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Science & Technology Development Journal, 23(1):479-483 Research Article Open Access Full Text Article Cominimax modules and generalized local cohomology modules Bui Thi Hong Cam, Nguyen Minh Tri* ABSTRACT Use your smartphone to scan this QR code and download this article The local cohomology theory plays an important role in commutative algebra and algebraic geometry. The I-cofiniteness of local cohomology modules is one of interesting properties which has been studied by many mathematicians. The I-cominimax modules is an extension of I-cofinite modules which was introduced by Hartshorne. An R -module M is I-cominimax if SuppR M ⊆ V (I) and ExtiR (R/I, M) is minimax for all i ≥ 0. The aim of this paper is to show some conditions such that the generalized local cohomology module HI′ (M, N) is I-cominimax for all i ≥ 0. We prove that HIi (M, K) if is I-cofinite for all i ≥ 0. We prove that if HIi (M, K) is I-cofinite for all i < t and all finitely generated R-module K, then HIi (M, N) is I-cominimax for all i < t and all minimax R-module N. If M is a finitely generated R-module, N is a minimax R-module and t is a non-negative integer such that dim SuppR HIi (M, N) ≤ 1 for all i < t , then HIi (M, N) is I-cominimax for all i < t . When dim R/I ≤ 1 and HIi (N) is I-cominimax for all i ≥ 0, then HI′ (M, N) is I-cominimax for all i ≥ 0. Key words: Generalized local cohomology, I-cominimax INTRODUCTION Department of Natural Science Education, Dong Nai University, Dong Nai, Vietnam Correspondence Nguyen Minh Tri, Department of Natural Science Education, Dong Nai University, Dong Nai, Vietnam Email: nguyenminhtri@dnpu.edu.vn History • Received: 2019-07-11 • Accepted: 2020-02-11 • Published: 2020-03-24 DOI : 10.32508/stdj.v23i1.1696 Copyright © VNU-HCM Press. This is an openaccess article distributed under the terms of the Creative Commons Attribution 4.0 International license. Let R be a local Noetherian ring, I an ideal of R and M a finitely generated R -module. It is well known that the local cohomology modules HIi (M) are not generally finitely generated for i > 0. In a 1970 paper Hartshorne 1 gave the concept of I-cofinite modules. An R-module K to be I-cofinite if SuppR K ⊆ V (I) j and ExtR (R/I, K) is finitely generated for all j ≥ 0. Hartshorne asked which rings R and ideals I the modules HIi (M) were I-cofinite for all i and all finitely generated modules M. In 1, if (R , m) is a complete regular local ring and M is a finitely generated R-module, then HIi (M) is I -cofinite in two cases: j-th generalized local cohomology module of M and N with respect to I is defined by ( ) j j HI (M, N) ∼ = lim ExtR (M/I n M, N) ⃗n j j We see that if M = R, then HI (M, N) = HI (N) the usual local cohomology module of Grothendieck 8 . Another similar question is: When is the module j HI (M, N)I-cofinite for all j ≥ 0? In 2001, Yassemi [ 9 , Theorem 2.8] showed that in a j Gorenstein ring, HI (M, N) is I -cofinite for all j ≥ 0 where I is non-zero principal ideal. In 2004, DivaaniAazar and Sazeedeh [ 10 , Theorem 2.8 and Theorem 2.9] have eliminated the Gorenstein hypothesis and showed that if either • I is a nonzero principal ideal, or • I is a prime ideal with dim R/I = 1 In 1991, Huneke and Koh 2 proved that if R is a complete local Gorenstein domain, I is a one dimension ideal of R and M is a finitely generated R-module, then Hli (M) is I-cofinite for all i. In 1997, Yoshida in 3 or Delfino and Marley in 4 extended (b) to all one dimension ideals I of an arbitrary local ring R. In 1998, Kawasaki 5 proved (a) in an arbitrary commutative Noetherian ring. The local condition in (b) has been removed by Bahmanpour and Naghipour in 6 . In 7 , Herzog gave a generalizations of the local cohomology theory. Let j be a non-negative integer and M a finitely generated R-module an N an R-module. The 1. I is principal, or 2. R is complete local and I is a prime ideal with j dim R/I = 1, then HI (M, N) is I-cofinite for all j ≥ 0. When I is a principal ideal, Cuong, Goto and Hoang j [ 11 , Theorem 1.1] gave another proof for HI (M, N) is I-cofinite for all j. They also showed that if dim M ≤ 2 j or dim N ≤ 2, then HI (M, N) is I-cofinite for all j. An extension of I-cofinite modules is I-cominimax modules which was introduced in 2009 12 . An Rmodule M is called I-comiminax if SuppR M ⊆ V (I) and ExtiR (R/I, M) is minimax for all i ≥ 0 (see [2, 3.1 and 2.2(ii)]). Naturally, we have a question: Cite this article : Hong Cam B T, Minh Tri N. Cominimax modules and generalized local cohomology modules. Sci. Tech. Dev. J.; 23(1):479-483. 479 Science & Technology Development Journal, 23(1):479-483 Question: When are the modules HIi (N) or HIi (M, N)I-cominimax for all i ≥ 0? In [2, 3.10], we see that if N is an I-minimax R-module and I is a principal ideal, then HIi (N) is I-cominimax for all i ≥ 0. In 2011, Mafi 13 proved that if N is a minimax R-module, then HIi (N) is I-cominimax for all i ≥ 0 when one of the following cases holds: 1. dim R/I ≤ 1; 2. cd(I) = 1; 3. dim R ≤ 2. In 14 , the authors showed that, if M is a minimax R( ) module with SuppR HIi (M) ≤ 1 for all i ≥ 0 and N is a finitely generated R-module with SuppR N ⊆ V (I), ) j( then ExtR N, Hli (M) is minimax for all i ≥ 0. In [18], the authors proved that in a local ring, if M is a finitely generated R-module and N, L are two minimax R-modules with SuppR L ⊆ V (I), then ) j( ExtR L, Hli (M, N) is minimax for all i and j when one of the following cases holds: 1. dim R/I ≤ 1; 2. cd(I) = 1; 3. dim R ≤ 2. The aim of this paper is to study the I-cominimaxness of HIi (M, N). Theorem 2.2 shows that if HIi (M, K) is Icofinite for all i < t and all finitely generated R-module K, then HIi (M, N) is I-cominimax for all i < t and all minimax R-module N. We will see in Theorem 2.4 that if M is a finitely generated R-module, N is a minimax R-module and t is a non-negative integer such that dim SuppR Hli (M, N) ≤ 1 for all i < t, then HIi (M, N) is I- cominimax for all i < t. When dim R/I ≤ 1, Theorem 2.9 shows that if HIi (N) is I-cominimax for all i ≥ 0, then Hli (M, N) is I-cominimax for all i ≥ 0. MAIN RESULTS In 15 , Zöschinger introduced the class of minimax modules. An R-module K is said to be a minimax module, if there is a finitely generated submodule T of K, such that K/T is Artinian. Remark 2.1 There are some elementary properties of minimax modules: 1. The class of minimax modules contains all finitely generated modules and all Artinian modules. 