# Bounds for cohomological Hilbert-functions of projective schemes over Artinian rings

Brodmann, M; Matteotti, C; Nguyen, D M (2000). Bounds for cohomological Hilbert-functions of projective schemes over Artinian rings. Vietnam Journal of Mathematics, 28(4):341-380.

## 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.

## 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.

## Statistics

### Citations

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics cohomological Hilbert functions; Castelnuovo bounds; Severi bounds English 2000 31 May 2010 15:25 06 Dec 2017 20:54 Springer 2305-221X http://www.math.ac.vn/publications/vjm/vjm_28/341.html http://www.ams.org/mathscinet-getitem?mr=1810157http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1008.13004