The weak Lefschetz property for vector bundle on P^2

Let R = K[x, y, z] be a standard graded polynomial ring where K is an algebraically closed field of characteristic zero. Let M = ⊕jMj 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 ×L : Mj → Mj+1 has maximal rank for every j. The main result of this paper is to show that if E is a locally free sheaf of rank 2 on P2 then the first cohomology module of E, H1 ∗ (P2, E), has the Weak Lefschetz Property