Levelness of toric rings arising from order and chain polytopes

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.