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 $\mathbb K$isanalgebraicallyclosedfieldofcharacteristiczero\mathrm{.}Let$M = \oplus_j M_j$beafinitelengthgraded$R$-module\mathrm{.}Wesaythat$M$ has the Weak Lefschetz Property if there is a homogeneous element $L$ of degree one in $R$ such that the multiplication map $\times L:M_j\to M_{j+1}$ 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 $\mathcal{E}$, $H_{\ast }^1({\mathbb{P}}^2,\mathcal{E})$, has the Weak Lefschetz Property.