Abstract. We show that if M and Hi I,J (M) are weakly Laskerian for all i < d, then AssR(Hom(RI, Hd I,J (M))) is finite. If Hi I,J (M) is (I, J)weakly cofinite for all i 6= d, then Hd I,J (M)) is also (I, J)weakly cofinite. We also study some properties of Hi I,J (M) concerning Serre subcategory.
SOME RESULTS ON LOCAL COHOMOLOGY MODULES WITH RESPECT TO A PAIR OF IDEALS TRAN TUAN NAM AND NGUYEN MINH TRI i Abstract. We show that if M and HI,J (M ) are weakly Lasked rian for all i < d, then AssR (Hom(R/I, HI,J (M ))) is finite. If i d (M )) HI,J (M ) is (I, J)-weakly cofinite for all i = d, then HI,J is also (I, J)-weakly cofinite. We also study some properties of i HI,J (M ) concerning Serre subcategory. Key words: local cohomology, weakly Laskerian, weakly cofinite. 2000 Mathematics subject classification: 13D45 Local cohomology. 1. Introduction Throughout this paper, R is a commutative Noetherian ring and I, J are two ideals of R. In [7] Takahashi, Yoshino and Yoshizawa introduced the definition of local cohomology modules with respect to a pair of ideals (I, J) which is a generalization of the definition of local cohomology modules with respect to an ideal I of Grothendieck. Let M be an R-module, the (I, J)-torsion submodule ΓI,J (M ) of M is the subset consisting of all elements x of M such that I n x ⊆ Jx for some i n ∈ N. For an integer i, the local cohomology functor HI,J with respect to a pair of ideals (I, J) is the i-th right derived functor of ΓI,J . Note i that if J = 0, then HI,J coincides with the ordinary local cohomology i functor HI of Grothendieck. In [5] Grothendieck gave a conjecture that: For any ideal I of R and any finitely generated R-module M, the module HomR (R/I, HIi (M )) is finitely generated, for all i. One year later, Hartshorne provided a counterexample to Grothendieck’s conjecture. He defined an R-module M to be I-cofinite if Supp(M ) ⊆ V (I) and ExtiR (R/I, M ) is finitely generated for all i and asked: For which rings R and ideals I are the modules HIi (M ) I-cofinite for all i and all finitely generated modules M? The organization of the paper is as follows. In next section, we will be concerned with Grothendieck’s conjecture. Theorem 2.2 shows i that if M and HI,J (M ) are weakly Laskerian R−modules for all i < d, 1 2 TRAN TUAN NAM, NGUYEN MINH TRI d then the set AssR (Hom(R/I, HI,J (M ))) is finite. Let denote W (I, J) = n {p ∈ Spec(R) | I ⊆ p + J for some positive integer n}. An R-module M is said to be (I, J)-weakly cofinite if Supp(M ) ⊆ W (I, J) and ExtiR (R/I, M ) is weakly Laskerian for all i ≥ 0. We prove in Theoi rem 2.6 that if HI,J (M ) is (I, J)-weakly cofinite for all i ≤ d, then i ExtR (R/I, M ) is weakly Laskerian for all i ≤ d. On the other hand, if i d HI,J (M ) is (I, J)-weakly cofinite for all i = d, then HI,J (M )) is also (I, J)-weakly cofinite (Theorem 2.7). i The last section is devoted to study some properties of HI,J (M ) i concerning Serre subcategory. Theorem 3.1 says that if HI,J (M ) ∈ S for all i < d, then ExtiR (R/I, M )) ∈ S for all i < d. In case (R, m) is a local ring and M is a finitely generated R−module we see that if d i (M )) ∈ S (Theorem (M ) ∈ S for all i < d, then HomR (R/m, HI,J HI,J 3.2). 