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

AbstractLet 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 {p0∈AssR0(HiR+(M)n)∣height(p0)⩽1} 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→−∞