Cominimax modules and generalized local cohomology modules

Science & Technology Development Journal, 23(1):479-483 Open Access Full Text Article Research Article Department of Natural Science Education, Dong Nai University, Dong Nai, Vietnam Correspondence NguyenMinh 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 open- access article distributed under the terms of the Creative Commons Attribution 4.0 International license. Cominimaxmodules and generalized local cohomologymodules Bui Thi Hong Cam, NguyenMinh Tri* Use your smartphone to scan this QR code and download this article ABSTRACT The local cohomology theory plays an important role in commutative algebra and algebraic ge- ometry. 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 -moduleM is I-cominimax if SuppRM 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 cohomologymoduleH 0I(M;N) is I-cominimax for all i 0. We prove that H iI(M;K) if is I-cofinite for all i 0. We prove that ifH iI(M;K) is I-cofinite for all i < t and all finitely generated R-module K, then H iI(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 dimSuppRH i I(M;N) 1 for all i < t , then H iI(M;N) is I-cominimax for all i < t . When dimR=I  1 and H iI(N) is I-cominimax for all i 0, then H 0I(M;N) is I-cominimax for all i 0. Key words: Generalized local cohomology, I-cominimax INTRODUCTION Let R be a local Noetherian ring, I an ideal of R andM a finitely generated R -module. It is well known that the local cohomology modules H iI(M) are not gen- erally finitely generated for i > 0. In a 1970 paper Hartshorne1 gave the concept of I-cofinite modules. An R-module K to be I-cofinite if SuppRK  V (I) and Ext jR(R=I;K) is finitely generated for all j  0. Hartshorne asked which rings R and ideals I themod- ulesH iI(M)were I-cofinite for all i and all finitely gen- erated modulesM. In 1, if (R , m) is a complete regular local ring and M is a finitely generated R-module, then H iI(M) is I -cofinite in two cases: • I is a nonzero principal ideal, or • I is a prime ideal with dimR=I = 1 In 1991, Huneke and Koh2 proved that if R is a com- plete local Gorenstein domain, I is a one dimension ideal ofR andM is a finitely generatedR-module, then H il (M) is I-cofinite for all i. In 1997, Yoshida in 3 or Delfino and Marley in 4 extended (b) to all one di- mension 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. In7, Herzog gave a generalizations of the local coho- mology theory. Let j be a non-negative integer andM a finitely generated R-module anN an R-module. The j-th generalized local cohomology module of M and N with respect to I is defined by H jI (M;N)= lim ~n  Ext jR (M=I nM;N)  We see that if M = R, then H jI (M;N) = H j I (N) the usual local cohomology module of Grothendieck8. Another similar question is: When is the module H jI (M;N)I-cofinite for all j  0? In 2001, Yassemi [9, Theorem 2.8] showed that in a Gorenstein ring, H jI (M;N) is I -cofinite for all j  0 where I is non-zero principal ideal. In 2004, Divaani- Aazar and Sazeedeh [10, Theorem 2.8 and Theorem 2.9] have eliminated the Gorenstein hypothesis and showed that if either 1. I is principal, or 2. R is complete local and I is a prime ideal with dimR=I = 1, then H jI (M;N) is I-cofinite for all j  0. When I is a principal ideal, Cuong, Goto and Hoang [11, Theorem 1.1] gave another proof for H jI (M;N) is I-cofinite for all j. They also showed that if dimM  2 or dimN  2, then H jI (M;N) is I-cofinite for all j. An extension of I-cofinite modules is I-cominimax modules which was introduced in 200912. An R- module M is called I-comiminax if SuppRM  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 H iI(N) or H iI(M;N)I-cominimax for all i 0? In [2, 3.10], we see that ifN is an I-minimaxR-module and I is a principal ideal, then H iI(N) is I-cominimax for all i  0. In 2011, Mafi13 proved that if N is a minimax R-module, then H iI(N) is I-cominimax for all i 0 when one of the following cases holds: 1. dimR=I  1; 2. cd(I) = 1; 3. dimR 2: In14, the authors showed that, if M is a minimax R- module with SuppR H iI(M)
 1 for all i  0 and N is a finitely generatedR-module withSuppRN V (I), then Ext jR N;H il (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 Ext jR L;H il (M;N)
isminimax for all i and jwhenone of the following cases holds: 1. dimR=I  1; 2. cd(I) = 1; 3. dimR 2: The aim of this paper is to study the I-cominimaxness ofH iI(M;N). Theorem 2.2 shows that ifH iI(M;K) is I- cofinite for all i < t and all finitely generatedR-module K, then H iI(M;N) is I-cominimax for all i < t and all minimaxR-moduleN.Wewill see inTheorem2.4 that ifM is a finitely generated R-module, N is a minimax R-module and t is a non-negative integer such that dimSuppRH i l (M;N)  1 for all i < t, then H iI(M;N) is I- cominimax for all i < t. When dimR=I  1, The- orem 2.9 shows that if H iI(N) is I-cominimax for all i 0, then H il (M;N) is I-cominimax for all i 0. MAIN RESULTS In15, 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 2.1There 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 ofR-modules. Then,M is minimax if and only if L and N are both minimax. Thus, any submod- ule and quotient of a minimax module is mini- max. Moreover, if N is finitely generated andM is minimax, then Ext jR(N;M) and TorRj (N;M) are minimax for all j  0. 3. The set of associated primes of any minimax R- module is finite. 4. If M is a minimax R-module and p is a non- maximal prime ideal of R , thenMp is a finitely generated Rp -module. Definition 2.1 (Azami, Naghipour and Vakili) An R- module M is I-cominimax if SuppRM  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 H iI(M;K) is I-cofinite for all i < t and all finitely generated R-module K.ThenH iI(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 ar- tinian. From the short exact sequence 0! K! N! N=K! 0 we get the following exact sequence