2. Weakly Laskerian modules and cofinite modules We begin by recalling the definition of weakly Laskerian modules ([3, 2.1]). An R-module M is said to be weakly Laskerian if the set of associated primes of any quotient module of M is finite. Lemma 2.1. ([3, 2.3]) i) Let 0 −→ L −→ M −→ N −→ 0 be an exact sequence of R−modules. Then M is weakly Laskerian if and only if L and N are both weakly Laskerian. Thus any subquotient of a weakly Laskerian module as well as any finite direct sum of weakly Laskerian modules is weakly Laskerian. ii) Let M and N be two R−modules. If M is weakly Laskerian and N is finitely generated, then ExtiR (N, M ) and TorR i (N, M ) are weakly Laskerian for all i ≥ 0. The following theorem answers the question concerning Grothendieck’s d conjecture: When is the set AssR (Hom(R/I; HI,J (M ))) finite?. Theorem 2.2. Let M be a weakly Laskerian R−module and d a noni negative integer. If HI,J (M ) is a weakly Laskerian R−module for all d i < d, then AssR (Hom(R/I; HI,J (M ))) is a finite set. Proof. Let us consider functors F = HomR (R/I, −) and G = ΓI,J (−). Then F G = Hom(R/I, −). We have a Grothendieck spectral sequence Some results on local cohomology modules with respect to a pair of ideals. 3 by [6, 11.38] q p+q E2p,q = ExtpR (R/I, HI,J (M )) ⇒ ExtR (R/I, M ). p We consider homomorphisms of the spectral d−k,d+k−1 d0,d Ek−k,d+k−1 −→ Ek0,d −→Ekk,d+1−k . 0,d Since Ek−k,d+k−1 = 0 for all k ≥ 2, Ker d0,d k = Ek+1 . It follows an exact sequence d0,d 0,d −→ Ek0,d −→Ekk,d+1−k . 0 −→ Ek+1 Hence 0,d Ass(Ek0,d ) ⊆ Ass(Ek+1 ) ∪ Ass(Ekk,d+1−k ). By iterating this for all k = 2, . . . , d + 1, we get d+1 Ass(E20,d ) Ass(Ekk,d+1−k )) ⊆( 0,d Ass( Ed+2 ). k=2 It is clear that 0,d 0,d 0,d Ed+2 = Ed+3 = ... = E∞ . Therefore d+1 Ass(E20,d ) Ass(Ekk,d+1−k )) ⊆( 0,d Ass(E∞ ). k=2 d+1−k For all k = 2, . . . , d + 1 as HI,J (M ) a weakly Laskerian R−module, k,d+1−k k d+1−k so is E2 = ExtR (R/I, HI,J (M )) by 2.1 (ii). As Ekk,d+1−k is a subquotient of E2k,d+1−k , it follows from 2.1 (i) that Ekk,d+1−k is a weakly d+1 Laskerian R−module. Thus Ass(Ekk,d+1−k ) is a finite set. k=2 0,d The proof is completed by showing that the set Ass(E∞ ) is finite. p+q p+q Indeed, there is a filtration Φ of H = ExtR (R/I, M ) with 0 = Φp+q+1 H p+q ⊆ Φp+q H p+q ⊆ . . . ⊆ Φ1 H p+q ⊆ Φ0 H p+q = Extp+q R (R/I, M ) and k,p+q−k E∞ = Φk H p+q /Φk+1 H p+q , 0 ≤ k ≤ p + q. p,q p,q It follows that E∞ is a weakly Laskerian R−module, so Ass(E∞ ) is 0,d finite for all p, q. In particular, Ass(E∞ ) is finite. Note that finitely generated modules or modules that have finite support are weakly Laskerian modules. So we have an immediate consequence. 4 TRAN TUAN NAM, NGUYEN MINH TRI Corollary 2.3. Let M be a finitely generated R−module and d a noni i negative integer. If HI,J (M ) is finitely generated or Supp(HI,J (M )) is d a finite set for all i < d, then AssR (HomR (R/I, HI,J (M ))) is finite. An R-module M is said (I, J)-cofinite if Supp(M ) ⊆ W (I, J) and ExtiR (R/I, M ) is finitely generated for all i ≥ 0 ([8]). The following definition is an extension of the definitions of (I, J)-cofinite modules and I−weakly cofinite modules ([4]). Definition 2.4. An R-module M is said (I, J)-weakly cofinite if Supp(M ) ⊆ W (I, J) and ExtiR (R/I, M ) is weakly Laskerian for all i ≥ 0. From the definition of (I, J)-weakly cofinite modules we have the following immediate consequence. Corollary 2.5. i) Every (I, J)-cofinite module is a (I, J)-weakly cofinite module. ii) If