Hilbert regularity of $\mathbb Z$-graded modules over polynomial rings

Abstract

Let $M$ be a finitely generated $\mathbb{Z} $-graded module over the standard graded polynomial ring $R=K[X_1, \ldots , X_d]$ 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.