# A bound on certain local cohomology modules and application to ample divisors

Albertini, C; Brodmann, M (2001). A bound on certain local cohomology modules and application to ample divisors. Nagoya Mathematical Journal, 163:87-106.

## Abstract

We consider a positively graded noetherian domain $R = \bigoplus_{n \in \BN_{0}} R_{n}$ for which $R_{0}$ is essentially of finite type over a perfect field $K$ of positive characteristic and we assume that the generic fibre of the natural morphism $\pi : Y = \Proj(R) \to Y_{0} = \Spec(R_{0})$ is geometrically connected, geometrically normal and of dimension $> 1$. Then we give bounds on the "ranks" of the $n$-th homogeneous part $H^{2}_{R_{+}} (R)_{n}$ of the second local cohomology module of $R$ with respect to $R_{+} := \bigoplus_{m > 0} R_{m}$ for $n < 0$. If $Y$ is in addition normal, we shall see that the $R_{0}$-modules $H^{2}_{R_{+}} (R)_{n}$ are torsion-free for all $n < 0$ and in this case our bounds on the ranks furnish a vanishing result. From these results we get bounds on the first cohomology of ample invertible sheaves in positive characteristic.

We consider a positively graded noetherian domain $R = \bigoplus_{n \in \BN_{0}} R_{n}$ for which $R_{0}$ is essentially of finite type over a perfect field $K$ of positive characteristic and we assume that the generic fibre of the natural morphism $\pi : Y = \Proj(R) \to Y_{0} = \Spec(R_{0})$ is geometrically connected, geometrically normal and of dimension $> 1$. Then we give bounds on the "ranks" of the $n$-th homogeneous part $H^{2}_{R_{+}} (R)_{n}$ of the second local cohomology module of $R$ with respect to $R_{+} := \bigoplus_{m > 0} R_{m}$ for $n < 0$. If $Y$ is in addition normal, we shall see that the $R_{0}$-modules $H^{2}_{R_{+}} (R)_{n}$ are torsion-free for all $n < 0$ and in this case our bounds on the ranks furnish a vanishing result. From these results we get bounds on the first cohomology of ample invertible sheaves in positive characteristic.

## Citations

2 citations in Web of Science®
2 citations in Scopus®

Detailed statistics

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Kodaira vanishing theorem; graded domain English 2001 27 May 2010 16:11 05 Apr 2016 13:25 Nagoya Daigaku 0027-7630 http://www.math.nagoya-u.ac.jp/en/journal/data/2001.html#163 http://projecteuclid.org/euclid.nmj/1114631622http://www.ams.org/mathscinet-getitem?mr=1854390http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1011.13011
Permanent URL: http://doi.org/10.5167/uzh-21985