# Artinianess of Graded Generalized Local Cohomology Modules

Document Type: Research Articles

Author

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
Hd􀀀2
m0 (Hs
R+(M;N)) is Artinian if and only if Hd
m0 (Hs􀀀1
R+
(M;N)) is Artinian.

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