Betti diagrams from graphs

TL;DR

This paper explores the combinatorial structures underlying Betti diagrams of 2-linear resolutions, establishing bijections with threshold graphs and anti-lecture hall compositions, and showing their realization via Stanley-Reisner rings.

Contribution

It extends previous results to 2-linear resolutions using combinatorial methods, connecting Betti diagrams with threshold graphs and lattice polytopes.

Findings

Bijective correspondences between Betti diagrams, threshold graphs, and anti-lecture hall compositions.

Betti diagrams of modules with 2-linear resolutions are realizable by Stanley-Reisner rings of threshold graphs.

Betti diagrams correspond to lattice points in a normal reflexive lattice polytope.

Abstract

The emergence of Boij-S\"oderberg theory has given rise to new connections between combinatorics and commutative algebra. Herzog, Sharifan, and Varbaro recently showed that every Betti diagram of an ideal with a k-linear minimal resolution arises from that of the Stanley-Reisner ideal of a simplicial complex. In this paper, we extend their result for the special case of 2-linear resolutions using purely combinatorial methods. Specifically, we show bijective correspondences between Betti diagrams of ideals with 2-linear resolutions, threshold graphs, and anti-lecture hall compositions. Moreover, we prove that any Betti diagram of a module with a 2-linear resolution is realized by a direct sum of Stanley-Reisner rings associated to threshold graphs. Our key observation is that these objects are the lattice points in a normal reflexive lattice polytope.