• Title, Summary, Keyword: local cohomology

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ON THE WEAK ARTINIANNESS AND MINIMAX GENERALIZED LOCAL COHOMOLOGY MODULES

  • Gu, Yan
    • 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.

FINITENESS PROPERTIES GENERALIZED LOCAL COHOMOLOGY WITH RESPECT TO AN IDEAL CONTAINING THE IRRELEVANT IDEAL

  • Dehghani-Zadeh, Fatemeh
    • Journal of the Korean Mathematical Society
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    • v.49 no.6
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    • pp.1215-1227
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    • 2012
  • The membership of the generalized local cohomology modules $H_a^i$(M,N) of two R-modules M and N with respect to an ideal a in certain Serre subcategories of the category of modules is studied from below ($i<t$). Furthermore, the behaviour of the $n$th graded component $H_a^i(M,N)_n$ of the generalized local cohomology modules with respect to an ideal containing the irrelevant ideal as $n{\rightarrow}-{\infty}$ is investigated by using the above result, in certain graded situations.

ARTINIANNESS OF LOCAL COHOMOLOGY MODULES

  • Abbasi, Ahmad;Shekalgourabi, Hajar Roshan;Hassanzadeh-lelekaami, Dawood
    • 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.

A Generalization of Formal Local Cohomology Modules

  • Rezaei, Shahram
    • 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 $Att_R\;{\mathfrak{F}}^{\text{dim }M}_{a,b}(M)$.

THE SET OF ATTACHED PRIME IDEALS OF LOCAL COHOMOLOGY

  • RASOULYAR, S.
    • 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.

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GENERALIZED LOCAL COHOMOLOGY AND MATLIS DUALITY

  • Abbasi, Ahmad
    • 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 $End_R(H^n_I(P,R)^{\vee}){\cong}Ext^n_R(H^n_I(P,R),P^*)^{\vee}$.

A NOTE ON ENDOMORPHISMS OF LOCAL COHOMOLOGY MODULES

  • Mahmood, Waqas;Zahid, Zohaib
    • 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]).

Results of Graded Local Cohomology Modules with respect to a Pair of Ideals

  • Dehghani-Zadeh, Fatemeh
    • Kyungpook Mathematical Journal
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    • v.58 no.1
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    • pp.9-17
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    • 2018
  • Let $R ={\oplus}_{n{\in}Z}R_n$ be a graded commutative Noetherian ring and let I be a graded ideal of R and J be an arbitrary ideal. It is shown that the i-th generalized local cohomology module of graded module M with respect to the (I, J), is graded. Also, the asymptotic behaviour of the homogeneous components of $H^i_{I,J}(M)$ is investigated for some i's with a specified property.