Some properties of generalized local cohomology modules with respect to a pair of ideals
aa r X i v : . [ m a t h . A C ] M a r SOME PROPERTIES OF GENERALIZED LOCALCOHOMOLOGY MODULES WITH RESPECT TO APAIR OF IDEALS
TRAN TUAN NAM AND NGUYEN MINH TRI
Abstract.
We introduce a notion of generalized local cohomol-ogy modules with respect to a pair of ideals (
I, J ) which is a gen-eralization of the concept of local cohomology modules with re-spect to (
I, J ) . We show that generalized local cohomology modules H iI,J ( M, N ) can be computed by the ˇCech cohomology modules.We also study the artinianness of generalized local cohomologymodules H iI,J ( M, N ) . Key words : generalized local cohomology, artinianness. : 13D45.1.
Introduction
Throughout this paper, R is a noetherian commutative (non-zeroidentity) ring. In [10], Takahashi, Yoshino and Yoshizawa introducedthe local cohomology modules with respect to a pair of ideals ( I, J ). Foran R -module M , the ( I, J )-torsion submodule of M is Γ I,J ( M ) = { x ∈ M | I n x ⊂ J x for some positive integer n } . Γ I,J is a covariant functorfrom the category of R -modules to itself. The i -th local cohomologyfunctor H iI,J with respect to ( I, J ) is defined to be the i -th right derivedfunctor of Γ I,J . When J = 0, the H iI,J coincides with the usual localcohomology functor H iI . For two R − modules M and N, we define Γ I,J ( M, N ) to be the (
I, J )-torsion submodule of Hom R ( M, N ) . For each R − module M, there isa covariant functor Γ I,J ( M, − ) from the category of R -modules to it-self. The i -th generalized local cohomology functor H iI,J ( M, − ) withthe respect to pair of ideals ( I, J ) is the i -th right derived functor ofΓ I,J ( M, − ) . This definition is really a generalization of the local coho-mology functors H iI,J with respect to ( I, J ) . This research is funded by Vietnam National Foundation for Science and Tech-nology Development (NAFOSTED).
The organization of the paper is as follows. In the next section,we study some elementary properties of generalized local cohomologymodules with respect to a pair of ideals (
I, J ) . We also show thatgeneralized local cohomology modules H iI,J ( M, N ) can be computedby ˇCech cohomology modules (Theorem 2.8).The last section is devoted to study the artinianness of local coho-mology modules H iI,J ( M, N ) . In Theorem 3.1 we prove that if
M, N are two finitely generated R -modules with p = pd( M ) and d = dim( N ) , then H r + dI,J ( M, N ) ∼ = Ext rR ( M, H dI,J ( N )) and H r + dI,J ( M, N ) is an artinian R − module. Theorem 3.2 shows that if M is a finitely generated R -modules and H iI,J ( N ) is artinian for all i < t, then H iI,J ( M, N ) isartinian for all i < t.
On the other hand, Ext iR ( R/ a , N ) is also artinianfor all i < t and for all a ∈ ˜ W ( I, J ) . Let
I, J be two ideals of thelocal ring ( R, m ) such that √ I + J = m and M, N are two finitelygenerated R -modules with dim( N ) < ∞ . If H iI,J ( M, N ) is an artinian R -module for all i > t, then H tI,J ( M, N ) /J H tI,J ( M, N ) is also an ar-tinian R -module (Theorem 3.4). This section is closed by Theorem 3.7which says that H iI,J ( M, N ) is artinian for all i ≥ M is afinitely generated R -module and N is an artinian R -module.2. Some basic properties of generalized localcohomology modules with respect to a pair of ideals
Let
I, J be two ideals of R. For an R -module M , the ( I, J )-torsionsubmodule of M isΓ I,J ( M ) = { x ∈ M | I n x ⊂ J x for some positive integer n } ([10]) . We introduce the following definition.
Definition 2.1.
