Evaluations of initial ideals and Castelnuovo-Mumford regularity (Q2781282)

Let \(S=k[x_1,\ldots, x_n]\) be a polynomial ring over a field \(k\) of arbitrary characteristic and let \(I\) be an arbitrary homogeneous ideal of \(S\). Buchberger's syzygy algorithm to compute reg\((S/I)\) is known, but the computations are often very big. From the theoretic point of view, if the characteristic of the field \(k\) is zero, reg\((S/I)\) is equal to the largest degree of the generators of the generic initial ideal of \(I\) with respect to the reverse lexicographic order, but is not easy to know when an initial ideal is generic. The simple method presented in this paper to compute reg\((S/I)\) which is based only on evaluations of in\((I)\), where in\((I)\) denotes the initial ideal of \(I\) with respect to the reverse lexicographic order is very interesting. This method is inspired by \textit{I. Bermejo} and \textit{P. Gimenez} [see Proc. Am. Math. Soc. 128, 1293--1299 (2000; Zbl 0944.13007)] which concerns the regularity of a saturated ideal \(I\) defining a projective curve. The author determines the regularity of \(S/I\) and the partial regularities of \(S/I\) which were introduced by \textit{D. Bayer}, \textit{H. Charalambous} and \textit{S. Popescu} [J. Algebra 221, 497--512 (1999; Zbl 0946.13008)] in terms of certain invariants \(c_i(I),\) \(i=0,\ldots, d\), and \(r(I)\), where \(d=\dim S/I,\) constructed from in\((I)\) by suitable evaluations. The numbers \(c_i(I)\) also allow us to determine the place at which reg\((S/I)\) is attained in the minimal free resolution of \(S/I\). The presented method is illustrated by an example.