Cohomological dimension, cofiniteness and Abelian categories of cofinite modules

Accepted Manuscript Cohomological dimension, cofiniteness and Abelian categories of cofinite modules

Kamal Bahmanpour

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S0021-8693(17)30260-0 http://dx.doi.org/10.1016/j.jalgebra.2017.04.019 YJABR 16207

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1 December 2016

Please cite this article in press as: K. Bahmanpour, Cohomological dimension, cofiniteness and Abelian categories of cofinite modules, J. Algebra (2017), http://dx.doi.org/10.1016/j.jalgebra.2017.04.019

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COHOMOLOGICAL DIMENSION, COFINITENESS AND ABELIAN CATEGORIES OF COFINITE MODULES KAMAL BAHMANPOUR Abstract. Let R be a commutative Noetherian ring, I ⊆ J be ideals of R and M be a finitely generated R-module. In this paper it is shown that q(J, M ) ≤ q(I, M )+cd(J, M/IM ). Furthermore, it is shown that, for any ideal I of R and any finitely generated R-module M with q(I, M ) ≤ 1, the local cohomology modules HIi (M ) are I-cofinite for all integers i ≥ 0. As a consequence of this result it is shown that, if q(I, R) ≤ 1, then for any finitely generated R-module M , the local cohomology modules HIi (M ) are I-cofinite for all integers i ≥ 0. Finally, it is shown that the category of all I-cofinite R-modules C (R, I)cof is an Abelian subcategory of the category of all R-modules, whenever (R, m) is a complete Noetherian local ring and I is an ideal of R with q(I, R) ≤ 1. These assertions answer affirmatively two questions raised by R. Hartshorne in [Affine duality and cofiniteness, Invent. Math. 9(1970), 145-164], in the some special case.

1. Introduction Throughout, let R denote a commutative Noetherian ring and I be an ideal of R. The local cohomology modules HIi (M ), i = 0, 1, 2, . . . , of an R-module M with respect to I were introduced by Grothendieck in [G]. They arise as the derived functors of the left exact functor ΓI (−), where for an R-module M , ΓI (M ) is the submodule of M consisting of all elements  n annihilated by some power of I, i.e., ∞ n=1 (0 :M I ). There is a natural isomorphism: HIi (M )  lim ExtiR (R/I n , M ). −→ n≥1

We refer the reader to [G] or [BS] for more details about local cohomology. For an R-module M , the notion cd(I, M ), the cohomological dimension of M relative to I, is defined to be the greatest integer i such that HIi (M ) = 0 if there exist such i s and −∞ otherwise. Divaani-Aazar et al [DNT] proved that if M and N are two finitely generated R-modules such Date: 1 December 2016. 2010 Mathematics Subject Classification. Primary 13D45; Secondary 14B15, 13E05 This research of the author was supported by a grant from IPM (No. 95130022). Key words and phrases. Abelian category, cofinit module, cohomological dimension, local cohomology, Noetherian ring, Serre category. 1

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that M is supported in Supp N , then cd(I, M ) ≤ cd(I, N ). Hartshorne [Ha2] has defined the notion q(I, R) as the greatest integer i such that HIi (R) is not Artinian. Dibaei and Yassemi [DY] extended this notion to arbitrary R-modules, to the effect that for any R-module N they defined q(I, N ) as the greatest integer i such that HIi (N ) is not Artinian if there exist such i s and −∞ otherwise. Among other things, they showed that if M and N are two finitely generated R-modules such that M is supported in Supp N , then q(I, M ) ≤ q(I, N ). Recall that, a subcategory M of the category of all R-modules is said to be a Serre category if in any short exact sequence of R-modules and Rhomomorphisms, the middle module is in M if and only if the two other modules are in M . Now let M be a Serre category of modules. Our objective in section 2 of this paper is to investigate some properties of a new generalization of the notions cd(I, M ) and q(I, M ). More precisely, we define the M -cohomological dimension of M relative to I, denoted by M -cd(I, M ), as the greatest integer i such that HIi (M ) is not in M if there exist such i s and −∞ otherwise. Note that with this new notion, M -cd(I, M ) denotes the notion cd(I, M ) whenever M is equal to the zero Serre category. Also, M -cd(I, M ) denotes the notion q(I, M ) whenever M is equal to the Serre category of all Artinian R-modules. In section 2, among other things, as a generalization of the results given in [DNT] and [DY], we show that if M and N are two finitely generated R-modules such that M is supported in Supp N , then M -cd(I, M ) ≤ M -cd(I, N ). Also, it shown that if I ⊆ J are two ideals of R, then q(J, M ) ≤ q(I, M ) + cd(J, M/IM ), for any finitely generated R-module M . Hartshorne in [Ha1] defined an R-module L to be I-cofinite, if Supp L ⊆ V (I) and ExtiR (R/I, L) is a finitely generated module for all i. Now, for each R-module M we define q(I, M ) = M - cd(I, M ), where M denotes the Serre category of all Artinian I-cofinite R-modules. Our main goal in section 3 of this paper is to establish some results for the torsion functors of the local cohomology modules HIt (M ) with t ∈ {q(I, M ), q(I, M )}. Hartshorne in [Ha1] also posed the following questions: (i) For which rings R and ideals I are the modules HIi (M ) I-cofinite for all i and all finitely generated modules M ?

COHOMOLOGICAL DIMENSION, COFINITENESS AND ABELIAN CATEGORIES 3

(ii) Whether the category C (R, I)cof of I-cofinite modules is an Abelian subcategory of the category of all R-modules? That is, if f : M −→ N is an R-homomorphism of I-cofinite modules, are ker f and cokerf I-cofinite? With respect to the question (i), Hartshorne in [Ha1] and later Chiriacescu in [Ch] showed that if R is a complete regular local ring and I is a prime ideal such that dim R/I = 1, then HIi (M ) is I-cofinite for any finitely generated R-module M . Also, Delfino and Marley [DM] and Yoshida [Yo] have eliminated the complete hypothesis entirely. Finally, the local condition on the ring has been removed in [BN]. With respect to the question (ii), Hartshorne gave a counterexample to show that this question has not an affirmative answer in general, (see [Ha1, §3]). On the positive side, Hartshorne proved that if I is a prime ideal of dimension one in a complete regular local ring R, then the answer to his question is yes. On the other hand, in [DM], Delfino and Marley extended this result to arbitrary complete local rings. Kawasaki in [K2] generalized the Delfino and Marley’s result for an arbitrary ideal I of dimension one in a local ring R. Finally, Melkersson in [Me1] generalized the Kawasaki’s result for all ideals of dimension one of any arbitrary Noetherian ring R. Recently, in [BNS] as a generalization of Melkersson’s result it is shown that for any ideal I in a commutative Noetherian ring R, the category of all I-cofinite R-modules M with dim M ≤ 1 is an Abelian subcategory of the category of all R-modules. Recall that, for any proper ideal I of R, the arithmetic rank of I, denoted by ara(I), is the least number of elements of I required to generate an ideal which has the same radical as I. Kawasaki in [K1] proved that if ara(I) = 1 then the category C (R, I)cof of I-cofinite modules is an Abelian subcategory of the category of all Rmodules. Pirmohammadi et al in [PAB] as a generalization of Kawasaki’s result proved that if I is an ideal of a Noetherian local ring with cd(I, R) ≤ 1, then the category C (R, I)cof of I-cofinite modules is an Abelian subcategory of the category of all R-modules. In section 4 of this paper, as a new approach to solving problem (i) in the some special case, it is shown that for any ideal I of a Noetherian ring R and any finitely generated R-module M with q(I, M ) ≤ 1, the local cohomology

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modules HIi (M ) are I-cofinite for all integers i ≥ 0. As a consequence of this result it is shown that if q(I, R) ≤ 1, then for any finitely generated R-module M , the local cohomology modules HIi (M ) are I-cofinite for all integers i ≥ 0. In section 5 of this paper we present a partially affirmative answer for the Question (ii). More precisely, it is shown that for any ideal I of a complete Noetherian local ring (R, m) with q(I, R) ≤ 1, the category C (R, I)cof of Icofinite modules is an Abelian subcategory of the category of all R-modules. For any R-module N , we use the notation ER (N ) for the injective envelope of the R-module N . For every R-module L, we denote by mAssR L the set of minimal elements of AssR L with respect to inclusion. For any ideal a of R, we denote {p ∈ Spec R : p ⊇ a} by V (a). Also, for any ideal b of R, the radical of b, denoted by Rad(b), is defined to be the set {x ∈ R : xn ∈ b  the for some n ∈ N}. For any Noetherian local ring (R, m), we denote by R m-adic completion of R. Finally, we denote by Max(R) the set of all maximal ideals of R. For any unexplained notation and terminology we refer the reader to [BS] and [Ma]. 2. Some properties of local cohomology functors Recall that according to the Gruson’s Theorem for a given finitely generated faithful module N over a commutative ring R, every finitely generated R-module admits a finite filtration of submodules whose factors are quotients of direct sums of copies of N . This theorem has an interesting corollary as follows: Corollary. Let R be a Noetherian ring and N and M be two finitely generated R-modules. If Supp M ⊆ Supp N , then there exists a finite filtration of M whose factors are quotients of direct sums of copies of N . Using an alternative easy proof we shall improve this consequence of the Gruson’s Theorem in lemma 2.2. To do this we start this section with following well known lemma. Lemma 2.1. Let R be a Noetherian ring and let M and N be two finitely generated R-modules. Then AssR HomR (N, M ) = AssR M ∩ Supp N.

