Homology and Combinatorics of Monomial Ideals

Abstract

This thesis is in combinatorial commutative algebra. It contains four papers, the first three of which concern homological properties and invariants of monomial ideals. In Publication I we examine a construction originally defined in complexity theory to reduce the isomorphism problem for arbitrary graphs to so-called Booth-Lueker graphs. The map associating to a graph G its Booth-Lueker graph BL(G) can be interpreted from an algebraic point of view as a construction that associates to a squarefree quadratic monomial ideal a squarefree quadratic monomial ideal with a 2-linear resolution. We study numerical invariants coming from the minimal resolutions of the edge ideals of Booth-Lueker graphs, in particular their Betti numbers and Boij-Söderberg coefficients. We provide very explicit formulas for these invariants. Publication II concerns a generalization of the construction in Publication I: starting from an arbitrary monomial ideal I we define its linearization Lin(I), which is an equigenerated monomial ideal with linear quotients, and hence in particular with a linear resolution. We moreover introduce another construction, called equification, that to an arbitrary monomial ideal associates an equigenerated monomial ideal. We study several properties of both constructions, with particular attention to their homological invariants. In Publication III we address the central open problem in the theory of edge ideals of describing their regularity. We prove new results in this direction by employing the methods of critical graphs. We introduce the concept of parabolic Betti number and provide structural descriptions for almost all graphs whose edge ideal has some parabolic Betti numbers equal to zero. For a parabolic Betti number in row r of the Betti table, we show that, for almost all graphs whose edge ideal has that Betti number equal to zero, the regularity of the edge ideal is r-1. Publication IV deals with separations (i.e., a generalization of the classical concept of polarization) of the Stanley-Reisner ideals of stacked simplicial complexes. We study combinatorial and algebraic properties of the Stanley-Reisner ideals of triangulated balls, and in particular those of triangulated polygons.