Artinianness of Composed Graded Local Cohomology Modules

Abstract

Let $R\,=\,{{\oplus }_{n\ge 0}}{{R}_{n}}$ be a graded Noetherian ring with local base ring $\left( {{R}_{0}},{{\text{m}}_{0}} \right)$ and let ${{R}_{+}}\,=\,{{\oplus }_{n>0}}{{R}_{n}}$ . Let $M$ and $N$ be finitely generated graded $R$ -modules and let $\mathfrak{a}\,=\,{{\mathfrak{a}}_{0}}\,+\,{{R}_{+}}$ an ideal of $R$ . We show that $H_{\mathfrak{b}0}^{j}\,\left( H_{\mathfrak{a}}^{i}\left( M,\,N \right) \right)$ and ${H_{\mathfrak{a}}^{i}\left( M,\,N \right)}/{{{\mathfrak{b}}_{0}}H_{\mathfrak{a}}^{i}\left( M,\,N \right)}\;$ are Artinian for some $i\text{ s}$ and $j\,\text{s}$ with a specified property, where ${{\mathfrak{b}}_{o}}$ is an ideal of ${{R}_{0}}$ such that ${{\mathfrak{a}}_{0}}\,+\,{{\mathfrak{b}}_{0}}$ is an ${{\mathfrak{m}}_{0}}$ -primary ideal.