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( HCMUE Journal of Science ) ( Vol 18, No 9 (2021) 1596 1602 ) ( TẠP CHÍ KHOA HỌC HO CHI MINH CITY UNIVERSITY OF EDUCATION TRƯỜNG ĐẠI HỌC SƯ PHẠM TP HỒ CHÍ MINH JOURNAL OF SCIENCE Tập 18, Số 9 (2021)[.]
TẠP CHÍ KHOA HỌCHO CHI MINH CITY UNIVERSITY OF EDUCATION TRƯỜNG ĐẠI HỌC SƯ PHẠM TP HỒ CHÍ MINHJOURNAL OF SCIENCE Tập 18, Số (2021):1596-1602 ISSN: 2734-9918 Vol 18, No (2021): 1596-1602 Website: Research Article THE ARTINIANESS AND (I , J ) -STABLE OF LOCAL HOMOLOGY MODULE WITH RESPECT TO A PAIR OF IDEALS Tran Tuan Nam1*, Do Ngoc Yen2 Ho Chi Minh City University of Education, Vietnam Posts and Telecommunications Institute of Technology, Ho Chi Minh City, Vietnam * Corresponding author: Tran Tuan Nam – Email: namtt@hcmue.edu.vn Received: June 22, 2021; Revised: June 29, 2021; Accepted: August 31, 2021 ABSTRACT The concept of I -stable modules was defined by Tran Tuan Nam (Tran, 2013), and the author used it to study the representation of local homology modules In this paper, we will introduce the concept of (I , J ) -stable modules, which is an extension of the I -stable modules We study the (I , J ) -stable for local homology modules with respect to a pair of ideals, these modules have been studied by Tran and Do (2020) We show some basic properties of (I , J ) -stable modules and use them to study the artinianess of local homology modules with respect to a pair of ideals Moreover, we also examine the relationship between the artinianess, (I , J ) -stable, and the varnishing of local homology module with respect to a pair of ideals Keywords: artinian module; I -stable module; local homology Introduction Throughout this paper, ( R, m) is a local noetherian ring with the maximal ideal m Let I , J be ideals of R In (Tran & Do, 2020) we defined the local homology module H I ,J (i M ) H I ,J i with respect to a pair of ideals (I , J ) by (M ) = lim Tor R (R / a, M ) a∈W( I ,J ) in which W (I , J ) i I n ⊆ a + J for some integer n This definition is dual to the generalized local cohomology as reported in a study by Takahashi, Yoshino, and Yoshizawa (2009) and an extension from the local homology module in a study by Nguyen and Tran (2001) We also studied some properties of these modules in a the set of ideals a of R such that Cite this article as: Tran Tuan Nam, & Do Ngoc Yen (2021) The artinianess and (I , J ) -stable of local homology module with respect to a pair of ideals Ho Chi Minh City University of Education Journal of Science, 18(9), 1596-1602 HCMUE Journal of Science Tran Tuan Nam et al study by Tran and Do (2020), especially, we established the relationship between these modules and local homology modules with respect to an ideal through the isomorphic H I ,J (M ) ≅ lim H a (M ) Tran (2013) introduced the definition of I -stable modules, and i a∈W ( I ,J ) i the author used it to study the representation of local homology modules In this paper, we will introduce the concept of (I , J ) -stable module, which is an extension of the concept I -stable in Tran (2013)’s study Also, we show some properties of artinian and (I , J ) -stable of local homology modules Hi I ,J (M ) The first main result is p pCoass(M ) Proposition 2.2, there is a b∈W (I , J such that b ⊆ where M is ) (I , J ) -separated artinian R -module Next, Theorem 2.7 gives us the equivalent properties on artinianess of the local homology module The last result gives the relationship between the artinianess, (I , J ) -stable, and the varnishing of