext0R (R/I, M̄ ) = 0). Now suppose, inductively, that t > 0 and the result has been
proved for t − 1. By Remark 3.8, it is easy to show that L satisfies in our inductive
t−1
t
hypothesis. Therefore, the assertion follows from HI,J
(M ) ∼
(L).

= HI,J
i
Corollary 3.12. Let M ∈ S and t ∈ N0 be such that HI,J
(M ) is (S, I, J)t
cominimax for all i 6= t. Then HI,J
(M ) is (S, I, J)-cominimax.

Proof. This is an immediate consequence of Lemma 2.8 (i) and Theorem 3.11. 
Corollary 3.13. Let I be a principal ideal and J be an arbitrary ideal of R. Let
i
M ∈ S. Then HI,J
(M ) is (S, I, J)-cominimax for all i ≥ 0.
0
Proof. For i = 0, since HI,J
(M ) is a submodule of M and M ∈ S, it turns out
0
that HI,J (M ) is (S, I, J)-cominimax, by Remark 3.4 (i). Now, let I = aR. By [22,
i
•
Definition 2.2 and Theorem 2.4], we have HI,J
(M ) ∼
⊗R M ) = 0 for all
= H i (CI,J
i > 1. Therefore the result follows from Theorem 3.11.


ON THE LOCAL COHOMOLOGY MODULES DEFINED BY A PAIR OF IDEALS

9

Now we are prepared to prove the main theorem of this section, which is a
generalization of one of the main results of [3, Theorem 2.2] and also [23, Theorem
2.3].
Theorem 3.14. Let a ∈ W̃ (I, J). Let t ∈ N0 be such that ExttR (R/a, M ) ∈ S and
i
ExtjR (R/a, HI,J
(M )) ∈ S for all i < t and all j ≥ 0. Then for any submodule N
t
t
of HI,J (M ) such that Ext1R (R/a, N ) ∈ S, we have HomR (R/a, HI,J
(M )/N ) ∈ S;
in particular, for a = I.
Proof. Considering the following long exact sequence
t
t
· · · → HomR (R/a, HI,J
(M )) → HomR (R/a, HI,J
(M )/N ) → Ext1R (R/a, N ) → . . . ,

t
since Ext1R (R/a, N ) ∈ S, it is enough to show that HomR (R/a, HI,J
(M )) ∈ S.
To do this, we use induction on t. When t = 0, since HomR (R/a, ΓI,J (M )) =
HomR (R/a, M ) ∈ S, the result is obtained. Next, we assume that t > 0 and that
the claim is true for t − 1. Let M̄ = M/ΓI,J (M ). Then, by the long exact sequence

· · · → ExtjR (R/a, M ) → ExtjR (R/a, M̄ ) → Extj+1
R (R/a, ΓI,J (M )) → . . . ,

and assumption, we conclude that ExtjR (R/a, M̄) ∈ S. Now, by using notation
of Remark 3.8, it is easy to see that L satisfies the inductive hypothesis. So that
t−1
t
we get HomR (R/a, HI,J
(L)) ∈ S and therefore, HomR (R/a, HI,J
(M )) ∈ S, as
required.

The main results of [4, Theorem 4.2] , [5, Lemma 2.2] , [2, Theorem 1.2] ,[16] ,
[12, Corollary 2.7], and [17, Corollary 2.3] are all special cases of next corollary, by
replacing various Serre classes with S and J = 0.

i
Corollary 3.15. Let t ∈ N0 be such that ExttR (R/I, M ) ∈ S and HI,J
(M ) ∈
t
C(S, I, J) for all i < t. Then for any submodule N of HI,J (M ) and any finite
R-module M ′ with Supp(M ′ ) ⊆ V (I) and Ext1R (M ′ , N ) ∈ S, we have HomR (M ′ ,
t
HI,J
(M )/N ) ∈ S.

Proof. Apply Theorem 3.14 and Corollary 2.11.



i
Proposition 3.16. Let t ∈ N0 be such that HI,J
(M ) ∈ C(S, I, J) for all i < t.
Then the following statements hold:
t
(i) If ExttR (R/I, M ) ∈ S, then HomR (R/I, HI,J
(M )) ∈ S.
t+1
1
t
(ii) If ExtR (R/I, M ) ∈ S, then ExtR (R/I, HI,J (M )) ∈ S.
t+1
(iii) If ExtiR (R/I, M ) ∈ S for all i ≥ 0, then HomR (R/I, HI,J
(M )) ∈ S iff
2
t
ExtR (R/I, HI,J (M )) ∈ S.

Proof. (i) Apply Corollary 3.15 or Theorem 3.14 .
(ii) We proceed by induction on t. If t = 0, then by the long exact sequence
(∗)
0 → Ext1R (R/I, ΓI,J (M )) → Ext1R (R/I, M ) → Ext1R (R/I, M̄ )
→ Ext2R (R/I, ΓI,J (M )) → Ext2R (R/I, M ) → Ext2R (R/I, M̄ )
..
.
→ ExtiR (R/I, ΓI,J (M )) → ExtiR (R/I, M ) → ExtiR (R/I, M̄ )
→ Exti+1
R (R/I, ΓI,J (M )) → · · ·

