• Title/Summary/Keyword: minimax module

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COMINIMAXNESS OF LOCAL COHOMOLOGY MODULES WITH RESPECT TO IDEALS OF DIMENSION ONE

  • Roshan-Shekalgourabi, Hajar
    • Honam Mathematical Journal
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    • v.40 no.2
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    • pp.211-218
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    • 2018
  • Let R be a commutative Noetherian ring, a be an ideal of R and M be an R-module. It is shown that if $Ext^i_R(R/a,M)$ is minimax for all $i{\leq}{\dim}\;M$, then the R-module $Ext^i_R(N,M)$ is minimax for all $i{\geq}0$ and for any finitely generated R-module N with $Supp_R(N){\subseteq}V(a)$ and dim $N{\leq}1$. As a consequence of this result we obtain that for any a-torsion R-module M that $Ext^i_R(R/a,M)$ is minimax for all $i{\leq}dim$ M, all Bass numbers and all Betti numbers of M are finite. This generalizes [8, Corollary 2.7]. Also, some equivalent conditions for the cominimaxness of local cohomology modules with respect to ideals of dimension at most one are given.

ON ARTINIANNESS OF GENERAL LOCAL COHOMOLOGY MODULES

  • Tri, Nguyen Minh
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.3
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    • pp.689-698
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    • 2021
  • In this paper, we show some results on the artinianness of local cohomology modules with respect to a system of ideals. If M is a 𝚽-minimax ZD-module, then Hdim M𝚽(M)/𝖆Hdim M𝚽(M) is artinian for all 𝖆 ∈ 𝚽. Moreover, if M is a 𝚽-minimax ZD-module, t is a non-negative integer and Hi𝚽(M) is minimax for all i > t, then Hi𝚽(M) is artinian for all i > t.

MINIMAXNESS AND COFINITENESS PROPERTIES OF GENERALIZED LOCAL COHOMOLOGY WITH RESPECT TO A PAIR OF IDEALS

  • Dehghani-Zadeh, Fatemeh
    • Communications of the Korean Mathematical Society
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    • v.31 no.4
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    • pp.695-701
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    • 2016
  • Let I and J be two ideals of a commutative Noetherian ring R and M, N be two non-zero finitely generated R-modules. Let t be a non-negative integer such that $H^i_{I,J}(N)$ is (I, J)-minimax for all i < t. It is shown that the generalized local cohomology module $H^i_{I,J}(M,N)$ is (I, J)-Cofinite minimax for all i < t. Also, we prove that the R-module $Ext^j_R(R/I,H^i_{I,J}(N))$ is finitely generated for all $i{\leq}t$ and j = 0, 1.

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.

ON THE LOCAL COHOMOLOGY OF MINIMAX MODULES

  • Mafi, Amir
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.6
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    • pp.1125-1128
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    • 2011
  • Let R be a commutative Noetherian ring, a an ideal of R, and M a minimax R-module. We prove that the local cohomology modules $H^j_a(M)$ are a-cominimax; that is, $Ext^i_R$(R/a, $H^j_a(M)$) is minimax for all i and j in the following cases: (a) dim R/a = 1; (b) cd(a) = 1, where cd is the cohomological dimension of a in R; (c) dim $R{\leq}2$. In these cases we also prove that the Bass numbers and the Betti numbers of $H^j_a(M)$ are finite.

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.

MINIMAXNESS OF LOCAL COHOMOLOGY MODULES DEFINED BY A PAIR OF IDEALS

  • Abbasi, A.;Roshan Shekalgourabi, H.
    • Honam Mathematical Journal
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    • v.34 no.2
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    • pp.161-169
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    • 2012
  • Let R be a commutative Noetherian ring and I, J be ideals of R. We introduced the notion of (I; J)-cominimax R-modules. For an integer $n$ and an R-module M, let $H^i_{I,J}(M)$ be an (I; J)-cominimax R-module for all $i<n$. The J-minimaxness of some Ext modules of $H^n_{I,J}(M)$ is investigated. Among of the obtaining results, there is a generalization of the main result of [1].

SOME FINITENESS RESULTS FOR CO-ASSOCIATED PRIMES OF GENERALIZED LOCAL HOMOLOGY MODULES AND APPLICATIONS

  • Do, Yen Ngoc;Nguyen, Tri Minh;Tran, Nam Tuan
    • Journal of the Korean Mathematical Society
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    • v.57 no.5
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    • pp.1061-1078
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    • 2020
  • We prove some results about the finiteness of co-associated primes of generalized local homology modules inspired by a conjecture of Grothendieck and a question of Huneke. We also show some equivalent properties of minimax local homology modules. By duality, we get some properties of Herzog's generalized local cohomology modules.

FINITENESS PROPERTIES OF EXTENSION FUNCTORS OF COFINITE MODULES

  • Irani, Yavar;Bahmanpour, Kamal
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.2
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    • pp.649-657
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    • 2013
  • Let R be a commutative Noetherian ring, I an ideal of R and T be a non-zero I-cofinite R-module with dim(T) ${\leq}$ 1. In this paper, for any finitely generated R-module N with support in V(I), we show that the R-modules $Ext^i_R$(T,N) are finitely generated for all integers $i{\geq}0$. This immediately implies that if I has dimension one (i.e., dim R/I = 1), then $Ext^i_R$($H^j_I$(M), N) is finitely generated for all integers $i$, $j{\geq}0$, and all finitely generated R-modules M and N, with Supp(N) ${\subseteq}$ V(I).

COLOCALIZATION OF GENERALIZED LOCAL HOMOLOGY MODULES

  • Hatamkhani, Marziyeh
    • Bulletin of the Korean Mathematical Society
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    • v.59 no.4
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    • pp.917-928
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    • 2022
  • Let R be a commutative Noetherian ring and I an ideal of R. In this paper, we study colocalization of generalized local homology modules. We intend to establish a dual case of local-global principle for the finiteness of generalized local cohomology modules. Let M be a finitely generated R-module and N a representable R-module. We introduce the notions of the representation dimension rI(M, N) and artinianness dimension aI(M, N) of M, N with respect to I by rI(M, N) = inf{i ∈ ℕ0 : HIi(M, N) is not representable} and aI(M, N) = inf{i ∈ ℕ0 : HIi(M, N) is not artinian} and we show that aI(M, N) = rI(M, N) = inf{rIR𝔭 (M𝔭,𝔭N) : 𝔭 ∈ Spec(R)} ≥ inf{aIR𝔭 (M𝔭,𝔭N) : 𝔭 ∈ Spec(R)}. Also, in the case where R is semi-local and N a semi discrete linearly compact R-module such that N/∩t>0ItN is artinian we prove that inf{i : HIi(M, N) is not minimax}=inf{rIR𝔭 (M𝔭,𝔭N) : 𝔭 ∈ Spec(R)\Max(R)}.