Supp(M ) ⊆ W (I, J) and M is weakly Laskerian, then M is (I, J)-weakly cofinite. Theorem 2.6. Let M be an R-module and d a non-negative intei ger such that HI,J (M ) is (I, J)-weakly cofinite for all i ≤ d. Then i ExtR (R/I, M ) is weakly Laskerian for all i ≤ d. Proof. We now proceed by induction on d. When d = 0, the short exact sequence 0 → ΓI,J (M ) → M → M/ΓI,J (M ) → 0 induces an exact sequence 0 → HomR (R/I, ΓI,J (M )) → HomR (R/I, M ) → HomR (R/I, M/ΓI,J (M )). Since M/ΓI,J (M ) is (I, J)-torsion free, it is also I-torsion free and then HomR (R/I, M/ΓI,J (M )) = 0. It follows HomR (R/I, M ) ∼ = HomR (R/I, ΓI,J (M )). Hence HomR (R/I, M ) is a weakly Laskerian R-module. i i Let d > 0. Note that HI,J (M ) ∼ (M/ΓI,J (M )) for all i > 0. Let = HI,J M = M/ΓI,J (M ) and E(M ) denote the injective hull of M . From the short exact sequence 0 → M → E(M ) → E(M )/M → 0 we get and ExtiR (R/I, E(M )/M ) ∼ = Exti+1 R (R/I, M ) i+1 i HI,J (E(M )/M ) ∼ (M ) = HI,J Some results on local cohomology modules with respect to a pair of ideals. 5 i for all i ≥ 0. It follows from the hypothesis that HI,J (E(M )/M ) is (I, J)-weakly cofinite for all i ≤ d − 1. By the inductive hypothesis ExtiR (R/I, E(M )/M ) is weakly Laskerian for all i ≤ d − 1 and then ExtiR (R/I, M ) is also weakly Laskerian for all i ≤ d. Now the short exact sequence 0 → ΓI,J (M ) → M → M → 0 gives rise to a long exact sequence · · · → ExtiR (R/I, ΓI,J (M )) → ExtiR (R/I, M ) → ExtiR (R/I, M ) → · · · . Since ΓI,J (M ) is (I, J)-weakly cofinite, ExtiR (R/I, ΓI,J (M )) is weakly Laskerian for all i ≥ 0. Finally, it follows from the long exact sequence that ExtiR (R/I, M ) is also weakly Laskerian for all i ≤ d. Theorem 2.7. Let M be R-module such that ExtiR (R/I, M ) is weakly i (M ) is (I, J)Laskerian for all i and d a non-negative integer. If HI,J d weakly cofinite for all i = d, then HI,J (M )) is also (I, J)-weakly cofinite. Proof. We use induction on d. When d = 0, set M = M/ΓI,J (M ), then the short exact sequence 0 → ΓI,J (M ) → M → M → 0 gives rise a long exact sequence · · · → ExtiR (R/I, ΓI,J (M )) → ExtiR (R/I, M ) → ExtiR (R/I, M ) → · · · . 0 i i (M ) = 0. From the (M ) ∼ (M ) for all i > 0 and HI,J We have HI,J = HI,J i hypothesis, HI,J (M ) is (I, J)-weakly cofinite for all i ≥ 0. It follows from 2.6 that ExtiR (R/I, M ) is weakly Laskerian for all i ≥ 0. Therefore, combining the long exact sequence and the hypothesis gives that 0 ExtiR (R/I, ΓI,J (M )) is weakly Laskerian. This implies that HI,J (M ) is (I, J)-weakly cofinite. Let d > 0. The short exact sequence 0 → M → E(M ) → E(M )/M → 0 yields ExtiR (R/I, E(M )/M ) ∼ = Exti+1 R (R/I, M ) and i+1 i (E(M )/M ) ∼ HI,J (M ) = HI,J i for all i ≥ 0. Then HI,J (E(M )/M ) is (I, J)-weakly cofinite for all i i = d − 1. Note that ExtR (R/I, ΓI,J (M )) is weakly Laskerian and then 6 TRAN TUAN NAM, NGUYEN MINH TRI ExtiR (R/I, E(M )/M ) is weakly Laskerian for all i ≥ 0. By the inducd−1 tive hypothesis HI,J (E(M )/M ) is (I, J)-weakly cofinite. Therefore d HI,J (M ) is (I, J)-weakly cofinite. Combining 2.1(ii) with 2.7 we obtain the following consequence. Corollary 2.8. Let M be a weakly Laskerian R-module and d a noni negative integer. If HI,J (M ) is (I, J)-weakly cofinite for all i = d, then d HI,J (M )) is also (I, J)-weakly cofinite. Corollary 2.9. Let I be a principal ideal of R and M a weakly Laskei rian module. Then HI,J (M ) is (I, J)-weakly cofinite for all i ≥ 0. i (M ) = 0 for all i > 1. MoreProof. It follows from [7, 4.11] that HI,J 0 0 over, HI,J (M ) is a weakly Laskerian R−module, since HI,J (M ) is a i submodule of M. That means HI,J (M ) is (I, J)-weakly cofinite for all i = 1. Now the conclusion follows from 2.7. 