ABSTRACT
Contents 4.1 The Partial Order and Some Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.2 Lex Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.3 Lex-Plus-Powers Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.4 The EGH Conjecture and Its Ramifications . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.5 The LPP Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.6 Equivalences and Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
In the last several decades, researchers interested in Hilbert functions and free resolutions have been trying to understand the relationship between these two invariants. It is easy to give examples, for instance, of two ideals with the same Hilbert function but different graded Betti numbers. This raises the question:
Question 4.0.1 Given a Hilbert function for a cyclic module (i.e., a polynomial ring modulo a homogeneous ideal), what graded Betti numbers actually occur for modules with that Hilbert function?
