Bounds for Cohomological HilbertFunctions 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 CastelnuovoSerre and
of SeveriEnriquesZariskiSerre. 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 $\Cal F$.
