Artinian level modules of embedding dimension two

We prove that a sequence of positive integers (h0,h1,…,hc) is the Hilbert function of an artinian level module of embedding dimension two if and only if hi−1−2hi+hi+1≤0 for all 0≤i≤c, where we assume that h−1=hc+1=0. This generalizes a result already known for artinian level algebras. We provide two proofs, one using a deformation argument, the other a construction with monomial ideals. We also discuss liftings of artinian modules to modules of dimension one.