COHOMOLOGICAL DIMENSION, COFINITENESS AND ABELIAN CATEGORIES 5

The following lemma is the improved consequence of Gruson’s Theorem. Lemma 2.2. Let R be a Noetherian ring and let M and N be two finitely generated R-modules such that Supp M ⊆ Supp N . Then there exists a finite filtration of M whose factors are quotients of N . Proof. Without loss of generality, we may assume M =  0. Then we have AssR M = ∅. So, using the fact that AssR M ⊆ Supp M ⊆ Supp N, it follows from Lemma 2.1 that ∅ = AssR M = AssR M ∩ Supp N = AssR HomR (N, M ). Thus, we have HomR (N, M ) = 0. Therefore, there exists an element f0 ∈ HomR (N, M ) such that f0 = 0. Set M0 = 0 and M1 = im(f0 ). Then we have M1 /M0  M1 = 0 and there is an isomorphism of R-modules N/ker(f0 )  M1 . Now if M1 = M then there is nothing to prove. So, we may assume M/M1 = 0. Then we have Supp M/M1 ⊆ Supp M ⊆ Supp N. Hence, repeating the same argument we see that HomR (N, M/M1 ) = 0. Therefore, there exists an element f1 ∈ HomR (N, M/M1 ) such that f1 = 0. Set M2 /M1 = im(f1 ). Then we have M2 /M1 = 0 and there is an isomorphism of R-modules N/ker(f1 )  M2 /M1 . Now, proceeding in the same way, since 0 = M0 ⊂ M1 ⊂ M2 ⊂ M3 ⊂ · · · , is an strictly ascending chain of submodules of M and by the hypothesis M is a Noetherian R-module it follows that after finitely many steps of repeating this argument for some positive integer k we must have Mk = M . So, 0 = M0 ⊂ M1 ⊂ · · · ⊂ Mk−1 ⊂ Mk = M is a finite filtration of M whose factors are quotients of N .



The following result is a generalization of [DNT, Theorem 2.2] and [DY, Theorem 3.2]. Theorem 2.3. Let I be a proper ideal of a Noetherian ring R and let M and N be finitely generated R-modules such that Supp M ⊆ Supp N. Let M be a Serre category of R-modules and let n ≥ 0 be an integer such that for each i ≥ n the local cohomology module HIi (N ) belongs to M . Then for each i ≥ n the local cohomology module HIi (M ) belongs to M . In particular M -cd(I, M ) ≤ M -cd(I, N ). Proof. By [BS, Corollary 3.3.3] the assertion is clear if n ≥ ara(I) + 1. So, without loss of generality we may assume n ≤ ara(I). We prove the assertion

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by descending induction on i with n ≤ i ≤ ara(I) + 1. For i = ara(I) + 1 there is nothing to prove. Now suppose n ≤ i ≤ ara(I). By lemma 2.2, there is a finite filtration 0 = M0 ⊂ M1 ⊂ · · · ⊂ Mk = M such that the factors Mj /Mj−1 are homomorphic images of N . By using short exact sequences the situation can be reduced to the case k = 1. Therefore for some finitely generated R-module L there exists an exact sequence 0 −→ L −→ N −→ M −→ 0. Thus we have the following long exact sequence · · · −→ HIi (L) −→ HIi (N ) −→ HIi (M ) −→ HIi+1 (L) −→ · · · . By the induction hypothesis HIi+1 (L) is in M . Since HIi (N ) is in M , we have that HIi (M ) is in M too.  Definition 2.4. Let R be a Noetherian ring, I be an ideal of R and let N be an R-module. Then we define q(I, N ) as the greatest integer i such that HIi (N ) is not an Artinian I-cofinite module if there exist such i s and −∞ otherwise. Remark. Let R be a Noetherian ring, I be an ideal of R and let N be an Rmodule. Then, it is clear that q(I, N ) ≤ q(I, N ) ≤ cd(I, N ). The following example shows that each of these inequalities can be strict. Example 2.5. (i) Let (R, m) be a Gorenstein local ring of dimension d ≥ 4. Let x1 , ..., xd be a system of parameters for R and J = (x1 , ..., xd−2 ) and K = (x3 , ..., xd ). Then for the ideal I = J ∩ K we have q(I, R) = cd(I, R) = d − 1 and q(I, R) = d − 2 and so q(I, R) > q(I, R). (ii) Let p denote the prime ideal of the Noetherian local domain (S, n) given in [BS, Exercise 8.2.13], with dim S = 2, dim S/ p = 1 and Hp2 (S) = 0. Then cd(p, S) = 2 and q(p, S) = q(p, S) = 1 and so cd(p, S) > q(p, S). Proof. (i) Using [BS, Theorem 3.2.3, Corollary 3.3.3 and Theorem 6.1.2] it follows that HIi (R) = 0 for each i ≥ d and hence q(I, R) ≤ cd(I, R) ≤ d − 1. Also, in view of the Mayer-Veitoris sequence, there exist isomorphisms: d HId−1 (R)  HJ+K (R) = Hmd (R)  ER (R/m).

As the R-module ER (R/m) is Artinian, it follows that q(I, R) ≤ d − 2. Furthermore, by localising at a prime ideal minimal over J, the Grothendieck’s Non-vanishing Theorem implies that HId−2 (R) is not Artinian and therefore q(I, R) = d − 2.

COHOMOLOGICAL DIMENSION, COFINITENESS AND ABELIAN CATEGORIES 7

Now on the contrary assume that the R-module HId−1 (R) is I-cofinite. Then by definition it follows that the R-module HomR (R/I, ER (R/m))  ER/I (R/m) is finitely generated. It yields that dim R/I = 0, which is a contradiction since dim R/I = 2. This means that q(I, R) = cd(I, R) = d − 1. (ii) In view of [Me2, Proposition 5.1], the non-zero S-module Hp2 (S) is Artinian and p-cofinite and so that using [BS, Theorem 6.1.2] it follows that cd(p, S) = 2 and q(p, S) < 2. Also, by [BS, Theorem 6.1.4] we have p ∈ Supp Hp1 (S) and hence the S-module Hp1 (S) is not Artinian. Thus, q(p, S) = q(p, S) = 1.  The following result, which will be quite useful in the proof of Lemma 4.2 and Proposition 4.4, is the first main result of this section. Theorem 2.6. Let I be a proper ideal of a Noetherian ring R and let M and N be two finitely generated R-modules such that Supp M ⊆ Supp N. Then q(I, M ) ≤ q(I, N ). Proof. It follows from [Me2, Proposition 4.1] that the class of all Artinian I-cofinite modules is a Serre category. So, the assertion follows from Theorem 2.3.  The following proposition is needed in the proof of Theorems 3.4 and 3.6. Note that the case I = J in Proposition 2.7 is treated in [Me2, Proposition 3.9] and the proof given here is quite similar to the proof of that result. Proposition 2.7. Let I ⊆ J be ideals of a Noetherian ring R and M be an R-module. Let M be a Serre category of R-modules such that the Rmodule ExtjR (R/J, HIi (M )) belongs to M , for each i ≥ 0 and each j ≥ 0 (respectively, for each 0 ≤ i ≤ n and each j ≥ 0). Then ExtiR (R/J, M ) belongs to M , for each i ≥ 0 (respectively, for each 0 ≤ i ≤ n). Proof. We use induction on n. The case n = 0 is clear, so let n > 0. We first reduce to the case ΓI (M ) = 0. This is possible, since if we let M = M/ΓI (M ), then we have the long exact sequence · · · −→ Exti−1 R (R/J, M ) −→ ExtiR (R/J, ΓI (M )) −→ ExtiR (R/J, M ) −→ ExtiR (R/J, M ) −→ Exti+1 R (R/J, ΓI (M )) −→ · · · , and the isomorphisms HI0 (M ) = 0 and HIi (M )  HIi (M ), for each i ≥ 1. So, let us assume that ΓI (M ) = 0. Let E be an injective hull of M and set

