Some finiteness properties of generalized graded local cohomology modules

Abstract

Let $R={⨁}_{n\in {\mathbb{N}}_0}{R}_n$ be a Noetherian homogeneous ring with local base ring $(R_0,{\mathfrak{m}}_0)$ and let $M$ and $N$ be finitely generated graded $R$-modules. Let $i,j\in {\mathbb{N}}_0$. In this paper we will study Artinianess of ${\Gamma }_{{\mathfrak{m}}_{0}R}(H_{R_{+}}^i(M,N)),H_{{\mathfrak{m}}_{0}R}^1(H_{R_{+}}^i(M,N)),H_{R_{+}}^i(M,N)/{\mathfrak{m}}_0{H}_{R_{+}}^i(M,N),H_{R_{+}}^j(M,H_{{\mathfrak{m}}_{0}R}^i(N)),H_{{\mathfrak{m}}_{0}R}^j(M,H_{R_{+}}^i(N))$, where $R_{+}$ denotes the irrelevant ideal of $R$.

8 pages