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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 $\lc^{i}_\Phi(M)$ is ${\rm FD_{\leq 1}}$ (or weakly Laskerian) for all $i<t$, then $\lc^{i}_\Phi(M)$ is $\Phi$-cofinite for all $i<t$ and for any ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$ of $\lc^t_\Phi(M)$, the $R$-modules $\Hom_R(R/I,\lc^t_\Phi(M)/N)$ and $\Ext^1_R(R/I,\lc^t_\Phi(M)/N)$ are finitely generated. Also it is shown that if $\cd I = 1$ or $\dim M/IM \leq 1$ {\rm(}e.g., $\dim R/I \leq 1${\rm)} for all $I \in \Phi$, then the local cohomology module $\lc^{i}_\Phi(M)$ is $\Phi$-cofinite for all $i\geq 0$. These generalize the main results of Aghapournahr and Bahmanpour \cite{AB}, Bahmanpour and Naghipour \cite{BN2, BN}. Also we study cominimaxness and weakly cofiniteness of local cohomology modules with respect to a system of ideals.
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참고문헌 (33)
참고문헌 신청
M. Aghapournahr / Cofiniteness of certain local cohomology modules for small dimen-sions / preprint
M. Aghapournahr / 2014 / Cofiniteness of weakly Laskerian local coho-mology modules / Bull. Math. Soc. Sci. Math. Roumanie (N.S) 57(105)(4):347~356
M. Aghapournahr / 2010 / A natural map in local cohomology / Ark. Mat 48(2):243~251
J. Asadollahi / 2003 / A generalization of the cofiniteness problem in local cohomology modules / J. Aust. Math. Soc 75(3):313~324
K. Bahmanpour / 2014 / On the category of weakly Laskerian cofinite modules / Math. Scand 115(1):62~68
M. Aghapournahr / 2014 / Cofiniteness of weakly Laskerian local coho-mology modules / Bull. Math. Soc. Sci. Math. Roumanie (N.S) 57(105)(4):347~356
M. Aghapournahr / 2010 / A natural map in local cohomology / Ark. Mat 48(2):243~251
J. Asadollahi / 2003 / A generalization of the cofiniteness problem in local cohomology modules / J. Aust. Math. Soc 75(3):313~324
K. Bahmanpour / 2014 / On the category of weakly Laskerian cofinite modules / Math. Scand 115(1):62~68
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