Dimensions of Betti cones on edge ideals (Q2125179)

Boij-Soderberg Theory allows to consider Betti tables of graded modules over a polynomial ring as vectors in a suitable infinite-dimensional \(\mathbb{Q}\)-vector space. In particular, it is possible to study the cones that these vectors generate (the so-called \textit{Betti cones}): they are interesting, because they provide a useful framework to study the structure of the Betti tables involved. In this paper, the author focus on Betti cones generated by the Betti tables of edge ideals. He obtains an explicit formula for the dimension of the cone generated by all edge ideals (Theorem 1.2) and an explicit formula for the dimension of the cone generated by edge ideals with fixed height (Theorem 1.3). The proof of these results is given by presenting both upper bounds (Sections 4.1 and 5.1) and lower bounds (Sections 4.2 and 5.2). It was crucial to understand the structure of the possible Betti tables of edge ideals (in particular the technical results Propositions 4.1 and 5.3 have their own independent interest). The interpretation of monomial ideals as Stanley-Reisner ideals of simplicial complexes (and in particular edge ideals as Stanley-Reisner ideals of independence complexes) is a part of the machinery. In fact, a key tools to get the goal is Hochster's formula (Proposition 2.1). Another ingredient in various passages is given by Herzog-Kuhl equations (Proposition 2.2). The well-known available formulas for particular edge ideals (Propositions 2.3, 2.4 and 2.5) and the formula for the graded Betti numbers of the Stanley-Reisner ring of the suspension of a simplicial complex (Proposition 2.6) play an important role in the proofs. Although not difficult, the proofs are absolutely not trivial. For this reason, I believe that this paper could be a forge of useful tools for people working on Boij-Soderberg theory.