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L := E/M . Then, also ΓI (E) = 0 and HomR (R/J, E) = 0. Hence, we get the isomorphisms HIi (L)  HIi+1 (M ) and ExtiR (R/J, L)  Exti+1 R (R/J, M ) for all integers i ≥ 0. Now, the assertion easily follows applying the inductive hypothesis to the R-module L. This completes the proof.  The following lemma will be quite useful in this section. Lemma 2.8. Let I ⊆ J be ideals of a Noetherian ring R and M be an R-module. Let M be a Serre category of R-modules such that the R-module HJj (HIi (M )) belongs to M , for each i ≥ 0 and each j ≥ 0. Then the local cohomology module HJi (M ) belongs to M , for each i ≥ 0. Proof. Assume the opposite. Then by the definition M -cd(J, M ) ≥ 0. Let C denote the class of all R-modules N such that the R-module HJj (HIi (N )) belongs to M , for each i ≥ 0 and each j ≥ 0 but M -cd(J, N ) ≥ 0. Set t = min{M -cd(J, N ) : N is an R-module in C }. Then there exists an R-module T in C such that M -cd(J, T ) = t. Since HJ0 (T )  HJ0 (HI0 (T )) it follows from the hypothesis that HJ0 (T ) belongs to M . Therefore, we have t > 0. Set T := T /ΓI (T ). Then there are the isomorphisms HI0 (T ) = 0 and HIi (T )  HIi (T ), for each i ≥ 1. Thus, the R-module HJj (HIi (T )) belongs to M , for each i ≥ 0 and each j ≥ 0. Moreover, the long exact sequence · · · −→ HJi (HI0 (T )) −→ HJi (T ) −→ HJi (T ) −→ HJi+1 (HI0 (T )) −→ · · · , implies that M -cd(J, T ) = t. So, T is in C . Now replacing T with T let us assume that ΓI (T ) = 0. Let E be an injective hull of T and set L := E/T . Then also ΓI (E) = 0 and therefore we get the isomorphisms HIi (L)  HIi+1 (T ) for all i ≥ 0. Thus, the R-module HJj (HIi (L))  HJj (HIi+1 (T )) belongs to M , for each i ≥ 0 and each j ≥ 0. Furthermore, we have ΓJ (E) = 0 and therefore we get the isomorphisms HJi (L)  HJi+1 (T ) for all i ≥ 0. So, the R-module L belongs to C and M -cd(J, L) = t − 1 < t, which is a contradiction.  The following consequence of Lemma 2.8 plays a key role in the proof of the next lemma.

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Corollary 2.9. Let I ⊆ J be ideals of a Noetherian ring R and M be an R-module. If q(J, M ) ≥ 0, then q(I, M ) ≥ 0 and 

cd(I,M )

q(J,

HIi (M )) = sup{q(J, HIi (M )) : i ∈ N0 } ≥ 0.

i=0

Proof. Set t := q(J, M ) ≥ 0. Then, from the definition it follows that the R-module HJt (M ) is not Artinian. Therefore, it follows from [AM, Theorem 2.9] that the R-module ExtjR (R/J, M ) is not Artinian for some j ≥ 0. Now since Supp R/J ⊆ Supp R/I, it follows from [AB2, Lemma 2.2] that the R-module ExtkR (R/I, M ) is not Artinian for some k ≥ 0. Hence, in view of [AM, Theorem 2.9] there exists an integer i ≥ 0 such that the R-module HIi (M ) is not Artinian and so q(I, M ) ≥ 0. In order to prove the second assertion, on the contrary assume that  ) q(J, cd(I,M HIi (M )) = −∞. Then, by the definition the R-module HJj (HIi (M )) i=0 is Artinian, for each i ≥ 0 and each j ≥ 0. So, in view of Lemma 2.8 the local cohomology module HJi (M ) is Artinian, for each i ≥ 0. Hence, we have q(J, M ) = −∞, which is a contradiction.  The following lemma is needed in the proof of Theorem 2.11. Lemma 2.10. Let R be a Noetherian ring and I ⊆ J be ideals of R and let M be an R-module. Then, 

cd(I,M )

q(J, M ) ≤ q(I, M ) + q(J,

HIi (M ))

i=0

= q(I, M ) + sup{q(J, HIi (M )) : i ∈ N0 }. Proof. If q(J, M ) = −∞, then the assertion is clear. So, let us assume that q(J, M ) ≥ 0. Now, we use induction on t := q(J, M ). For t = 0 the assertion follows from Corollary 2.9. Now, let t > 0 and assume that the result has been proved for t − 1. Then, by corollary 2.9 we have q(I, M ) ≥ 0. Also, it follows from the exact sequence 0 → ΓI (M ) → M → M/ΓI (M ) → 0. that t = q(J, M ) ≤ sup{q(J, ΓI (M )), q(J, M/ΓI (M ))}. Now, if t ≤ q(I, M )+q(J, ΓI (M )), then there is nothing to prove. Therefore, we may assume that t > q(I, M ) + q(J, ΓI (M )). Then, we have sup{q(J, ΓI (M )), q(J, M/ΓI (M ))} = q(J, M/ΓI (M )).

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Furthermore, since t = q(J, M ) > q(I, M ) + q(J, ΓI (M )) ≥ q(J, ΓI (M )), it follows from the long exact sequence · · · → HJi (ΓI (M )) → HJi (M ) → HJi (M/ΓI (M )) → HJi+1 (ΓI (M )) → · · · , that q(J, M ) = t = q(J, M/ΓI (M )). Set N := M/ΓI (M ). Then, ΓJ (N ) ⊆ ΓI (N ) = 0 and q(J, N ) = t ≥ 1. Also, from the exact sequence 0 → N → ER (N ) → ER (N )/N → 0. it follows that q(J, ER (N )/N ) = q(J, N ) − 1 = t − 1 ≥ 0. Hence, it follows from the inductive hypothesis that q(J, ER (N )/N ) ≤ q(I, ER (N )/N ) + sup{q(J, HIi (ER (N )/N )) : i ∈ N0 }. Thus, t − 1 ≤ q(I, ER (N )/N ) + sup{q(J, HIi (N )) : i ∈ N} Therefore, t − 1 ≤ q(I, ER (N )/N ) + sup{q(J, HIi (M )) : i ∈ N}. But, since q(J, ER (N )/N ) = t − 1 ≥ 0 it follows from Corollary 2.9 that q(I, ER (N )/N ) ≥ 0 and so q(I, ER (N )/N ) = q(I, N ) − 1 ≥ 0. Therefore, q(J, M ) = t ≤ 1 + q(I, ER (N )/N ) + sup{q(J, HIi (M )) : i ∈ N} = q(I, N ) + sup{q(J, HIi (M )) : i ∈ N} ≤ q(I, M ) + sup{q(J, HIi (M )) : i ∈ N0 }. This completes the inductive step and the proof.



Now we are ready to state and prove the second main result of this section which is an analogue result of [GBA, Theorem 2.5]. Theorem 2.11. Let R be a Noetherian ring and I ⊆ J be ideals of R and let M be a finitely generated R-module. Then q(J, M ) ≤ q(I, M ) + cd(J, M/IM ). Proof. If q(J, M ) = −∞, then the assertion is clear. So, let us assume that q(J, M ) ≥ 0. Then in view of Lemma 2.10, we have  q(J, M ) ≤ q(I, M ) + q(J, HIi (M )). i≥0

COHOMOLOGICAL DIMENSION, COFINITENESS AND ABELIAN CATEGORIES 11

Set k := q(J,

 i≥0

HIi (M )). Then, by the definition we have  HIi (M )) = 0. HJk ( i≥0

But as the direct limits commute with the local cohomology functors and each R-module is the direct limit of the family of all finitely generated submodules of it, it follows that there exists a finitely generated submodule  L of the R-module i≥0 HIi (M ), such that HJk (L) = 0. But  HIi (M )) ⊆ Supp(M/IM ). Supp(L) ⊆ Supp( i≥0

Therefore, it follows from [DNT, Theorem 2.2], that cd(J, M/IM ) ≥ cd(J, L) ≥ k. Consequently, q(J, M ) ≤ q(I, M ) + q(J,



HIi (M ))

i≥0

≤ q(I, M ) + cd(J, L) ≤ q(I, M ) + cd(J, M/IM ). This completes the proof.



3. Torsion functors of local cohomology modules The main goal of this section is to establish some results for the torsion functors of the local cohomology modules HIt (M ) with t ∈ {q(I, M ), q(I, M )}. The following two well known lemmata will be quite useful in this section. Lemma 3.1. Let R be a Noetherian ring, I be an ideal of R and M be an R-module. Then the following conditions are equivalent: (i) (ii) (iii) (iv) (v)

ExtiR (R/I, M ) = 0, for all integers i ≥ 0, ExtiR (R/I, M ) = 0, for all integers 0 ≤ i ≤ cd(I, R), TorR i (R/I, M ) = 0, for all integers i ≥ 0, TorR i (R/I, M ) = 0, for all integers 0 ≤ i ≤ cd(I, R), i HI (M ) = 0, for all integers i ≥ 0.

Proof. For (i)⇔(ii) and (i)⇔(v), see [AM, Theorem 2.9]. Also, for (i)⇔(iii), see [PAB, Proposition 3.2]. Finally, for (iii)⇔(iv), see [PAB, Proposition 3.6]. 