For two R − modules M, N we denote by Γ
I,J ( M, N )the following moduleΓ
I,J ( M, N ) = Γ
I,J (Hom R ( M, N )) . In the special case M = R, Γ I,J ( R, N ) = Γ
I,J ( N ) the ( I, J )-torsionsubmodule of N. Note that an element f ∈ Γ I,J ( M, N ) if and only ifthere is an integer n > I n f ( x ) ⊂ J f ( x ) for all x ∈ M .For each R -module M , Γ I,J ( M, − ) is a left exact covariant functorfrom the category of R -modules to itself.Let us denote by H iI,J ( M, − ) the i -th right derived functor of Γ I,J ( M, − )and call the i -th generalized local cohomology functor with the respectto pair of ideals ( I, J ). ome properties of generalized local cohomology modules... 3 Theorem 2.2.
Let M be a finitely generated R -module and N an R − module. Then Γ I,J ( M, N ) = Hom R ( M, Γ I,J ( N )) . Proof. If f ∈ Γ I,J ( M, N ), there exists an integer n > I n f ( x ) ⊂ J f ( x ) for all x ∈ M . Since f ( x ) ∈ N , we get f ( x ) ∈ Γ I,J ( N )for all x ∈ M and then f ∈ Hom R ( M, Γ I,J ( N )) . Let f ∈ Hom R ( M, Γ I,J ( N )). Assume that x , x , . . . , x m are genera-tors of M .Since f ( x i ) ∈ Γ I,J ( N ), there exist an integer n i such that I n i f ( x i ) ⊂ J f ( x i ) for i = 1 , , . . . , m. Set n = n n . . . n m , then I n f ( x i ) ⊂ J f ( x i ) for all i = 1 , , . . . , m. It follows I n f ( x ) ⊂ J f ( x ) for all x ∈ M . So I n f ⊂ J f and then f ∈ Γ I,J (Hom R ( M, N )) = Γ
I,J ( M, N ) . (cid:3) Note that in [11] Zamani introduced an other definition of local co-homology functors H iI,J as follow H iI,J ( M, N ) = H i (Hom R ( M, Γ I,J ( E • )))for all i ≥ , where E • is an injective resolution of R -module N. Thusfrom 2.2 we see that our definition is coincident with Zamani’s one.We have a property of the set of associated primes of Γ
I,J ( M, N ). Corollary 2.3.
Let M be a finitely generated R -module and N an R − module. Then Ass(Γ
I,J ( M, N )) = Supp( M ) ∩ Ass( N ) ∩ W ( I, J ) . Proof.
Since M is a finitely generated R -module, Ass(Hom R ( M, K )) =Supp( M ) ∩ Ass( K ) for all R -modules K. By 2.2, we haveAss(Γ
I,J ( M, N )) = Ass(Γ
I,J (Hom R ( M, N )))= Ass(Hom R ( M, Γ I,J ( N )))= Supp( M ) ∩ Ass(Γ
I,J ( N ))= Supp( M ) ∩ Ass( N ) ∩ W ( I, J )as required. (cid:3)
The following proposition is an extension of [10, 1.4].
Proposition 2.4.
Let M be a finitely generated R -module and N an R − module. Let I, I ′ , J, J ′ be ideals of R . Then (i) Γ I,J (Γ I ′ ,J ′ ( M, N )) = Γ I ′ ,J ′ (Γ I,J ( M, N )) . (ii) If I ⊆ I ′ , then Γ I,J ( M, N ) ⊇ Γ I ′ ,J ( M, N ) . TRAN TUAN NAM, NGUYEN MINH TRI (iii) If J ⊆ J ′ , then Γ I,J ( M, N ) ⊆ Γ I,J ′ ( M, N ) . (iv) Γ I,J (Γ I ′ ,J ( M, N )) = Γ I + I ′ ,J ( M, N ) . (v) Γ I,J (Γ I,J ′ ( M, N )) = Γ
I,JJ ′ ( M, N ) = Γ
I,J ∩ J ′ ( M, N ) . Moreover, H iI,JJ ′ ( M, N ) = H iI,J ∩ J ′ ( M, N ) for all i. (vi) If J ′ ⊆ J , then Γ I + J ′ ,J ( M, N ) = Γ
I,J ( M, N ) . Moreover, Γ I + J,J ( M, N ) = Γ
I,J ( M, N ) and H iI + J,J ( M, N ) = H iI,J ( M, N ) for all i. (vii) If √ I = √ I ′ , then H iI,J ( M, N ) = H iI ′ ,J ( M, N ) for all i. Inparticular, H iI,J ( M, N ) = H i √ I,J ( M, N ) for all i. (viii) If √ J = √ J ′ , then H iI,J ( M, N ) = H iI,J ′ ( M, N ) for all i .Proof. We only prove (i), the others are similar.Combining [10, 1.4] and 2.2, we haveΓ
