Hilbert regularity of \(\mathbb Z\)-graded modules over polynomial rings (Q2396354)

Let \(M\) be a finitely generated \(d\)-dimensional \(\mathbb{Z}\)-graded module over the standard graded polynomial ring \(R = K[X_1,\dots,X_d]\) with \(K\) a field, and let \(H_M(t) = Q_M(t)/(1-t)^d\) be the Hilbert series of \(M\). The paper under review may be considered as part of a program that aims at estimating numerical invariants of a graded module \(M\) over a polynomial ring \(R\) in terms of the Hilbert series \(H_M(t)\). Well-known examples of such estimates are the bounds of \textit{A. M. Bigatti} [Commun. Algebra 21, No. 7, 2317--2334 (1993; Zbl 0817.13007)] and \textit{H. A. Hulett} [Commun. Algebra 21, No. 7, 2335--2350 (1993; Zbl 0817.13006)] on the Betti numbers or the bounds of \textit{J. Elias} et al. [Nagoya Math. J. 123, 39--76 (1991; Zbl 0714.13016)] on the number of generators for ideals primary to \(\mathfrak m = (X_1,\dots,X_d)\). A more recent result is the upper bound on \(\mathrm{Depth }M\) (or, equivalently, a lower bound on \(\operatorname{projdim} M\)) given by the third author [Manuscr. Math. 132, No. 1--2, 159--168 (2010; Zbl 1198.13016)], namely, the \textit{Hilbert depth} \(\mathrm{Hdepth }M\). It is defined as the maximum value of \(\mathrm{Depth }N\) for a module \(N\) with \(H_M(t) = H_N(t)\). The authors introduce the Hilbert regularity of \(M\) as the lowest possible value of the Castelnuovo-Mumford regularity for an \(R\)-module with Hilbert series \(H_M\). They show that Hilbert regularity of \(M\) is the smallest \(k\) such that the power series \(Q_M(1-t)/(1-t)^k\) has no negative coefficients, see Theorems 4.7 and 4.10. Their main tool for the analysis of Hilbert series is \[ H_M(t)=\sum_{i=0}^{k-1}\frac{f_it^i}{(1-t)^n}+\frac{ct^k}{(1-t)^n}+\sum_{j=0}^{d-n-1}\frac{g_{j}t^k}{(1-t)^{d-j}} \] which they call \((n,k)\)-\textit{boundary presentations} since the pairs of exponents \((u, v)\) occurring in the numerator and denominator of the terms \(t^i/ (1-t)^n\), \(t^k/(1-t)^n\), and \(t^k/(1-t)^{d-j}\) occupy the lower and the right boundary of a rectangle in the \(u\)-\(v\)-plane whose right lower corner is \((k, n)\). Finally, they give an algorithm for the computation of the Hilbert regularity and the Hilbert depth of an \(R\)-module.