Attached primes and annihilators of top local cohomology modules defined by a pair of ideals

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Дата:	2020
Автори:	Karimi, S., Payrovi, Sh.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2020
Назва видання:	Algebra and Discrete Mathematics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/188516
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Цитувати:	Attached primes and annihilators of top local cohomology modules defined by a pair of ideals / S. Karimi, Sh. Payrovi // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 2. — С. 211–220. — Бібліогр.: 13 назв. — англ.

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spelling	irk-123456789-1885162023-03-04T01:27:06Z Attached primes and annihilators of top local cohomology modules defined by a pair of ideals Karimi, S. Payrovi, Sh. 2020 Article Attached primes and annihilators of top local cohomology modules defined by a pair of ideals / S. Karimi, Sh. Payrovi // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 2. — С. 211–220. — Бібліогр.: 13 назв. — англ. 1726-3255 DOI:10.12958/adm429 2010 MSC: 13D45, 14B15 http://dspace.nbuv.gov.ua/handle/123456789/188516 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
format	Article
author	Karimi, S. Payrovi, Sh.
spellingShingle	Karimi, S. Payrovi, Sh. Attached primes and annihilators of top local cohomology modules defined by a pair of ideals Algebra and Discrete Mathematics
author_facet	Karimi, S. Payrovi, Sh.
author_sort	Karimi, S.
title	Attached primes and annihilators of top local cohomology modules defined by a pair of ideals
title_short	Attached primes and annihilators of top local cohomology modules defined by a pair of ideals
title_full	Attached primes and annihilators of top local cohomology modules defined by a pair of ideals
title_fullStr	Attached primes and annihilators of top local cohomology modules defined by a pair of ideals
title_full_unstemmed	Attached primes and annihilators of top local cohomology modules defined by a pair of ideals
title_sort	attached primes and annihilators of top local cohomology modules defined by a pair of ideals
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2020
url	http://dspace.nbuv.gov.ua/handle/123456789/188516
citation_txt	Attached primes and annihilators of top local cohomology modules defined by a pair of ideals / S. Karimi, Sh. Payrovi // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 2. — С. 211–220. — Бібліогр.: 13 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT karimis attachedprimesandannihilatorsoftoplocalcohomologymodulesdefinedbyapairofideals AT payrovish attachedprimesandannihilatorsoftoplocalcohomologymodulesdefinedbyapairofideals
first_indexed	2025-07-16T10:36:34Z
last_indexed	2025-07-16T10:36:34Z
