Graded Betti numbers and $h$-vectors of level modules

We study $h$-vectors and graded Betti numbers of level modules up to multiplication by a rational number. Assuming a conjecture on the possible graded Betti numbers of Cohen-Macaulay modules we get a description of the possible $h$-vectors of level modules up to multiplication by a rational number. We also determine, again up to multiplication by a rational number, the cancellable $h$-vectors and the $h$-vectors of level modules with the weak Lefschetz property. Furthermore, we prove that level modules of codimension three satisfy the upper bound of the Multiplicity conjecture of Herzog, Huneke and Srinivasan, and that the lower bound holds if the module, in addition, has the weak Lefschetz property.