Trees and Cohen-Macaulay Monomial Ideals

Abstract

In this thesis we give a structure theorem for Cohen-Macaulay monomial ideals of codimension 2, and describe all possible relation matrices of such ideals. We also study the set T (I) of all relation trees of a Cohen–Macaulay monomial ideal of codimension 2. We show that T (I) is the set of bases of a matroid. In case that the ideal has a linear resolution, the relation matrices can be identified with the spanning trees of a connected chordal graph with the property that each distinct pair of maximal cliques of the graph has at most one vertex in common. We give the equivalent conditions for a squarefree monomial ideal to be a com- plete intersection. Then we study the set of Cohen–Macaulay monomial ideals with a given radical. Among this set of ideals are the so-called Cohen–Macaulay modifica- tions. Not all Cohen–Macaulay squarefree monomial ideals admit nontrivial Cohen– Macaulay modifications. It is shown that if there exists one such modification, then there exist indeed infinitely many. We also present classes of Cohen–Macaulay squarefree monomial ideals with infinitely many nontrivial Cohen–Macaulay modi- fications.