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

Abstract

Let R = ⊕n≥0 Rn be a homogeneous noetherian ring and let M be a finitely generated graded R-module. Let HR+i (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 HR+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 nth graded component HR+i (M)n of HR+i (M) for n → ∞: if R0 is a domain or essentially of finite type over a field, the set {p0 ∈ AssR0 (HR+i (M n) height(p0) ≤ 1} is asymptotically stable for n → -∞. If R0 is semilocal and of dimension 2, the modules HR+i (M) are tame. If R0 is in addition a domain or essentially of finite type over a field, the set AssR0 (HR+i (M)n) is asymptotically stable for n → -∞. © 2004 Elsevier Inc. All rights reserved.