HILBERT REGULARITY OF Z-GRADED MODULES OVER POLYNOMIAL RINGS

Abstract

Let M be a finitely generated Z-graded module over the standard graded polynomial ring R = K[X-1, ... ,X-d] with K a field, and let H-M(t) = Q(M)(t)/(1 - t)(d) be the Hilbert series of M. We introduce the Hilbert regularity of M as the lowest possible value of the Castelnuovo-Mumford regularity for an R-module with Hilbert series H-M. Our main result is an arithmetical description of this invariant which connects the Hilbert regularity of M to the smallest k such that the power series Q(M)(1 - t)/(1 - t)(k) has no negative coefficients. Finally, we give an algorithm for the computation of the Hilbert regularity and the Hilbert depth of an R-module.