On the weak Lefschetz property for vector bundles on [formula omitted].

Let R = K [ x , y , z ] be a standard graded polynomial ring where K is an algebraically closed field of characteristic zero. Let M = ⊕ 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 × L : M j → M j + 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 P 2 then the first cohomology module of E , H ⁎ 1 (P 2 , E) , has the Weak Lefschetz Property. [ABSTRACT FROM AUTHOR]