Let M be a finitely generated graded module over a Noetherian homogeneous ring R with local base ring (R0, m0). Then, the nth graded component HR+i (M)n of the ith local cohomology module of M with respect to the irrelevant ideal R+ of R is a finitely generated R0-module which vanishes for all n ≫ 0. In various situations we show that, for an m0-primary ideal q0 ⊆ R0, the multiplicity eq0 (HR+i (M)n) of HR+i (M)n) is antipolynomial in n of degree less than i. In particular we consider the following three cases: (a) i < g(M), where g(M) is the so-called cohomological finite length dimension of M; (b) i = g(M) (c) dim (R0) = 2. In cases (a) and (b) we express the degree and the leading coefficient of the representing polynomial in terms of local cohomological data of M (e.g. the sheaf induced by M) on Proj (R). We also show that the lengths of the graded components of various graded submodules of (HR+i (M) are antipolynomial of degree less than i and prove invariance results on these degrees. © 2004 Elsevier B.V. All rights reserved.

Brodmann, M; Rohrer, F; Sazeedeh, R (2005). *Multiplicities of graded components of local cohomology modules.* Journal of Pure and Applied Algebra, 197(1-3):249-278.

## Abstract

Let M be a finitely generated graded module over a Noetherian homogeneous ring R with local base ring (R0, m0). Then, the nth graded component HR+i (M)n of the ith local cohomology module of M with respect to the irrelevant ideal R+ of R is a finitely generated R0-module which vanishes for all n ≫ 0. In various situations we show that, for an m0-primary ideal q0 ⊆ R0, the multiplicity eq0 (HR+i (M)n) of HR+i (M)n) is antipolynomial in n of degree less than i. In particular we consider the following three cases: (a) i < g(M), where g(M) is the so-called cohomological finite length dimension of M; (b) i = g(M) (c) dim (R0) = 2. In cases (a) and (b) we express the degree and the leading coefficient of the representing polynomial in terms of local cohomological data of M (e.g. the sheaf induced by M) on Proj (R). We also show that the lengths of the graded components of various graded submodules of (HR+i (M) are antipolynomial of degree less than i and prove invariance results on these degrees. © 2004 Elsevier B.V. All rights reserved.

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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 |

Language: | English |

Date: | 2005 |

Deposited On: | 08 Feb 2010 09:23 |

Last Modified: | 05 Apr 2016 13:24 |

Publisher: | Elsevier |

ISSN: | 0022-4049 |

Free access at: | Related URL. An embargo period may apply. |

Publisher DOI: | https://doi.org/10.1016/j.jpaa.2004.08.034 |

Related URLs: | http://www.math.uzh.ch/fileadmin/math/preprints/07-04.pdf |

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