Local cohomology modules with respect to an ideal containing the irrelevant ideal

Let R=⨁n≥0RnR=⨁n≥0Rn be a homogeneous Noetherian ring, let MM be a finitely generated graded RR-module and let R+=⨁n>0RnR+=⨁n>0Rn. Let b≔b0+R+b≔b0+R+, where b0b0 is an ideal of R0R0. In this paper, we first study the finiteness and vanishing of the nn-th graded component Hbi(M)n of the ii-th local cohomology module of MM with respect to bb. Then, among other things, we show that the set AssR0(Hbi(M)n) becomes ultimately constant, as n→−∞n→−∞, in the following cases: (i)dim(R0b0)≤1 and (R0,m0)(R0,m0) is a local ring;(ii)dim(R0)≤1dim(R0)≤1 and R0R0 is either a finite integral extension of a domain or essentially of finite type over a field;(iii)i≤gb(M)i≤gb(M), where gb(M)gb(M) denotes the cohomological finite length dimension of MM with respect to bb. Also, we establish some results about the Artinian property of certain submodules and quotient modules of Hbi(M).