local homology module H iI ,J (M ) Some properties Lemma 2.1 Let M be an artinian R-module Then x ∈b such that xM = M for some b∈W (I , J ) H I ,J (M ) = if and only if there is 0 Proof According to Tran and Do (2020), H I ,J (M ) ≅Λ there is b∈W (I , J ) such that ΛI ,J (M ) ≅ M / I ,J (M ) and by M is artinian so Therefore, H bM only if M bM = I ,J (M ) = if and and by (Macdonald, 1973) if and only if M xM = for x ∈b We recall the concept of (I , J ) -separated The module M is called (I , J ) -separated aW ( I ,J ) aM if Proposition 2.2 If M is (I , J ) -separated artinian R-module Then there is a b∈W (I , J ) such that b ⊆p pCoass( M ) Proof M is (I , J ) -separated, by (Tran & Do, 2020) M ≅ ΛI ,J (M ) ≅ M / bM for HCMUE Journal of Science som b∈W (I , J hence e ) bM = It implies that btM = 0, Tran Tuan Nam et al so M is b -separated It follows Tran (2013) that b ⊆ p pCoass( M ) Corollary 2.3 Let M is an artinian R-module If b∈W (I , J ) such that b ⊆ H I ,J (M ) ≠ 0, i p p∈Coass( Hi I ,J ( M )) then there is a HCMUE Journal of Science Proof Vol 18, No (2021): 15961602 I ,J (M ) Accordin H g to Tran and Do (2020), Propositi on 2.2, we have the conclusio n HCMUE Journal of Science is (I , J ) -separated, and hence by i Corollary 2.4 Let M is an artinian R-module If ) i R then there is an ideal b∈W (I , J ) such that b An n ⊆ H I ,J (M ) R Proof According to Brodmann (1998), p p∈Att( Hi hand, Vol 18, No (2021): 15961602 I ,J H (M is an artinian R-module, I ,J = Ann H I ,J (M ( M )) ) On the other H I ,J (M ) is a representable, so Att(H I ,J (M )) = Coass(H I ,J (M )), by i i (Yassemi, 1995) Now the conclusion follows from Corollary 2.3 The concept of I -stable modules was defined in (Tran, 2009) An R-module N is called I -stable if for each x∈ I, there is a positive integer n such that element n Now we will give an extension concept of the I -stable for all t xt N = xnN Definition 2.5 M is called (I , J ) -stable if there is an ideal aM = bM a∈W ( I ,J ) When J = , we have bM = it is n such that I n ⊆ b So t >0 It M aM I t M Since b∈W (I , J and aW ( I ,J )t 0 ) I tM = I n M = I nM , then J = then M is I -stable Lemma 2.6 Let →M →N g→P →0 b∈W (I , J such that ) for all t > J =0, Hence, n when be a short exact sequence in which the module M , N, P are (I , J ) -separated Then module N is (I , J ) -stable if and s only if module M , P are (I , J ) -stable s Proof Assume that ideal bN = ∩ aN = 0, N is (I , J ) -stable Then there is N is (I , J ) -saparated, bM ⊆ b∈W (I , J such that ) bN = , so M is (I , J ) -stable We HCMUE Journal of Science Vol 18, No (2021): 1596have bP ≅ (bN + Kerg) / Kerg = = ∩bP, so P is 1602 (I , J ) -stable Otherwise, suppose that M and P are (I , J ) -stable, then there are ideals d = a ∩b , then dM = dP = 0, a,b such that bP = aM = Let (Brodmann, 1998), d n N = 0, so N is (I , J ) -stable Proposition 2.7 Let M be an artinian R-module and t a positive integer Then the following statements are equivalent i) H iI ,J (M ) is an artinian for all i < t; ii) There is an ideal b∈W (I , J ) such that b ⊆ Rad(Ann(H iI ,J (M ))) for all i < t HCMUE Journal of Science Proof (i ⇒ii) I ,J H ) Tran Tuan Nam et al is artinian, hence according to Tran and Do (2020), there is (M i b∈W (I , J ) such that bH I ,J (M ) = aH I ,J (M ) = Therefore, b ⊆ Rad(Ann(H I ,J (M for all ))) i i i a∈W ( I ,J ) i < t We use induction on t When t = 1, (ii ⇒i) H I ,J (M ) is artinian t> 1, Let0 I ,J (M ) ≅ M / bM , so according to Tran and Do (2020), we can replace M by aM