On the linearity of the syzygies of Hibi rings

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 $p\ge 4$ 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.

arXiv comment23 pages, 13 figures. The previous version "Green-Lazarsfeld property $N_p$ for Hibi rings" has been divided into two parts. This version describes the results on Green-Lazarsfeld property $N_p$ for Hibi rings for $p\ge 2$ while the second part is about the Green-Lazarsfeld property $N_p$ for Segre product of Hibi rings