Finiteness of graded generalized local cohomology modules

We consider two finitely generated graded modules over a homogeneous Noetherian ring $R = oplus _{n in mathbb{N}_0 } R_n$ with a local base ring (R ₀, m₀) and irrelevant ideal R ₊ of R. We study the generalized local cohomology modules H _b ⁱ (M,N) with respect to the ideal b = b₀ + R ₊, where b₀ is an ideal of R ₀. We prove that if dimR ₀/b₀ ≤ 1, then the following cases hold: for all i ≥ 0, the R-module H _b ⁱ (M,N)/a₀ H _b ⁱ (M,N) is Artinian, where $sqrt {mathfrak{a}_0 + mathfrak{b}_0 } = mathfrak{m}_0$ ; for all i ≥ 0, the set $Ass_{R_0 } left( {H_mathfrak{b}^i left( {M,N} right)_n } right)$ is asymptotically stable as n→?∞. Moreover, if H _b ⁱ (M,N) _n is a finitely generated R ₀-module for all n ≤ n ₀ and all j < i, where n ₀ ∈ ? and i ∈ ?₀, then for all n ≤ n ₀, the set $Ass_{R_0 } left( {H_mathfrak{b}^i left( {M,N} right)_n } right)$ is finite.