Tameness and Artinianness of Graded Generalized Local Cohomology Modules

Abstract

Let R=⊕n ≥ 0 Rn be a standard graded ring, 𝔞 ⊇ ⊕n > 0 Rn an ideal of R, and M, N two finitely generated graded R-modules. This paper studies the homogeneous components of graded generalized local cohomology modules. We show that for any i ≥ 0, the n-th graded component [Formula: see text] of the i-th generalized local cohomology module of M and N with respect to 𝔞 vanishes for all n ≫ 0. Some sufficient conditions are proposed to satisfy the equality [Formula: see text]. Also, some sufficient conditions are proposed for the tameness of [Formula: see text] such that [Formula: see text] or i= cd 𝔞(M,N), where [Formula: see text] and cd 𝔞(M,N) denote the R+-finiteness dimension and the cohomological dimension of M and N with respect to 𝔞, respectively. Finally, we consider the Artinian property of some submodules and quotient modules of [Formula: see text], where j is the first or last non-minimax level of [Formula: see text].