!H iI(M;K) fi! H iI(M;N) gi! H iI(M;N=K) hi! H i+1I (M;K)!


Now, the short exact sequence 0! lm fi ! H iI(M;N)! Imgi ! 0 induces a long exact sequence


! Ext jR (R=I; Im fi)! Ext jR  R=I;H iI(M;N)  ! Ext jR (R=I; Imgi)! Ext j+1R (R=I; Im fi)!


gives rise to a long exact sequence I -cofinite


! Ext jR (R=I; Imhi1)! Ext jR  R=I;H iI(M;K)  ! Ext jR (R=I; Im fi)! Ext j+1R (R=I; Imhi1)!


By the hypothesis, H iI(M;K) is I-cofinite for all i 0. Hence Ext jR R=I;H iI(M;K)
is finitely generated for all i < t; j  0. Since N/K is artinian, it follows from [16, 2.6] that H iI(M;N=K) is artinian for all i  0. It is easy to see that Ext jR (R=I; Imhi1) is Artinian for all i; j  0. Consequently, Ext jR (R=I; Im fi) is mini- max for all i < t; j  0. Since Imgi is a submodule of H iI(M;N=K), it follows that Ext j R (R=I; Imgi) is ar- tinian for all i; j  0. Thus Ext jR R=I;H iI(M;N)
is minimax for all i< t; j  0. Before showing a consequence ofTheorem 2.2, we re- call 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 H iI(M) = 0 for all R- modulesM, and for all i > n. We show some conditions such that the module H iI(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 480 Science & Technology Development Journal, 23(1):479-483 1. I is principal, or 2. dimM  2, or 3. dimN  2, or 4. cd(I) = 1, 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 I- cominimax for all i  0: 4. follows from [16, 2.2] andTheorem 2.2. Next, we will show some results concerning to small dimensions and R is an arbitrary (not local) commu- tative 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 dimSuppR H iI(M;N)
 1 for all i < t. Then H iI(M;N) is I-cominimax for all i < t and Hom R=I;HtI (M;N)
is minimax. Proof. SinceN 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.


! H iI(M;K) fi! H iI(M;N) gi! H iI(M;N=K) hi! H i+1I (M;K)!


Since N/K is artinian, it follows from [ 17, 2.6] that H iI(M;N=K) is artinian for all i  0. By the assump- tion, we induce dimSuppR H iI(M;K)
 1 for all i < t. It follows from [11, 1.2] that H iI(M;K) is I-cofinite for all i < t. Now, the short exact sequence 0! lm fi ! H iI(M;N)! Imgi ! 0 induces a long exact sequence


! Ext jR (R=I; Im fi)! Ext jR  R=I;H iI(M;N)  ! Ext jR R=I; lmg j
! Ext j+1R (R=I; Im fi)!


