Graded Betti Numbers and Hilbert Functions of Graded Cohen-Macaulay Modules

Abstract: In this thesis we study graded Cohen-Macaulay modules and their possible Hilbert functions and graded Betti numbers. In most cases the Cohen-Macaulay modules we study are level modules.In order to use dualization to study Hilbert functions of artinian level algebras we extend the notion of level sequences and cancellable sequences, introduced by Geramita and Lorenzini, to include Hilbert functions of certain level modules. As in the case of level algebras, a level sequence is cancellable, but now by dualization its reverse is also cancellable which gives a new condition on level sequences. We also give a characterization of the cancellable sequences.We prove that a sequence of positive integers (h0, h1, . . . ,hc) is the Hilbert function of an artinian level module of codimension 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.Zanello gives a lower bound for Hilbert functions of generic level quotients of artinian level algebras. We give a new and more straightforward proof of Zanello’s result.Conjectures on the possible graded Betti numbers of Cohen-Macaulay modules up to multiplication by a positive rational number are given. The idea is that the Betti diagrams should be non-negative linear combinations of pure diagrams. The conjectures are verified for modules of codimension two, for Gorenstein algebras of codimension three and for complete intersections. The motivation for the conjectures comes from the Multiplicity conjecture of Herzog, Huneke and Srinivasan.The h-vectors and graded Betti numbers of level modules up to multiplication by a rational number are studied. Assuming the conjecture, mentioned above, on the set of 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 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 and that the lower bound holds if the module, in addition, has the weak Lefschetz property.