3. On Serre subcategory Recall that the class S of R-modules is a Serre subcategory of the category of R-modules if it is closed under taking submodules, quotients and extensions. Theorem 3.1. Let M be an R−module and d a non-negative integer. i If HI,J (M ) ∈ S for all i < d, then ExtiR (R/I, M )) ∈ S for all i < d. Proof. We begin by considering functors F = HomR (R/I, −) and G = ΓI,J (−). It is clear that F G = Hom(R/I, −). By [6, 11.38] we have a Grothendieck spectral sequence q p+q E2p,q = ExtpR (R/I, HI,J (M )) ⇒ ExtR (R/I, M ). p E2p,q i Since HI,J (M ) ∈ S for all i < d, ∈ S for all p ≥ 0, 0 ≤ q < d. We consider homomorphisms of the spectral for all p ≥ 0, 0 ≤ t < d and i ≥ 2 dp−i,t+i−1 dp,t i Eip−i,t+i−1 i −→ Eip,t −→ Eip+i,t−i+1 . p−i+1,t+i−2 Note that Eip,t = Ker dp,t and Eip,j = 0 for all j < 0. i−1 / Im di−1 p,t−p ∼ p,t−p ∼ This implies Ker dp,t−p for all 0 ≤ p ≤ t. We = ... ∼ = E∞ t+2 = Et+2 t t now have a filtration Φ of H = ExtR (R/I, M ) such that 0 = Φt+1 H t ⊆ Φt H t ⊆ . . . ⊂ Φ1 H t ⊆ Φ0 H t = ExttR (R/I, M ) and i,t−i Φi H t /Φi+1 H t = E∞ Some results on local cohomology modules with respect to a pair of ideals. 7 for all 0 ≤ i ≤ t. Then there is a short exact sequence i,t−i 0 → Φi+1 H t → Φi H t → E∞ → 0. i,t−i i,t−i i,t−i ∼ From the proof above we have E∞ = Et+2 ∼ = Ker dt+2 a subquotient i,t−i ∈ S for of E2i,t−i and E2i,t−i ∈ S for all 0 ≤ i ≤ t. It follows that E∞ i t all 0 ≤ i ≤ t. By induction on i we get Φ H ∈ S for all 0 ≤ i ≤ t. Finally ExttR (R/I, M ) ∈ S for all t < d. Theorem 3.2. Let M be a finitely generated module over a local ring i (R, m) and d a non-negative integer. If HI,J (M ) ∈ S for all i < d, then d HomR (R/m, HI,J (M )) ∈ S. Proof. The proof is by induction on d. When d = 0, since M is finitely 0 0 generated, so is HI,J (M ). Hence HomR (R/m, HI,J (M ))) has finite length 0 and then HomR (R/m, HI,J (M ))) ∈ S by [1, 2.11]. i i Let d > 0. It follows from [7, 1.13 (4)] that HI,J (M ) ∼ (M/ΓI,J (M )) = HI,J for all i > 0. Thus we can assume, by replacing M with M/ΓI,J (M ), that M is (I, J)-torsion-free. Since ΓI (M ) ⊆ ΓI,J (M ) = 0, it follows that M is also I-torsion-free. Hence, there exists an element x ∈ I which is non-zerodivisor on M. Set M = M/xM, the short exact sequence .x 0→M →M →M →0 gives rise to a long exact sequence g f .x .x d−1 d−1 d−1 d d (M ) → HI,J (M ) → · · · . · · · → HI,J (M ) → HI,J (M ) → HI,J (M ) → HI,J i i As HI,J (M ) ∈ S for all i < d, HI,J (M ) ∈ S for all i < d − 1. Then d−1 HomR (R/m, HI,J (M )) ∈ S by the inductive hypothesis. Applying the functor HomR (R/m, −) to the short exact sequence d−1 (M ) → Im g → 0 0 → Im f → HI,J we get a long exact sequence d−1 0 → HomR (R/m, Im f ) → HomR (R/m, HI,J (M )) → → HomR (R/m, Im g) → Ext1R (R/m, Im f ) → · · · . Note that Ext1R (R/m, Im f ) ∈ S, so HomR (R/m, Im g)) ∈ S. Now from the exact sequence .x d d (M ) 0 → Im g → HI,J (M ) → HI,J we obtain an exact sequence .x d d 0 → HomR (R/m, Im g) → HomR (R/m, HI,J (M )) → HomR (R/m, HI,J (M )). 