Artinianess of Graded Generalized Local Cohomology Modules

Subject Areas : Applied Mathematics

Sh. Tahamtan ¹ (Department of Mathematics, Islamic Azad University, Borujerd-Branch, Borujerd, iran.)

Received: 2015-09-30 Accepted : 2015-09-30 Published : 2013-05-01

Keywords: Artinian module, Generalized local cohomology module, Minimax module,

Abstract :

Let R = L n2N0Rn be a Noetherian homogeneous graded ring with local basering (R0;m0) of dimension d . Let R+ = Ln2NRn denote the irrelevant idealof R and let M and N be two nitely generated graded R-modules. Lett = tR+(M;N) be the rst integer i such that HiR+(M;N) is not minimax.We prove that if i  t, then the set AssR0 (HiR+(M;N)n) is asymptoticallystable for n 􀀀! 􀀀1 and Hjm0 (HiR+(M;N)) is Artinian for 0  j  1. More-over, let s = sR+(M;N) be the largest integer i such that HiR+(M;N) is notminimax. For each i  s, we prove that R0m0R0HiR+(M;N) is Artinian andthat Hjm0 (HiR+(M;N)) is Artinian for d 􀀀 1  j  d. Finally we show thatHd􀀀2m0 (HsR+(M;N)) is Artinian if and only if Hdm0 (Hs􀀀1R+(M;N)) is Artinian.

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