On the cofiniteness of local cohomology modules

Document Type : Research Paper

Author

Department of Basic Sciences, Arak University of Technology, P. O. Box 38135-1177, Arak, Iran.

Abstract

Let $R$ be a commutative Noetherian ring with identity, $I$ be an ideal of $R$ and $M$ be an $R$-module such that $Ext^j_R(R/I, M)$ is finitely generated for all $j$. We prove that if $\dim H^i_I(M)\leq 1$ for all $i$, then for any $i \geq 0$ and for any submodule $N$ of $H^i_I(M)$ that is either $I$-cofinite or minimax, the $R$-module $H^i_I(M)/N$ is $I$-cofinite. This generalizes the main result of Bahmanpour and Naghipour [8, Theorem 2.6]. As a consequence, the Bass numbers and Betti numbers of $H^i_I (M)$ are finite for all $i \geq 0$. Also, among other things, we show that if either $\dim R/I\leq 2$ or $\dim M\leq 2$, then for each finitely generated $R$-module $N$, the $R$-module $Ext^j_R (N, H^i_I(M))$ is $I$-weakly cofinite, for all $i \geq 0$ and $j\geq 0$. This generalizes [1, Corollary 2.8].

Keywords

References

[1] R. Abazari and K. Bahmanpour, Co_niteness of extension functors of cofinite modules, J. Algebra, 330 No. 1 (2011) 507-516.
[2] A. Abbasi and H. Roshan Shekalgourabi, Serre subcategory properties of generalized local cohomology modules, Korean J. Math., 28 No. 1 (2011) 25-37.
[3] M. Aghapournahr and K. Bahmanpour, Cofiniteness of weakly laskerian local cohomology modules, Bull. Math. Soc. Sci. Math. Roumanie, 105 No. 4 (2014) 347-356.
[4] M. Aghapournahr and K. Bahmanpour, Cofiniteness of general local cohomology modules for small dimensions, Bull. Korean Math. Soc., 53 No. 5 (2016) 1341-1352.
[5] J. Asadollahi, K. Khashyarmanesh, and Sh. Salarian, A generalization of the cofiniteness problem in local cohomology modules, J. Aust. Math. Soc., 75 (2003) 313-324.
 [6] K. Bahmanpour, On the category of weakly laskerian co_nite modules, Math. Scand., 115 No. 1 (2014) 62-68.
[7] K. Bahmanpour and R. Naghipour, On the cofiniteness of local cohomology modules, Proc. Amer. Math. Soc., 136 No. 7 (2008) 2359-2363.
[8] K. Bahmanpour and R. Naghipour, Cofiniteness of local cohomology modules for ideals of small dimension, J. Algebra, 321 (2009) 1997-2011.
[9] K. Bahmanpour, R. Naghipour, and M. Sedghi, Minimaxness and cofiniteness properties of local cohomology modules, Comm. Algebra, 41 (2013) 2799-2814.
[10] M. P. Brodmann and F. A. Lashgari, A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc., 128 (2000) 2851-2853.
[11] 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.
[12] L. Burch, Codimension and analytic spread, Math. Proc. Camb. Phil. Soc., 72 No. 3 (1972) 369-373.
[13] G. Chiriacescu, Cofiniteness of local cohomology modules, Bull. London Math. Soc., 32 (2000) 1-7.
[14] D. Delfino, On the coiniteness of local cohomology modules, Math. Proc. Camb. Phil. Soc., 115 (1994) 79-84.
[15] D. Delfino and T. Marley, Cofinite modules and local cohomology, J. Pure Appl. Algebra, 121 No. 1 (1997) 45-52.
[16] M. T. Dibaei and S. Yassemi, Associated primes and cofiniteness of local cohomology modules, Manuscripta Math., 117 No. 2 (2005) 199-205.
[17] M. T. Dibaei and S. Yassemi, Finiteness of extension functors of local cohomology modules, Comm. Algebra, 34 (2006) 3097-3101.
[18] K. Divaani-Aazar and A. Mafi, Associated primes of local cohomology modules, Proc. Amer. Math. Soc., 133 No. 3 (2005) 655-660.
[19] K. Divaani-Aazar and A. Mafi, Associated primes of local cohomology modules of weakly Laskerian modules, Comm. Algebra, 34 (2006) 681-690.
[20] A. Grothendieck, Cohomologie locale des faisceaux et theoremes de lefshetz locaux et globaux (SGA2), 1968.
[21] R. Hartshorne, Affine duality and cofiniteness, Invent. Math., 9 No. 2 (1970) 145-164.
[22] C. Huneke and J. Koh, Cofiniteness and vanishing of local cohomology modules, Math. Proc. Camb. Phil. Soc., 110 No. 3 (1991) 421-429.
[23] C. Huneke and R.Y. Sharp, Bass numbers of local cohomology modules, Trans. Amer. Soc., 339 No. 17 (1993) 765-779.
[24] K.I. Kawasaki, On the finiteness of bass numbers of local cohomology modules, Proc. Amer. Math. Soc., 124 (1996) 3275-3279.
[25] K. Khashyarmanesh and Sh. Salarian, On the associated primes of local cohomology modules, Comm. Algebra, 27 No. 12 (1999) 6191-6198.
[26] T. Marley, The associated primes of local cohomology modules over rings of small dimension, Manuscripta Math., 104 (2001) 519-525.
[27] H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, UK, 1986.
[28] L. Melkersson, On asymptotic stability for sets of prime ideals connected with the powers of an ideal, Math. Proc. Camb. Phil. Soc., 107 (1990) 267-271.
[29] L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra, 285 (2005) 649-668.
[30] P. H. Quy, On the finiteness of associated primes of local cohomology modules, Proc. Amer. Math. Soc., 6 (2010) 1965-1968.
[31] K. I. Yoshida, Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J., 147 (1997) 179-191.
[32] T. Yoshizawa, Subcategories of extension modules by serre subcategories, Proc. Amer. Math. Soc., 140 (2012) 2293-2305.
[33] H. Zöschinger, Minimax-moduln, J. Algebra, 102 (1986) 1-32.
Algebraic Structures and Their Applications
Volume 9, Issue 1
February 2022
Pages 81-92

Files

Share

How to cite

Statistics