# COFINITENESS OF GENERAL LOCAL COHOMOLOGY MODULES FOR SMALL DIMENSIONS

• Aghapournahr, Moharram (Moharram Aghapournahr Department of Mathematics Faculty of Science Arak University) ;
• Bahmanpour, Kamal (Department of Mathematics Faculty of Mathematical Sciences University of Mohaghegh Ardabili)
• Published : 2016.09.30

#### Abstract

Let R be a commutative Noetherian ring, ${\Phi}$ a system of ideals of R and $I{\in}{\Phi}$. In this paper among other things we prove that if M is finitely generated and $t{\in}\mathbb{N}$ such that the R-module $H^i_{\Phi}(M)$ is $FD_{{\leq}1}$ (or weakly Laskerian) for all i < t, then $H^i_{\Phi}(M)$ is ${\Phi}$-cofinite for all i < t and for any $FD_{{\leq}0}$ (or minimax) submodule N of $H^t_{\Phi}(M)$, the R-modules $Hom_R(R/I,H^t_{\Phi}(M)/N)$ and $Ext^1_R(R/I,H^t_{\Phi}(M)/N)$ are finitely generated. Also it is shown that if cd I = 1 or $dimM/IM{\leq}1$ (e.g., $dim\;R/I{\leq}1$) for all $I{\in}{\Phi}$, then the local cohomology module $H^i_{\Phi}(M)$ is ${\Phi}$-cofinite for all $i{\geq}0$. These generalize the main results of Aghapournahr and Bahmanpour [2], Bahmanpour and Naghipour [6, 7]. Also we study cominimaxness and weakly cofiniteness of local cohomology modules with respect to a system of ideals.

