Finiteness of Graded Generalized Local Cohomology Modules

We consider two finitely generated graded modules over a homogeneous Noetherian ring $R=\bigoplus_{n\in\mathbb{N}_0}R_n$ with a local base ring $(R_0,\mathfrak{m}_0)$ and irrelevant ideal $R_{+}$ of $R$. We study the generalized local cohomology modules $H_\mathfrak{b}^i(M,N)$ with respect to the ideal $\mathfrak{b}=\mathfrak{b}_0+{R}_+$, where $\mathfrak{b}_0$ is an ideal of $R_0$. We prove that if $\operatorname{dim} R_0/\mathfrak{b}_0\le 1$, then the following cases hold: for all $i\ge 0$, the $R$-module $H_\mathfrak{b}^i(M,N)/{\mathfrak{a}_0H_\mathfrak{b}^i(M,N)}$ is Artinian, where $\sqrt{\mathfrak{a}_0+\mathfrak{b}_0}=\mathfrak{m}_0$; for all $i\ge 0$, the set $\operatorname{Ass}_{R_0}(H_\mathfrak{b}^i(M,N)_n)$ is asymptotically stable as $n\to{-\infty}$. Moreover, if $H_{\mathfrak{b}}^j(M,N)_n$ is a finitely generated $R_0$-module for all $n\le n_0$ and all $j<i$, where $n_0\in\mathbb{Z}$ and $i\in\mathbb{N}_0$, then for all $n\le n_0$, the set $\operatorname{Ass}_{R_0}(H_{\mathfrak{b}}^i(M,N)_n)$ is finite.