Green's theorem and Gorenstein sequences (Q2403982)

Let \(R = k[x_0,\dots,x_n]\) denote the polynomial ring with \(I \subset R\) a homogeneous ideal such that \(A = R/I\) is a graded Artinian \(k\)-algebra. For two positive integers \(r,i\) let \(r_{i}\) the \(i\)-th binomial expansion of \(r\) (see e.g [\textit{W. Bruns} and \textit{J. Herzog}, Cohen-Macaulay rings. Cambridge: Cambridge University Press (1993; Zbl 0788.13005)]) Then define \(r_{(i)}|^a_b\) as in ([loc.cit.] Section 4.2 and 4.3). Let \(h_d\) be the entry of degree \(d\) of the Hilbert function of \(R/I\) and let \(\ell_d\) be the degree \(d\) entry of the Hilbert function of \(R/(I, L)\), where \(L\) is a general linear form of \(R\). Then there are the following results: (1) Macaulay's Theorem \(h_{d+1} \leq ((h_d)_{(d)})|^1_1\). (2) Green's Hyperplane Restriction Theorem \(\ell_d \leq ((h_d)_{(d)})^{-1}_0\). The first interest of the authors is the the study of consequences of the extremality for Green's Theorem and an analysis of the equivalence between this extremality and that for Macaulay's Theorem. As a main result they generalize Green's Theorem 4 (see [\textit{M. Green}, Lect. Notes Math. 1389, 76--86 (1989; Zbl 0717.14002)]) to the case of a hypersurface in a linear space. This is applied to the study of Gorenstein sequences of socle degree \(4\). It is shown that \((1,19,17,19,1)\) is not a Gorenstein sequence and sequences of the form \((1,a,a-2,a,1)\) that are Gorenstein are classified. Moreover they study a certain numerical assumption as well as a certain Lefschetz assumption in order to conclude that the two different kinds of extremal behavior are equivalent.