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 R0 , the component of R of degree 0. It is known that the n-th 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.

Brodmann, M; Katzman, M; Sharp, R (2002). *Associated primes of graded components of local cohomology modules.* Transactions of the American Mathematical Society, 354(11):4261-4283 (electronic).

## 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 R0 , the component of R of degree 0. It is known that the n-th 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 |
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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: | 05 Apr 2016 13:25 |

Publisher: | American Mathematical Society |

ISSN: | 0002-9947 |

Additional Information: | First published in [Trans. Amer. Math. Soc. 354 (2002)], published by the American Mathematical Society |

Publisher DOI: | https://doi.org/10.1090/S0002-9947-02-02987-2 |

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