Macaulay Constants and Vanishing of Cohomology

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.