Note that Ext jR (R=I; Imgi) is artinian for all i; j  0. Let i < t, the short exact sequence 0! lmhi1 ! H iI(M;K)! Im fi ! 0 induces a long exact sequence


! Ext jR (R=I; Imhi1)! Ext jR  R=I;H iI(M;K)  ! Ext jR (R=I; Im fi)! Ext j+1R (R=I; Imhi1)!


: Since H iI(M;K) is I-cofinite, it follows that Ext jR R=I;H iI(M;K)
is finitely generated for all j  0. By the artinianness of Ext jR (R=I; Imhi1), we can conclude that Ext jR (R=I; Im fi) is minimax for all j  0. Therefore Ext jR R=I;H iI(M;N)
is minimax for all j  0. We have two following exact sequences 0! HomR (R=I; Im ft)! HomR R=I;HtI (M;N)
! HomR (R=I; Imgt) and HomR R=I;HtI (M;K)
! HomR (R=I; Im ft)! Ext1R (R=I; Imht1)!


Since dimSuppR H iI(M;K)
 1 for all i < t, it follows from [4, Theorem 1.2] that HomR R=I;HtI (M;K)
is finitely generated. On the other hand, Ext1R (R=I; Imht1) is an artinian R-module. Therefore HomR (R=I; Im ft) is minimax. We see that HomR (R=I; Imgt) is artinian and then HomR R=I;HtI (M;N)
is minimax. In [18, 3.1 and 3.2], the authors showed thatH iI(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 inte- ger. Assume that dimM=IM  1 or dimN  1 or dimR=I  1. Then H iI(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 H iI(M;N)
is finite for all i < t. Then H iI(M;N) is I-cominimax for all i < t andHomR R=I;HtI (M;N)
is minimax. In particular, Ass HtI (M;N)
is a finite set. Proof. Since SuppR H iI(M;N)
is a finite set, we can conclude that dimSuppR H iI(M;N)
 1. It fol- lows fromTheorem 2.4 thatH iI(M;N) is I-cominimax for all i < t and HomR R=I;HtI (M;N)
is minimax. Moreover, we have Ass HtI (M;N)
= Ass HomR R=I;HtI (M;N)

By Remark 2.1.3, Ass HtI (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 dimSuppRH iI(N)  1 for all i < t . Then the fol- lowing statements hold: 1. the R-modules H iI(N) are I-cominimax for all i < t; 2. the R-module HomR R=I;HtI (N)
is minimax. Lemma 2.8 Let M be a finitely generated R-module such that SuppRM  V (I) and N an I-cominimax R- module. 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 ofM. 0=M0 M1  : : :Mk =M such that M j=M j1 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 j1 !Mi !M j=M j1 ! 0 Thefirst 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 posi- tive 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 j1;N
is minimax for all 1 i k . Now, the exact sequence 0! HomR M j=M j1;N
! HomR M j;N
! HomR M j1;N
!


deduces that HomR M j;N
is minimax for all j and thenHomR(M;N) isminimax. 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


! Exti1R (K;N)! ExtiR (M1;N)! ExtiR (R=I)t ;N



By the inductive hypothesis, Exti1R (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 mini- max. Analysis similar to the above proof, we have ExtiR (Mk;N) is minimax and which completes the proof. The following result shows a connection on the I- cominimaxness ofH iI(N) andH iI(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 dimR=I= 1 and t a non-negative integer such thatH iI(N) is I-cominimax for all i < t. ThenH iI(M;N) is I-cominimax for all i < t. Proof. We prove by induction on p= pd(M). If p = 0, thenM is a projective R-module. It follows from [19, 2.5] that H iI(M;N) = HomR M;H iI(N)
for all i  0. By [20, 10.65], we have Ext jR  R=I;HomR  M;H iI(N) = Ext jRM=IM;H iI(N) for all i< t; j  0. Therefore Ext jR R=I;H iI(M;N)
= Ext jR  M=IM;H jI (N)  where Ext jR M=IM;H iI(N)
is minimax for all j  0 by Lemma 2.8 and then the as- sertion follows. Let p > 0 and the statement is true for all finitely gen- erated 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 H iI(K;N) is I-cominimax for all i < t. On the other hand, there is a long exact sequence


