Let R = n 0 R n be a homogeneous noetherian ring and let M be a finitely generated graded R-module. Let H i R + (M) denote the ith local cohomology module of M with respect to the irrelevant ideal R + := n>0 R n of R. We show that if R 0 is a domain, there is some s ∈ R 0 \{0} such that the (R 0 ) s -modules H i 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 nth graded component H i R + (M) n of H i R + (M) for n → −∞: if R 0 is a domain or essentially of finite type over a field, the set {p 0 ∈ Ass R 0 (H i R + (M) n ) | height(p 0 ) 1} is asymptotically stable for n → −∞. If R 0 is semilocal and of dimension 2, the modules H i R + (M) are tame. If R 0 is in addition a domain or essentially of finite type over a field, the set Ass R 0 (H i R + (M) n ) is asymptotically stable for n → −∞. 2004 Elsevier Inc. All rights reserved.
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