• Bulletin of the Korean Mathematical Society
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• v.60 no.1
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• pp.149-160
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• 2023

Let 𝖆 and 𝖇 be ideals of a commutative Noetherian ring R and M a finitely generated R-module of finite dimension d > 0. In this paper, we obtain some results about the annihilators and attached primes of top local cohomology and top formal local cohomology modules. In particular, we determine Ann(𝖇 H^d_𝖆(M)), Att(𝖇 H^d_𝖆(M)), Ann(𝖇𝔉^d_𝖆(M)) and Att(𝖇𝔉^d_𝖆(M)).

• Kyungpook Mathematical Journal
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• v.63 no.1
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• pp.37-43
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• 2023

Let 𝔞 and 𝔟 be ideals of a commutative Noetherian ring R and M be a finitely generated R-module of dimension d > 0. We prove some results concerning the top local cohomology and top formal local cohomology modules. Among other things, we determine Supp_R(𝔟 H^d_𝔞(M)) and Supp_R(𝔟𝔉^d_𝔞(M)). Also, we obtain some relations between Ann_R(𝔟 H^d_𝔞(M)), Att_R(𝔟 H^d_𝔞(M)) and Supp_R(𝔟 H^d_𝔞(M)) and we get similar results for 𝔟𝔉^d_𝔞(M).

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1855-1861
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• 2013

Let R be a commutative Noetherian ring, I an ideal of R, M and N two R-modules. We characterize the least integer i such that $H^i_I(M,N)$ is not weakly Artinian by using the notion of weakly filter regular sequences. Also, a local-global principle for minimax generalized local cohomology modules is shown and the result generalizes the corresponding result for local cohomology modules.

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.2013-2020
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• 2013

We study some properties of representable generalized local homology modules. By duality, we get some properties of good generalized local cohomology modules.

• Communications of the Korean Mathematical Society
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• v.33 no.4
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• pp.1049-1053
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• 2018

It is shown that any equidimensional local ring which has finite Cousin cohomology modules with respect to the dimension filtration has a uniform local cohomological annihilator and is universally catenary.

• Honam Mathematical Journal
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• v.38 no.2
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• pp.295-304
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• 2016

In this paper we investigate the Artinianness of certain local cohomology modules $H^i_I(N)$ where N is a minimax module over a commutative Noetherian ring R and I is an ideal of R. Also, we characterize the set of attached prime ideals of $H^n_I(N)$, where n is the dimension of N.

• Kyungpook Mathematical Journal
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• v.56 no.3
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• pp.737-743
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• 2016

Let a and b be two ideals of a commutative Noetherian ring R, M a finitely generated R-module and i an integer. In this paper we study formal local cohomology modules with respect to a pair of ideals. We denote the i-th a-formal local cohomology module M with respect to b by ${\mathfrak{F}}^i_{a,b}(M)$. We show that if ${\mathfrak{F}}^i_{a,b}(M)$ is artinian, then $a{\subseteq}{\sqrt{(0:{\mathfrak{F}}^i_{a,b}(M))$. Also, we show that ${\mathfrak{F}}^{\text{dim }M}_{a,b}(M)$ is artinian and we determine the set $Att_R\;{\mathfrak{F}}^{\text{dim }M}_{a,b}(M)$.

• Honam Mathematical Journal
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• v.23 no.1
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• pp.1-4
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• 2001

In [2, 7.3.2], the set of attached prime ideals of local cohomology module $H_m^n(M)$ were calculated, where (A, m) be Noetherian local ring, M finite A-module and $dim_A(M)=n$, and also in the special case in which furthermore A is a homomorphic image of a Gornestien local ring (A', m') (see [2, 11.3.6]). In this paper, we shall obtain this set, by another way in this special case.

• Honam Mathematical Journal
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• v.30 no.3
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• pp.513-519
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• 2008

Let (R, m) be a Noetherian local ring with maximal ideal m, E := $E_R$(R/m) and let I be an ideal of R. Let M and N be finitely generated R-modules. It is shown that $H^n_I(M,(H^n_I(N)^{\vee})){\cong}(M{\otimes}_RN)^{\vee}$ where grade(I, N) = n = $cd_i$(I, N). We also show that for n = grade(I, R), one has $End_R(H^n_I(P,R)^{\vee}){\cong}Ext^n_R(H^n_I(P,R),P^*)^{\vee}$.

• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.319-329
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• 2017

Let I denote an ideal of a Noetherian local ring (R, m). Let M denote a finitely generated R-module. We study the endomorphism ring of the local cohomology module $H^c_I(M)$, c = grade(I, M). In particular there is a natural homomorphism $$Hom_{\hat{R}^I}({\hat{M}}^I,\;{\hat{M}}^I){\rightarrow}Hom_R(H^c_I(M),\;H^c_I(M))$$, $where{\hat{\cdot}}^I$ denotes the I-adic completion functor. We provide sufficient conditions such that it becomes an isomorphism. Moreover, we study a homomorphism of two such endomorphism rings of local cohomology modules for two ideals $J{\subset}I$ with the property grade(I, M) = grade(J, M). Our results extends constructions known in the case of M = R (see e.g. [8], [17], [18]).