10

KH. AHMADI-AMOLI AND M.Y. SADEGHI

and Ext1R (R/I, M̄ ) ∈ S, the result follows. Suppose that t > 0 and the assertion
is true for t − 1. Since ΓI,J (M ) ∈ C(S, I, J), so ExtiR (R/I, ΓI,J (M )) ∈ S for all
i ≥ 0, and so by (∗), Extt+1
R (R/I, M̄ ) ∈ S. Now, by the notations of Remark
3.8, it is easy to see that R-module L satisfies the inductive hypothesis and so
t−1
t−1
t
Ext1R (R/I, HI,J
(M )) ∈ S. Now, the result follows from HI,J
(L) ∼
(M ).
= HI,J
(iii) (⇒) We use induction on t. Let t = 0. Then considering the long exact
sequence (∗), it is enough to show that Ext1R (R/I, M̄ ) ∈ S. By Remark 3.8, we
have
Ext1R (R/I, M̄ ) ∼
= HomR (R/I, L)
∼
= HomR (R/I, ΓI,J (L))
1
∼
(M )),
= HomR (R/I, HI,J

as required. Suppose t > 0 and the assertion is true for t − 1. Since ΓI,J (M ) ∈
C(S, I, J), we have ExtiR (R/I, ΓI,J (M )) ∈ S for all i ≥ 0. Therefore the exactness
of sequence (∗) implies that ExtiR (R/I, M̄ ) ∈ S for all i ≥ 0. Again by using
the notations of Remark 3.8, we get ExtiR (R/I, L) ∈ S, for all i ≥ 0, and also
t+1
t
HomR (R/I, HI,J
(L)) ∼
(M )) ∈ S. Now, by inductive hypothe= HomR (R/I, HI,J
t−1
2
t
sis, ExtR (R/I, HI,J (L)) ∈ S and hence Ext2R (R/I, HI,J
(M )) ∈ S, as required.
(⇐) This part can be proved by the same method of (⇒), using induction on t, the
following exact sequence
Ext1R (R/I, M ) → Ext1R (R/I, M̄ ) → Ext2R (R/I, ΓI,J (M )),
and Remark 3.8.



4. Finiteness Properties Of Associated Primes
In this short section, we obtain some results, as some applications of previous
sections, about associated prime ideals of local cohomology modules and also finiteness properties of them.
i
Proposition 4.1. Let t ∈ N0 be such that ExttR (R/I, M ) ∈ SI,J and HI,J
(M ) ∈
t
1
C(SI,J , I, J) for all i < t. Let N be a submodule of HI,J (M ) such that ExtR (R/I, N )
t
t
∈ SI,J . If Supp(HI,J
(M )/N ) ⊆ V (I), then Gdim(HI,J
(M )/N ) < ∞ and so
t
HI,J
(M )/N has finitely many associated primes.

Proof. By using Theorem 3.14, for the Serre class SI,J , we have HomR (R/I,
t
t
HI,J
(M )/N ) ∈ SI,J . Hence, by Lemma 2.8 (ii), HomR (R/I, HI,J
(M )/N ) ∈ S0 , as
required.

i
Corollary 4.2. Let t ∈ N0 be such that ExttR (R/I, M ) and HI,J
(M ) are (I, J)t
minimax R-modules for all i < t. Let N be a submodule of HI,J
(M ) such that
t
t
Supp(HI,J
(M )/N ) ⊆ V (I) and Ext1R (R/I, N ) is (I, J)-minimax. Then HI,J
(M )/N
t
(M )/N ) is a finite set; in particular
has finite Goldie dimension and so Ass(HI,J
t
(M ).
for N = JHI,J

Proof. For the first part, apply Remark 3.4 and proposition 4.1. Since by [22,
i
t
Corollary 1.9], HI,J
(M )/JHI,J
(N ) is I-torsion, so the last part immediately follows
from the first.


ON THE LOCAL COHOMOLOGY MODULES DEFINED BY A PAIR OF IDEALS

11

Corollary 4.3. (See [21, Theorem 4]) Let t ∈ N0 be such that ExttR (R/I, M ) is a
finite R-module. If HIi (M ) is I-cofinite for all i < t and HIt (M ) is minimax, then
HIt (M ) is I-cofinite and so Ass((HIt (M )) is a finite set.
Proof. In Proposition 3.16 (i), apply S as the class of finite R-modules, and J =
0. Therefore, HomR (R/I, HIt (M )) is finite R-module. Now, use [18, Proposition
3.4].

Corollary 4.4. Let the situation be as in Corollary 4.3. Then the following statements hold:
t+1
(i) If Extt+1
(M )) and Ext1R (R/I, HIt (M ))
R (R/I, M ) is finite, then HomR (R/I, HI
t+1
are finite and so Ass(HI (M )) is a finite set.
t
(ii) If ExtiR (R/I, M ) is finite for all i ≥ 0, then Ext2R (R/I, HI,J
(M )) is finite.
Proof. (i) By Corollary 4.3, we conclude that HIt (M ) is I-cofinite. So HIi (M ) is
I-cofinite for all i < t + 1. Now, using Proposition 3.16 (i), (ii).
(ii) The result follows from (i) and Proposition 3.16 (iii).

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Department of Mathematics, Payame Noor university, Tehran, 19585-83133, Iran
Current address: Department of Mathematics, Payame Noor university, Tehran, 19585-83133,
Iran
E-mail address: khahmadi@pnu.ac.ir
Department of Mathematics, Payame Noor university, Tehran, 19585-83133, Iran
E-mail address: sadeghi@phd.ac.ir

