On the linearity of the syzygies of Hibi rings

Abstract

In this paper, we prove necessary conditions for Hibi rings to satisfy Green–Lazarsfeld property [Formula: see text] for [Formula: see text] and [Formula: see text]. We also show that if a Hibi ring satisfies property [Formula: see text], then it is a polynomial ring or it has a linear resolution. Therefore, it satisfies property [Formula: see text] for all [Formula: see text] 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.