FINITENESS RESULT FOR GENERALIZED LOCAL COHOMOLOGY MODULES

Abstract

Let $R$ be a Noetherian ring, let $M$ and $N$ be finitely generated modules and let $\mathfrak{a}$ and $\mathfrak{b}$ be ideals of $R$. Let $s$ be an integer such that $\mathfrak{b}_{\mathfrak{p}} \subseteq \sqrt{\mbox{Ann} \mbox{H}^i_{\mathfrak{a}_{\mathfrak{p}}}(M_{\mathfrak{p}} ,N_{\mathfrak{p}})}$ for all $i \le s$ and all prime ideal $\mathfrak{p}$ of $R$. Then we show the following statements hold:

(1) If $\mbox{H}^i_{\mathfrak{b}}(N) = 0$ for all $i \lt s$, then $\mbox{H}_{\mathfrak{a}}^{i}(M,N)$ is finitely generated for all $i \leq s$.

(2) $\mathfrak{b} \subseteq \sqrt{\mbox{Ann} \mbox{H}_{\mathfrak{a}}^{2}(M,N)}$.

These statements generalize the corresponding results which are shown in [6] and [1] for standard local cohomology module.