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.

Abstract. Let R = n≥0 Rn be a homogeneous Noetherian ring with local base ring (R 0 , m 0 ) and let M be a finitely generated graded R-module. Let

Let Hilb p be the Hilbert scheme parametrizing the closed subschemes of P n K with Hilbert polynomial p ∈ Q[t] over a field K of characteristic zero. By bounding below the cohomological Hilbert functions of the points of Hilb p we define locally closed subspaces of the Hilbert scheme. The aim of this paper is to show that some of these subspaces are connected. For this we exploit the edge ideals constructed by D. Mall in [D. Mall, Connectedness of Hilbert function strata and other connectedness results, J. Pure Appl. Algebra 150 (2000) . It turns out that these ideals are sequentially Cohen-Macaulay and that their initial ideals with respect to the reverse lexicographic term order are generic initial ideals.