A Problem of Local Cohomology Modules

Let R = ⨁ n∈ℕ 0 R n be a positively graded Noetherian commutative ring. Set R +: = ⨁ n∈ℕ R n . Let N = ⨁ n∈ℤ N n be a nonzero finitely generated graded R-module. Here, ℕ 0, ℕ, and ℤ denote the set of non-negative, positive, and all integers, respectively. Let (P) denote the properties: (i) for all i ∈ ℕ 0 and all n ∈ ℤ, the R 0-module is finitely generated; (ii) for all i ∈ ℕ 0, , where end(T) = Sup{r | T r  ≠ 0}. In this note, we study the following question: For a graded ideal I contained in R +, when does have the properties (P)? Further, we study the tameness of .