Monomial Ideals, Almost Complete Intersections and the Weak Lefschetz Property

has maximal rank, i.e. it is injective or surjective. In this case, the linear form L is called a Lefschetz element of A. (We will often abuse notation and say that the corresponding ideal has the WLP.) The Lefschetz elements of A form a Zariski open, possibly empty, subset of (A)1. Part of the great interest in the WLP stems from the fact that its presence puts severe constraints on the possible Hilbert functions (see [6]), which can appear in various disguises (see, e.g., [12]). Though many algebras are expected to have the WLP, establishing this property is often rather difficult. For example, it is open whether every complete intersection of height four over a field of characteristic zero has the WLP. (This is true if the height is at most 3 by [6].) In some sense, this note presents a case study of the WLP for monomial ideals and almost complete intersections. Our results illustrate how subtle the WLP is. In particular, we investigate its dependence on the characteristic of the ground field K. The following example (Example 7.7) illustrates the surprising effect that the characteristic can have on the WLP. Consider the ideal I = (x, y, z, xyz) ⊂ R = K[x, y, z]. Our methods show that R/I fails to have the WLP in characteristics 2, 3 and 11, but possesses it in all other characteristics. One starting point of this paper has been Example 3.1 in [4], where Brenner and Kaid show that, over an algebraically closed field of characteristic zero, any ideal of the form (x, y, z, f(x, y, z)), with deg f = 3, fails to have the WLP if and only if