As M is artinian, there is a aM H 0I ,J (M ) ≅ Λ aW ( I ,J ) b∈W (I , J ) such that bM = aW ( I ,J ) Therefore, we can assume that M = bM according to MacDonal (1973), there is an , element x ∈b such that M = xM By the hypothesis, there is a positive integer s such that x s H I ,J (M ) = for all i < t Then the short exact sequence i s s x →(0 :M x ) →M →M →0 gives rise the exact sequence →H I ,J (M ) →H I ,J (0 : i+1 for all i i < t −1 b ⊆ Rad(Ann(H I ,J (0 i : artinian for all It M xs ) →H I ,J (M ) →0 i follows a study by Brodmann (1998) xs )) and by the inductive hypothesis that H I ,J (0 ) : i < t −1 Thus H I ,J (M ) is artinian for all i < t i that xs ) is i We now recall the concept of the Noetherian dimension of an R -module M denoted by Ndim M Note that the notion of the Noetherian dimension was introduced first by Roberts (1975) by the name Krull dimension Later, Kirby (1990)changed this terminology of Roberts and referred to the Noetherian dimension to avoid confusion with the well7 HCMUE Journal of Science Tran Tuan Nam et known Krull dimension of finitely generated modules Let Mal be an R -module When M = , we put Ndim M = Then by induction, for any ordinal −1 Ndim M = α when (i) α Ndim M < α, is false, and (ii) for every ascending chain M ⊆ M1 ⊆ …of submodules of M , there exists a positive integer m0 Ndim(Mm+1 / M m ) < α we put such that for all m ≥ m0 Thus M is non-zero and finitely generated if and only if Ndim M = HCMUE Journal of Science Vol 18, No (2021): 15961602 Theorem 2.8 Let M be an artinian R-module and s an integer Then the following statements are equivalent H iI ,J (M ) is (I , J ) -stable for all i > s; i) ii) H iI ,J (M ) is artinian for all i > s; iii) Ass(H Ii ,J (M )) ⊆{m} for all i > s; iv) H iI ,J (M ) = for all i > s Proof (i ⇒ii) We use induction on d = Ndim M If d = 0, Hi I ,J (M ) = for all , d> so H I ,J (M ) is artinian Let 0, i hence we may assume bM M= i>0 aM we can replace M by aW ( I ,J ) for some b∈W (I , J ) and and M is artinian; is the minimum in the set bM {aM | a ∈W (I , J )} tha t cH I ,J (M ) = i c ∈W (I , such H I ,J (M is (I , J ) -stable so there is an ideal J) ) b, i aH I ,J (M ) = Le d = the dM = bM = M hence t n c , i a∈W ( I ,J ) there is d x∈ xM = M an d , such that xH I ,J (M ) = We have the short exact i sequence →(0 :M x) →M →M →0 gives rise to the exact sequence →H I,J (M ) →H I,J (0 : i+1 Because for all i M x) →H I,J (M ) →0 i H I ,J (M ) is (I , J ) -stable for all i > s , H iI ,J (0 so : x) is (I , J ) -stable i i > s −1 By the induction H I ,J (0 : hypothesis Therefore, (ii ⇒iii) i H I ,J i (M ) is artinian for all i > s (Yassemi, 1995), Supp(H iI ,J (M )) ⊇ Cosupp(H I ,J (M )) ∩ Max(R) = {m} Hence Ass(H I ,J (M )) ⊆ {m} (iii ⇒iv) x) is artinian for all i > s −1 i HCMUE Journal of Science We use induction on Vol 18, No (2021): 15961602 When d = (Tran & Do, 2020), d = Ndim M H I ,J (M ) = for all i > 0i Now, let d> 0, 0, we may assume that xM M= for and b∈W (I , J ) From the short exact sequence →(0 :M x) →M →M →0 rise to the exact sequence x ∈b gives H I ,J (M ) →H I ,J (0 : x) →H I ,J (M ) →H I ,J (M ) i+1 i Ass(H I ,J (0 i : H I ,J (0 : i M i i x)) ⊆{m} and Ndim(0 :M x) x) = for all By the induction hypothesis d 1 i> s From 10 that, we have the exact sequence HCMUE Journal of Science →H I ,J (M ) .x →H I ,J (M ) i Tran Tuan Nam et al If H I ,J (M ) ≠ 0, i Ass(H I ,J (M )) ={m}, there is an am = 0, so xa = 0, all i>s, then i a ∈ H I ,J (M such that m = Ann( a) ) it elementi implies that for hence