Bounds for cohomological Hilbert-functions of projective schemes over Artinian rings. (English) Zbl 1008.13004

Given a projective scheme \(X\) over an Artin commutative ring and a coherent sheaf \(F\) on that scheme, one defines the \(i\)th cohomological Hilbert function of an integer argument \(n\) measuring the length of the \(i\)th cohomology module of \(X\) with coefficients in the \(n\)th twist of \(F\). Because of the various vanishing theorems, like those of Grothendieck, Castelnuovo-Serre, and Severi-Enriques-Zariski-Serre, there are obvious constraints on the values of the cohomological Hilbert functions. There are also constraints on the rate of growth of those functions. The goal of the paper is to find further bounds on those functions, which would only depend on their diagonal values. The authors find that such bounds naturally break into two groups: bounds of Castelnuovo type and bounds of Severi type. Those bounds are given in terms of recursively defined bounding functions. The results are first established for finitely generated graded modules and then transferred to sheaves using the Serre - Grothendieck correspondence.


13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14B15 Local cohomology and algebraic geometry
13D45 Local cohomology and commutative rings
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)