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Lemma 3.2. Let (R, m) be a Noetherian local ring, I be an ideal of R and M be an R-module. Then the following conditions are equivalent: (i) The R-modules TorR i (R/I, M ) are finitely generated for all i ≥ 0, (ii) The R-modules TorR i (R/I, M ) are finitely generated for all 0 ≤ i ≤ cd(I, R). 

Proof. See [PAB, Proposition 2.2].

The following proposition plays a key role in the proof of Theorem 3.4. Proposition 3.3. Let R be a Noetherian ring, I be an ideal of R and M be an R-module. Then the following conditions are equivalent: (i) (ii) (iii) (iv)

The R-module ExtiR (R/I, M ) is Artinian, for each integer i ≥ 0, The R-module TorR i (R/I, M ) is Artinian, for each integer i ≥ 0, i The R-module HI (M ) is Artinian, for each integer i ≥ 0, q(I, M ) = −∞.

Proof. (i)⇔(iii) Follows from [AM, Theorem 2.9]. (iii)⇔(iv) Is trivial. (iii)⇒(ii) Since by the hypothesis the R-module HIi (M ) is Artinian for each i ≥ 0, it follows that the R-module HIi (M ) has finite support contained in Max(R) for each i ≥ 0. Moreover, by [BS, Corollary 3.3.3], HIi (M ) = 0 for all integers i > ara(I). Hence, the set  A := Supp HIi (M ) i≥0

is a finite subset of Max(R). For every p ∈ Spec(R)\A it is clear that i HIR (Mp )  (HIi (M ))p = 0 for each i ≥ 0. Therefore, Lemma 3.1 implies p Rp that (TorR i (R/I, M ))p  Tori (Rp /IRp , Mp ) = 0 for each i ≥ 0. Thus,  p ∈ i≥0 Supp TorR i (R/I, M ). Hence,  Supp TorR i (R/I, M ) ⊆ A ⊆ Max(R). i≥0

If A = ∅ then there is nothing to prove. So, without loss of generality, we may assume A = ∅. Assume that A = {m1 , ..., mt }. Then for each i ≥ 0 there is an isomorephism TorR i (R/I, M )



t  j=1

Γmj (TorR i (R/I, M )).

COHOMOLOGICAL DIMENSION, COFINITENESS AND ABELIAN CATEGORIES 13

Thus, for each i ≥ 0 it is clear that the R-module TorR i (R/I, M ) is Artinian if and only if the Rmj -module Rmj

(TorR i (R/I, M ))mj  Tori

(Rmj /IRmj , Mmj )  Γmj (TorR i (R/I, M ))

is Artinian for each 1 ≤ j ≤ t. So, localizing at each of the elements of A, without loss of generality, we may assume that R is a Noetherian local ring with unique maximal ideal m. Now, since the R-module HIi (M ) is Artinian,  for each integer i ≥ 0 it follows that the R-module   H i  (M ⊗R R)  HIi (M ) ⊗R R IR is Artinian, for each integer i ≥ 0. Furthermore, it is clear that the R-module  TorR i (R/I, M ) is Artinian for each i ≥ 0 if and only if the R-module  R     TorR i (R/I, M ) ⊗R R  Tori (R/I R, M ⊗R R)

is Artinian, for each i ≥ 0. Thus, without loss of generality, we may assume that (R, m) is a complete Noetherian local ring. Let T := D(M ), where D(−) denotes the Matlis dual functor. Then using the implication (iii)⇒(i) and by the adjointness we deduce that the R-module D(ExtiR (R/I, M ))  TorR i (R/I, T ) is finitely generated for each i ≥ 0. So, by [Me2, Theorem 2.1] the R-module ExtiR (R/I, T ) is finitely generated for each i ≥ 0. On the other hand, from the adjointness it follows that ExtiR (R/I, T )  D(TorR i (R/I, M )) for each i ≥ 0. i Hence, the R-module D(D(TorR i (R/I, M )))  D(ExtR (R/I, T )) is Artinian for each i ≥ 0. But, in view of [Ma, Theorem 18.6(i)] the canonical map R θ : TorR i (R/I, M ) −→ D(D(Tori (R/I, M ))) defined by θ(x)(ϕ) = ϕ(x) is injective for all i ≥ 0. Now it is clear that the R-module TorR i (R/I, M ) is Artinian, for each integer i ≥ 0.

(ii)⇒(i) Since the Artinian R-module TorR i (R/I, M ) has finite support contained in Max(R) for each i ≥ 0 it follows that the set 

cd(I,R)

B :=

Supp TorR i (R/I, M )

i≥0

is a finite subset of Max(R). For every p ∈ Spec(R)\B, Lemma 3.1 yields that R

p (TorR i (R/I, M ))p  Tori (Rp /IRp , Mp ) = 0

14

K. BAHMANPOUR

for each i ≥ 0. Therefore, it follows from Lemma 3.1 that ExtiRp (Rp /IRp , Mp ) =  0 for each i ≥ 0. Thus p ∈ i≥0 Supp ExtiR (R/I, M ). Hence  Supp ExtiR (R/I, M ) ⊆ B ⊆ Max(R). i≥0

If B = ∅ then the assertion is clear. So, without loss of generality, we may assume B = ∅. Assume that B = {n1 , ..., nk }. Then for each i ≥ 0 there is an isomorphism ExtiR (R/I, M )



k 

Γnj (ExtiR (R/I, M )).

j=1

Thus, for each i ≥ 0 it is clear that the R-module ExtiR (R/I, M ) is Artinian if and only if the Rnj -module (ExtiR (R/I, M ))nj  ExtiRnj (Rnj /IRnj , Mnj )  Γnj (ExtiR (R/I, M )) is Artinian for each 1 ≤ j ≤ k. So, localizing at each of the elements of B, without loss of generality, we may assume that R is a Noetherian local ring with unique maximal ideal n. Since, for each integer i ≥ 0, the R-module TorR i (R/I, M ) is Artinian, it  follows that the R-module  R     TorR i (R/I, M ) ⊗R R  Tori (R/I R, M ⊗R R)

is Artinian, for each integer i ≥ 0. On the other hand, it is clear that the R-module ExtiR (R/I, M ) is Artinian for each i ≥ 0 if and only if the the  R-module  R,  M ⊗R R)   Exti (R/I  ExtiR (R/I, T ) ⊗R R R is Artinian, for each i ≥ 0. So, without loss of generality, we may assume that (R, n) is a complete Noetherian local ring. Let T := D(M ), where D(−) denotes the Matlis dual functor. Then by the adjointness the R-module ExtiR (R/I, T )  D(TorR i (R/I, M )) is finitely generated for each i ≥ 0. So, by [Me2, Theorem 2.1] the R-module TorR i (R/I, T ) is finitely generated for each i ≥ 0. On the other hand, the adjointness yields the isomorphisms D(ExtiR (R/I, M ))  TorR i (R/I, T ) for each i ≥ 0. Hence the R-module D(D(ExtiR (R/I, M )))  D(TorR i (R/I, T )) is Artinian for each i ≥ 0. Now using [Ma, Theorem 18.6(i)] it follows that the R-module ExtiR (R/I, M ) is Artinian, for each integer i ≥ 0. 

COHOMOLOGICAL DIMENSION, COFINITENESS AND ABELIAN CATEGORIES 15

The following result is the first main result of this section. Theorem 3.4. Let R be a Noetherian ring and let I ⊆ J be two ideals of R. Let M be an R-module and t ≥ 2 be an integer such that the R-module i TorR j (R/J, HI (M )) is Artinian for each i > t and each j ≥ 0. Then the t R-module TorR j (R/J, HI (M )) is Artinian for j = 0, 1. Proof. Let ε

f0

f1

f2

0 −→ M −→ E0 −→ E1 −→ E2 −→ · · · be a minimal injective resolution for M . Set Ki := kerfi for i ≥ 0 and N := Kt−1 /ΓI (Kt−1 ). Then by the proof of [PAB, Lemma 3.4] we have ΓI (N ) = 0, N ⊗R R/J = 0 and HIi (N )  HIt+i−1 (M ), for each i ≥ 1. By [BS, Remark 2.2.7] there is an exact sequence 0 −→ N −→ DI (N ) −→ HI1 (N ) −→ 0. (3.4.1) Therefore, by the hypothesis the R-module R t+i−1 i (M )) TorR j (R/J, HI (N ))  Torj (R/J, HI

is Artinian for each i ≥ 2 and each j ≥ 0. Thus, by Proposition 3.3 the Rmodule ExtjR (R/J, HIi (N )) is Artinian for each i ≥ 2 and each j ≥ 0. Also, by [BS, Corollary 2.2.8] there is an isomorphism HIi (DI (N ))  HIi (N ), for i ≥ 2. So, the R-module ExtjR (R/J, HIi (DI (N )))  ExtjR (R/J, HIi (N )) is Artinian for each i ≥ 2 and each j ≥ 0. Also, it follows from [BS, Corollary 2.2.8] that HIi (DI (N )) = 0, for i = 0, 1. So, the R-module ExtjR (R/J, HIi (DI (N ))) is Artinian for each i ≥ 0 and each j ≥ 0. Therefore, by Proposition 2.7 the R-module ExtjR (R/J, DI (N )), is Artinian for each j ≥ 0. Thus, by Proposition 3.3 the R-module TorR i (R/J, DI (N )) is Artinian for each i ≥ 0. On the other hand, the exact sequence (3.4.1) induces the exact sequence R 1 TorR 1 (R/J, DI (N )) −→ Tor1 (R/J, HI (N )) −→ R R 1 TorR 0 (R/J, N ) −→ Tor0 (R/J, DI (N )) −→ Tor0 (R/J, HI (N )) −→ 0, 1 which implies that the R-module TorR j (R/J, HI (N )) is Artinian for j = 0, 1. Thus, the R-module R t 1 TorR j (R/J, HI (M ))  Torj (R/J, HI (N ))

is Artinian for j = 0, 1. The following result is a generalization of [ADT, Corollary 3.2].



