Gorenstein on the punctured spectrum and nearly Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph

TL;DR

This paper characterizes when the Ehrhart ring of the stable set polytope of an h-perfect graph is nearly Gorenstein, linking it to Gorenstein properties of components and clique number differences, and explores Gorenstein on the punctured spectrum for certain ring products.

Contribution

It provides a criterion for the nearly Gorenstein property of Ehrhart rings of stable set polytopes of h-perfect graphs, connecting component Gorenstein conditions and clique number constraints.

Findings

Nearly Gorenstein Ehrhart rings correspond to Gorenstein components with clique number differences at most one.

Segre product of Cohen-Macaulay Gorenstein-on-punctured-spectrum rings retains the property under certain grading conditions.

Abstract

In this paper, we give a criterion of the nearly Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph: the Ehrhart ring of the stable set polytope of an h-perfect graph $G$ with connected components $G^{(1)}, \ldots, G^{(\ell)}$ is nearly Gorenstein if and only if (1) for each $i$, the Ehrhart ring of the stable set polytope of $G^{(i)}$ is Gorenstein and (2) $|\omega(G^{(i)})-\omega(G^{(j)})|\leq 1$ for any $i$ and $j$, where $\omega(G^{(i)})$ is the clique number of $G^{(i)}$. We also show that the Segre product of Cohen-Macaulay graded rings with linear non-zerodivisor which are Gorenstein on the punctured spectrum is also Gorenstein on the punctured spectrum if all but one rings are standard graded.