Nagoya Mathematical Journal

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

Claudia Albertini and Markus Brodmann

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We consider a positively graded noetherian domain $R = \bigoplus_{n \in \mathbb{N}_{0}} R_{n}$ 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 $\pi : Y = \rm{Proj}(R) \to Y_{0} = \rm{Spec}(R_{0})$ 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.

Article information

Nagoya Math. J., Volume 163 (2001), 87-106.

First available in Project Euclid: 27 April 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14B15: Local cohomology [See also 13D45, 32C36]
Secondary: 13D45: Local cohomology [See also 14B15]


Albertini, Claudia; Brodmann, Markus. A bound on certain local cohomology modules and application to ample divisors. Nagoya Math. J. 163 (2001), 87--106.

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