Levelness of toric rings arising from order and chain polytopes

TL;DR

This paper investigates the algebraic properties of toric rings associated with order and chain polytopes of finite posets, establishing their Betti number equivalences and conditions for being level rings.

Contribution

It proves that the Betti numbers of the toric rings of order and chain polytopes are equal and characterizes when these rings are level.

Findings

Betti numbers of $K[ ext{Order}]$ and $K[ ext{Chain}]$ are equal for all j.

$K[ ext{Order}]$ is level if and only if $K[ ext{Chain}]$ is level.

Provides a criterion linking the levelness of these toric rings.

Abstract

Let $K[\mathcal{O}(P)]$ denote the toric ring of the order polytope $\mathcal{O}(P)$ of a finite partially ordered set $P$ and $K[\mathcal{C}(P)]$ that of the chain polytope $\mathcal{C}(P)$. It will be shown that $\beta_{p, p+j}(K[\mathcal{O}(P)]) = \beta_{p, p+j}(K[\mathcal{C}(P)])$ for all $j \geq 0$, where $p$ is the projective dimension of $K[\mathcal{O}(P)]$ (and that of $K[\mathcal{C}(P)]$). In particular, $K[\mathcal{O}(P)]$ is level if and only if $K[\mathcal{C}(P)]$ is level.