摘要
Let R = ⊙n〉0 Rn be a standard graded ring, a ∩ ⊙n〉0 Rn an ideal of R, and M, N two finitely generated graded R-modules. This paper studies the homogeneous components of graded generalized local cohomology modules. We show that for any i 〉 0, the n-th graded component Hiα（M, N）n of the i-th generalized local cohomology module of M and N with respect to a vanishes for all n 〉〉 0. Some sufficient conditions are pro- posed to satisfy the equality sup{end（Hiα （M, N）） [ i _〉 0} = sup{end（HiR＋ （M, N）） ｜ i 〉 0}. Also, some sufficient conditions are proposed for the tameness of Hiα（M, N） such that i = fRα＋（M,N） or i = cdα（M,g）, where fRα＋（M,N） and cdα（M,g） denote the R＋- finiteness dimension and the cohomological dimension of M and N with respect to a, respectively. Finally, we consider the Artinian property of some submodules and quotient modules of Hjα（M, N）, where j is the first or last non-minimax level of Hiα（M, N）.
Let R = ⊙n〉0 Rn be a standard graded ring, a ∩ ⊙n〉0 Rn an ideal of R, and M, N two finitely generated graded R-modules. This paper studies the homogeneous components of graded generalized local cohomology modules. We show that for any i 〉 0, the n-th graded component Hiα（M, N）n of the i-th generalized local cohomology module of M and N with respect to a vanishes for all n 〉〉 0. Some sufficient conditions are pro- posed to satisfy the equality sup{end（Hiα （M, N）） [ i _〉 0} = sup{end（HiR＋ （M, N）） ｜ i 〉 0}. Also, some sufficient conditions are proposed for the tameness of Hiα（M, N） such that i = fRα＋（M,N） or i = cdα（M,g）, where fRα＋（M,N） and cdα（M,g） denote the R＋- finiteness dimension and the cohomological dimension of M and N with respect to a, respectively. Finally, we consider the Artinian property of some submodules and quotient modules of Hjα（M, N）, where j is the first or last non-minimax level of Hiα（M, N）.
