Graded Betti numbers of Cohen–Macaulay modules and the multiplicity conjecture

We give conjectures on the possible graded Betti numbers of Cohen–Macaulay modules up to multiplication by positive rational numbers. The idea is that the Betti diagrams should be non‐negative linear combinations of pure diagrams. The conjectures are verified in the cases where the structure of resolutions is known, that is: for modules of codimension two, for Gorenstein algebras of codimension three and for complete intersections. The motivation for proposing the conjectures comes from the Multiplicity conjecture of Herzog, Huneke and Srinivasan.