I,J (Γ I ′ ,J ′ ( M, N )) = Γ
I,J (Hom R ( M, Γ I ′ .J ′ ( N )))= Hom R ( M, Γ I,J (Γ I ′ ,J ′ ( N )))= Hom R ( M, Γ I ′ ,J ′ (Γ I,J ( N )))= Γ I ′ ,J ′ (Γ I,J ( M, N ))as required. (cid:3)
Lemma 2.5. If E is an injective R -module, then Γ I,J ( E ) is also in-jective.Proof. From [10, 3.2] we haveΓ
I,J ( E ) ∼ = lim −→ a ∈ ˜ W ( I,J ) Γ a ( E ) , where ˜ W ( I, J ) is the set of ideals a of R such that I n ⊂ a + J for someinteger n. Since E is an injective R -module, Γ a ( E ) is also injective. Moreover, R is a Noetherian ring, then the direct limit of injective modules isinjective. Therefore we have the conclusion. (cid:3) It is well-known that H iI ( M, N ) ∼ = Ext iR ( M, N ) , where N is an I -torsion R -module. The following proposition gives a similar result when N is ( I, J )-torsion.
Proposition 2.6.
Let N be an ( I, J ) -torsion R -module. Then H iI,J ( M, N ) ∼ = Ext iR ( M, N ) for all i ≥ . ome properties of generalized local cohomology modules... 5 Proof.
From [10, 1.12] there exists an injective resolution E • of N suchthat each term is an ( I, J )-torsion R -module. Then we have by 2.2 H iI,J ( M, N ) ∼ = H i (Hom R ( M, Γ I,J ( E • ))) ∼ = H i (Hom R ( M, E • ))= Ext iR ( M, N )for all i ≥ . (cid:3) When N is a J -torsion R -module, we have the following proposition. Proposition 2.7. If N is a J -torsion R -module, then H iI,J ( M, N ) ∼ = H iI ( M, N ) for all i ≥ . Proof.
It is obvious that Γ I ( N ) ⊂ Γ I,J ( N ) . Let x ∈ Γ I,J ( N ), there existintegers n, k such that I n x ⊂ J x and J k x = 0. Hence I nk x = 0 andthen x ∈ Γ I ( N ). Thus Γ I,J ( N ) = Γ I ( N ) . It remains to prove that Γ
I,J ( M, N ) ∼ = Γ I ( M, N ). From 2.2 we haveΓ
I,J ( M, N ) = Hom R ( M, Γ I,J ( N ))= Hom R ( M, Γ I ( N )) ∼ = Γ I ( M, N ) . By the property of derived functors, we obtain H iI,J ( M, N ) ∼ = H iI ( M, N )for all i ≥ . (cid:3) Let J be an ideal of R. For an element a ∈ R the set S a,J = { a n + j | n ∈ N , j ∈ J } is a multiplicatively closed subset of R ([10, 2.1]). Let M be a finitelygenerated R -module. Denote by M a,J the module of fractions of the R − module M with respect to S a,J . The complex C • a,J was given by C • a,J : 0 → R → R a,J → . For a sequence a = { a , a , . . . , a r } of elements of R , the ˇCech com-plex C • a ,J was defined as C • a ,J = r O i =1 C • a i ,J = (cid:16) → R → r Y i =1 R a i ,J → Y i In [10, 2.4], there is a natural isomorphism H iI,J ( M ) ∼ = H i ( C • a ,J ⊗ R M ) , where a = { a , a , . . . , a r } is a sequence of elements of R thatgenerates I. Let F • : · · · −→ F −→ F −→ F −→ M −→ M with the finitely generated free modules.Apply the functor Hom R ( − , N ) to above resolution, we have a com-plexHom R ( F • , N ) : 0 → Hom R ( M, N ) → Hom R ( F , N ) → Hom R ( F , N ) → .... Then there is a bicomplex C • a ,J ⊗ R Hom R ( F • , N ) = { C p a ,J ⊗ R Hom R ( F q , N ) } , where C p a ,J is the p -th position in the ˇCech complex C • a ,J . Thus we geta total complex Tot( M, N ) of bicomplex C • a ,J ⊗ R Hom R ( F • , N ) whereTot( M, N ) n = M p + q = n C p a ,J ⊗ R Hom R ( F q , N ) . Theorem 2.8. Let M be a finitely generated R -module. Then for all R -modules N and n ≥ ,H nI,J ( M, N ) ∼ = H n (Tot( M, N )) . Proof. It is clear that { H n (Tot( M, − )) } n and { H nI,J ( M, − ) } n are exactconnected right sequences of functors.We next prove that H (Tot( M, N )) ∼ = H I,J ( M, N ) . Consider thehomomorphism d : Tot( M, N ) → Tot( M, N ) . As C a ,J = R, we get H (Tot( M, N )) ∼ = Ker(Hom R ( F , N ) d → Hom R ( F , N ) ⊕ ( C a ,J ⊗ R Hom R ( F , N ))) . Thus H (Tot( M, N )) = Hom R ( M, N ) \ Γ I,J (Hom R ( F , N ))= Γ I,J (Hom R ( M, N )) = H I,J ( M, N )by [10, 2.3(5)].The proof is completed by showing that H n (Tot( M, E )) = 0 for all n > R -module E. It follows from [8, 10.18] aspectral sequence E p,q = H ′′ p H ′ q ( C • a ,J ⊗ R Hom R ( F • , E )) ⇒ p H n (Tot( M, E )) . Note that E p,q = H q ( C • a ,J ⊗ Hom R ( F p , E )) . ome properties of generalized local cohomology modules... 7 From the proof of [10, 2.4], H i ( C • a ,J ⊗ R E ) = 0 for all i > R − module E . Note that Hom R ( F q , E ) is also an injective R -module for all q ≥ . Hence E p,q = , q > R ( F p , E ) → r Q i =1 Hom R ( F p , E ) a i ,J ) , q = 0 . Combining [10, 2.3(5)] with 2.2 yieldsKer(Hom R ( F p , E ) → r Y i =1 Hom R ( F p , E ) a i ,J ) ∼ = Γ I,J (Hom R ( F p , E )) ∼ = Hom R ( F p , Γ I,J ( E )) . It follows E p,q = ( , q > H p (Hom R ( F • , Γ I,J ( E ))) , q = 0 . As Γ I,J ( E ) is an injective R − module, the following sequence is exactHom R ( F • , Γ I,J ( E )) : 0 → Hom R ( M, Γ I,J ( E )) → Hom R ( F , Γ I,J ( E )) →→ Hom R ( F , Γ I,J ( E )) → .... Thus E p, = 0 for all p > . From [8, 10.21 (ii)] we get H n (Tot( M, E )) ∼ = E n, = 0for all n > . The proof is complete. (cid:3) On artinianness of generalized local cohomologymodules with respect to a pair of ideals We have the following theorem. Theorem 3.1. Assume that ( R, m ) is a local ring. Let M, N be twofinitely generated R -modules with r = pd( M ) and d = dim( N ) . Then H r + dI,J ( M, N ) ∼ = Ext rR ( M, H dI,J ( N )) . Moreover H r + dI,J ( M, N ) is an artinian R − module. TRAN TUAN NAM, NGUYEN MINH TRI Proof. Let G ( − ) = Γ I,J ( − ) and F ( − ) = Hom R ( M, − ) be functors fromcategory of R -modules to itself. Then F G = Γ I,J ( M, − ) and F is leftexact. For any injective module ER i F ( G ( E )) = R i Hom R ( M, Γ I,J ( E )) = 0for all i > , as Γ I,J ( E ) is an injective R − module. By [8, 10.47] thereis a Grothendieck spectral sequence E pq = Ext pR ( M, H qI,J ( N )) = ⇒ p H p + qI,J ( M, N ) . We now consider the homomorphisms of the spectral E r − k,d + k − k → E r,dk → E r + k,d +1 − kk . We have H qI,J ( N ) = 0 for all q > d by [10, 4.7]. Then E pq = 0 for all p > r or q > d. Thus E r − k,d + k − k = E r + k,d +1 − kk = 0 for all k ≥ , so E r,d = E r,d = ... = E r,d ∞ . It remains to prove that E r,d ∞ ∼ = H r + dI,J ( M, N ) . Indeed, there is a filtra-tion Φ of H r + d = H r + dI,J ( M, N ) such that0 = Φ r + d +1 H r + d ⊆ Φ r + d H r + d ⊆ . . . ⊆ Φ H r + d ⊆ Φ H r + d = H r + dI,J ( M, N )and E i,r + d − i ∞ = Φ i H r + d / Φ i +1 H r + d , ≤ i ≤ r + d. From the above proof we have E i,r + d − i = Ext iR ( M, H r + d − iI,J ( N )) = 0 forall i = r. HenceΦ r +1 H r + d = Φ r +2 H r + d = . . . = Φ r + d +1 H r + d = 0and Φ r H r + d = Φ r − H r + d = . . . = Φ H r + d = H r + dI,J ( M, N ) . This gives E r,d ∞ ∼ = Φ r H r + d / Φ r +1 H r + d ∼ = H r + dI,J ( M, N ) . Thus Ext rR ( M, H dI,J ( N )) ∼ = H r + dI,J ( M, N ) . It follows from [2, 2.1] that H dI,J ( N ) is an artinian R − module. Therefore H r + dI,J ( M, N ) is also anartinian R − module. (cid:3) Next theorem, we show the connection between the artinianness of H iI,J ( N ) and H iI,J ( M, N ). Theorem 3.2. Let M be a finitely generated R -modules and N an R − module. Let t be a positive integer. If H iI,J ( N ) is artinian for all i < t, then ome properties of generalized local cohomology modules... 9 (i) H iI,J ( M, N ) is artinian for all i < t. (ii) Ext iR ( R/ a , N ) is artinian for all i < t and for all a ∈ ˜ W ( I, J ) . Proof. (i) We use induction on t . When t = 1 , by 2.2 we haveΓ I,J ( M, N ) = Hom R ( M, Γ I,J ( N )) . Since Γ I,J ( N ) is artinian, the statement is true in this case.Let t > t − R -module N . Denote by E ( N ) the injective hull of N . Apply-ing the functors Γ I,J ( − ) and Γ I,J ( M, − ) to the following short exactsequence 0 → N → E ( N ) → E ( N ) /N → H iI,J ( E ( N ) /N ) ∼ = H i +1 I,J ( N )and H iI,J ( M, E ( N ) /N ) ∼ = H i +1 I,J ( M, N )for all i > . From the hypothesis, H iI,J ( N ) is artinian for all i < t. Itfollows that H iI,J ( E ( N ) /N ) is also artinian for all i < t − 1. By theinductive hypothesis on E ( N ) /N , H iI,J ( M, E ( N ) /N ) is artinian for all i < t − 1. We conclude from the second isomorphism that H iI,J ( M, N )is artinian for all i < t. (ii) The proof is by induction on t . When t = 1 , the short exactsequence 0 → Γ a ( N ) → N → N/ Γ a ( N ) → . deduces an exact sequence → Hom R ( R/ a , Γ a ( N )) → Hom R ( R/ a , N ) → Hom R ( R/ a , N/ Γ a ( N )) → · · · As N/ Γ a ( N ) is a -torsion-free, we have Hom R ( R/ a , N/ Γ a ( N )) = 0and then Hom R ( R/ a , N ) ∼ = Hom R ( R/ a , Γ a ( N )). Note that Γ a ( N ) ⊂ Γ I,J ( N ) . By the hypothesis, Γ a ( N ) is an artinian R -module and thenHom R ( R/ a , Γ a ( N )) is also an artinian R -module.The proof for t > (cid:3) In [2, 2.4], when ( R, m ) is a local ring and N is a finitely generated R -module, there is an equalityinf { i | H iI,J ( N ) is not artinian } = inf { depth N p | p ∈ W ( I, J ) \ { m }} . We have the following consequence. Corollary 3.3. Let ( R, m ) be a local ring. If M and N are two finitelygenerated R -modules, then inf { depth N p | p ∈ W ( I, J ) \{ m }} ≤ inf { i | H iI,J ( M, N ) is not artinian } . Proof. From 3.2, We have the following inequalityinf { i | H iI,J ( N ) is not artinian } ≤ inf { i | H iI,J ( M, N ) is not artinian } . Thus the conclusion follows from [2, 2.4]. (cid:3) Theorem 3.4. Let I, J be two ideals of the local ring ( R, m ) such that √ I + J = m . Assume that M, N are two finitely generated R -moduleswith dim( N ) < ∞ and t is an non-negative integer. If H iI,J ( M, N ) isan artinian R -module for all i > t, then H tI,J ( M, N ) /J H tI,J ( M, N ) isalso an artinian R -module.Proof. Combining 2.4(vi) with 2.4(vii) we conclude that H iI,J ( M, N ) = H i m ,J ( M, N ) for all i ≥ , as √ I + J = m . Thus, without loss of gener-ality we can assume that I = m . We now use induction on dim( N ) = d. When d = 0 , N is m -torsion and then N is ( m , J )-torsion. From 2.6, there is an isomorphism H i m ,J ( M, N ) ∼ = Ext iR ( M, N ) for all i ≥ . Since N is artinian, it followsthat H i m ,J ( M, N ) is an artinian R -module for all i ≥ . Therefore thestatement is true in this case.Let d > . The short exact sequence0 → Γ J ( N ) → N → N/ Γ J ( N ) → . induces a long exact sequence H t m ,J ( M, Γ J ( N )) α → H t m ,J ( M, N ) β → H t m ,J ( M, N/ Γ J ( N )) γ → · · · Since Γ J ( N ) is a J -torsion R -module, there is an isomorphism H i m ,J ( M, Γ J ( N )) ∼ = H i m ( M, Γ J ( N ))by 2.7. From [4, 2.2] H i m ,J ( M, Γ J ( N )) is artinian for all i. From the long exact sequence, we get two short exact sequences0 → Im α → H t m ,J ( M, N ) → Im β → → Im β → H t m ,J ( M, N/ Γ J ( N )) → Im γ → . Two above exact sequences deduce long exact sequences · · · Im α/J Im α → H t m ,J ( M, N ) /J H t m ,J ( M, N ) → Im β/J Im β → · · · → Tor R ( R/J, Im γ ) → Im β/J Im β → ome properties of generalized local cohomology modules... 11 → H t m ,J ( M, N/ Γ J ( N )) /J H t m ,J ( M, N/ Γ J ( N )) → Im γ/J Im γ → . Note that Im α and Im γ are artinian R -modules. The proof is com-pleted by showing that H t m ,J ( M, N/ Γ J ( N )) /J H t m ,J ( M, N/ Γ J ( N )) is anartinian R − module.Let N = N/ Γ J ( N ), then N is J -torsion-free. Thus there exists anelement x ∈ J that is a non-zerodivisor on N . The short exact sequence0 → N .x → N → N /xN → · · · → H t m ,J ( M, N ) f → H t m ,J ( M, N /xN ) g → H t +1 m ,J ( M, N ) → · · · From the hypothesis, we get that H i m ,J ( M, N /xN ) is artinian for all i > t. As dim( N /xN ) ≤ d − H t m ,J ( M, N /xN ) /J H t m ,J ( M, N /xN ) isartinian by the inductive hypothesis.We now consider two exact sequences0 → Im f → H t m ,J ( M, N /xN ) → Im g → H t m ,J ( M, N ) .x → H t m ,J ( M, N ) → Im f → . They gives two long exact sequencesTor R ( R/J, Im g ) → Im f /J Im f →→ H t m ,J ( M, N /xN ) /J H t m ,J ( M, N /xN ) → Im g/J Im g → H t m ,J ( M, N ) /J H t m ,J ( M, N ) .x → H t m ,J ( M, N ) /J H t m ,J ( M, N ) →→ Im f /J Im f → . Since x ∈ J, we obtain from the exact sequence that H t m ,J ( M, N ) /J H t m ,J ( M, N ) ∼ = Im f /J Im f. On other hand, Tor R ( R/J, Im g ) is artinian, as Im g ⊂ H t +1 m ,J ( M, N )an artinian R − module. Hence Im f /J Im f is an artinian R − moduleand the proof is complete. (cid:3) Theorem 3.5. Let M, N be two finitely generated R -modules and t apositive integer such that H tI,J ( M, R/ p ) is artinian for all p ∈ Supp( N ) .Then H tI,J ( M, N ) is also artinian. Proof. As N is finitely generated, there is a chain of submodules of N N ⊂ N ⊂ N ⊂ . . . ⊂ N k = N such that N i /N i − ∼ = R/ p i for some p i ∈ Supp( N ) . For each 1 ≤ i ≤ k , the short exact sequence0 → N i − → N i → R/ p i → · · · → H tI,J ( M, N i − ) → H