! H iI(K;N)! H i+1I (M;N)! H i+1I (P;N)!


in which H iI(K;N) and H iI(P;N) are I-cominimax for all i 0. It follows from [21, 2.6] thatH iI(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 in- terest. AUTHOR CONTRIBUTION Bui Thi Hong Cam has contributed the Theorem 2.2 and has written the manuscript. Nguyen Minh Tri has contributed theTheorem 2.4, 2.9 and revising the manuscript. ACKNOWLEDGMENTS The authors would like to thank the referees for his or her substantial comments. This work was supported by Dong Nai University. 482 Science & Technology Development Journal, 23(1):479-483 REFERENCES 1. Hartshorne R. Affine duality and cofiniteness. Inventiones Mathematicae. 1970;9(2):145–164. 2. HunekeC, Koh J. Cofiniteness and vanishingof local cohomol- ogy modules. Mathematical Proceedings of the Cambridge Philosophical Society. 1991;110(3):421–429. 3. Yoshida KI. Cofiniteness of local cohomologymodules for ide- als of dimension one. Nagoya Math J. 1997;147:179–191. 4. Delfino D, Marley T. Cofinite modules and local cohomology. Journal of Pure and Applied Algebra. 1997;121(1):45–52. 5. Kawasaki KI. Cofiniteness of local cohomology modules for principal ideals. Bull London Math Soc. 1998;30(3):241–246. 6. Bahmanpour K, Naghipour R. Cofiniteness of local cohomol- ogymodules for ideals of small dimension. Journal of Algebra. 2009;321(7):1997–2011. 7. Herzog J. Komplexe. Auflösungen und dualität in der localen Algebra. 1970;Habilitationsschrift, Universität Regensburg. 8. Grothendieck A. Cohomologie local des faisceaux coherents et theorèmes de Lefschetz locaux et globaux (SGA2); 1968. 9. Yassemi S. Cofinite modules. Communications in Algebra. 2001;29(6):2333–2340. 10. Divaani-Aazar K, Sazeedeh R. Cofiniteness of generalized local cohomology modules. Colloquium Mathematicum. 2004;99(2):283–290. 11. Cuong NT, Goto S, Hoang NV. On the cofiniteness of general- ized local cohomology modules. Kyoto Journal of Mathemat- ics. 2015;55(1):169–185. 12. Azami J, Naghipour R, Vakili B. Finiteness properties of local cohomologymodules for a-minimaxmodules. Proceedings of the American Mathematical Society. 2009;137(02):439–448. 13. Mafi A. On the local cohomology of minimax modules. Bul- letin of the Korean Mathematical Society. 2011;48(6):1125– 1128. 14. Abbasi A, Roshan-Shekalgourabi H, Hassanzadeh-Lelekaami D. Some results on the local cohomology of minimax mod- ules. Czechoslovak Mathematical Journal. 2014;64(2):327– 333. 15. Zöschinger H. Minimax-Moduln. Journal of Algebra. 1986;102(1):1–32. 16. Marley T, Vassilev JC. Cofiniteness and associated primes of local cohomology modules. Journal of Algebra. 2002;256(1):180–193. 17. Nam TT. On the non-vanishing and artinianness of general- ized local cohomology modules. Journal of Algebra and Its Applications. 2013;12(4):1250203. 18. Roshan-Shekalgourabi H, Hassanzadeh-Lelekaami D. On the generalized local cohomology of minimax modules. Journal of Algebra and Its Applications. 2016;15(08):1650147. 19. Hassanzadeh SH, Vahidi A. On Vanishing and Cofiniteness of Generalized Local CohomologyModules. Communications in Algebra. 2009;37(7):2290–2299. 20. Rotman J. An introduction to Homological Algebra. 2009;. 21. Irani Y. Cominimaxness with respect to ideal of dimension one. Bull Korean Math Soc. 2017;54(1):289–298. 483