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fulltext	“adm-n2” — 2020/7/8 — 8:15 — page 211 — #71 © Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 29 (2020). Number 2, pp. 211–220 DOI:10.12958/adm429 Attached primes and annihilators of top local cohomology modules defined by a pair of ideals S. Karimi and Sh. Payrovi Communicated by V. Lyubashenko Abstract. Assume that R is a complete Noetherian local ring and M is a non-zero finitely generated R-module of dimension n = dim(M) > 1. It is shown that any non-empty subset T of Assh(M) can be expressed as the set of attached primes of the top local cohomology modules Hn I,J (M) for some proper ideals I, J of R. Moreover, for ideals I, J = ⋂ p∈AttR(Hn I (M)) p and J ′ of R it is proved that T = AttR(H n I,J(M)) = AttR(H n I,J′(M)) if and only if J ′ ⊆ J . Let Hn I,J(M) 6= 0. It is shown that there exists Q ∈ Supp(M) such that dim(R/Q) = 1 and Hn Q(R/p) 6= 0, for each p ∈ AttR(H n I,J(M)). In addition, we prove that if I and J are two proper ideals of a Noetherian local ring R, then AnnR(H n I,J (M)) = AnnR(M/TR(I, J,M)), where TR(I, J,M) is the largest submodule of M with cd(I, J, TR(I, J,M)) < cd(I, J,M), here cd(I, J,M) is the cohomological dimension of M with respect to I and J . This result is a generalization of [1, Theorem 2.3] and [2, Theorem 2.6]. Introduction Assume that R is a Noetherian ring and I, J are two ideals of R and M is an R-module. As a generalization of the usual local cohomology modules, the local cohomology modules with respect to a system of ideals was introduced, in [3]. As a special case of these extended modules, in [13], 2010 MSC: 13D45, 14B15. Key words and phrases: associated prime ideals, attached prime ideals, top local cohomology modules. https://doi.org/10.12958/adm429 “adm-n2” — 2020/7/8 — 8:15 — page 212 — #72 212 Attached primes of local cohomology the local cohomology modules with respect to a pair of ideals is defined. To be more precise, let W (I, J) = {p ∈ Spec(R)\|In ⊆ p+ J for some positive integer n}. The (I, J)-torsion submodule ΓI,J (M) of M , which consists of all elements x of M with Supp(Rx) ⊆ W (I, J), is considered. For an integer i, the local cohomology functor H i I,J with respect to (I, J) is defined to be the i-th right derived functor of ΓI,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). Recall that for an R-module K, a prime ideal p of R is said to be an attached prime ideal of K if p = Ann(K/N) for some submodule N of K. The set of attached prime ideals of K is denoted by AttR(K). When K has a secondary representation, this definition agrees with the usual definition of attached primes in [12]. Let R be a Noetherian local ring with maximal ideal m and M be a finitely generated R-module of dimension n. The main theorem in Section 2, shows that if R is complete with respect to m-adic topology, then for any non-empty subset T of Assh(M) there exist ideals I, J of R such that T = AttR(H n I (M)) = AttR(H n I,J (M)) which is an another version of Theorem 2.8 in [8]. Moreover we show that for each p ∈ AttR(H n I,J(M)) there exists Q ∈ Supp(M) such that dim(R/Q) = 1 and Hn Q(R/p) 6= 0. Let R be a Noetherian ring, I, J be two ideals of R and M be a non- zero finitely generated R-module of dimension n. Let cd(I, J,M) denote the supremum of all integers r for which Hr I,J (M) 6= 0. We call this integer the cohomological dimension of M with respect to ideals I, J , see [7]. In Section 3, first we define TR(I, J,M) the largest submodule of M such that cd(I, J, TR(I, J,M)) < cd(I, J,M) and we show that TR(I, J,M) = Γa(M) = ⋂ cd(I,J,R/pj)=c Nj , where 0 = ⋂n j=1Nj denotes a reduced primary decomposition of the zero submodule of M , Nj is a pj-primary submodule of M and a = ∏ cd(I,J,R/pj) 6=c pj . We show AnnR(H n I,J(M)) = AnnR(M/TR(I, J,M)), which is a generalization of [1, Theorem 2.3] and [2, Theorem 2.6] and some applications of this theorem are given. “adm-n2” — 2020/7/8 — 8:15 — page 213 — #73 S. Karimi, Sh. Payrovi 213 1. Attached prime ideals of top local cohomology modules In this section we assume that (R,m) is local and complete with respect to m-adic topology, M is a non-zero finitely generated R-module of dimension n > 1 and T is a non-empty subset of Assh(M). Definition 1. Let K be an R-module, a prime ideal p of R is said to be an attached prime ideal of K if p = Ann(K/N) for some submodule N of K. The set of attached prime ideals of K is denoted by AttR(K). Lemma 1. [1, Lemma 3.2] Let K be an R-module. Then the set of minimal elements of V (AnnR(K)) coincides with that of AttR(K). In particular, √ AnnR(K) = ⋂ p∈AttR(K) p. Theorem 1. Let M be a non-zero finitely generated R-module of dimen- sion n and T be a non-empty subset of Assh(M). Then the following statements are true: (i) If T ⊆ AttR(H n I (M)) for some ideal I, then T = AttR(H n I,J(M)), where J = ⋂ p∈T p. (ii) T = AttR(H n I (M)) = AttR(H n I,J(M)), where I, J are ideals of R and J = √ AnnR(Hn I (M)). Proof. (i) By assumption T is a non-empty subset of Assh(M). Set J := ⋂ p∈T p. We show that T = AttR(H n I,J(M)). Assume that q ∈ AttR(H n I,J(M)). Then by [6, Theorem 2.1] it follows that J ⊆ q. Thus p ⊆ q for some p ∈ T . Hence, this fact that p, q are in Assh(M) shows that p = q and so q ∈ T . Now, let q ∈ T . Then J ⊆ q and also q ∈ AttR(H n I (M)). Therefore, q ∈ AttR(H n I,J(M)) by [6, Theorem 2.1]. (ii) In view of [8, Theorem 2.8] there exists an ideal I of R such that T = AttR(H n I (M)). Thus by (i) and Lemma 1 the result follows. Corollary 1. Let M be a non-zero finitely generated R-module of dimen- sion n and let I1, I2, J1, J2 be ideals of R. Then the following statements hold: (i) If T ⊆ AttR(H n I1,J1 (M))∪AttR(H n I2,J2 (M)) is a non-empty set and J = ⋂ p∈T p, then AttR(H n I1+I2,J (M)) = T . (ii) If T = AttR(H n I1,J1 (M)) ∪ AttR(H n I2,J2 (M)) is a non-empty set and J = ⋂ p∈T p, then AttR(H n I1+I2,J (M)) = AttR(H n I1,J1 (M)) ∪ AttR(H n I2,J2 (M)). (iii) If T = AttR(H n I1,J1 (M)) ∩ AttR(H n I2,J2 (M)) is a non-empty set and J = ⋂ p∈T p, then AttR(H n I1+I2,J (M)) = AttR(H n I1,J1 (M)) ∩ AttR(H n I2,J2 (M)). “adm-n2” — 2020/7/8 — 8:15 — page 214 — #74 214 Attached primes of local cohomology Proof. Let p ∈ T and p ∈ AttR(H n I1,J1 (M)). Then p ∈ AttR(H n I1 (M)), by [6, Theorem 2.1]. Thus dimR/p = n and by Lichtenbaum-Hartshorne Vanishing Theorem dimR/(I1 + p) = 0. Since I1 + p ⊆ I1 + I2 + p, it follows that dimR/(I1 + I2 + p) = 0 and so Hn I1+I2 (R/p) 6= 0. Thus p ∈ AttR(H n I1+I2 (M)), by [9, Theorem A]. Therefore, T ⊆ AttR(H n I1+I2 (M)) and the result follows by Theorem 1(i). Corollary 2. Let I, J be ideals of R and let M be a non-zero finitely generated R-module of dimension n. If T = AttR(H n I (M)) and J ′ = J + ⋂ p∈T p, then AttR(H n I,J(M)) = AttR(H