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


Brodmann, M; Fumasoli, S; Lim, C (2004). Low-codimensional associated primes of graded components of local cohomology modules. Journal of Algebra, 275(2):867-882.

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.

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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:homogeneous noetherian ring; Local cohomology modules; Graded components; Associated primes
Language:English
Date:2004
Deposited On:27 May 2010 07:05
Last Modified:06 Dec 2017 20:47
Publisher:Elsevier
ISSN:0021-8693
Publisher DOI:https://doi.org/10.1016/j.jalgebra.2003.12.003
Related URLs:http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1086.13006

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