On graded local cohomology modules defined by a pair of ideals (1502.04970v1)

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