Remarks on Hilbert series of graded modules over polynomial rings
type: Item
instance of: scholarly article
title: Remarks on Hilbert series of graded modules over polynomial rings
zbMATH Open document ID: 1198.13016
author: Jan Uliczka
published in: Manuscripta Mathematica
publication date: 2010-04-30T00:00:00Z
review text: An integer Laurent function \(f\) is said to be of polynomial type \(d\) if there is a polynomial \(P\in\mathbb{Z}[t]\) of degree \(d\) such that \(f(t)=P(t)\) for \(t\gg 0\); \(f\in\mathbb{Z}[t,t^{-1}]\) is called \textit{non-negative} if all its coefficients are non-negative. The technical heart of the paper under review is the following result: If an integer Laurent function \(f\) is of polynomial type \(d-1\) such that \(f(n)\geq 0\) for all \(n\in\mathbb{Z}\), then there exist non-negative \(Q_0,\dots,Q_d\in\mathbb{Z}[t,t^{-1}]\) such that the associated Laurent series \(H_f\), which is defined by \(H_f(t)=\sum_{k\in\mathbb{Z}}f(k)t^k\), can be written as \[ H_f(t)=\sum^d_{j=0}\frac{Q_j(t)}{(1-t)^j}. \] An immediate consequence of this result is a characterization of Hilbert functions of finitely generated graded modules over the polynomial ring \(R=\mathbb{K}[x_1,\dots,x_m]\) with the standard grading over a field \(\mathbb{K}\): An integer Laurent function \(f\) is the Hilbert function of a finitely generated graded \(R\)-module if and only if \(f\) is of polynomial type such that \(f(n)\geq 0\) for all \(n\in\mathbb{Z}\). The paper under review also proves a characterization of the numerator of a Hilbert series: Let \(f\in\mathbb{Z}[t,t^{-1}]\) be a nonzero Laurent polynomial. Then there exist a finitely generated graded \(R\)-module M and a natural number \(d\) such that the Hilbert series \(H_M\) of \(M\) can be written as \(H_M(t)=\frac{f(t)}{(1-t)^d}\) if and only if \(f(x)>0\) for all \(x\in(0,1)\). The maximal depth of a finitely generated graded \(R\)-module with a given Hilbert series is also discussed.
reviewed by: Wenliang Zhang
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 5702988
zbMATH Keywords: integer Laurent function
zbMATH Keywords: Hilbert series
MaRDI profile type: Publication
full work available at URL: s00229-010-0341-9
OpenAlex ID: W2081635100
cites work: Q5317677
cites work: Hilbert functions of graded algebras
DOI: 10.1007/S00229-010-0341-9
Recommended article: Hilbert series of modules over positively graded polynomial rings
Recommended article: Hilbert series and applications to graded rings
Recommended article: Hilbert regularity of \(\mathbb Z\)-graded modules over polynomial rings
Recommended article: On Hilbert series for commutative and noncommutative graded algebras
Recommended article: Hilbert coefficients and the associated graded rings
Recommended article: Graded algebras with cyclotomic Hilbert series
Recommended article: Hilbert depth of graded modules over polynomial rings in two variables
Recommended article: Multigraded Hilbert series of noncommutative modules
Recommended article: Graded Complexes Over Power Series Rings
Recommended article: Hilbert functions of graded algebras over Artinian rings
description: scientific article; zbMATH DE number 5702988