MMN-608
Weakly Laskerian modules and weak cofiniteness
Abstract
Let $R$ be a commutative Noetherian ring, $a$ an ideal of R. It is shown that if $a = (x_1,\dots, x_t)$, and $M$ is an $R$-module, then ${\mathop{\mbox{\rm Ext}}}_i^R(R/a,M)$ is weakly Laskerian for all $i$ iff ${\mathop{\mbox{\rm Tor}}}^R_i(R/a,M)$ is weakly Laskerian for all $i$ iff The Koszul cohomology module
$H^i(x_1,\dots, x_t;M)$ is weakly Laskerian for all $i$. Furthermore, each of coditions imply
that $M/a^nM$ is weakly Laskerian for all $n \in \mathbb{N}$. In section 3, we show that if $M$ is an
R-module with $Supp M\subseteq V(a)$, then $M$ is $a$-weakly cofinite, in the following cases:
\begin{enumerate}
\item[a)] there exists $x\in a $ such that $0:_M x$ and $M/xM$ are both $a$-weakly cofinite.
\item[b)] there exists $x\in\sqrt{a}$ such that $0:_M x$ and $M/xM$ are both weakly Laskerian.
\end{enumerate}
Vol. 15 (2014), No. 2, pp. 761-770