16

K. BAHMANPOUR

Corollary 3.5. Let R be a Noetherian ring and let I ⊆ J be two ideals of R. Let M be an R-module such that q(I, M ) = t ≥ 2. Then the R-module t TorR j (R/J, HI (M )) is Artinian for j = 0, 1. Proof. In view of the definition the R-module HIi (M ) is Artinian for each i i > t. So, the R-module TorR j (R/J, HI (M )) is Artinian for each i > t and each j ≥ 0. Hence, the assertion follows from Theorem 3.4.  The following result is our second main result in this section. Theorem 3.6. Let R be a Noetherian ring and let I ⊆ J be two ideals of R. Let M be an R-module and t ≥ 2 be an integer such that the R-module i TorR j (R/J, HI (M )) is finitely generated for each i > t and each j ≥ 0. Then t the R-module TorR j (R/J, HI (M )) is finitely generated for j = 0, 1. Proof. Let ε

f0

f1

f2

0 −→ M −→ E0 −→ E1 −→ E2 −→ · · · be a minimal injective resolution for M . Set Ki := kerfi for i ≥ 0 and N := Kt−1 /ΓI (Kt−1 ). Then by the proof of [PAB, Lemma 3.4] we have ΓI (N ) = 0, N ⊗R R/J = 0 and HIi (N )  HIt+i−1 (M ), for each i ≥ 1. By [BS, Remark 2.2.7] there is an exact sequence 0 −→ N −→ DI (N ) −→ HI1 (N ) −→ 0. (3.6.1) By the hypothesis the R-module R t+i−1 i (M )) TorR j (R/J, HI (N ))  Torj (R/J, HI

is finitely generated for each i ≥ 2 and each j ≥ 0. Thus, by [Me2, Theorem 2.1] the R-module ExtjR (R/J, HIi (N )) is finitely generated for each i ≥ 2 and each j ≥ 0. Also, by [BS, Corollary 2.2.8] there is an isomorphism HIi (DI (N ))  HIi (N ), for i ≥ 2. So, the R-module ExtjR (R/J, HIi (DI (N )))  ExtjR (R/J, HIi (N )) is finitely generated for each i ≥ 2 and each j ≥ 0. Also, it follows from [BS, Corollary 2.2.8] that HIi (DI (N )) = 0, for i = 0, 1. Therefore, the R-module ExtjR (R/J, HIi (DI (N ))) is finitely generated for each i ≥ 0 and each j ≥ 0. Hence, by Proposition 2.7 the R-module ExtjR (R/J, DI (N )), is finitely generated for each j ≥ 0. Thus by [Me2, Theorem 2.1] the R-module TorR i (R/J, DI (N )) is finitely generated for each i ≥ 0. On the other hand, the exact sequence (3.6.1) induces the exact sequence R 1 TorR 1 (R/J, DI (N )) −→ Tor1 (R/J, HI (N )) −→

COHOMOLOGICAL DIMENSION, COFINITENESS AND ABELIAN CATEGORIES 17 R R 1 TorR 0 (R/J, N ) −→ Tor0 (R/J, DI (N )) −→ Tor0 (R/J, HI (N )) −→ 0,

which implies that the R-module 1 TorR j (R/J, HI (N ))

is finitely generated for j = 0, 1. Therefore, the R-module R t 1 TorR j (R/J, HI (M ))  Torj (R/J, HI (N ))

is finitely generated for j = 0, 1.



The following theorem is an immediate consequence of Theorems 3.4 and 3.6. Theorem 3.7. Let R be a Noetherian ring and let I ⊆ J be two ideals of R. Let M be an R-module and t ≥ 2 be an integer such that the R-module i TorR j (R/J, HI (M )) has finite length for each i > t and each j ≥ 0. Then t the R-module TorR j (R/J, HI (M )) has finite length for j = 0, 1. Proof. Since by the hypothesis R is a Noetherian ring it follows that a given R-module has finite length if and only if it is finitely generated and Artinian. Using this remark the assertion follows from Theorems 3.4 and 3.6.  The following result is a generalization of [AT, Corollary 3.2]. Corollary 3.8. Let R be a Noetherian ring ring and let I ⊆ J be two ideals of R. Let M be a non-zero finitely generated R-module of finite dimension n. Then the following statements hold: n−1 (i) If n > 2 then the R-module TorR (M )) has finite length for j (R/J, HI j = 0, 1, (ii) If n > 1 then there exist a positive integer k such that I k HIn−1 (M ) = I k+1 HIn−1 (M ) and the R-module HIn−1 (M )/I k HIn−1 (M ) is of finite length.

Proof. (i) In view of the Grothendieck’s Vanishing Theorem HIi (M ) = 0 for all i > n. Also, by [Me2, Proposition 5.1] the R-module HIn (M ) is Artinian and I-cofinite. So by the definition the R-module ExtjR (R/I, HIi (M )) is finitely generated for each i > n − 1 and each j ≥ 0. Hence by [DM, Theorem 1] the R-module ExtjR (R/J, HIi (M )) is finitely generated for each i > n − 1 and each j ≥ 0. Therefore, using [Me2, Theorem 2.1] it follows i that the R-module TorR j (R/J, HI (M )) is finitely generated and Artinian for each i > n − 1 and each j ≥ 0. So, the assertion follows from Theorem 3.7.

18

K. BAHMANPOUR

(ii) Set H := HIn−1 (M ) and assume that n > 1. By [AT, Corollary 3.2] the R-module H/IH is finitely generated. So, there is a finitely generated submodule L of H such that H = L + IH. Since, the local cohomology module H is I-torsion it follows that L is a finitely generated I-torsion module. Thus, there exists a positive integer k such that I k L = 0. Therefore, I k H = I k L + I k+1 H = I k+1 H. Also, by [AT, Corollary 3.2] the R-module HIn−1 (M )/I k HIn−1 (M ) has finite length.  Corollary 3.9. Let R be a Noetherian ring and let I ⊆ J be two ideals of R. Let M be an R-module such that q(I, M ) = t ≥ 2. Then the R-module t TorR j (R/J, HI (M )) has finite length for j = 0, 1. Proof. Follows applying the method used in the proof of Corollary 3.8. 

4. Cofiniteness of local cohomology modules In this section we establish several results concerning the cofiniteness of local cohomology modules. In this direction we solve a problem posed by Hartshorne in [Ha1] in the some special case. More precisely, it is shown that, for any ideal I of R and any finitely generated R-module M with q(I, M ) ≤ 1, the local cohomology modules HIi (M ) are I-cofinite for all integers i ≥ 0. As a consequence of this result it is shown that if q(I, R) ≤ 1, then for any finitely generated R-module M , the local cohomology modules HIi (M ) are I-cofinite for all integers i ≥ 0. We start this section with the following easy lemma, which is needed in the proofs of Lemma 4.2, Proposition 4.4, Lemma 4.6 and Theorem 5.3. Lemma 4.1. Let R be a Noetherian ring and I be an ideal of R. Then the following conditions are equivalent: (i) q(I, R) ≤ 0, (ii) q(I, R) ≤ 0, (iii) dim R/(I+ΓI (R)) = 0, whenever I+ΓI (R) = R and dim R/(I+p) = 0, for each p ∈ mAssR R with cd(I, R/ p) > 0, (iv) q(I, R) = q(I, R) ≤ 0. Proof. (i)⇒(ii) Is clear.

