Low-codimensional associated primes of graded components of local cohomology modules

Let R= n 0Rn be a homogeneous noetherian ring and let M be a finitely generated graded R-module. Let HiR+(M) denote the ith local cohomology module of M with respect to the irrelevant ideal R+:= n>0Rn of R. We show that if R0 is a domain, there is some s R0 {0} such that the (R0)s-modules HiR+(M)s are torsion-free (or vanishing) for all i. On use of this, we can deduce the following results on the asymptotic behaviour of the nth graded component HiR+(M)n of HiR+(M) for n→−∞: if R0 is a domain or essentially of finite type over a field, the set is asymptotically stable for n→−∞. If R0 is semilocal and of dimension 2, the modules HiR+(M) are tame. If R0 is in addition a domain or essentially of finite type over a field, the set AssR0(HiR+(M)n) is asymptotically stable for n→−∞.