• Bulletin of the Korean Mathematical Society
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• v.61 no.1
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• pp.273-280
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• 2024

Let I be an ideal of a commutative Noetherian semi-local ring R and M be an R-module. It is shown that if dim M ≤ 2 and Supp_R M ⊆ V (I), then M is I-weakly cofinite if (and only if) the R-modules Hom_R(R/I, M) and Ext¹_R(R/I, M) are weakly Laskerian. As a consequence of this result, it is shown that the category of all I-weakly cofinite modules X with dim X ≤ 2, forms an Abelian subcategory of the category of all R-modules. Finally, it is shown that if dim R/I ≤ 2, then for each pair of finitely generated R-modules M and N and each pair of the integers i, j ≥ 0, the R-modules Tor^R_i(N, H^j_I(M)) and Extⁱ_R(N, H^j_I(M)) are I-weakly cofinite.

• Bulletin of the Korean Mathematical Society
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• v.60 no.3
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• pp.637-647
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• 2023

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, M an arbitrary R-module and X a finite R-module. We prove a characterization for Hⁱ_𝔞(M) and Hⁱ_𝔞(X, M) to be 𝔞-weakly cofinite for all i, whenever one of the following cases holds: (a) ara(𝔞) ≤ 1, (b) dim R/𝔞 ≤ 1 or (c) dim R ≤ 2. We also prove that, if M is a weakly Laskerian R-module, then Hⁱ_𝔞(X, M) is 𝔞-weakly cofinite for all i, whenever dim X ≤ 2 or dim M ≤ 2 (resp. (R, m) a local ring and dim X ≤ 3 or dim M ≤ 3). Let d = dim M < ∞, we prove an equivalent condition for top local cohomology module H^d_𝔞(M) to be weakly Artinian.

• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1341-1352
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• 2016

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.