Permanent URL to this publication: http://dx.doi.org/10.5167/uzh21932
Brodmann, M; Katzman, M; Sharp, R (2002). Associated primes of graded components of local cohomology modules. Transactions of the American Mathematical Society, 354(11):42614283 (electronic).

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Abstract
The ith local cohomology module of a ﬁnitely generated graded module M over a standard positively graded commutative Noetherian ring R, with respect to the irrelevant ideal R+ , is itself graded; all its graded components are ﬁnitely generated modules over R0 , the component of R of degree 0. It is known that the nth component Hi R+ (M )n of this local cohomology module Hi R+ (M ) is zero for all n >> 0. This paper is concerned with the asymptotic behaviour of AssR 0 (Hi R+ (M )n ) as n → −∞. The smallest i for which such study is interesting is the ﬁniteness dimension f of M relative to R+ , deﬁned as the least integer j for which Hj R+ (M ) is not ﬁnitely generated. Brodmann and Hellus have shown that AssR 0 (H f R+ (M )n ) is constant for all n << 0 (that is, in their terminology, AssR 0 (H f R+ (M )n ) is asymptotically stable for n → −∞). The ﬁrst main aim of this paper is to identify the ultimate constant value (under the mild assumption that R is a homomorphic image of a regular ring): our answer is precisely the set of contractions to R0 of certain relevant primes of R whose existence is conﬁrmed by Grothendieck’s Finiteness Theorem for local cohomology. Brodmann and Hellus raised various questions about such asymptotic behaviour when i > f . They noted that Singh’s study of a particular example (in which f = 2) shows that AssR 0 (H3 R+ (R)n ) need not be asymptotically stable for n → −∞. The second main aim of this paper is to determine, for Singh’s example, AssR 0 (H3 R+ (R)n ) quite precisely for every integer n, and, thereby, answer one of the questions raised by Brodmann and Hellus.
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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:  Graded commutative Noetherian ring, graded local cohomology module, associated prime ideal, ideal transform, regular ring, Gr\"{o}bner bases 
Language:  English 
Date:  2002 
Deposited On:  27 May 2010 15:59 
Last Modified:  28 Jan 2014 15:13 
Publisher:  American Mathematical Society 
ISSN:  00029947 
Additional Information:  First published in [Trans. Amer. Math. Soc. 354 (2002)], published by the American Mathematical Society 
Publisher DOI:  10.1090/S0002994702029872 
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