COHOMOLOGICAL DIMENSION, COFINITENESS AND ABELIAN CATEGORIES 19

(ii)⇒(iii) Assume that q(I, R) ≤ 0 and I + ΓI (R) = R. Set R := R/ΓI (R). Then there are isomorphisms HI0 (R)  ΓI (R) = 0 and HIi (R)  HIi (R) for each integer i ≥ 1. So, it follows from the hypothesis that the R-module HIi (R) is Artinian for each i ≥ 0. Therefore, Supp HIi (R) ⊆ Max(R) for each i ≥ 0. Since, by the hypothesis I + ΓI (R) = R it follows that dim R/(I + ΓI (R)) ≥ 0. Now, on the contrary assume that dim R/(I + ΓI (R)) > 0. Then, there is p ∈ mAssR R/(I + ΓI (R)) such that p ∈ Max(R). Then, it follows from the Grothendieck’s Non-vanishing Theorem that p ∈ Supp Hpt (R), where t = height p R. Therefore, t (R))p (HIt (R))p  (HIR t  HIR (Rp ) p t = HRad(IR (Rp ) p)

= Hpt Rp (Rp )  Hpt Rp (Rp )  (Hpt (R))p = 0, and so p ∈ Supp HIt (R) ⊆ Max(R), which is a contradiction. Now, let p ∈ mAssR R and cd(I, R/ p) > 0. We claim that p ∈ V (I). Assume the opposite. Then R/ p is an I-torsion R-module and hence cd(I, R/ p) = 0, which is a contradiction. Since p ∈ V (I) it follows that HI0 (R/ p) = 0. Also, by [DY, Theorem 3.2] we have q(I, R/ p) ≤ 0 which implies that q(I, R/ p) = −∞. So, the R-module HIi (R/ p) is Artinian for each i ≥ 0. Now, repeating the same argument applied for the case dim R/(I + ΓI (R)) = 0 it follows that dim R/(I + p) = 0.

 i≥1

(iii)⇒(iv) It follows from the hypothesis that  Supp HIi (R) = Supp HIi (R/ΓR (R)) ⊆ Supp R/(I + ΓI (R)) ⊆ Max(R). i≥1

Moreover, since HI0 (R/ΓI (R)) = 0 it follows that  Supp HIi (R/ΓR (R)) ⊆ Max(R). i≥0

Therefore, it follows from [BN, Proposition 2.2 and Corollary 2.3] that the R-module HIi (R)  HIi (R/ΓI (R)) is Artinian and I-cofinite for each i ≥ 1. So, we have q(I, R) ≤ q(I, R) ≤ 0. Since, the finitely generated R-module HI0 (R) is I-cofinite it follows that HI0 (R) is Artinian and I-cofinite if and

20

K. BAHMANPOUR

only if HI0 (R) is Artinian. Now, it is clear that q(I, R) = q(I, R) ≤ 0. 

(iv)⇒(i) Is obvious.

The following lemma, which is needed in the proof of Theorem 4.8, is an immediate consequence of Lemma 4.1. Lemma 4.2. Let R be a Noetherian ring and I be an ideal of R. Then for any finitely generated R-module M the following conditions are equivalent: (i) q(I, M ) ≤ 0, (ii) q(I, M ) ≤ 0, (iii) q(I, M ) = q(I, M ) ≤ 0. Proof. Since, Supp M = Supp AnnRR M it follows from Theorem 2.6, [BS, Theorem 4.2.1] and [DY, Theorem 3.2] that q(I, M ) = q(I,

R ) and AnnR M

I + AnnR M R R ) = q( , ). AnnR M AnnR M AnnR M Also, using [Me2, Proposition 4.1] and [BS, Theorem 4.2.1] it follows that q(I, M ) = q(I,

q(I,

R I + AnnR M R ) = q( , ). AnnR M AnnR M AnnR M

Now the assertion follows applying Lemma 4.1 for the ideal Noetherian ring AnnRR M .

I+AnnR M AnnR M

of the 

The following well known lemma plays a key role in the proof of Proposition 4.4. Lemma 4.3. Let R be a Noetherian ring and I ⊆ J be two ideals of R. Let t K be an R-module such that cd(I, K) = t ≥ 2. Then TorR i (R/J, HI (K)) = 0, for i = 0, 1. Proof. See [PAB, Corollary 3.5].



The following proposition is crucial for the proof of our next main results. Proposition 4.4. Let (R, m) be a complete Noetrherian local ring and I be an ideal of R. Then the following conditions are equivalent: (i) q(I, R) = 1, (ii) q(I, R) = 1,

COHOMOLOGICAL DIMENSION, COFINITENESS AND ABELIAN CATEGORIES 21

(iii) q(I, R) = 1 and mAssR R = (A ∪ B ∪ C), where A = mAssR R ∩ V (I), B = {p ∈ mAssR R : cd(I, R/ p) = 1} and C = {p ∈ mAssR R : Rad(I + p) = m}, (iv) q(I, R) = q(I, R) = 1. Proof. (i)⇒(ii) By the definition q(I, R) ≤ q(I, R) = 1. Now, on the contrary assume that q(I, R) ≤ 0. Then by Lemma 4.1 it follows that q(I, R) ≤ 0, which is a contradiction. (ii)⇒(iii) It is clear that (A ∪ B ∪ C) ⊆ mAssR R. So, it is enough to prove mAssR R ⊆ (A ∪ B ∪ C). To do this, it is enough to prove mAssR R\(A ∪ B) ⊆ C.  Let p ∈ mAssR R\(A ∪ B) . Then, from the hypothesis q(I, R) = 1 it follows that cd(I, R) ≥ 1 and so I = R. Since, p ∈ A it follows that cd(I, R/ p) > 0. Now, since p ∈ B it follows that cd(I, R/ p) > 1. Set t := cd(I, R/ p). In view of [DY, Theorem 3.2], we have q(I, R/ p) ≤ q(I, R) = 1 < t. Therefore, the R-module HIt (R/ p) is Artinian. Set M := D(HIt (R/ p)), where D(−) denotes the Matlis dual functor. Then, M is a finitly generated R-module. Furthermore, it follows from Lemma 4.3 and adjointness that t ExtiR (R/I, M )  D(TorR i (R/I, HI (R/ p))) = 0, for i = 0, 1.

This means that grade(I, M ) ≥ 2 and hence HIi (M ) = 0 for i = 0, 1. On the hand, it follows from [DY, Theorem 3.2] that the R-module HIi (M ) is Artinian for each i ≥ 2. Therefore, the R-module HIi (M ) is Artinian for each i ≥ 0. Thus, by the definition we have q(I, M ) = −∞. Now, it follows from Lemma 4.2 that q(I, M ) = −∞. Therefore, by the definition the R-module HIt (M ) is Artinian and I-cofinite. Since p ⊆ AnnR HIt (R/ p) = AnnR D(HIt (R/ p)) = AnnR M, it follows that Supp M ⊆ Supp R/ p and by [DNT, Theorem 2.2] we have cd(I, M ) ≤ cd(I, R/ p) = t. On the other hand, using [BS, Theorem 4.2.1], it follows that I +p , R/ p) = cd(I, R/ p) = t. cd( p

22

K. BAHMANPOUR

Whence, using [BS, Theorem 4.2.1 and Exercise 6.1.8] it follows that D(HIt (M ))  D(H tI+p (M )) p

 D(H tI+p (R/ p) ⊗R/ p M ) p

= HomR (HIt (R/ p) ⊗R/ p M, ER (R/ m))  HomR (HIt (R/ p) ⊗R M, ER (R/ m))  HomR (M, HomR (HIt (R/ p), ER (R/ m)))  HomR (M, M ) = 0. Thus, HIt (M ) = 0 and hence cd(I, M ) = t. Since, the R-module HIt (M ) is Artinian and I-cofinite and R is a complete ring it follows from [AB1, Lemma 2.1] that Rad(I + AnnR HIt (M )) = m. Furthermore, since AnnR HIt (R/ p) = AnnR M = AnnR HomR (M, M ) = AnnR D(HIt (M )) = AnnR HIt (M ), it follows that Rad(I + AnnR HIt (R/ p)) = Rad(I + AnnR HIt (M )) = m. So, by [AB1, Lemma 2.1] the R/ p-module HIt (R/ p)  H tI+p (R/ p) p

I+p -cofinite p

module. Hence, by [B, Theorem 2.7] we have is an Artinian t AnnR HI (R/ p) = p, which implies that Rad(I + p) = Rad(I + AnnR HIt (R/ p)) = m . Therefore, p ∈ C. (iii)⇒(iv) By the definition we have 1 = q(I, R) ≤ q(I, R). On the other hand for each p ∈ C the R-module HIi (R/ p) is Artinian for each i ≥ 2 and Rad(I + AnnR HIi (R/ p)) ⊇ Rad(I + p) = m. Therefore, by [AB1, Lemma 2.1] the R-module HIi (R/ p) is Artinian and I-cofinite for each i ≥ 2. Also, for each p ∈ (A ∪ B) we have HIi (R/ p) = 0 for each i ≥ 2. Hence, for the R-module  N := R/ p p∈mAssR R

we have Supp N = Spec R = Supp R and the R-module HIi (N ) is Artinian and I-cofinite for each i ≥ 2. Thus, in view of Theorem 2.6 we have q(I, R) = q(I, N ) ≤ 1. So, q(I, R) = q(I, R) = 1.

