Artinianess of Graded Generalized Local Cohomology Modules

Let R = L n∈N0 Rn be a Noetherian homogeneous graded ring with local base ring (R0, m0) of dimension d . Let R+ = L n∈N Rn denote the irrelevant ideal of R and let M and N be two finitely generated graded R-modules. Let t = tR+ (M, N) be the first integer i such that Hi R+ (M, N) is not minimax. We prove that if i ≤ t, then the set AssR0 (Hi R+ (M, N)n) is asymptotically stable for n −→ −∞ and H j m0 (Hi R+ (M, N)) is Artinian for 0 ≤ j ≤ 1. Moreover, let s = sR+ (M, N) be the largest integer i such that Hi R+ (M, N) is not minimax. For each i ≥ s, we prove that R0 m0 ⊗ R0Hi R+ (M, N) is Artinian and that H j m0 (Hi R+ (M, N)) is Artinian for d − 1 ≤ j ≤ d. Finally we show that Hd−2 m0 (Hs R+ (M, N)) is Artinian if and only if Hd m0 (H s−1 R+ (M, N)) is Artinian.