On the Artinianness of Graded Local Cohomology Modules

Abstract

Let R = ⨁n≥ 0 Rn be a homogeneous noetherian ring with local base ring [Formula: see text], and N a finitely generated graded R-module. Let [Formula: see text] be the i-th local cohomology module of N with respect to R+ := ⨁n > 0 Rn. Let t be the largest integer such that [Formula: see text] is not minimax. We prove that [Formula: see text] is [Formula: see text]-coartinian for any i > t, and [Formula: see text] is artinian. Let s be the first integer such that [Formula: see text] is not minimax. We show that for any i ≤ s, the graded module [Formula: see text] is artinian.