COHOMOLOGICAL DIMENSION, COFINITENESS AND ABELIAN CATEGORIES 23



(iv)⇒(i) Is trivial.

The following consequence of Proposition 4.4 is needed in the proof of the lemma 4.6. Corollary 4.5. Let (R, m) be a local Noetherian ring and I be an ideal of R. Then the following conditions are equivalent: (i) q(I, R) = 1, (ii) q(I, R) = 1, (iii) q(I, R) = q(I, R) = 1.  R)  it follows that q(I, R) = Proof. Since, by Lemma 3.4, q(I, R) = q(I R,  R)  and hence the assertion follows from Proposition 4.4 q(I R,  In order to prove Lemma 4.7, we need the following lemma. Lemma 4.6. Let R be a Noetherian ring and I be an ideal of R. Then the following conditions are equivalent: (i) q(I, R) = 1, (ii) q(I, R) = 1, (iii) q(I, R) = q(I, R) = 1. Proof. (i)⇒(ii) By the definition q(I, R) ≤ q(I, R) = 1. Now, on the contrary assume that q(I, R) ≤ 0. Then by Lemma 4.1 it follows that q(I, R) ≤ 0, which is a contradiction. (ii)⇒(iii) and (ii)⇒(i) By the definition the R-module HIi (R) is Artinian for each i ≥ 2. Hence, the R-module HIi (R) has finite support contained in Max(R) for each i ≥ 2. Also, in view of [BS, Corollary 3.3.3], HIi (R) = 0  for each i > ara(I). Therefore, the set i≥2 Supp HIi (R) is a finite subset of Max(R). Assume that  Supp HIi (R) = {m1 , ..., mn }. i≥2

Then it is clear that Supp HomR (R/I, HIi (R)) ⊆ {m1 , ..., mn } for each i ≥ 2. Also, by Corollary 4.5 and Lemma 4.1 the Rmk -module i (HomR (R/I, HIi (R)))mk  HomRmk (Rmk /IRmk , HIR (Rmk )) m k

is finitely generated for each i ≥ 2 and each 1 ≤ k ≤ n. Now, it is clear that the R-module HomR (R/I, HIi (R)) has finite length for each i ≥ 2.

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K. BAHMANPOUR

Therefore, by [Me2, Proposition 4.1] the R-module HIi (R) is Artinian and I-cofinite for each i ≥ 2 and so that q(I, R) ≤ 1. On the other hand, by the definition 1 = q(I, R) ≤ q(I, R). Therefore, we have q(I, R) = 1, as required. (iii)⇒(ii) Is trivial.



The following lemma plays a key role in the proof of Theorem 4.8. Lemma 4.7. Let R be a Noetherian ring and I be an ideal of R. Then for any finitely generated R-module M the following conditions are equivalent: (i) q(I, M ) = 1, (ii) q(I, M ) = 1, (iii) q(I, M ) = q(I, M ) = 1. Proof. Applying the method used in the proof of Lemma 4.2 the assertion follows from Lemma 4.6.  We are now in a position to state and prove the main results of this section. Theorem 4.8. Let R be a Noetherian ring and I be an ideal of R. Then, q(I, M ) = q(I, M ), for any finitely generated R-module M with q(I, M ) ≤ 1. Proof. Follows from Lemmata 4.2 and 4.7.



Theorem 4.9. Let R be a Noetherian ring, I be an ideal of R and let M be a finitely generated R-module with q(I, M ) ≤ 1. Then the R-modules HIi (M ) are I-cofinite for all i ≥ 0. Proof. Since, by the hypothesis q(I, M ) ≤ 1 it follows from Theorem 4.8 that q(I, M ) ≤ 1. Therefore, by the definition the R-module HIi (M ) is Icofinite for each i ≥ 2. Also, since the R-module HI0 (M ) is finitely generated with support in V (I) it follows that HI0 (M ) is I-cofinite. Therefore, for each i = 1 the R-module HIi (M ) is I-cofinite. Hence, by [Me2, Proposition 3.11] the R-module HI1 (M ) is I-cofinite too. 

Theorem 4.10. Let R be a Noetherian ring and I be an ideal of R with q(I, R) ≤ 1. Then for any finitely generated R-module M , the R-module HIi (M ) is I-cofinite for each i ≥ 0.

COHOMOLOGICAL DIMENSION, COFINITENESS AND ABELIAN CATEGORIES 25

Proof. Since q(I, R) ≤ 1 and for each finitely generated R-module M we have Supp M ⊆ Spec R = Supp R it follows from [DY, Theorem 3.2] that q(I, M ) ≤ q(I, R) ≤ 1. Therefore, the assertion follows from Theorem 4.9.  The following results give some partially affirmative answers to the Lynch’s conjecture in [L], (See also, [B]). Corollary 4.11. Let R be a Noetherian ring and I be an ideal of R with  q(I, R) ≤ 1 and cd(I, R) = t. Then AnnR HIt (R) ⊆ p∈mAssR R p. Proof. Follows from Theorem 4.10 and [B, Theorem 2.6].



Corollary 4.12. Let R be a Noetherian domain and I be an ideal of R with q(I, R) ≤ 1 and cd(I, R) = t > 0. Then AnnR HIt (R) = 0. Proof. Follows from Corollary 4.11.



Remark. Let R be a Noetherian ring and I be an ideal of R. In general q(I, R) = 2 does not imply q(I, R) = q(I, R). For instance, in Example 2.5(i) get dim R = 3. Then we have 3 = q(I, R) > q(I, R) = 2. 5. An Abelian category of cofinite modules We begin this section with the following conjecture, which is a special case of Hartshore’s general question in [Ha1]. Conjecture: Let I be an ideal of a Noetherian ring R with q(I, R) ≤ 1. Then the category C (R, I)cof of I-cofinite modules is an Abelian subcategory of the category of all R-modules. In this paper we are not able to prove this conjecture in the general case. But as the main result of this section, we present an affirmative answer to this conjecture, whenever (R, m) is a complete Noetherian local ring. In order to prove the main result of this section, we need the following lemma. Lemma 5.1. Let (R, m) be a Noetherian local ring and I be an ideal of R. Let M be an R-module and J be an ideal with JM = 0. Then the following conditions are equivalent:

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(i) For every i ≥ 0, the R-module TorR i (R/I, M ) is finitely generated, (ii) For every i ≥ 0, the R-module ExtiR (R/I, M ) is finitely generated, (iii) For every i ≥ 0, the S-module TorSi (S/IS, M ) is finitely generated, where S = R/J, (iv) For every i ≥ 0, the S-module ExtiS (S/IS, M ) is finitely generated, where S = R/J. Proof. (i)⇔(ii) and (iii)⇔(iv) follow from [Me2, Theorem 2.1]. (i)⇒(iii) By the adjointness for each i ≥ 0, there exists an isomorphism ExtiR (R/I, DR (M ))  DR (TorR i (R/I, M )), where DR (−) denotes the Matlis dual functor DR (−) = HomR (−, ER (R/ m)). So, the R-module ExtiR (R/I, DR (M )) is Artinian for each i ≥ 0. Then, in view of [AM, Theorem 2.9], the R-module HIi (DR (M )) is Artinian for each i ≥ 0. Since, by the hypothesis JM = 0 it follows that JDR (M ) = 0 and so both of the R-modules M and DR (M ) has S-module structure. Therei fore, from the isomorphism HIi (DR (M ))  HIS (DR (M )), it follows that the i S-module HIS (DR (M )) is Artinian for each i ≥ 0. Hence, it follows from [AM, Theorem 2.9] that the S-module ExtiS (S/IS, DR (M )) is Artinian for each i ≥ 0. By the adjointness for each i ≥ 0 there exists an isomorphism of S-modules ExtiS (S/IS, DS (M ))  DS (TorSi (S/IS, M )), where DS (−) = HomS (−, ES (R/ m)) denotes the Matlis dual functor. Then, for any R-module X with JX = 0 we have DR (X) = HomR (X, ER (R/ m))  HomR (X/JX, ER (R/ m))  HomR (X ⊗R R/J, ER (R/ m))  HomR (X, HomR (R/J, ER (R/ m)))  HomR (X, ER/J (R/ m))  HomS (X, ES (R/ m)) = DS (X). So, there exists an isomorphism of S-modules DR (M )  DS (M ). Thus, for each i ≥ 0 there exists an isomorphism of S-modules ExtiS (S/IS, DR (M ))  DS (TorSi (S/IS, M )) and the S-module ExtiS (S/IS, DR (M )) is Artinian for each i ≥ 0. So, the S-module DS (TorSi (S/IS, M )) is Artinian for each i ≥ 0. Now, it follows

COHOMOLOGICAL DIMENSION, COFINITENESS AND ABELIAN CATEGORIES 27

from [KLW, Lemma 1.15(a)] that, the S-module TorSi (S/IS, M ) is finitely generated for each i ≥ 0, as required. (iii)⇒(i) Follows applying the method used in the proof of (i)⇒(iii).  The following result plays a key role in the proof of Theorem 5.3. Lemma 5.2. Let I be an ideal of a Noetherian local ring (R, m) such that cd(I, R) ≤ 1. Let C (R, I)cof denote the category of I-cofinite R-modules. Then C (R, I)cof is an Abelian subcategory of the category of all R-modules. Proof. See [PAB, Theorem 2.4].



