# A cohomological stability result for projective schemes over surfaces - Zurich Open Repository and Archive

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 −∞.

## 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

3 citations in Web of Science®
4 citations in Scopus®

## Altmetrics

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English June 2007 10 Apr 2009 11:54 05 Apr 2016 13:12 De Gruyter 0075-4102 Related URL. An embargo period may apply. https://doi.org/10.1515/CRELLE.2007.040 http://www.math.uzh.ch/fileadmin/math/preprints/22-05.pdf (Author)