On the combinatorics of resolutions of monomial ideals

Abstract

In this thesis, we investigate the relation between invariants of minimal free resolutions of monomial ideals and combinatorial properties of simplicial complexes. We provide a sufficient combinatorial condition for monomial ideals to have nonzero Betti numbers and show that such a condition completely characterizes Betti numbers of facet ideals of simplicial forests. We also present a new approach to computing Betti numbers of path ideals of certain graph classes.