Hartshorne's questions and weakly cofiniteness

Abstract

Let $R$ be a commutative Noetherian ring, $\fa$ be an ideal of $R$ and $M$ be an $R$-module. The main purpose of this paper is to answer the Hartshorn's questions in the class of weakly Laskerian modules. It is shown that if $s\geq 1$ is a positive integer such that $\Ext^j_R(R/\fa, M)$ is weakly Laskerian for all $j\leq s$ and the $R$-module $H^i_\fa(M)$ is $FD_{\leq 1}$ for all $i < s$, then the $R$-module $H^i_\fa(M)$ is $\fa$-weakly cofinite for all $i <s$. In addition, we show that the category of all $\fa$-weakly cofinite $FD_{\leq 1}$ $R$-modules is an Abelian subcategory of the category of all $R$-modules. Also, we prove that if $\Ext^i_R(R/\fa,M)$ is weakly Laskerian for all $i\leq \dim M$, then the $R$-module $\Ext^i_R(N,M)$ is weakly Laskerian for all $i\geq 0$ and for any finitely generated $R$-module $N$ with $\Supp_R(N) \subseteq V (\fa)$ and $\dim N \leq 1$.