Abstract
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, H⁎1(P2,E), has the Weak Lefschetz Property.
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