COFINITENESS OF LOCAL COHOMOLOGY MODULES IN THE CLASS OF MODULES IN DIMENSION LESS

. Let n be a non-negative integer, R a commutative Noetherian ring with dim( R ) ≤ n + 2, a an ideal of R , and X an arbitrary R -module. In this paper, we ﬁrst prove that X is an (FD <n , a )-coﬁnite R -module if X is an a -torsion R -module such that Hom R (cid:0) R a ,X (cid:1) and Ext 1 R (cid:0) R a ,X (cid:1) are FD <n R -modules. Then, we show that H i a ( X ) is an (FD <n , a )-coﬁnite R -module and { p ∈ Ass R (H i a ( X )) : dim (cid:0) R p (cid:1) ≥ n } is a ﬁnite set for all i when Ext iR (cid:0) R a ,X (cid:1) is an FD <n R -module for all i ≤ n + 2. As a consequence, it follows that Ass R (H i a ( X )) is a ﬁnite set for all i whenever R is a semi-local ring with dim( R ) ≤ 3 and X is an FD < 1 R -module. Finally, we observe that the category of (FD <n , a )-coﬁnite R -modules forms an Abelian subcategory of the category of R -modules.

R a , X and Ext 1 R R a , X are FD<n R-modules. Then, we show that H i a (X) is an (FD<n, a)-cofinite R-module and {p ∈ Ass R (H i a (X)) : dim R p ≥ n} is a finite set for all i when Ext i R R a , X is an FD<n R-module for all i ≤ n + 2. As a consequence, it follows that Ass R (H i a (X)) is a finite set for all i whenever R is a semi-local ring with dim(R) ≤ 3 and X is an FD <1 R-module. Finally, we observe that the category of (FD<n, a)-cofinite R-modules forms an Abelian subcategory of the category of R-modules.

Introduction
We adopt throughout the following notation: let R denote a commutative Noetherian ring with non-zero identity, a and b ideals of R, M a finite (i.e., finitely generated) R-module, X an arbitrary R-module which is not necessarily finite, and n a non-negative integer. We refer the reader to [7,8,23] for basic results, notations, and terminology not given in this paper.
Hartshorne, in [14], defined an a-torsion R-module X to be a-cofinite if the R-module Ext i 192 A. VAHIDI AND M. PAPARI-ZAREI There have been many attempts in the literature to study the above questions. Hartshorne in [14,Proposition 7.6 and Corollary 7.7] showed that the answer to these questions is yes if R is a complete regular local ring and a is a prime ideal of R with dim R a ≤ 1. Huneke  show that the answer to Questions 1.4-1.6 is also yes if dim(R) ≤ n + 2. As a consequence, we provide an affirmative answer to Question 1.3 for the case that R is a semi-local ring with dim(R) ≤ 3. This result is a generalization of Marley's result in [19] where he showed that the answer to Question 1.3 is yes if R is a local ring with dim(R) ≤ 3 (see [19, Proposition 1.1 and Corollary 2.5]).
In the main result of Section 2, we observe that if dim(R) ≤ n + 2 and X is an a-torsion R-module such that Hom R R a , X and Ext 1 R R a , X are FD <n R-modules, then X is an (FD <n , a)-cofinite R-module. Section 3 is devoted to the study of Questions 1.5 and 1.6. We show that H i a (X) is an (FD <n , a)-cofinite R-module and Ass R (H i a (X)) ≥n is a finite set for all i whenever dim(R) ≤ n + 2 and Ext i R R a , X is an FD <n R-module for all i ≤ n+2 (e.g., X is an FD <n R-module).

It follows that if R is a semi-local ring with dim(R) ≤ 3 and Ext
is an aweakly cofinite R-module and Ass R (H i a (X)) is a finite set for all i. Recall that X is said to be an a-weakly cofinite R-module if X is an a-torsion R-module and the set of associated prime ideals of any quotient module of Ext i R R a , X is finite for all i (see [12,Definition 2.1] and [13,Definition 2.4]). In Section 4, with respect to Question 1.4, we prove that when dim(R) ≤ n + 2, the category of (FD <n , a)-cofinite R-modules forms an Abelian subcategory of the category of R-modules.

A criterion for cofiniteness
The following two lemmas will be useful in the proof of the main result of this section. Note that when bX = 0, X is an FD <n R-module if and only if X is an FD <n R b -module.

