Finiteness of certain local cohomology modules

Abstract

Cofiniteness of the generalized local cohomology modules <br />$H^{i}_{mathfrak{a}}(M,N)$ of two $R$-modules $M$ and $N$ with<br />respect to an ideal $mathfrak{a}$ is studied for some $i^{,}s$ with<br />a specified property. Furthermore, Artinianness of<br />$H^{j}_{mathfrak{b}_{0}}(H_{mathfrak{a}}^{i}(M,N))$ is<br />investigated by using the above result, in certain graded situations, where $mathfrak{b}_{0}$ is an ideal of $R_{0}$ and<br />$mathfrak{a}=mathfrak{a}_{0}+R_{+}$ such that<br />$mathfrak{b}_{0}+mathfrak{a}_{0}$ is an $mathfrak{m}_{0}$-primary ideal.