On the Weak Lefschetz Property for Vector Bundles on $\mathbb P^2$

Let $R=mathbb K[x,y,z]$ be a standard graded polynomial ring where $mathbb K$ is an algebraically closed field of characteristic zero. Let $M = oplus_j M_j$ be a finite length graded $R$-module. We say that $M$ has the Weak Lefschetz Property if there is a homogeneous element $L$ of degree one in $R$ such that the multiplication map $times L : M_j rightarrow M_{j+1}$ has maximal rank for every $j$. The main result of this paper is to show that if $mathcal E$ is a locally free sheaf of rank 2 on $mathbb P^2$ then the first cohomology module of $mathcal E$, $H^1_*(mathbb P^2, mathcal E)$, has the Weak Lefschetz Property.