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

Abstract

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.

Some properties of graded local cohomology modules

“…When R 0 is semilocal of dimension at most 2, Brodmann, Fumasoli, and Lim [1] proved that H i R + (M) is tame for all i. Assuming that M is Cohen-Macaulay, we eliminate the condition that R 0 is semilocal.…”

Asymptotic behaviour of cohomology: tameness, supports and associated primes

“…A) Let M be a finitely generated graded R-module and let p ∈ Proj(R). We define the (R + • In [2,Theorem 3.7] it is shown that "Asymptotic Stability of Associated Primes" holds "in codimension ≤ 1" if the base ring R 0 is essentially of finite type over a field. We now shall prove that under certain additional assumptions "Asymptotic Stability of Associated Primes holds in codimension ≤ 2".…”

“…Therefore, the set Ass [2,Corollary (4.8)] for all p 0 ∈ S. So, by the local flat base change property, the set Ass …”

Asymptotic Behaviour and Artinian Property of Graded Local Cohomology Modules

“…Now, according to [5, 15.1.5], for each i ≥ 0, the R 0 -module H i R + (M) n is finitely generated for all n ∈ Z and vanishes for all sufficiently large values of n. The cohomological dimension of M with respect to R + is denoted by cd(R + , M). Thus cd(R + , M) is the largest non-negative integer i such that (M) n when n → −∞ constitute lots of interest (see for example [1], [2], [3], [4] and [7]). As a basic reference in this topic we recommend [1].…”