The weak Lefschetz property for ${\mathfrak{m}}$-full ideals and componentwise linear ideals

Abstract

We give a necessary and sufficient condition for a standard graded Artinian ring of the form $K[x_{1},\ldots,x_{n}]/I$, where $I$ is an ${\mathfrak{m}}$-full ideal, to have the weak Lefschetz property in terms of graded Betti numbers. This is a generalization of a theorem of Wiebe for componentwise linear ideals. We also prove that the class of componentwise linear ideals and that of completely ${\mathfrak{m}}$-full ideals coincide in characteristic zero and in positive characteristic, with the assumption that $\mathrm{Gin}(I)$ w.r.t. the graded reverse lexicographic order is stable.