Artinian level modules of embedding dimension two

We prove that a sequence of positive integers (h₀,h₁,…,h_c) is the Hilbert function of an artinian level module of embedding dimension two if and only if h_i−1−2h_i+h_i+1≤0 for all 0≤i≤c, where we assume that h₋₁=h_c+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.