n m,J ′(M)). Proof. In view of [6, Theorem 2.1] and Lichtenbaum-Hartshorne Vanishing Theorem, we have AttR(H n I,J(M)) = {p ∈ Supp(M) ∩ V (J) : √ I + p = m}. Let p ∈ AttR(H n I,J(M)). Then p ∈ AttR(H n I (M)) and 0 6= Hn I (R/p) ∼= Hn I+p(R/p) ∼= Hn m(R/p). Hence, p ∈ AttR(H n m,J ′(M)). The proof of the opposite inclusion is similar. Theorem 2. Let M be a non-zero finitely generated R-module of di- mension n and let I, J = ⋂ p∈AttR(Hn I (M)) p and J ′ be ideals of R. Then AttR(H n I,J(M)) = AttR(H n I,J ′(M)) if and only if J ′ ⊆ J . Proof. Let AttR(H n I,J(M)) = AttR(H n I,J ′(M)). Then Theorem 1 shows that AttR(H n I (M)) = AttR(H n I,J(M)). Hence, by [6, Theorem 2.1] J ′ ⊆ ⋂ p∈AttR(Hn I,J′ (M)) p = ⋂ p∈AttR(Hn I,J (M)) p = ⋂ p∈AttR(Hn I (M)) p = J. Conversely, if J ′ ⊆ J , then by [6, Theorem 2.1] we have AttR(H n I (M)) = AttR(H n I,J(M)) ⊆ AttR(H n I,J ′(M)) ⊆ AttR(H n I (M)). So the result follows. Theorem 3. Let M be a non-zero finitely generated R-module of dimen- sion n and let I, I ′ and J = ⋂ p∈AttR(Hn I (M)) p be ideals of R such that I ⊆ I ′. Then Att(Hn I,J(M)) = Att(Hn I′,J(M)) Proof. Assume that p ∈ Att(Hn I,J(M)). Thus [6, Theorem 2.1] shows that p ∈ Att(Hn I (M)) and J ⊆ p. By assumption and [10, Proposition “adm-n2” — 2020/7/8 — 8:15 — page 215 — #75 S. Karimi, Sh. Payrovi 215 1.6], AttR(H n I (M)) ⊆ AttR(H n I′(M)) so that p ∈ Att(Hn I′(M)) and p ∈ Att(Hn I′,J(M)). Thus Att(Hn I,J(M)) ⊆ Att(Hn I′,J(M)). Therefore, J ⊆ ⋂ p∈Att(Hn I′,J (M)) p ⊆ ⋂ p∈Att(Hn I,J (M)) p = J which shows that Att(Hn I,J(M)) = Att(Hn I′,J(M)). Theorem 4. Let M be a finitely generated R-module of dimension n and I, J be ideals of R such that Hn I,J (M) 6= 0. Then there exists Q ∈ Supp(M) such that dim(R/Q) = 1 and Hn Q(R/p) 6= 0, for each p ∈ AttR(H n I,J (M)). Proof. By assumption T = AttR(H n I,J(M)) 6= ∅. Then in view of [8, Theorem 2.8] we have T = AttRH n a (M) for some ideal a of R. Now, [8, Proposition 2.1] shows that there exists an integer r such that for all 1 6 i 6 r there exists Qi ∈ Supp(M) with dim(R/Qi) = 1 such that ⋂ p∈T p 6⊆ Qi. In addition, we may assume that a = ⋂r i=1Qi. Let p ∈ AttRH n I,J(M). Then p ∈ AttRH n a (M) and so Hn⋂r i=1 Qi (R/p) 6= 0. Now, by setting b = ⋂r−1 i=1 Qi and c = Qr we have the following long exact sequence Hn b+c(R/p) → Hn b (R/p)⊕Hn c (R/p) → Hn b∩c(R/p) → 0, where Hn b∩c(R/p) = Hn Q1∩···∩Qr (R/p) 6= 0. So Hn b (R/p) ⊕Hn c (R/p) 6= 0. Therefore Hn b (R/p) 6= 0 or Hn c (R/p) 6= 0. If Hn c (R/p) 6= 0 we are done. Otherwise, one can set b = ⋂r−2 i=1 Qi and c = Qr−1 and with repeat this method, to get the result. Corollary 3. Let M be a finitely generated R-module of dimension n > 1 and I, J be ideals of R such that Hn I,J(M) 6= 0. Then there exists Q ∈ Supp(M) such that dim(R/Q) = 1 and AttR(H n Q,J(M)) 6= ∅. 