Artinianess of Graded Generalized Local Cohomology Modules

Abstract

Let R = L n2N0<br />Rn be a Noetherian homogeneous graded ring with local base<br />ring (R0;m0) of dimension d . Let R+ = Ln2N<br />Rn denote the irrelevant ideal<br />of R and let M and N be two nitely generated graded R-modules. Let<br />t = tR+(M;N) be the rst integer i such that Hi<br />R+(M;N) is not minimax.<br />We prove that if i t, then the set AssR0 (Hi<br />R+(M;N)n) is asymptotically<br />stable for n 􀀀! 􀀀1 and Hj<br />m0 (Hi<br />R+(M;N)) is Artinian for 0 j 1. More-<br />over, let s = sR+(M;N) be the largest integer i such that Hi<br />R+(M;N) is not<br />minimax. For each i s, we prove that R0<br />m0<br />R0Hi<br />R+(M;N) is Artinian and<br />that Hj<br />m0 (Hi<br />R+(M;N)) is Artinian for d 􀀀 1 j d. Finally we show that<br />Hd􀀀2<br />m0 (Hs<br />R+(M;N)) is Artinian if and only if Hd<br />m0 (Hs􀀀1<br />R+<br />(M;N)) is Artinian.