2. Let 0 → L → M → N → 0 be an exact sequence of R-modules. Then, M is minimax if and only if L and N are both minimax. Thus, any submodule and quotient of a minimax module is minimax. Moreover, if N is finitely generated and M j is minimax, then ExtR (N, M) and TorRj (N, M) are minimax for all j ≥ 0. 480 3. The set of associated primes of any minimax Rmodule is finite. 4. If M is a minimax R-module and p is a nonmaximal prime ideal of R , then M p is a finitely generated R p -module. Definition 2.1 (Azami, Naghipour and Vakili) An Rmodule M is I-cominimax if SuppR M ⊆ V (I) and ExtiR (R/I, M) is minimax for all i ≥ 0 The following result is a generalization of [14, 2.3]. Theorem 2.2 Let t be a non-negative integer. Assume that HIi (M, K) is I-cofinite for all i < t and all finitely generated R-module K. Then HIi (M, N) is I-cominimax for all i < t and all minimax R-module N. Proof. Since N is a minimax R-module, there is a finitely generated R-module K of such that N/K is artinian. From the short exact sequence 0 → K → N → N/K → 0 we get the following exact sequence fi gi h · · · →HIi (M, K) → HIi (M, N) → HIi (M, N/K) →i HIi+1 (M, K) → · · · Now, the short exact sequence 0 → lm fi → HIi (M, N) → Im gi → 0 induces a long exact sequence ( ) j j · · · → ExtR (R/I, Im fi ) → ExtR R/I, HIi (M, N) → j j+1 ExtR (R/I, Im gi ) → ExtR (R/I, Im fi ) → · · · gives rise to a long exact sequence I -cofinite ( ) j j · · · → ExtR (R/I, Im hi−1 ) → ExtR R/I, HIi (M, K) j j+1 → ExtR (R/I, Im fi ) → ExtR (R/I, Im hi−1 ) → · · · By the hypothesis, HIi (M, K) is I-cofinite for all i ≥ 0. ) j( Hence ExtR R/I, HIi (M, K) is finitely generated for all i < t, j ≥ 0. Since N/K is artinian, it follows from [16, 2.6] that HIi (M, N/K) is artinian for all i ≥ 0. It j is easy to see that ExtR (R/I, Im hi−1 ) is Artinian for j all i, j ≥ 0. Consequently, ExtR (R/I, Im fi ) is minimax for all i < t, j ≥ 0. Since Im gi is a submodule j of HIi (M, N/K), it follows that ExtR (R/I, Imgi ) is ar) j( tinian for all i, j ≥ 0. Thus ExtR R/I, HIi (M, N) is minimax for all i < t, j ≥ 0. Before showing a consequence of Theorem 2.2, we recall the concept of the local cohomology dimension of an ideal. Definition 2.3 The cohomological dimension of I in R, denoted by cd(I) is the smallest integer n such that the local cohomology modules HIi (M) = 0 for all Rmodules M, and for all i > n. We show some conditions such that the module HIi (M, N) is I-cominimax for all i ≥ 0. Corollary 2.4 Let M be a finitely generated R-module and N a minimax R-module. If either Science & Technology Development Journal, 23(1):479-483 1. 2. 3. 4. ) j( j ≥ 0. Therefore ExtR R/I, HIi (M, N) is minimax for all j ≥ 0. We have two following exact sequences I is principal, or dim M ≤ 2, or dim N ≤ 2, or cd(I) = 1, ( ) 0 → HomR (R/I, Im ft ) → HomR R/I, HIt (M, N) → then HIi (M, N) is I-cominimax for all i ≥ 0. Proof. (1), (2) and (3), Combining heorem 2.2 with [11, 1.1] or [11 , 1.3], it follows that HIi(M, N) is Icominimax for all i ≥ 0. Next, we will show some results concerning to small dimensions and R is an arbitrary (not local) commutative Noetherian