RESULTS OF GENERALIZED LOCAL COHOMOLOGY WITH RESPECT TO A PAIR OF IDEALS

Let (I, J) be a pair of ideals of a commutative Noetherian local ring R, and M a finitely generated module. Let t be a positive integer. We prove that (i) ifH i I,J(M) is minimax for all i < t, thenH i I,J(M) is (I, J)-cofinite for all i < t and Hom ( R/I,H I,J(M) ) is finitely generated; (ii) if a ∈ W̃ (I, J) and H i I,J(M) is minimax for all i < t, then Ext i R(R/a, T ) is minimax for all i < t. We also prove that if SuppH i I,J(M) = {m} for all i < t, then H i I,J(M) is Artinian and (I, J)-cofinite for all i < t. AMS Subject Classification: 13D45, 13E05, 14B15


Introduction
Throughout this paper, R is denoted a commutative Noetherian ring, I, and J are denoted two ideals of R, and M is a finitely generated R-module.For notations and terminologies that is not given in this paper, the reader is referred to [1].
As a generalization of the ordinary local cohomology modules, Takahashi, Yoshino and Yoshizawa [7] introduced the local cohomology modules with respect to a pair of ideals (I, J).To be more precise, let W (I, J) = {p ∈ Spec(R) | Received: October 22, 2014 c 2015 Academic Publications, Ltd. url: www.acadpubl.euI n ⊆ p + J for some positive integer n}.Then for an R-module M , the (I, J)torsion submodule Γ I,J (M ) of M , which consists of all elements x of M with SuppRx ⊆ W (I, J), is considered.It is known, Γ I,J ( ) is a left exact additive functor from the category of all R-modules and R-homomorphism to itself.For all integer i, the i-th local cohomology functor H i I,J with respect to (I, J) is defined to be the i-th right derived functor of Γ I,J ( ).The i-th local cohomology module of M with respect to (I, J) is denoted by H i I,J (M ).When J = 0, then H i I,J coincides with the usual local cohomology functor H i I with the support in the closed subset V (I).
There are many questions about ordinary local cohomology modules.In particular, Hunke [5] proposed the following question: for an integer i, when is H i I (M ) Artinian?In [3] Grothendieck conjectured that for any finite Rmodule M , Hom R (R/I, H i I (M )) is finite for all i.Hartshorne [4] later refined this conjecture, and proposed the following one: Let M be a finite R-module, and let I be an ideal of R. Then is finite, for every i ≥ 0 and j ≥ 0. The purpose of this paper is to investigate a similar question as above for this generalized version of local cohomology.
Obviously, these results are true in the category of all graded R-modules and homogeneous R-homomorphisms.

