A generalization of the finiteness problem of the local cohomology modules

Abstract

Let R be a commutative Noetherian ring and $\mathfrak{a}$ an ideal of R. We introduce the concept of $\mathfrak{a}$-weakly Laskerian R-modules, and we show that if M is an $\mathfrak{a}$-weakly Laskerian R-module and s is a non-negative integer such that Ext ^j _R $(R/\mathfrak{a},H_\mathfrak{a}^i (M))$ is $\mathfrak{a}$-weakly Laskerian for all i < s and all j, then for any $\mathfrak{a}$-weakly Laskerian submodule X of $H_\mathfrak{a}^s (M)$, the R-module $Hom_R (R/\mathfrak{a},H_\mathfrak{a}^s (M)/X)$ is $\mathfrak{a}$-weakly Laskerian. In particular, the set of associated primes of $H_\mathfrak{a}^s (M)/X$ is finite. As a consequence, it follows that if M is a finitely generated R-module and N is an $\mathfrak{a}$-weakly Laskerian R-module such that $H_\mathfrak{a}^i (N)$(N) is $\mathfrak{a}$-weakly Laskerian for all i < s, then the set of associated primes of $H_\mathfrak{a}^s (M,N)$(M,N) is finite. This generalizes the main result of S. Sohrabi Laleh, M.Y. Sadeghi, and M.Hanifi Mostaghim (2012).