Algebraic Properties of Monomial Ideals

Abstract

In this thesis we first study a special class of squarefree monomial ideals, namely, path ideals. We give a formula to compute all graded Betti numbers of the path ideal of a cycle and a path. As a consequence we can give new and short proofs for the known formulas of regularity and projective dimensions of path ideals of path graphs and cycles. We also study the Rees algebra of squarefree monomial ideals. In 1995 Villarreal gave a combinatorial description of the equations of Rees algebras of quadratic squarefree monomial ideals. His description was based on the concept of closed even walks in a graph. In this thesis we will generalize his results to all squarefree monomial ideals by using a definition of even walks in a simplicial complex. We show that simplicial complexes with no even walks have facet ideals that are of linear type, generalizing Villarreal’s work.