The weak Lefschetz property for Artinian graded rings and basic sequences

Abstract

The basic sequence of a homogeneous ideal $I\sset R=k[\seq{x}{1}{r}]$ defining an Artinian graded ring $A=R/I$ not having the weak Lefschetz property has the property that the first term of the last part is less than the last term of the penultimate part. For a general linear form $\ell$ in $\seq{x}{1}{r}$, this fact affects in a certain way the behavior of the $r-1$ square matrices in $k[\ell ]$ which represent the multiplications of the elements of $A$ by $\seq{x}{1}{r-1}$ through a minimal free presentation of $A$ over $k[\ell ]$. Taking advantage of it, we consider some modules over an algebra generated over $k[\ell ]$ by the square matrices mentioned above. In this manner, for the case $r=3$, we prove that an Artinian \Gor\ graded ring $A=k[x_1,x_2,x_3]/I$ has the weak Lefschetz property if $\mathrm{ch}k=0$ and the number of the minimal generators of $0{:}_A\ell$ over $k[x_1,x_2,x_3]$ is two.

Cited by in corpus (2)

• A tour of the Weak and Strong Lefschetz Properties
• On the Weak-Lefschetz property for Artinian Gorenstein algebras