Macaulay Constants and Vanishing of Cohomology

TL;DR

This paper introduces Macaulay constants for finitely generated graded modules, providing bounds on local cohomology degrees, Castelnuovo-Mumford regularity, and characterizing Hilbert polynomials, extending previous theories to modules and sheaves.

Contribution

It generalizes Macaulay constants to all finitely generated graded modules, linking them to cohomology bounds and regularity, and extends Gotzmann's theorem.

Findings

Macaulay constants give upper bounds for local cohomology degrees.

Bounds on Castelnuovo-Mumford regularity are established.

Characterization of Hilbert polynomials using Macaulay constants.

Abstract

Dub\'e introduced cone decompositions and their Macaulay constants and used them to obtain an upper bound on the degrees of the generators in a Gr\"obner basis of an ideal. Liang extended the theory to submodules of a free module. In this paper, Macaulay constants of any finitely generated graded module $M$ over a polynomial ring are introduced by adapting the concept of a cone decomposition to $M$. It is shown that these constants provide upper bounds for the degrees in which the local cohomology modules of $M$ are not zero. The results include an upper bound on the Castelnuovo-Mumford regularity of $M$ and a generalization of Gotzmann's Regularity Theorem from ideals to modules. As an application, an upper bound on the Castelnuovo-Mumford regularity of any coherent sheaf on projective space is established. The mentioned bounds are sharp even for cyclic modules. Furthermore, Macaulay constants are utilized to provide a characterization of Hilbert polynomials of finitely generated graded modules.