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 H R+(M) denote the i-th 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 H R+(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 n-th graded component H R+(M)n of H i R+ (M) for n→ −∞: If R0 is a domain or essentially of finite type over a field, the set {p0 ∈ AssR0 ( H R+(M)n ) | height(p0) ≤ 1} is asymptotically stable for n→ −∞. If R0 is semilocal and of dimension 2, the modules H R+(M) are tame. If R0 is in addition a domain or essentially of finite type over a field, the set AssR0 ( H R+(M)n ) is asymptotically stable for n→ −∞.