Triangulations of polygons and stacked simplicial complexes: separating their Stanley–Reisner ideals

A triangulation of a polygon has an associated Stanley–Reisner ideal. We obtain a full algebraic and combinatorial understanding of these ideals and describe their separated models. More generally, we do this for stacked simplicial complexes, in particular for stacked polytopes.publishedVersio

The regularity of almost all edge ideals

A fruitful contemporary paradigm in graph theory is that almost all graphs that do not contain a certain subgraph have common structural characteristics. The "almost" is crucial, without it there is no structure. In this paper we transfer this paradigm to commutative algebra and make use of deep graph theoretic results. A key tool are the critical graphs introduced by Balogh and Butterfield. We consider edge ideals $I_G$ of graphs and their Betti numbers. The numbers of the form $\beta_{i,2i+2}$ constitute the "main diagonal" of the Betti table. It is well known that any Betti number $\beta_{i,j}(I_G)$ below (or equivalently, to the left of) this diagonal is always zero. We identify a certain "parabola" inside the Betti table and call parabolic Betti numbers the entries of the Betti table bounded on the left by the main diagonal and on the right by this parabola. Let $\beta_{i,j}$ be a parabolic Betti number on the $r$-th row of the Betti table, for $r\ge3$. Our main results state that almost all graphs $G$ with $\beta_{i,j}(I_G)=0$ can be partitioned into $r-2$ cliques and one independent set, and in particular for almost all graphs $G$ with $\beta_{i,j}(I_G)=0$ the regularity of $I_G$ is $r-1$.Comment: Edit: improved proofs and statements, added future directions, no new main results. 31 pages. Accepted for publication in Advances in Mathematic

Homology and Combinatorics of Monomial Ideals

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