^{}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.

Article Title [Persian]

Artinianess of Graded Generalized Local
Cohomology Modules

Authors [Persian]

Sh. Tahamtan

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

Abstract [Persian]

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.

Keywords [Persian]

Artinian module، Generalized local cohomology module، Minimax module

References

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[5] K. Khashayarmanesh, Associated primes of graded components of generalized local cohomology modules, Comm. Algebra. 33(9) (2005), 3081-3090. [6] D. Kirby, Artinian modules and Hilbert polynomials, Quarterly Journal Mathe- matics Oxford (2) 24 (1973), 47-57. [7] R. Sazeedeh, Finiteness of graded local cohomology modules, J. Pure Appl. Alg. 212(1) (2008), 275-280. [8] Sh. Tahamtan, H. Zakeri, A note on Artinianess of certain generalized local cohomology modules, Journal of sciences, Islamic republic of Iran 19(3):265- 272(2008).