ON GRADED LOCAL COHOMOLOGY MODULES DEFINED BY A PAIR OF IDEALS

Abstract

Let $R = bigoplus_{n in mathbb{N}_{0}} R_{n}$ be a standard<br />graded ring, $M$ be a finitely generated graded $R$-module and $J$<br />be a homogenous ideal of $R$. In this paper we study the graded<br />structure of the $i$-th local cohomology module of $M$ defined by a<br />pair of ideals $(R_{+},J)$, i.e. $H^{i}_{R_{+},J}(M)$. More<br />precisely, we discuss finiteness property and vanishing of the<br />graded components $H^{i}_{R_{+},J}(M)_{n}$.<br /><br />Also, we study the Artinian property and tameness of certain<br />submodules and quotient modules of $H^{i}_{R_{+},J}(M)$.