^{}Department of Mathematics, Islamic Azad University, Borujerd-Branch, Borujerd, iran.

Abstract

Let R = L n2N0 Rn be a Noetherian homogeneous graded ring with local base ring (R0;m0) of dimension d . Let R+ = Ln2N Rn denote the irrelevant ideal of R and let M and N be two nitely generated graded R-modules. Let t = tR+(M;N) be the rst 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 ! 1 and Hj m0 (Hi R+(M;N)) is Artinian for 0 j 1. More- over, 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 Hj m0 (Hi R+(M;N)) is Artinian for d 1 j d. Finally we show that Hd2 m0 (Hs R+(M;N)) is Artinian if and only if Hd m0 (Hs1 R+ (M;N)) is Artinian.

References

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