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 |
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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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