Cofiniteness of local cohomology modules for ideals of dimension one

Abstract

Let $R$ denote a commutative Noetherian (not necessarily local) ring, $M$ an arbitrary $R$-module and $I$ an ideal of $R$ of dimension one. It is shown that the $R$-module $\Ext^i_R(R/I,M)$ is finitely generated (resp. weakly Laskerian) for all $i\le \mathrm{cd}(I,M)+1$ if and only if the local cohomology module $H_I^i(M)$ is $I$-cofinite (resp. $I$-weakly cofinite) for all $i$. Also, we show that when $I$ is an arbitrary ideal and $M$ is finitely generated module such that the $R$-module $H_I^i(M)$ is weakly Laskerian for all $i\le t-1$, then $H_I^i(M)$ is $I$-cofinite for all $i\le t-1$ and for any minimax submodule $K$ of $H_I^t(M)$, the $R$-modules $\Hom_R(R/I, H^{t}_I(M)/K)$ and $\Ext^{1}_R(R/I, H^{t}_I(M)/K)$ are finitely generated, where $t$ is a non-negative integer. This generalizes the main result of Bahmanpour-Naghipour \cite{BN} and Brodmann and Lashgari \cite{BL}.

7 pages

References in corpus (1)

• On the finiteness of Bass numbers of local cohomology modules and Cominimaxness