Graded local cohomology: Attached and associated primes, asymptotic behaviors

Assume that R = circle plus(i epsilon N0) R-i is a homogeneous graded Noetherian ring, and that M is a Z-graded R-module, where N-0 (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, H-R+(c) (M), with respect to the irrelevant ideal R+ := circle plus(i >= 1) R-i and the set of associated primes of H-R+(i)(M) is studied. The asymptotic behavior of Hom(R) (R/R+, H-R+(s) (M)) for s >= f (M) is discussed, where f (M) is the finiteness dimension of M. It is shown that H-R+(h) (M) is tame if H-R+(i) (M) is Artinian for all i > h.