Finiteness of graded generalized local cohomology modules

We consider two finitely generated graded modules over a homogeneous Noetherian ring \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R = \oplus _{n \in \mathbb{N}_0 } R_n$$\end{document} with a local base ring (R0, m0) and irrelevant ideal R+ of R. We study the generalized local cohomology modules Hbi (M,N) with respect to the ideal b = b0 + R+, where b0 is an ideal of R0. We prove that if dimR0/b0 ≤ 1, then the following cases hold: for all i ≥ 0, the R-module Hbi(M,N)/a0Hbi(M,N) is Artinian, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt {\mathfrak{a}_0 + \mathfrak{b}_0 } = \mathfrak{m}_0$$\end{document}; for all i ≥ 0, the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ass_{R_0 } \left( {H_\mathfrak{b}^i \left( {M,N} \right)_n } \right)$$\end{document} is asymptotically stable as n→−∞. Moreover, if Hbi(M,N)n is a finitely generated R0-module for all n ≤ n0 and all j < i, where n0 ∈ ℤ and i ∈ ℕ0, then for all n ≤ n0, the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ass_{R_0 } \left( {H_\mathfrak{b}^i \left( {M,N} \right)_n } \right)$$\end{document} is finite.