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 , . . . , xn]/I, where I is an 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 m-full ideals coincide in characteristic zero and in positive characteristic, with the assumption that Gin(I) w.r.t. the graded reverse lexicographic order is stable.

Completely -full ideals and componentwise linear ideals

“…The following is the main theorem. The "if" part was proved in Proposition 18 of [6]. So we will show the "only if" part.…”

“…, x 1 ) has the complete m-full property.Proof. The "if" part follows from Example 17 in[6]. So we show the "only if" part.…”

“…These ideals are two important classes of ideals having various interesting properties. In [6] the authors proved that these notion are equivalent in the class of graded ideals provided that their generic initial ideals with respect to the graded reverse lexicographic order are stable, and further conjectured that these notions are equivalent without adding the assumption on generic initial ideals. The purpose of this paper is to prove that the conjecture is true.…”