Let X be a projective scheme over a field K and let F be a coherent sheaf of OX-modules. We show that the cohomological postulation numbers νFi of F, e.g., the ultimate places at which the cohomological Hilbert functions n dimK (Hi (X, F(n))) =: hFi (n) start to be polynomial for n ≪ 0, are (polynomially) bounded in terms of the cohomology diagonal (hFi (-i) i=0dim(F) of F. As a consequence, we obtain that there are only finitely many different cohomological Hilbert functions hFi if F runs through all coherent sheaves of OX-modules with fixed cohomology diagonal. In order to prove these results, we extend the regularity bound of Bayer and Mumford [Computational Algebraic Geometry and Commutative Algebra, Proc. Cortona, 1991, Cambridge Univ. Press, 1993, pp. 1-48] from graded ideals to graded modules. Moreover, we prove that the Castelnuovo-Mumford regularity of the dual FV of a coherent sheaf of OℙrK, -modules F is (polynomially) bounded in terms of the cohomology diagonal of F. © 2003 Published by Elsevier Inc.

Brodmann, M; Lashgari, A (2003). *A diagonal bound for cohomological postulation numbers of projective schemes.* Journal of Algebra, 265(2):631-650.

## Abstract

Let X be a projective scheme over a field K and let F be a coherent sheaf of OX-modules. We show that the cohomological postulation numbers νFi of F, e.g., the ultimate places at which the cohomological Hilbert functions n dimK (Hi (X, F(n))) =: hFi (n) start to be polynomial for n ≪ 0, are (polynomially) bounded in terms of the cohomology diagonal (hFi (-i) i=0dim(F) of F. As a consequence, we obtain that there are only finitely many different cohomological Hilbert functions hFi if F runs through all coherent sheaves of OX-modules with fixed cohomology diagonal. In order to prove these results, we extend the regularity bound of Bayer and Mumford [Computational Algebraic Geometry and Commutative Algebra, Proc. Cortona, 1991, Cambridge Univ. Press, 1993, pp. 1-48] from graded ideals to graded modules. Moreover, we prove that the Castelnuovo-Mumford regularity of the dual FV of a coherent sheaf of OℙrK, -modules F is (polynomially) bounded in terms of the cohomology diagonal of F. © 2003 Published by Elsevier Inc.

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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: | Cohomology of projective schemes; Cohomological Hilbert functions; Cohomological postulation numbers; Castelnuovo–Mumford regularity |

Language: | English |

Date: | 2003 |

Deposited On: | 27 May 2010 08:35 |

Last Modified: | 05 Apr 2016 13:24 |

Publisher: | Elsevier |

ISSN: | 0021-8693 |

Publisher DOI: | https://doi.org/10.1016/S0021-8693(03)00234-5 |

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