Journal of Commutative Algebra

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

Winfried Bruns, Julio José Moyano-Fernández, and Jan Uliczka

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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.

Article information

J. Commut. Algebra, Volume 9, Number 2 (2017), 157-184.

First available in Project Euclid: 3 June 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
Secondary: 13D45: Local cohomology [See also 14B15]

Hilbert regularity boundary presentation of a rational function nonnegative power series Hilbert depth


Bruns, Winfried; Moyano-Fernández, Julio José; Uliczka, Jan. Hilbert regularity of $\mathbb Z$-graded modules over polynomial rings. J. Commut. Algebra 9 (2017), no. 2, 157--184. doi:10.1216/JCA-2017-9-2-157.

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