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[X1,…,Xd] with K a field, and let HM(t)=QM(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 HM. 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 QM(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.