The strong Lefschetz property for quadratic reverse lexicographic ideals (Q6572993)

Let \(R = k[x_1,\dots,x_n]\), where \(k\) is a field of characteristic zero. Let \(A = R/I\) be a standard graded artinian algebra. Specifically, the author considers ideals of the form \(I = (x_1^2,\dots,x_n^2) + \hbox{RLex}(x_ix_j)\), where \(\hbox{RLex}(x_i x_j)\) is the ideal generated by all squarefree monomials which are greater than or equal to \(x_i x_j\) in the reverse lexicographic order. The author is interested in the shape of the Hilbert function of \(R/I\) and especially in the question of whether \(R/I\) has the so-called Strong Lefschetz Property (SLP). Recall that the latter means that for each \(i\) and each \(j\), there exists a linear form \(\ell\) so that the homomorphism \(\times \ell^i : [A]_j \rightarrow [A]_{j+i}\) has maximal rank. The main result of this paper is that such an algebra does indeed have the SLP. In fact, the author gives a slightly more general result by allowing the characteristic to be positive under some restrictions. As a consequence, he derives some properties of the Hilbert function of such an algebra. His approach is by thinking of \(I\) in terms of the graph associated to it. An important tool is a theorem of Lindsey about the behavior of SLP under certain tensor products. Another consequence is that for any possible number of minimal generators for an artinian quadratic ideal there exists such an ideal minimally generated by that many monomials and defining an algebra with the SLP.