A note on graded generalized local cohomology modules

Abstract

Let R = ⊕_n∈N0R_n be a positively graded commutative Noetherian ring with irrelevant ideal R₊ = ⊕_n∈NR_n and let a be a graded ideal contained in R₊. Let M, N be two finitely generated graded R-modules. In this paper, we study the finiteness and vanishing of the n-th graded component Hⁱ_a (M;N)n of the i-th generalized local cohomology module of M and N with respect to a. Also, we show that the least i such that R₊ ⊊ √AnnR(Hⁱ_a (M;N)) is equal to the least integer i for which Hⁱ_a (M;N) is not finitely graded, where a graded module is finitely graded which is non-zero in only finitely many graded pieces.

Keywords: Local cohomology modules, generalized local cohomology modules, graded modules