#### References

1. M. Aghapournahr, Cofiniteness of certain local cohomology modules for small dimensions, preprint.
2. M. Aghapournahr and K. Bahmanpour, Cofiniteness of weakly Laskerian local coho- mology modules, Bull. Math. Soc. Sci. Math. Roumanie (N.S) 57(105) (2014), no. 4, 347-356.
3. M. Aghapournahr, L. Melkersson, A natural map in local cohomology, Ark. Mat. 48 (2010), no. 2, 243-251. https://doi.org/10.1007/s11512-009-0115-3
4. J. Asadollahi, K. Khashyarmanesh, and Sh. Salarian, A generalization of the cofiniteness problem in local cohomology modules, J. Aust. Math. Soc. 75 (2003), no. 3, 313-324. https://doi.org/10.1017/S1446788700008132
5. K. Bahmanpour, On the category of weakly Laskerian cofinite modules, Math. Scand. 115 (2014), no. 1, 62-68. https://doi.org/10.7146/math.scand.a-18002
6. K. Bahmanpour and R. Naghipour, On the cofiniteness of local cohomology modules, Proc. Amer. Math. Soc. 136 (2008), no. 7, 2359-2363. https://doi.org/10.1090/S0002-9939-08-09260-5
7. K. Bahmanpour and R. Naghipour, Cofiniteness of local cohomology modules for ideals of small dimension, J. Algebra 321 (2009), no. 7, 1997-2011. https://doi.org/10.1016/j.jalgebra.2008.12.020
8. K. Bahmanpour, R. Naghipour, and M. Sedghi, On the category of cofinite modules which is Abelian, Proc. Amer. Math. Soc. 142 (2014), no. 4, 1101-1107. https://doi.org/10.1090/S0002-9939-2014-11836-3
9. R. Belshoff, S. P. Slattery, and C. Wickham, The local cohomology modules of Matlis reflexive modules are almost cofinite, Proc. Amer. Math. Soc. 124 (1996), no. 9, 2649- 2654. https://doi.org/10.1090/S0002-9939-96-03326-6
10. R. Belshoff, S. P. Slattery, and C. Wickham, Finiteness properties of Matlis reflexive modules, Comm. Algebra 24 (1996), no. 4, 1371-1376. https://doi.org/10.1080/00927879608825640
11. M. H. Bijan-Zadeh, Torsion theories and local cohomology over commutative Noetherian ring, J. Lond. Math. Soc. (2) 19 (1979), no. 3, 402-410.
12. M. H. Bijan-Zadeh, A common generalization of local cohomology theories, Glasgow Math. J. 21 (1980), no. 2, 173-181. https://doi.org/10.1017/S0017089500004158
13. M. H. Bijan-Zadeh, On the Artinian property of certain general local cohomology, J. Lond. Math. Soc. (2) 35 (1985), no. 3, 399-403.
14. M. P. Brodmann and A. Lashgari, A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc. 128 (2000), no. 10, 2851-2853. https://doi.org/10.1090/S0002-9939-00-05328-4
15. M. P. Brodmann and R. Y. Sharp, Local Cohomology: An algebraic introduction with geometric applications, Cambridge. Univ. Press, 1998.
16. W. Bruns and J. Herzog, Cohen Macaulay Rings, Cambridge Studies in Advanced Mathematics, Vol. 39, Cambridge Univ. Press, Cambridge, UK, 1993.
17. D. Delfino and T. Marley, Cofinite modules and local cohomology, J. Pure Appl. Algebra 121 (1997), no. 1, 45-52. https://doi.org/10.1016/S0022-4049(96)00044-8
18. M. T. Dibaei and S. Yassemi, Associated primes and cofiniteness of local cohomology modules, Manuscripta Math. 117 (2005), no. 2, 199-205. https://doi.org/10.1007/s00229-005-0538-5
19. M. T. Dibaei and S. Yassemi, Associated primes of the local cohomology modules, Abelian groups, rings, modules and homological algebra, 51-58, Chapman and Hall/CRC, 2006.
20. K. Divaani-Aazar and A. Mafi, Associated primes of local cohomology modules, Proc. Amer. Math. Soc. 133 (2005), no. 3, 655-660. https://doi.org/10.1090/S0002-9939-04-07728-7
21. A. Grothendieck, Cohomologie locale des faisceaux coherents et theoremes de Lefschetz locaux et globaux (SGA2), North-Holland, Amsterdam, 1968.
22. R. Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1970), 145-164. https://doi.org/10.1007/BF01404554
23. P. Hung Quy, On the finiteness of associated primes of local cohomology modules, Proc. Amer. Math. Soc. 138 (2010), no. 6, 1965-1968. https://doi.org/10.1090/S0002-9939-10-10235-4
24. K. Khashyarmanesh and Sh. Salarian, On the associated primes of local cohomology modules, Comm. Algebra 27 (1999), no. 12, 6191-6198. https://doi.org/10.1080/00927879908826816
25. A. Mafi and H. Saremi, On the cofiniteness properties of certain general local cohomology modules, Acta Sci. Math. (Szeged) 74 (2008), no. 3-4, 501-507.
26. T. Marley, The associated primes of local cohomology modules over rings of small dimension, Manuscripta Math. 104 (2001), no. 4, 519-525. https://doi.org/10.1007/s002290170024
27. T. Marley and J. C. Vassilev, Cofiniteness and associated primes of local cohomology modules, J. Algebra 256 (2002), no. 1, 180-193. https://doi.org/10.1016/S0021-8693(02)00151-5
28. H. Matsumura, Commutative Ring Theory, Cambridge Univ. Press, Cambridge, UK, 1986.
29. L. Melkersson, Some applications of a criterion for artinianness of a module, J. Pure Appl. Algebra 101 (1995), no. 3, 291-303. https://doi.org/10.1016/0022-4049(94)00059-R
30. L. Melkersson, Properties of cofinite modules and applications to local cohomology, Math. Proc. Cambridge Philos. Soc. 125 (1999), no. 3, 417-423. https://doi.org/10.1017/S0305004198003041
31. L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra 285 (2005), no. 2, 649-668. https://doi.org/10.1016/j.jalgebra.2004.08.037
32. T. Yoshizawa, Subcategories of extension modules by Serre subcategories, Proc. Amer. Math. Soc. 140 (2012), no. 7, 2293-2305. https://doi.org/10.1090/S0002-9939-2011-11108-0
33. H. Zoschinger, Minimax Moduln, J. Algebra 102 (1986), no. 1, 1-32. https://doi.org/10.1016/0021-8693(86)90125-0

#### Cited by

1. Stable under specialization sets and cofiniteness pp.1793-6829, 2018, https://doi.org/10.1142/S0219498819500154
2. Weakly cofiniteness of local cohomology modules pp.1793-6829, 2018, https://doi.org/10.1142/S0219498819500907