Lemma 2.1. Let t be a non-negative integer and let X be an
Proof. We prove this by using induction on t. The case t = 0 is clear from the isomorphisms Suppose that t > 0 and that t−1 is settled. It is enough to show that Ext t [23,Theorem 11.65], there is a spectral sequence Let r ≥ 2 and set B t,0 a+b , X is an R-module. Thus, as we noted at the beginning of this section, Ext t R R a+b , X is an FD <n R b -module and hence φ t H t is an FD <n R b -module. Therefore E t,0 Let t be a non-negative integer and let X be an R-module such that bX = 0 and Ext i Proof. From [23,Theorem 11.65], there is a spectral sequence There exists a finite filtration We are now ready to state and prove the main result of this section, which plays an important role in Sections 3 and 4 to study Questions 1.4-1.6. Theorem 2.3. Suppose that dim(R) ≤ n + 2 and X is an a-torsion R-module such that Hom R R a , X and Ext 1 R R a , X are FD <n R-modules. Then X is an (FD <n , a)-cofinite R-module.
Proof. Assume that a is nilpotent. Then a t = 0 for some integer t. By [15,Proposition 3.4], Hom R R a t , X is an FD <n R-module and hence X = (0 : X a t ) is an (FD <n , a)-cofinite R-module. Now, assume that a is not nilpotent. Since Γ a (R) is finite, there is an integer t such that (0 : R a t ) = Γ a (R). Set b := (0 : R a t ) and Y := X (0: X a t ) . It is easy to see that bY = 0, Y is an (a + b)-torsion R-module, and dim R a+b ≤ n + 1.  (FD <n , a)-cofinite R-module from the above short exact sequence.
The following corollary is an immediate application of the above theorem.  (FD <n , a)-cofinite R-module.
Proof. By the short exact sequence

Cofiniteness and associated primes of local cohomology modules
The following is the main result of this section; it shows that the answer to Questions 1.5 and 1.6 is yes if dim(R) ≤ n + 2. (iii)⇒(i). We first show that if t is a non-negative integer such that Ext i R R a , X is an FD <n R-module for all i ≤ t+1, then H i a (X) is an (FD <n , a)-cofinite R-module for all i ≤ t. We prove this by using induction on t. The case t = 0 follows from Corollary 2.4. Suppose that t > 0 and that t−1 is settled. It is enough to show that H t a (X) is an (FD <n , a) Suppose that dim(R) ≤ n + 2, X is an arbitrary R-module, and t is a non-negative integer such that Ext i R R a , X is an FD <n R-module for all i ≤ t + 1 (resp. for all i ≤ n + 2). Then H i a (X) is an (FD <n , a)-cofinite R-module for all i ≤ t (resp. for all i). In particular, Ass R (H i a (X)) ≥n is a finite set for all i ≤ t (resp. for all i).
Proof. The first assertion follows from the proof of Theorem 3.1. The last assertion follows by the first one and [8, Exercise 1.2.28].
We have the following corollaries by taking n = 0 in Theorem 3.1 and Corollary 3.2. [21,Theorem 7.10]). Suppose that dim(R) ≤ 2 and X is an arbitrary R-module. Then the following statements are equivalent:

Corollary 3.3 (see
If R is a local ring with dim R a ≤ 2, then the answer to Question 1.3 is yes by Bahmanpour-Naghipour's result [6, Theorem 3.1] (see also [20,Theorem 3.3(c)]). In [24,Corollary 5.6], the first author and Morsali generalized this result to arbitrary semi-local rings. In the next result, by putting n = 1 in Corollary 3.2, we provide an affirmative answer to Question 1.3 for the case that R is a semi-local ring with dim(R) ≤ 3. Note that our result is a generalization of Marley's result in [19], where he showed that if R is a local ring with dim(R) ≤ 3 and M is a finite R-module, then Ass R (H i a (M )) is a finite set for all i (see [19, Proposition 1.1 and Corollary 2.5]). Note also that, if R is a semi-local ring and X is an (FD <1 , a)cofinite R-module, then X is an a-weakly cofinite R-module by [5,Theorem 3.3].

Corollary 3.5.
Suppose that R is a semi-local ring with dim(R) ≤ 3, X is an arbitrary R-module, and t is a non-negative integer such that Ext i R R a , X is an FD <1 R-module for all i ≤ t + 1 (resp. for all i ≤ 3). Then H i a (X) is an a-weakly cofinite R-module for all i ≤ t (resp. for all i). In particular, Ass R (H i a (X)) is a finite set for all i ≤ t (resp. for all i).

Abelianness of the category of cofinite modules
The following theorem is the main result of this section; it shows that the answer to Question 1.4 is also yes if dim(R) ≤ n + 2. Proof. The proof is similar to that of [24,Theorem 3.1]. We bring it here for the sake of completeness. Assume that X and Y are (FD <n , a)-cofinite R-modules and f : X → Y is an R-homomorphism. We show that ker f , im f , and coker f are (FD <n , a) As an immediate application of the above theorem, we have the following corollary.

Corollary 4.2.
Suppose that dim(R) ≤ n + 2, N is a finite R-module, and X is an (FD <n , a)-cofinite R-module. Then Ext j R (N, X) and Tor R j (N, X) are (FD <n , a)cofinite R-modules for all j.
We have the following results by taking n = 0 in Theorem 4.1 and Corollary 4.2.