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

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