Graded Local Cohomology: Attached and Associated Primes, Asymptotic Behaviors

Assume that R = ⨁ i∈ℕ0 R i is a homogeneous graded Noetherian ring, and that M is a ℤ-graded R-module, where ℕ0 (resp. ℤ) denote the set all non-negative integers (resp., integers). The set of all homogeneous attached prime ideals of the top nonvanishing local cohomology module of a finitely generated module M, , with respect to the irrelevant ideal R +: =⨁ i≥1 R i and the set of associated primes of is studied. The asymptotic behavior of for s ≥ f(M) is discussed, where f(M) is the finiteness dimension of M. It is shown that is tame if is Artinian for all i > h.