The Results
Definition 1.An R-module T is called (I, J)-cofinite if SuppT ⊆ W (I, J) and Ext i R (R/I, T ) is a finite R-modules, for every i ≥ 0. Lemma 2. Let T be an arbitrary R-module.Then the following statements hold: (iv) Let SuppT ⊆ V (I).If there is an element x ∈ I, such that (0 : T x) is Artinian and (I, J)-cofinite, then T is Artinian and (I, J)-cofinite.
(iii) Since, in view of the hypothesis, Hom(R/I, T ) has finite length and SuppT ⊆ V (m) ⊆ V (I).By using [6, Proposition 4.1], we can get Ext i R (R/I, T ) is finitely generated for all i ≥ 0 thus the proof is completed.
(iv) In view of (ii), (0 : T x) is Artinian and I-cofinite.Therefore, according to [6, Proposition 4.1] T is Artinian and I-cofinite; and hence the result follows.
Therefore, in view of lemma (2, ii) and [6,Corollary 4.4], the class of (I, J)cofinite Artinian modules is closed under taking submodules, quotients and extensions.
Definition 3.An R-module T is said to be minimax module, if there is a finitely generated submodule T 1 of T such that the quotient module T /T 1 is Artinian.The class of minimax modules includes all finite and all Artinian modules.Theorem 4. Let T be an arbitrary R-module and t a positive integer and let a ∈ W (I, J), where W (I, J) denote the set of ideals a of R such that I n ⊆ a+J for some integer n.If H i I,J (T ) is Artinian (minimax) for all 0 ≤ i < t, then Ext i R (R/a, T ) is Artinian (minimax)for all 0 ≤ i < t.
Proof.We prove by induction on t.The case where t = 1 yields Γ I,J (T ) is Artinian (minimax).In view of proof [7 ,Theorem 3.2], Γ a (T ) ⊆ Γ I,J (T ).Thus Γ a (T ) is Artinian (minimax).In particular, Hom(R/a, T ) is Artinian (minimax).Assume that t ≥ 2 and the Theorem holds true for t − 1.It follows from [7,Corollary 1.13] that H i I,J (T ) ∼ = H i I,J (T /Γ I,J (T )) for all i > 0. Also, T /Γ I,J (T ) is an (I, J)-torsion-free R-module.Hence we can (and do) assume that M is an (I, J)-torsion-free R-module.Thus Γ I,J (T ) = 0 implies that Γ a (T ) = 0. Then there exists x ∈ a such that x is an T -sequence.Now, we may consider the exact sequence 0 −→ T x −→ T −→ T /xT −→ 0 to obtain the exact sequences Now, the above exact sequences are used in conjunction with the inductive hypothesis to see that the R-modules H i I,J (T /xT ) and Ext i R (R/a, T ) and Ext i R (R/a, T /xT ) are Artinian (minimax) for all i < t − 1.It is enough to show that Ext t−1 R (R/a, T ) is Artinian (minimax).Finally, note that x ∈ a, and ( * ) exact sequence yields that the sequence ) is Artinian (minimax), as required.
Lemma 5. Let T is a minimax R-module with Supp in W (I, J).Then the following hold: (ii) If there is an element x ∈ I, such that (0 : T x) is minimax and (I, J)cofinite, then T is minimax and (I, J)-cofinite.
Proof.(i) Let T 1 be a finitely generated submodule of T , such that T 2 = T /T 1 is Artinian and suppose that Hom( R I , T ) is finitely generated.The exactness of implies that Hom(R/I, T 2 ) is finitely generated.Hence we get from Lemma (2, iii) and [2, Lemma 2.1] that T 2 is Artinian and (I, J)-cofinite, therefore T is also (I, J)-cofinite.
(ii) If (o : T x) minimax and (I, J)-cofinite, then Hom(R/I, (0 : T x) ∼ = Hom(R/I, T ) is finitely generated.It is clear by (i).Theorem 6.Let t be a non-negative integer, such that H i I,J (M ) is (I, J)cofinite minimax for all i < t.Then, the R-module Hom R/I, H i I,J (M ) is finitely generated for all i ≤ t.
Proof.Since H i I,J (M ) is (I, J)-cofinite, Hom R/I, H i I,J (M ) is finitely generated for all i < t.So it is enough to prove that Hom R/I, H t I,J (M ) is finitely generated.We prove by induction on t ≥ 0. If t = 0 then the result is clear.Assume that t > 0 and the result holds true for t − 1.It follows from [7, Corollary 1.13] that H i I,J (M ) ∼ = H i I,J M/Γ I,J (M ) for all i > 0. Also M/Γ I,J (M ) is (I, J)-torsion-free R-module.Since Γ I (M ) ⊆ Γ I,J (M ).We can assume that M is an I-torsion-free R-module.Then there exists an element x ∈ I which is M -regular.The exact sequence 0 is Artinian and (I, J)-cofinite for all i < t − 1.Therefore, by Theorem (6) Hom R/I, H t−1 I,J (M ) is finitely generated R-module.Since SuppH t−1 I,J (M ) = {m}, it follows that it yields from Lemma (2, iv) that, H t−1 I,J (M ) is Artinian and (I, J)-cofinite.Now assertion follows from Theorem (6).