On graded local cohomology modules defined by a pair of ideals

Abstract

Let $R={⨁}_{n\in {\mathbb{N}}_0}{R}_n$ 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. $H_{R_{+},J}^i(M)$. More precisely, we discuss finiteness property and vanishing of the graded components $H_{R_{+},J}^i(M{)}_n$. Also, we study the Artinian property and tameness of certain submodules and quotient modules of $H_{R_{+},J}^i(M)$.

are welcome!