Bounds for Cohomological Hilbert-Functions of Projective Schemes
Over Artinian Rings
M. Brodmann, C. Matteotti, and Nguyen Duc Minh Abstract
\centerline{$h^i_{X, \Cal F} : \Bbb Z \to \Bbb N_0, n \mapsto h^i_{X,
\Cal F} (n) =
of $\Cal F$. Our main interest is to bound these functions in
terms of the so called cohomology diagonal $(h^j_{X, \Cal F} (-j))^{\dim
(\Cal F)}_{j = 0}$ of $\Cal F$. Our results present themselves as
quantitative versions of the vanishing theorems of Castelnuovo-Serre and
of Severi-Enriques-Zariski-Serre. In particular we get polynomial
bounds for the (Castelnuovo) regularity at arbitrary levels and for the
(Severi) coregularity at any level below the global subdepth $\delta (\Cal
F) := \min{depth(\Cal F_x) | x \in X, x closed}$ of $\Cal F$.
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