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

Let R = circle plus(ngreater than or equal to0)R(n) be a homogeneous noetherian ring and let M be a finitely generated graded R-module. Let H-R+(i) (M) denote the i th local cohomology module of M with respect to the irrelevant ideal R+ := circle plus(n>0)R(n) of R. We show that if R-0 is a domain, there is some s is an element of R-0\{0} such that the (R-0)(s)-modules H-R+(i) (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+(i) (M)(n) of H-R+(i)(M) for n -->-infinity: if R-0 is a domain or essentially of finite type over a field, the set {p(0) is an element of Ass(R0)(H-R+(i)(M)(n)) \ height(p(0)) less than or equal to 1} is asymptotically stable for n -->-infinity. If R-0 is semilocal and of dimension 2, the modules H-R+(i)(M) are tame. If R-0 is in addition a domain or essentially of finite type over a field, the set AssR(0)(H-R+(i)(M)(n)) is asymptotically stable for n-->-infinity. (C) 2004 Elsevier Inc. All rights reserved.