8 TRAN TUAN NAM, NGUYEN MINH TRI .x d d It is clear that Im(HomR (R/m, HI,J (M )) → HomR (R/m, HI,J (M ))) = d ∼ 0. Hence HomR (R/m, Im g) = HomR (R/m, HI,J (M )) and then d HomR (R/m, HI,J (M )) ∈ S. The proof is complete. It should be mentioned that if M is a finitely generated module over a local ring (R, m) with Supp(M ) ⊆ {m}, then M is artinian. From 3.2 we obtain the following consequence. Corollary 3.3. Let M be a finitely generated module over a local ring i (R, m) and d a non-negative integer. If HI,J (M ) is finitely generated d for all i < d, then HomR (R/m, HI,J (M )) has finite length. d (M )) is finitely generProof. It follows from 3.2 that HomR (R/m, HI,J d ated. Moreover Supp(HomR (R/m, HI,J (M ))) ⊆ {m}. Therefore d (M )) is an artinian R-module and then it has finite HomR (R/m, HI,J length. Acknowledgments. We would like to express our gratitude to Professor Nguyen Tu Cuong for his support and advice. The first author is partially supported by the Vietnam Institute for Advanced Study in Mathematics (VIASM), Hanoi, Vietnam. References [1] M. Asgharzadeh, M. Tousi, ”A unified approach to local cohomology modules using Serre classes”, Canad. Math. Bull. 53 (2010) 577-586. [2] N. T. Cuong, N. V. Hoang, ”On the vanishing and the finiteness of supports of generalized local cohomological modules,” Manuscripta Math. 126 (2008) 59-72. [3] K. Divaani-Aazar, A. Mafi, ”Associated primes of local cohomology modules”, Proc. Amer. Math. Soc, 133 (2004) 655-660. [4] K. Divaani-Aazar and A. Ma, ”Associated primes of local cohomology modules of weakly Laskerian modules”, Communications in Algebra, 34 (2006) 681-690. [5] A. Grothendieck, Cohomologie local des faisceaux coherents et thormes de Lefschetz locaux et globaux (SGA2), North-Holland, Amsterdam, 1968. [6] J. Rotman, An introduction to homological algebra, 2nd edition, Springer, 2009. [7] R. Takahashi, Y. Yoshino, T. Yoshizawa, ”Local cohomology based on a nonclosed support defined by a pair of ideals”, J. Pure Appl. Algebra, 213 (2009) 582-600. [8] A. Tehranian, A. Pour Eshmanan Talemi, ”Cofinite of local cohomology based on a non-closed support defined by a pair of ideals”, Bulletin of the Iranian Mathematical Society Vol. 36 No. 2 (2010), 145-155. [9] N. Zamani, ”Results on local cohomology of weakly Laskerian modules”, Journal of Algebra and Its Applications, Vol. 10, No. 2 (2011) 303-308. Some results on local cohomology modules with respect to a pair of ideals. 9 Department of Mathematics-Informatics, Ho Chi Minh University of Pedagogy, Ho Chi Minh city, Viet Nam. E-mail address: namtuantran@gmail.com Department of Natural Science Education, Dong Nai University, Dong Nai, Viet Nam. E-mail address: triminhng@gmail.com ... Yoshizawa, Local cohomology based on a nonclosed support defined by a pair of ideals , J Pure Appl Algebra, 213 (2009) 582-600 [8] A Tehranian, A Pour Eshmanan Talemi, ”Cofinite of local cohomology. .. Laskerian modules , Journal of Algebra and Its Applications, Vol 10, No (2011) 303-308 Some results on local cohomology modules with respect to a pair of ideals Department of Mathematics-Informatics,... Divaani-Aazar and A Ma, ”Associated primes of local cohomology modules of weakly Laskerian modules , Communications in Algebra, 34 (2006) 681-690 [5] A Grothendieck, Cohomologie local des faisceaux