M. Brodmann,  C. Matteotti, and Nguyen Duc Minh

Abstract 
Let X be a projective scheme over an artinian commutative ring $R_0$.  Let $\Cal F$ be a coherent sheaf of $\Cal O_X$-modules.  We present a sample of bounding results for the so called cohomological Hilbert functions

\centerline{$h^i_{X, \Cal F} : \Bbb Z \to \Bbb N_0, n \mapsto h^i_{X, \Cal F} (n) = 
length_{R_0} H^i (X, \Cal F (n))$}

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 $\Cal F$. 
We also show that the cohomology diagonal of $\Cal F$ provides minimal bounding systems for the mentioned regularities and coregularities.
As a fundamental tool we use an extended version of the method of linear systems of general hyperplane sections.