Non-Gorenstein loci of Ehrhart rings of chain and order polytopes

TL;DR

This paper investigates the non-Gorenstein loci of Ehrhart rings of order and chain polytopes of finite posets, revealing their dimension equality and characterizing nearly Gorenstein conditions for chain polytope rings.

Contribution

It establishes the equality of non-Gorenstein locus dimensions for Ehrhart rings of order and chain polytopes and characterizes when chain polytope rings are nearly Gorenstein based on poset structure.

Findings

Non-Gorenstein loci of Ehrhart rings have the same dimension.

Chain polytope rings are nearly Gorenstein iff the poset is a union of pure posets with rank differences at most 1.

The structure of the poset determines the Gorenstein properties of the associated Ehrhart rings.

Abstract

Let $P$ be a finite poset, $K$ a field, and $O(P)$ (resp. $C(P)$) the order (resp. chain) polytope of $P$. We study the non-Gorenstein locus of $E_K[O(P)]$ (resp. $E_K[C(P)]$), the Ehrhart ring of $O(P)$ (resp. $C(P)$) over $K$, which are each normal toric rings associated $P$. In particular, we show that the dimension of non-Gorenstein loci of $E_K[O(P)]$ and $E_K[C(P)]$ are the same. Further, we show that $E_K[C(P)]$ is nearly Gorenstein if and only if $P$ is the disjoint union of pure posets $P_1, \ldots, P_s$ with $|\mathrm{rank} P_i-\mathrm{rank} P_j|\leq 1$ for any $i$ and $j$.