ring. Theorem 2.4 Let M be a finitely generated R-module, N a minimax R-module and t a non-negative inte( ) ger such that dim SuppR HIi (M, N) ≤ 1 for all i < t. Then HIi (M, N) is I-cominimax for all i < t and ( ) Hom R/I, HIt (M, N) is minimax. Proof. Since N is minimax, there is a finitely generated R-module K of such that N/K is artinian. The short exact sequence 0 → K → N → N/K → 0 gives rise to a long exact sequence. gi h · · · → HIi (M, K) → HIi (M, N) → HIi (M, N/K) →i HIi+1 (M, K) → · · · Since N/K is artinian, it follows from [ 17 , 2.6] that HIi (M, N/K) is artinian for all i ≥ 0. By the assump( ) tion, we induce dim SuppR HIi (M, K) ≤ 1 for all i < t. It follows from [ 11 , 1.2] that HIi (M, K) is I-cofinite for all i < t. Now, the short exact sequence 0 → lm fi → HIi (M, N) → Im gi → 0 induces a long exact sequence ( ) j j · · · → ExtR (R/I, Im fi ) → ExtR R/I, HIi (M, N) ) j( j+1 → ExtR R/I, lm g j → ExtR (R/I, Im fi ) → · · · j Note that ExtR (R/I, Imgi ) is artinian for all i, j ≥ 0. Let i < t, the short exact sequence 0 → lm hi−1 → HIi (M, K) → Im fi and ( ) HomR R/I, HIt (M, K) → HomR (R/I, Im ft ) → Ext1R (R/I, Im ht−1 ) → · · · 4. follows from [ 16 , 2.2] and Theorem 2.2. fi HomR (R/I, Im gt ) →0 induces a long exact sequence j · · · → ExtR (R/I, Im hi−1 ) → ( ) j j i ExtR R/I, HI (M, K) → ExtR (R/I, Im fi ) → j+1 ExtR (R/I, Im hi−1 ) → · · · . Since HIi (M, K) is I-cofinite, it follows that ) j( ExtR R/I, HIi (M, K) is finitely generated for all j j ≥ 0. By the artinianness of ExtR (R/I, Im hi−1 ), we j can conclude that ExtR (R/I, Im fi ) is minimax for all ( ) Since dim SuppR HIi (M, K) ≤ 1 for all i < t, it follows ( ) from [4, Theorem 1.2] that HomR R/I, HIt (M, K) is finitely generated. On the other hand, 1 ExtR (R/I, Im ht−1 ) is an artinian R-module. Therefore HomR (R/I, Im ft ) is minimax. We see that HomR (R/I, Im gt ) is artinian and then ( ) HomR R/I, HIt (M, N) is minimax. In [ 18 , 3.1 and 3.2], the authors showed that HIi (M, N) is I-cominimax for all i ≥ 0 when dim R/I ≤ 1 where R is a local ring. Now we consider that R is not a local ring. Corollary 2.5 Let M be a finitely generated R-module, N a minimax R-module and t a non-negative integer. Assume that dim M/IM ≤ 1 or dim N ≤ 1 or dim R/I ≤ 1. Then HIi (M, N) is I-cominimax for all i ≥ 0. Corollary 2.6 Let M be a finitely generated R-module, N a minimax R-module and t a non-negative inte( ) ger. Assume that SuppR HIi (M, N) is finite for all i < t. Then HIi (M, N) is I-cominimax for all i < t ( ) and HomR R/I, HIt (M, N) is minimax. In particular, ( t ) Ass HI (M, N) is a finite set. ( ) Proof. Since SuppR HIi (M, N) is a finite set, we ( i ) can conclude that dim SuppR HI (M, N) ≤ 1. It follows from Theorem 2.4 that HIi (M, N) is I-cominimax ( ) for all i < t and HomR R/I, HIt (M, N) is minimax. Moreover, we have ( ) ( ( )) Ass HIt (M, N) = Ass HomR R/I, HIt (M, N) ( ) By Remark 2.1.3, Ass HIt (M, N) is a finite set. Corollary 2.7 Let N be a non-zero minimax R-module and I an ideal of R. Let t be a non-negative integer such that dim SuppR HIi (N) ≤ 1 for all i < t . Then the following statements hold: 1. the R-modules HIi (N) are I-cominimax for all i < t; ( ) 2. the R-module HomR R/I, HIt (N) is minimax. Lemma 2.8 Let M be a finitely generated R-module such that SuppR M ⊆ V (I) and N an I-cominimax Rmodule. Then ExtiR (M, N) is minimax for all i ≥ 0. 