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It is known that graded cyclic modules over $S=K[x,y]$ have the WeakLefschetz Property (WLP). This is not true for non-cyclic modules over $S$. Thepurpose of this note is to study which conditions on $S$-modules ensure theWLP. We give an algorithm to test the WLP for graded modules with fixed Hilbertfunction. In particular, we prove that indecomposable graded modules over $S$with the Hilbert function $(h_0,h_1)$ have the WLP.

On the weak Lefschetz Property of graded modules over $K[x,y]$