On the linearity of the syzygies of Hibi rings

TL;DR

This paper investigates the algebraic properties of Hibi rings, establishing conditions for their syzygies and resolutions, and characterizing certain classes of these rings based on their combinatorial structures.

Contribution

It provides necessary conditions for Hibi rings to satisfy Green-Lazarsfeld property N_p for p=2,3, and characterizes when they have linear resolutions or are polynomial rings.

Findings

Hibi rings satisfy N_p for all p≥4 if they satisfy N_4.

Distributive lattices with chordal comparability graphs are characterized by their join-irreducible subposets.

Complete intersection Hibi rings are characterized.

Abstract

In this article, we prove necessary conditions for Hibi rings to satisfy Green-Lazarsfeld property $N_p$ for $p=2$ and $3$. We also show that if a Hibi ring satisfies property $N_4$, then it is a polynomial ring or it has a linear resolution. Therefore, it satisfies property $N_p$ for all $p\geq 4$ as well. As a consequence, we characterize distributive lattices whose comparability graph is chordal in terms of the subposet of join-irreducibles of the distributive lattice. Moreover, we characterize complete intersection Hibi rings.