Remarks on Hilbert series of graded modules over polynomial rings
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.
scientific article; zbMATH DE number 5702988
scientific article