The weak Lefschetz property for Artinian graded rings and basic sequences

Details The weak Lefschetz property for Artinian graded rings and basic sequences The weak Lefschetz property for Artinian graded rings and basic sequences

2011-09-12

arxiv.org/abs/1109.2365v1

Mathematics

Commutative Algebra

Scientific paper

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.

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