2. Annihilators of top local cohomology modules In this section (R,m) is a Noetherian local ring with maximal ideal m and I, J are two proper ideals of R. Let M be a non-zero finitely generated R-module and let cd(I, J,M) denote the supremum of all integers r for which Hr I,J(M) 6= 0. We call this integer the cohomological dimension of M with respect to ideals I, J . When J = 0, we have cd(I, 0,M) = cd(I,M) which is just the supremum of all integers r for which Hr I (M) 6= 0. In [7, Corollary 3.3] a characterization of cd(I, J,M) is provided. “adm-n2” — 2020/7/8 — 8:15 — page 216 — #76 216 Attached primes of local cohomology Lemma 2. [7, Proposition 3.2] Let M and N be two finitely generated R- modules such that Supp(N) ⊆ Supp(M). Then cd(I, J,N) 6 cd(I, J,M). Definition 2. Let M be a non-zero finitely generated R-module of coho- mological dimension c. We denote by TR(I, J,M) the largest submodule of M such that cd(I, J, TR(I, J,M)) < cd(I, J,M). It is easy to check that TR(I, J,M) = ∪{N : N 6 M, cd(I, J,N) < cd(I, J,M)}. When J = 0, this definition coincides with that of [1, Defi- nition 2.2]. Lemma 3. Let M be a non-zero finitely generated R-module of dimension n such that n = cd(I, J,M). Then TR(m,M) ⊆ TR(I,M) ⊆ TR(I, J,M). Proof. For the first inclusion let x /∈ TR(I,M). Then cd(I, J,Rx) = n and so Hn I (Rx) 6= 0. Thus dim(Rx) = n. Hence, by Grothendieck’s Vanishing Theorem Hn m(Rx) 6= 0 and x /∈ TR(m,M). Let x /∈ TR(I, J,M). Then cd(I, J,Rx) = n and so Hn I,J(Rx) 6= 0. Thus AttR(H n I,J(Rx)) 6= ∅ and AttR(H n I (Rx)) 6= ∅ by [6, Theorem 2.1]. Hence, Hn I (Rx) 6= 0 and cd(I, Rx) = n. Therefore, x /∈ TR(I,M). Theorem 5. Let M be a non-zero finitely generated R-module with coho- mological dimension c = cd(I, J,M). Then TR(I, J,M) = Γa(M) = ⋂ cd(I,J,R/pj)=c Nj . Here 0 = ⋂n j=1Nj is a reduced primary decomposition of the zero submodule of M , Nj is a pj-primary submodule of M and a = ∏ cd(I,J,R/pj) 6=c pj. Proof. First we show the equality Γa(M) = ⋂ cd(I,J,R/pj)=cNj . To do this, the inclusion ⋂ cd(I,J,R/pj)=cNj ⊆ Γa(M) follows easily by the proof of [11, Theorem 6.8(ii)]. In order to prove the opposite inclusion, suppose, the con- trary is true. Then there exists x ∈ Γa(M) such that x /∈ ⋂ cd(I,J,R/pj)=cNj . Thus there exists an integer t such that x /∈ Nt and cd(I, J,R/pt) = c. Now, as x ∈ Γa(M), it follows that there is an integer s > 1 such that asx = 0, and so asx ⊆ Nt. Because of x /∈ Nt and Nt is a pt-primary submodule, it yields that a ⊆ pt. Hence, there is an integer j such that pj ⊆ pt and cd(I, J,R/pj) 6 c − 1. Therefore, in view of Lemma 2, we have cd(I, J,R/pt) 6 cd(I, J,R/pj) 6 c− 1, “adm-n2” — 2020/7/8 — 8:15 — page 217 — #77 S. Karimi, Sh. Payrovi 217 which is a contradiction. Now, we show that TR(I, J,M) = Γa(M). Let x ∈ TR(I, J,M). Then in view of Lemma 2, cd(I, J,Rx) 6 c− 1. Let p be a minimal prime ideal of AnnR(Rx), it follows that cd(I, J,R/p) 6 c− 1. So a ⊆ ⋂ cd(I,J,R/pj)6c−1 pj ⊆ ⋂ p∈AssR(Rx) p = √ AnnR(Rx). Thus there exists an integer k > 1 such that ak ⊆ AnnR(Rx). Hence, akx = 0 and x ∈ Γa(TR(I, J,M)). Thus TR(I, J,M) = Γa(TR(I, J,M)). Now, we have TR(I, J,M) = Γa(TR(I, J,M)) ⊆ Γa(M). We show that Γa(M) ⊆ TR(I, J,M), to do this, we show cd(I, J,Γa(M)) 6 c − 1. Let p ∈ Supp(Γa(M)). Then a ⊆ p and there exists pj ⊆ p such that cd(I, J,R/pj) 6 c − 1. Thus by Lemma 2, cd(I, J,R/p) 6 c − 1. Hence, cd(I, J,Γa(M)) 6 c − 1 by [7, Theorem 3.1]. Therefore, TR(I, J,M) = Γa(M). Corollary 4. Let M be a non-zero finitely generated R-module of dimen- sion n with cohomological dimension c = cd(I, J,M). Then the