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Keywords:
local cohomology module; Krull dimension; minimax module; cofinite module; weakly Laskerian module; associated primes
Summary:
Let $R$ be a commutative Noetherian ring with identity and $I$ an ideal of $R$. It is shown that, if $M$ is a non-zero minimax $R$-module such that $\dim \mathop {\rm Supp} H^i_I (M) \leq 1$ for all $i$, then the $R$-module $H^i_I(M)$ is $I$-cominimax for all $i$. In fact, $H^i_I(M)$ is $I$-cofinite for all $i\geq 1$. Also, we prove that for a weakly Laskerian $R$-module $M$, if $R$ is local and $t$ is a non-negative integer such that $\dim \mathop {\rm Supp} H^i_I (M)\leq 2$ for all $i<t$, then ${\rm Ext}^j_R (R/I, H^i_I (M))$ and ${\rm Hom}_R(R/I, H^t_I(M))$ are weakly Laskerian for all $i<t$ and all $j \geq 0$. As a consequence, the set of associated primes of $H^i_I (M)$ is finite for all $i\geq 0$, whenever $\dim R/I \leq 2$ and $M$ is weakly Laskerian.
References:
[1] Azami, J., Naghipour, R., Vakili, B.: Finiteness properties of local cohomology modules for $\mathfrak a$-minimax modules. Proc. Am. Math. Soc. 137 (2009), 439-448. DOI 10.1090/S0002-9939-08-09530-0 | MR 2448562
[2] Bahmanpour, K.: On the category of weakly Laskerian cofinite modules. Math. Scand. 115 (2014), 62-68. MR 3250048
[3] Bahmanpour, K., Naghipour, R.: On the cofiniteness of local cohohomology modules. Proc. Am. Math. Soc. 136 (2008), 2359-2363. DOI 10.1090/S0002-9939-08-09260-5 | MR 2390502
[4] Bahmanpour, K., Naghipour, R.: Cofiniteness of local cohomology modules for ideals of small dimension. J. Algebra 321 (2009), 1997-2011. DOI 10.1016/j.jalgebra.2008.12.020 | MR 2494753 | Zbl 1168.13016
[5] Brodmann, M. P., Sharp, R. Y.: Local Cohomology: An Algebraic Introduction with Geometric Applications. Cambridge Studies in Advanced Mathematics 60, Cambridge University Press, Cambridge (1998). MR 1613627 | Zbl 0903.13006
[6] Chiriacescu, G.: Cofiniteness of local cohomology modules over regular local rings. Bull. Lond. Math. Soc. 32 (2000), 1-7. DOI 10.1112/S0024609399006499 | MR 1718769 | Zbl 1018.13009
[7] Delfino, D.: On the cofiniteness of local cohomology modules. Math. Proc. Camb. Philos. Soc. 115 (1994), 79-84. DOI 10.1017/S0305004100071929 | MR 1253283 | Zbl 0806.13005
[8] Delfino, D., Marley, T.: Cofinite modules and local cohomology. J. Pure Appl. Algebra 121 (1997), 45-52. DOI 10.1016/S0022-4049(96)00044-8 | MR 1471123 | Zbl 0893.13005
[9] Divaani-Aazar, K., Mafi, A.: Associated primes of local cohomology modules. Proc. Am. Math. Soc. (electronic) 133 (2005), 655-660. DOI 10.1090/S0002-9939-04-07728-7 | MR 2113911 | Zbl 1103.13010
[10] Divaani-Aazar, K., Mafi, A.: Associated primes of local cohomology modules of weakly Laskerian modules. Commun. Algebra 34 (2006), 681-690. DOI 10.1080/00927870500387945 | MR 2211948 | Zbl 1097.13021
[11] Enochs, E.: Flat covers and flat cotorsion modules. Proc. Am. Math. Soc. 92 (1984), 179-184. DOI 10.1090/S0002-9939-1984-0754698-X | MR 0754698 | Zbl 0522.13008
[12] Hartshorne, R.: Affine duality and cofiniteness. Invent. Math. 9 (1970), 145-164. DOI 10.1007/BF01404554 | MR 0257096 | Zbl 0196.24301
[13] Huneke, C., Koh, J.: Cofiniteness and vanishing of local cohomology modules. Math. Proc. Camb. Philos. Soc. 110 (1991), 421-429. DOI 10.1017/S0305004100070493 | MR 1120477 | Zbl 0749.13007
[14] Mafi, A.: A generalization of the finiteness problem in local cohomology modules. Proc. Indian Acad. Sci., Math. Sci. 119 (2009), 159-164. DOI 10.1007/s12044-009-0016-1 | MR 2526419 | Zbl 1171.13011
[15] Melkersson, L.: Modules cofinite with respect to an ideal. J. Algebra 285 (2005), 649-668. DOI 10.1016/j.jalgebra.2004.08.037 | MR 2125457 | Zbl 1093.13012
[16] Quy, P. H.: On the finiteness of associated primes of local cohomology modules. Proc. Am. Math. Soc. 138 (2010), 1965-1968. DOI 10.1090/S0002-9939-10-10235-4 | MR 2596030 | Zbl 1190.13010
[17] Robbins, H. R.: Finiteness of Associated Primes of Local Cohomology Modules. Ph.D. Thesis, University of Michigan (2008). MR 2712251
[18] Yoshida, K.-I.: Cofiniteness of local cohomology modules for ideals of dimension one. Nagoya Math. J. 147 (1997), 179-191. MR 1475172 | Zbl 0899.13018
[19] Zink, T.: Endlichkeitsbedingungen für Moduln über einem Noetherschen Ring. German Math. Nachr. 64 (1974), 239-252. DOI 10.1002/mana.19740640114 | MR 0364223 | Zbl 0297.13015
[20] Zöschinger, H.: Minimax modules. German J. Algebra 102 (1986), 1-32. DOI 10.1016/0021-8693(86)90125-0 | MR 0853228 | Zbl 0593.13012
[21] Zöschinger, H.: On the maximality condition for radically full submodules. German Hokkaido Math. J. 17 (1988), 101-116. MR 0928469 | Zbl 0653.13011
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