Now we are ready to state and prove the main result of this section. Theorem 5.3. Let I be an ideal of a complete Noetherian local ring (R, m) such that q(I, R) ≤ 1. Let C (R, I)cof denote the category of I-cofinite Rmodules. Then C (R, I)cof is an Abelian subcategory of the category of all R-modules. Proof. Let M, N ∈ C (R, I)cof and let f : M −→ N be an R-homomorphism. It is enough to show that the R-modules kerf and cokerf are I-cofinite. Set A := {p ∈ mAssR R : cd(I, R/ p) ≤ 1} and B := {p ∈ mAssR R : cd(I, R/ p) ≥ 2}. We consider the following three cases:  Case 1. Assume that B = ∅. Then T := p∈mAssR R R/ p is a finitely generated R-module with Supp T = Spec R = Supp R. So, it follows from [DNT, Theorem 2.2] that cd(I, R) = cd(I, T ) = Max{cd(I, R/ p) : p ∈ A} ≤ 1. Thus, the assertion follows from Lemma 5.2. Case 2. Assume that B = ∅ and A = ∅. Then since in view of Lemma 4.1 and Proposition 4.4 for each p ∈ B we have Rad(I + p) = m it follows that for each p ∈ mAssR R we have Rad(I +p) = m. So Rad(I) = m and hence in view of [Me2, Proposition 4.1] the I-cofinite modules M and N are Artinian. Also, it follows from [Me2, Proposition 4.1] that the R-module kerf is Icofinite. Now, the assertion follows from the following exact sequences 0 −→ kerf −→ M −→ imf −→ 0,

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and 0 −→ imf −→ N −→ cokerf −→ 0.

Case 3. Assume that B = ∅ and A = ∅. Set J := p∈B p and K := ΓJ (R). Then the Lemma 4.1 and Proposition 4.4 yield that Rad(I +p) = m for each p ∈ B. Hence it is clear that Rad(I + J) = m. Moreover, since Supp K ⊆ V (J) it follows that the R-module K/IK has finite length. Therefore, we have Supp HomR (K/IK, M ) ⊆ {m}. Furthermore, as the R-module M is I-cofinite it follows that the R-module HomR (K/IK, M )  HomR (K ⊗R R/I, M )  HomR (K, HomR (R/I, M )) is finitely generated. Thus, the R-module Hom(R/I, HomR (K, M ))  HomR (R/I ⊗R K, M )  HomR (K/IK, M ) is of finite length. On the other hand, it is clear that Supp HomR (K, M ) ⊆ Supp M ⊆ V (I). Therefore, [Me2, Proposition 4.1] implies that the R-module HomR (K, M ) is Artinian and I-cofinite. Now the exact sequence 0 −→ K −→ R −→ R/K −→ 0,

(5.3.1)

induces the exact sequence h

0 −→ HomR (R/K, M ) −→ M −→ HomR (K, M ). (5.3.2) From the exact sequence (5.3.2) we get the short exact sequence 0 −→ HomR (R/K, M ) −→ M −→ im h −→ 0.

(5.3.3)

By [Me2, Corollary 4.4] the R-module im h is I-cofinite and hence the exact sequence (5.3.3) implies that the R-module HomR (R/K, M ) is Icofinite. Now set S = R/K. Since, K HomR (R/K, M ) = 0 it follows from Lemma 5.1 that the S-module HomR (R/K, M ) is IS-cofinite. By a similar method we deduce that the S-module HomR (R/K, N ) is IS-cofinite. Since, AssR R/K = AssR R\V (J) it follows that mAssS S = {p S : p ∈ A}. Whence, we can deduce that cd(IS, S) ≤ 1. Therefore, it follows from Lemma 5.2 that the S-module ker HomR (R/K, f ) is IS-cofinite. Hence in view of Lemma 5.1 the R-module ker HomR (R/K, f ) is I-cofinite. The exact sequence (5.3.1) yields the commutative diagram with exact rows

COHOMOLOGICAL DIMENSION, COFINITENESS AND ABELIAN CATEGORIES 29

o

/ HomR (R/K, M )

/ HomR (R, M )

/ HomR (K, M )

o

 / HomR (R/K, N )

 / HomR (R, N )

 / HomR (K, N )

and hence by Snake Lemma we get the exact sequence λ

0 −→ ker HomR (R/K, f ) −→ ker HomR (R, f ) −→ ker HomR (K, f ).

(5.3.4)

Since, the R-module HomR (K, M ) is Artinian and I-cofinite it follows from [Me2, Corollary 4.4] that the R-module im λ is I-cofinite. Moreover, the exact sequence (5.3.4) yields a short exact sequence 0 −→ ker HomR (R/K, f ) −→ kerf −→ im λ −→ 0, which implies that the R-module kerf is I-cofinite. Now, the remaining part of the proof follows from the exact sequences 0 −→ kerf −→ M −→ imf −→ 0, and 0 −→ imf −→ N −→ cokerf −→ 0.  Corollary 5.4. Let I be an ideal of a complete Noetherian local ring (R, m) such that q(I, R) ≤ 1. Let C (R, I)cof denote the category of I-cofinite modules over R. Let fi

f i+1

X • : · · · −→ X i −→ X i+1 −→ X i+2 −→ · · · , be a complex such that X i ∈ C (R, I)cof for all i ∈ Z. Then for each i ∈ Z the ith cohomology module H i (X • ) is in C (R, I)cof . Proof. The assertion follows from Theorem 5.3.



Corollary 5.5. Let (R, m) be a complete Noetherian local ring, I be an ideal of R with q(I, R) ≤ 1 and let M be an I-cofinite R-module. Then, i the R-modules TorR i (N, M ) and ExtR (N, M ) are I-cofinite, for all finitely generated R-modules N and all integers i ≥ 0. Proof. Since N is finitely generated it follows that, N has a free resolution with finitely generated free R-modules. Now the assertion follows using i Corollary 5.4 and computing the R-modules TorR i (N, M ) and ExtR (N, M ), using this free resolution. 

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Remark. Let (R, m) be a complete Noetherian local ring and I be an ideal of R. Then it is clear that if cd(I, R) ≤ 1 then q(I, R) ≤ 1. But in general q(I, R) ≤ 1 does not imply cd(I, R) ≤ 1. For instance, in the Example 2.5(ii) we have q(p, S) = 1 and cd(p, S) = 2. Now, it is clear that for the ideal p S  n S)  we have q(p S,  S)  = 1 and of the complete Noetrherian local ring (S,  S)  = 2. cd(p S, Acknowledgements The author is deeply grateful to the referee for a very careful reading of the manuscript and many valuable suggestions. Also the author would like to thank from School of Mathematics, Institute for Research in Fundamental Sciences (IPM) for its financial support. References [AB1] N. Abazari and K. Bahmanpour, A note on the Artinian cofinite modules, Comm. Algebra, 42 (2014), 1270–1275. [AB2] N. Abazari, K. Bahmanpour, Extension functors of local cohomology modules and Serre categories of modules, Taiwan. J. Math. 19 (2015), 211–220. [AM] M. Aghapournahr and L. Melkersson, Local cohomology and Serre subcategories, J. Algebra, 320 (2008), 1275–1287. [ADT] M. Asgharzadeh, K. Divaani-Aazar and M. Tousi, The finiteness dimension of local cohomology modules and its dual notion, J. Pure Appl. Algebra, 213 (2009), 321–328. [AT] M. Asgharzadeh and M. Tousi, A unified approach to local cohomology modules using Serre classes, Canad. Math. Bull. 53 (2010), 577–586. [B] K. Bahmanpour, A note on Lynch’s conjecture, Comm. Algebra, 45 (2017), 2738–2745. [BN] K. Bahmanpour and R. Naghipour, Cofiniteness of local cohomology modules for ideals of small dimension, J. Algebra, 321 (2009), 1997– 2011. [BNS] K. Bahmanpour, R. Naghipour and M. Sedghi, On the category of cofinite modules which is Abelian, Proc. Amer. Math. Soc. 142 (2014), 1101–1107. [BS] M.P. Brodmann and R.Y. Sharp, Local cohomology; an algebraic introduction with geometric applications, Cambridge University Press, Cambridge,1998. [Ch] G. Chiriacescu, Cofiniteness of local cohomology modules, Bull. London Math. Soc. 32 (2000), 1–7.

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Kamal Bahmanpour, Faculty of Sciences, Department of Mathematics, University of Mohaghegh Ardabili, 56199-11367 Ardabil, Iran;, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), 19395-5746 Tehran, Iran., E-mail:[email protected]