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 $\ch{k}=0$ and the number of the minimal generators of $0:_A\ell$ over $k[x_1,x_2,x_3]$ is two.