tI,J ( M, N i ) → H tI,J ( M, R/ p i ) → · · · In particular, H tI,J ( M, N ) ∼ = H tI,J ( M, R/ p ) . From the exact sequence,it follows that H tI,J ( M, N i ) is artinian for all 1 ≤ i ≤ k. This finishesthe proof. (cid:3) From Theorem 3.5 we have the following immediate consequence. Corollary 3.6. Let M, N be finitely generated R -modules and t apositive integer. Assume that H tI,J ( M, R/ p ) is artinian for all p ∈ Supp( N ) . (i) If L is a finitely generated R -module such that Supp( L ) ⊂ Supp( N ) , then H tI,J ( M, L ) is artinian.(ii) If a is an ideal of R such that V ( a ) ⊂ Supp( N ) , then H tI,J ( M, R/ a ) is artinian. In the following theorem, we study the artinianness of H iI,J ( M, N )when N is artinian. Theorem 3.7. Let M be a finitely generated R -module and N an ar-tinian R -module. Then H iI,J ( M, N ) is artinian for all i ≥ . Proof. We use induction on i . When i = 0 , we have Γ I,J ( M, N ) =Hom R ( M, Γ I,J ( N )) is artinian, as Γ I,J ( N ) ⊂ N. Let i > , denote by E ( N ) the injective hull of N . Note that, if N ⊂ K is an essential submodule, then N is artinian if and only if K is artinian. Hence E ( N ) is also artinian.Now the short exact sequence0 → N → E ( N ) → E ( N ) /N → · · · → H i − I,J ( M, E ( N ) /N ) → H iI,J ( M, N ) → H iI,J ( M, E ( N )) → · · · Since H iI,J ( M, E ( N )) = 0 for all i > 0, there are isomorphims H i − I,J ( M, E ( N ) /N ) ∼ = H iI,J ( M, N ) ome properties of generalized local cohomology modules... 13 for all i > .H i − I,J ( M, E ( N ) /N ) is artinian by inductive hypothesis. Therefore H iI,J ( M, N ) is also artinian. (cid:3) References [1] M. P. Brodmann, R. Y. Sharp, Local cohomology: an algebraic introductionwith geometric applications , Cambridge University Press, 1998.[2] L. Chu, Q. Wang, ”Some results on local cohomology modules defined by apair of ideals”, J. Math. Kyoto Univ , 49:193-200, 2009.[3] N. T. Cuong, N. V. Hoang, ”On the vanishing and the finiteness of supportsof generalized local cohomological modules,” Manuscripta Math. Algebra Colloquium , Vol. 12, No. 2 (2005) 213-218.[5] S. H. Hassanzadeh, A. Vahidi, ”On vanishing and cofiniteness of generalizedlocal cohomology modules”, Communication of Algebra , 37: 2290-2299,2009[6] L. Melkersson, ”On asymptotic stability for sets of prime ideals connected withthe powers of an ideal,” Math. Proc. Camb. Phil. Soc., 107 (1990), 267-271.[7] T. T. Nam, ”On the non-vanishing and the artinianness of generalized localcohomology modules”, Journal of Algebra and Its Applications (to appear).[8] J. Rotman, An introduction to homological algebra, 2nd edition , Springer, 2009.[9] N. Suzuki, ”On the generalized local cohomology and its duality,” J. Math.Kyoto Univ. (JAKYAZ), J. Pure Appl. Algebra , 213: 582-600,2009.[11] N. Zamani, ”Generalized local cohomology relative to ( I, J )”, Southeast AsianBullettin of Mathematics , 35: 1045-1050, 2011. Department of Mathematics-Informatics, Ho Chi Minh Universityof Pedagogy, Ho Chi Minh city, Viet Nam. E-mail address : [email protected] Department of Natural Science Education, Dong Nai University,Dong Nai, Viet Nam. E-mail address ::