following statements are true: (i) AssR(TR(I, J,M)) = {p ∈ AssR(M) : cd(I, J,R/p) 6 c− 1}, (ii) AssR(M/TR(I, J,M)) = {p ∈ AssR(M) : cd(I, J,R/p) = c}. If n = c, then AssR(M/TR(I, J,M)) = AttR(H n I,J(M)). Proof. By Theorem 5, TR(I, J,M) = Γa(M), where ∏ cd(I,J,R/pj)6c−1 pj = a. So by [4, Section 2.1, Proposition 10] we have AssR(TR(I, J,M)) = AssR(M) ∩ V (a). Now (i) follows from Lemma 2. In order to show (ii), use [5, Exercise 2.1.12] and [6, Theorem 2.1]. Corollary 5. Let M be a non-zero finitely generated R-module of dimen- sion n such that n = cd(I, J,M). Then there exists a positive integer t such that J tM ⊆ TR(I, J,M). Proof. Let 0 = ⋂n j=1Nj denote a reduced primary decomposition of the zero submodule of M , where Nj is a pj-primary submodule of M . In view of Theorem 5, TR(I, J,M) = ⋂ cd(I,J,R/pj)=nNj . If cd(I, J,R/pj) = n, then Hn I,J(R/pj) 6= 0. Thus J ⊆ pj = √ AnnR(M/Nj) by [13, Theorem 4.3]. Hence, there exists a positive integer tj such that J tjM ⊆ Nj . Let t = max{tj : cd(I, J,R/pj) = n}. Then J tM ⊆ ⋂ cd(I,J,R/pj)=nNj = TR(I, J,M). “adm-n2” — 2020/7/8 — 8:15 — page 218 — #78 218 Attached primes of local cohomology Theorem 6. Let M be a non-zero finitely generated R-module of dimen- sion n = cd(I, J,M). Then AnnR(H n I,J(M)) = AnnR(M/TR(I, J,M)). Proof. By Corollary 5, J tM ⊆ TR(I, J,M) for some integer t > 1 and by [13, Proposition 1.4(8)], H i I,J(M) ∼= H i I,Jt(M) for all i > 0. Then we can assume that JM ⊆ TR(I, J,M). First we show that TR(I,M/JM) = TR(I, J,M)/JM . Let x ∈ M and consider the exact sequence 0 → Rx ∩ JM → Rx → Rx Rx ∩ JM → 0 that induce an exact sequence · · · → Hn I,J(Rx ∩ JM) → Hn I,J(Rx) → Hn I,J( Rx Rx ∩ JM ) → 0. (∗) If x + JM ∈ TR(I,M/JM), then Hn I,J(Rx/Rx ∩ JM) ∼= Hn I (R(x + JM)) = 0, by [13, Corollary 2.5]. As Rx ∩ JM ⊆ TR(I, J,M) it follows that Hn I,J (Rx ∩ JM) = 0. Hence, Hn I,J (Rx) = 0. So that x ∈ TR(I, J,M). If x ∈ TR(I, J,M), then Hn I,J(Rx) = 0. Thus Hn I,J(Rx/Rx ∩ JM) = Hn I (R(x+ JM)) = 0 by (∗). Therefore, x+ JM ∈ TR(I,M/JM). Now, from the exact sequence 0 → JM → M → M JM → 0 we have the exact sequence · · · → Hn I,J(JM) → Hn I,J(M) → Hn I,J( M JM ) → 0. Since JM ⊆ TR(I, J,M), it follows that Hn I,J(JM) = 0 and so we have Hn I,J(M) ∼= Hn I,J(M/JM). Thus Hn I,J(M) ∼= Hn I (M/JM). Therefore, AnnR(H n I,J(M)) = AnnR(H n I (M/JM)) = AnnR( M/JM TR(I,M/JM) ) = AnnR( M/JM TR(I, J,M)/JM ) = AnnR(M/TR(I, J,M)), see [1, Theorem 2.3]. Corollary 6. Let M be a non-zero finitely generated R-module of di- mension n = cd(I, J,M) and JM ⊆ TR(I,M). Then AnnR(H n I (M)) = AnnR(H n I,J(M)). “adm-n2” — 2020/7/8 — 8:15 — page 219 — #79 S. Karimi, Sh. Payrovi 219 Proof. By a similar argument to that of Theorem 6, one can show that TR(I,M/JM) = TR(I,M)/JM . Also, by Lemma 3 we have TR(I,M) ⊆ TR(I, J,M). Thus JM ⊆ TR(I, J,M) and so Hn I,J(JM) = 0. Hence, it follows by (∗) that Hn I,J(M) ∼= Hn I (M/JM). Therefore, AnnR(H n I,J(M)) = AnnR(H n I (M/JM)) = AnnR( M/JM TR(I,M/JM) ) = AnnR( M/JM TR(I,M)/JM ) = AnnR(M/TR(I,M)). Now, the result follows by [1, Theorem 2.3] and Theorem 6. Corollary 7. Let M be a non-zero finitely generated R-module of dimen- sion n = cd(I, J,M). Then the following statements hold: (i) √ AnnR(Hn I,J(M)) = ⋂ p∈AssRM,cd(I,J,R/p)=n p, (ii) V (AnnR(H n I,J(M))) = Supp(M/TR(I, J,M)), (iii) If TR(I, J,M) = 0, then Supp(M) = V (AnnR(H n I,J(M))). Proof. (i) It follows by Lemma 1 and [6, Theorem 2.1]. To prove (ii) use Theorem 6. (iii) It follows from (ii). Corollary 8. Let d = dimR = cd(I, J,R). Then the following statements hold: (i) cd(I, J,AnnR(H d I,J(R))) < dimR. (ii) If d > 1, then dimR = dimR/AnnR(H d I,J(R)) = dimR/ΓI,J(R). Proof. (i) It follows from Theorem 6. (ii) By [13, Corollary 1.13 (4)], Hd I,J(R) ∼= Hd I,J(R/ΓI,J(R)). So that ΓI,J(R) ⊆ AnnR(H d I,J(R)). Corollary 9. If R is a domain of dimR = d and Hd I,J(R) 6= 0, then AnnR(H d I,J(R)) = 0. Proof. It follows by Theorems 5 and 6. 3. Acknowledgment The authors are deeply grateful to the S. Babaei and the referee for careful reading of the manuscript, very helpful suggestions and insightful comments. “adm-n2” — 2020/7/8 — 8:15 — page 220 — #80 220 Attached primes of local cohomology References [1] A. Atazadeh, M. Sedghi and R. Naghipour, On the annihilators and attached primes of top local cohomology modules, Arch. Math., 102 (2014), 225- 236. [2] K. Bahmanpour, J. A’zami and Gh. Ghasemi, On the annihilators of local coho- mology modules, J. Algebra, 363 (2012), 8- 13. [3] M. H. Bijan-Zadeh, Torsion theories and local cohomology over commutative Noethe- rian rings, J. London Math. Soc., 19 (1979), 402- 410. [4] N. Bourbaki, Commutative algebra, Herman, Paris 1972. [5] M. P. Brodmann and R. Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge studies in advanced mathematics, 60, Cambridge University Press, Cambridge, (1998). [6] L. Chu, Top local cohomology modules with respect to a pair of ideals, Proc. Amer. Math. Soc., 139 (2011), 777-782. [7] L. Chu and Q. Wang, Some results on local cohomology modules defined by a pair of ideals, J. Math. Kyoto Univ., 49 (2009), 193-200. [8] M. T. Dibaei and R. Jafari, Top local cohomology modules with specified attached primes, Algebra Colloq., 15 (2008), 341-344. [9] M. T. Dibaei and S. Yassemi, Attached primes of the top local cohomology modules with respect to an ideal, Arch. Math., 84 (2005), no. 4, 292-297. [10] M. T. Dibaei and S. Yassemi, Attached primes of the top local cohomology modules with respect to an ideal (II), arXiv: 0501063 v1 [math AC] 5 Jan 2005. [11] H. Matsumura, Commutative ring theory, Cambridge University Press, 1986. [12] I. G. MacDonald, Secondary representations of modules over a commutative rings, Symp. Math. Vol., XI (1973), 23-43. [13] R. Takahashi, Y. Yoshino and T. Yoshizawa, Local cohomology based on a nonclosed support defined by a pair of ideals, J. Pure Appl. Algebra, 213 (2009), 582-600. Contact information Susan Karimi Department of Mathematics, Payame Noor University, 19395-3679, Tehran, Iran E-Mail(s): susan_karimi2@yahoo.com Shiroyeh Payrovi Department of Mathematics, Imam Khomeini International University, 34149-1-6818, Qazvin, Iran E-Mail(s): shpayrovir@sci.ikiu.ac.ir Received by the editors: 13.03.2017. S. Karimi, Sh. Payrovi

Attached primes and annihilators of top local cohomology modules defined by a pair of ideals

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