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Associated primes of graded components of local cohomology modules

Brodmann, Markus P., Katzman, Mordechai and Sharp, Rodney Y. (2002) Associated primes of graded components of local cohomology modules. Transactions of the American Mathematical Society, 354 (11). pp. 4261-4283. ISSN 0002-9947

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Abstract

The i-th 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 R-0, the component of R of degree 0. It is known that the n-th component H-R+(i) (M)(n) of this local cohomology module H-R+(i) (M) is zero for all nmuch greater than0. This paper is concerned with the asymptotic behaviour of Ass(R0)(H-R+(i) (M)(n)) as n--> -infinity.

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 H-R+(j) (M) is not finitely generated. Brodmann and Hellus have shown that AssR(0)(H-R+(f) (M)(n)) is constant for all nmuch less than0 ( that is in their terminology AssR(0)(H-R+(f) (M)(n)) is asymptotically stable for n--> -infinity). 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 R-0 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)(H-R+(3) (R)(n)) need not be asymptotically stable for n--> -infinity. The second main aim of this paper is to determine, for Singh's example, AssR(0)(H-R+(3) (R)(n)) quite precisely for every integer n and, thereby answer one of the questions raised by Brodmann and Hellus.

Item Type: Article
Copyright, Publisher and Additional Information: © 2002 American Mathematical Society. Reproduced in accordance with the publisher's self-archiving policy.
Keywords: graded commutative Noetherian ring; graded local cohomology module; associated prime ideal; ideal transform; regular ring; Grobner bases
Institution: The University of Sheffield
Academic Units: The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield)
Depositing User: Miss Anthea Tucker
Date Deposited: 18 Nov 2009 12:20
Last Modified: 10 Jun 2014 00:49
Published Version: http://www.jstor.org/stable/3072898
Status: Published
Publisher: American Mathematical Society
Refereed: Yes
URI: http://eprints.whiterose.ac.uk/id/eprint/10175

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