Let π : X → X0 be a projective morphism of schemes such that X0 is noetherian and essentially of finite type over a field K. Let i N0, let F be a coherent sheaf of -modules and let L be an ample invertible sheaf over X. We show that the set of associated points of the higher direct image sheaf ultimately becomes constant if n tends to −∞, provided X0 has dimensione 2. If , this stability result need not hold any more.

To prove this, we show that the set of associated primes of the n-th graded component of the i-th local cohomology module of a finitely generated graded module M over a homogeneous noetherian ring which is essentially of finite type over a field becomes ultimately constant in codimension 2 if n tends to −∞.

Brodmann, M (2007). *A cohomological stability result for projective schemes over surfaces.* Journal für die Reine und Angewandte Mathematik, 606:179-192.

## Abstract

Let π : X → X0 be a projective morphism of schemes such that X0 is noetherian and essentially of finite type over a field K. Let i N0, let F be a coherent sheaf of -modules and let L be an ample invertible sheaf over X. We show that the set of associated points of the higher direct image sheaf ultimately becomes constant if n tends to −∞, provided X0 has dimensione 2. If , this stability result need not hold any more.

To prove this, we show that the set of associated primes of the n-th graded component of the i-th local cohomology module of a finitely generated graded module M over a homogeneous noetherian ring which is essentially of finite type over a field becomes ultimately constant in codimension 2 if n tends to −∞.

## Citations

## Altmetrics

## Additional indexing

Item Type: | Journal Article, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | June 2007 |

Deposited On: | 10 Apr 2009 11:54 |

Last Modified: | 05 Apr 2016 13:12 |

Publisher: | De Gruyter |

ISSN: | 0075-4102 |

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

Publisher DOI: | https://doi.org/10.1515/CRELLE.2007.040 |

Related URLs: | http://www.math.uzh.ch/fileadmin/math/preprints/22-05.pdf (Author) |

## Download

Full text not available from this repository.View at publisher

TrendTerms displays relevant terms of the abstract of this publication and related documents on a map. The terms and their relations were extracted from ZORA using word statistics. Their timelines are taken from ZORA as well. The bubble size of a term is proportional to the number of documents where the term occurs. Red, orange, yellow and green colors are used for terms that occur in the current document; red indicates high interlinkedness of a term with other terms, orange, yellow and green decreasing interlinkedness. Blue is used for terms that have a relation with the terms in this document, but occur in other documents.

You can navigate and zoom the map. Mouse-hovering a term displays its timeline, clicking it yields the associated documents.