a= 0, it is a contraction Therefore, H I ,J (M ) = for all i > s i (iv ⇒i) It is clear Conclusion In this paper, we gave the concept of the (I , J ) -stable module We studied the properties of the (I , J ) -stable of local homology module with respect to a pair of ideals (I , J ) Moreover, we showed the relationship between of the artinianess and the (I , J ) -stable of local homology module with respect to a pair of ideals Conflict of Interest: Authors have no conflict of interest to declare REFERENCES Brodmann, M P., & Sharp, R Y (1998) Local cohomology: an algebraic introduction with geometric applications Cambridge University Press Kirby, D (1990) Dimension and length of artinian modules Quart, J Math Oxford, 41, 419-429 Macdonald, I G (1973) Secondary representation of modules over a commuatative ring Symposia Mathematica, 11, 23-43 Nguyen, T C., & Tran, T N (2001) The I -adic completion and local homology for Artinian modules Math Proc Camb Phil Soc., 131, 61-72 Robert, R N (1975) Krull dimension for artinian modules over quasi-local commutative rings Quart J Math., 26, 269-273 Takahashi R., Yoshino Y., & Yoshizawa T (2009) Local cohomology based on a nonclosed support defined by a pair of ideals J Pure Appl Algebra, 213, 582-600 Tran, T N (2009) A finiteness result for co-associated and associated primes of generalized local homology and cohomology module Communications in Algebra, 37, 1748-1757 11 HCMUE Journal of Science Tran Tuan Nam et Tran, T N (2013) Some properties of local homology and local cohomology modules Studia al Scientiarum Mathematicarum Hungarica, 50, 129-141 Tran, T N., & Do, N Y (2020) Local homology with respect to a pair of ideal, reprint Yassemi, S (1995) Coassociated primes Comm Algebra, 23, 1473-1498 12 HCMUE Journal of Science Vol 18, No (2021): 15961602 TÍNH ARTIN VÀ TÍNH (I , J ) -ỔN ĐỊNH CỦA MÔĐUN ĐỒNG ĐIỀU ĐỊA PHƯƠNG TƯƠNG ỨNG VỚI MỘT CẶP IĐÊAN Trần Tuấn Nam1*, Đỗ Ngọc Yến2 Trường Đại học Sư phạm Thành phố Hồ Chí Minh, Việt Nam Học viên Cơng nghệ Bưu Viễn thơng, Thành phố Hồ Chí Minh, Việt Nam * Tác giả liên hệ: Trần Tuấn Nam – Email: namtt@hcmue.edu.vn Ngày nhận bài: 22-6-2021; ngày nhận sửa: 29-6-2021; ngày duyệt đăng: 31-8-2021 TĨM TẮT Khái niệm mơđun I -ổn định đưa Tran Tuan Nam báo (Tran, 2013) tác giả sử dụng cơng cụ để nghiên cứu tính biểu diễn lớp môđun đồng điều địa phương Trong báo này, giới thiệu lớp môđun (I , J ) -ổn định, xem khái niệm mở rộng thực từ khái niệm I -ổn định Chúng tơi nghiên cứu tính (I , J ) -ổn định cho lớp môđun đồng điều địa phương theo cặp iđêan, lớp môđun nghiên cứu (Tran & Do, 2020) Các tính chất mơđun (I , J ) -ổn định nghiên cứu sử dụng để nghiên cứu tính artin lớp mơđun đồng điều địa phương theo cặp iđêan Hơn nữa, chúng tơi đưa mối liên hệ tính artin, tính (I , J ) -ổn định tính triệt tiêu lớp môđun đồng điều địa phương theo cặp iđêan Từ khóa: mơđun artin; mơđun I -ổn định; đồng điều địa phương 13 ... tính artin lớp môđun đồng điều địa phương theo cặp iđêan Hơn nữa, đưa mối liên hệ tính artin, tính (I , J ) -ổn định tính triệt tiêu lớp mơđun đồng điều địa phương theo cặp iđêan Từ khóa: mơđun artin; ... HCMUE Journal of Science Vol 18, No (2021): 15961602 TÍNH ARTIN VÀ TÍNH (I , J ) -ỔN ĐỊNH CỦA MÔĐUN ĐỒNG ĐIỀU ĐỊA PHƯƠNG TƯƠNG ỨNG VỚI MỘT CẶP IĐÊAN Trần Tuấn Nam1*, Đỗ Ngọc Yến2 Trường Đại học Sư... niệm I -ổn định Chúng tơi nghiên cứu tính (I , J ) -ổn định cho lớp môđun đồng điều địa phương theo cặp iđêan, lớp môđun nghiên cứu (Tran & Do, 2020) Các tính chất môđun (I , J ) -ổn định nghiên