481 Science & Technology Development Journal, 23(1):479-483 Proof. The proof is by induction on i. Since N is an I-cominimax R-module, the module ExtiR (R/I, N) is minimax for all i ≥ 0. By Gruson’s theorem, there is a chain of submodules of M. 0 = M0 ⊆ M1 ⊆ . . . ⊆ Mk = M such that M j /M j−1 is a homomorphic image of (R/I)t for some positive integer t . We consider short exact sequences 0 → K → (R/I)m → M1 → 0 and 0 → M j−1 → Mi → M j /M j−1 → 0 The first exact sequence induces a long exact sequence 0 → HomR (M1 , N) → HomR ((R/I)m , N) → HomR (K, N) → · · · where K is a submodule of (R/I)m for some positive integer number m. Since HomR ((R/I)m , N) ∼ = HomR (R/I, N)m , it follows that HomR (M1 , N) is minimax. By similar arguments, we also get that ( ) HomR M j /M j−1 , N is minimax for all 1 ≤ i ≤ k . Now, the exact sequence ( ) ( ) 0 → HomR M j /M j−1 , N → HomR M j , N → ( ) HomR M j−1 , N → · · · ( ) deduces that HomR M j , N is minimax for all j and then HomR (M, N) is minimax. Therefore, we have the conclusion when i = 0. Let i > 0. The short exact sequence 0 → K → (R/I)m → M1 → 0 gives rise to a long exact sequence i · · · → Exti−1 R (K, N) → ExtR (M1 , N) → ( ) ExtiR (R/I)t , N · · · By the inductive hypothesis, Exti−1 R (K, N) is a minimax R-module. Since ExtiR ((R/I)m , N) ∼ = ExtiR (R/I, N)m , it follows that ExtiR (M1 , N) is minimax. Analysis similar to the above proof, we have ExtiR (Mk , N) is minimax and which completes the proof. 482 The following result shows a connection on the Icominimaxness of HIi (N) and HIi (M, N) when R is not a local ring and N is an arbitrary R-module. Theorem 2.9 Let M be a finitely generated R-module with pd(M) < ∞ and N an R- module. Let I be an ideal of R with dim R/I = 1 and t a non-negative integer such that HIi (N) is I-cominimax for all i < t. Then HIi (M, N) is I-cominimax for all i < t. Proof. We prove by induction on p = pd(M). If p = 0, then M is a projective R-module. It follows from [ 19 , ( ) 2.5] that HIi (M, N) ∼ = HomR M, HIi (N) for all i ≥ 0. By [ 20 , 10.65], we have ) ( ( ) ( j j ExtR R/I, HomR M, HIi (N) ∼ = ExtR M/IM, HIi (N) ) j( for all(i < t, j ≥ 0. Therefore ExtR R/I, HIi (M, N) ∼ = ) ) j j j( i ExtR M/IM, HI (N) where ExtR M/IM, HI (N) is minimax for all j ≥ 0 by Lemma 2.8 and then the assertion follows. Let p > 0 and the statement is true for all finitely generated R-module with projective dimension less than p. There is a short exact sequence 0 → K → P → M → 0, where K is finitely generated, P is projective finitely generated. Note that pd(K) = p − 1 and then by the inductive hypothesis HIi (K, N) is I-cominimax for all i < t. On the other hand, there is a long exact sequence · · · → HIi (K, N) → HIi+1 (M, N) → HIi+1 (P, N) → · · · in which HIi (K, N) and HIi (P, N) are I-cominimax for all i ≥ 0. It follows from [ 21 , 2.6] that HIi (M, N) is also I-comiminax for all i ≥ 0 and the proof is complete. COMPETING INTERESTS The authors declare that they have no conflicts of interest. 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