On the diameter of dual graphs of Stanley-Reisner rings with Serre $(S_2)$ property and Hirsch type bounds on abstractions of polytopes

TL;DR

This paper establishes bounds on the diameter of Hochster-Huneke graphs of Stanley-Reisner rings with Serre's $(S_2)$ property, linking algebraic properties to combinatorial bounds similar to polyhedral graph diameters.

Contribution

It provides the first explicit bounds on the diameter of these dual graphs for $(S_2)$ Stanley-Reisner rings, connecting algebraic and polyhedral combinatorics.

Findings

Bounds depend on variables and dimension

Graphs are a natural abstraction of polyhedral 1-skeletons

Implications for Hirsch-type bounds on polytopes

Abstract

Let $R$ be a Noetherian commutative ring of positive dimension. The Hochster-Huneke graph of $R$ (sometimes called the dual graph of Spec $R$ and denoted by $\mathcal{G} (R)$) is defined as follows: the vertices are the minimal prime ideals of $R$, and the edges are the pairs of prime ideals $(P_1,P_2)$ with height $(P_1 + P_2) = 1$. If $R$ satisfies Serre's property $(S_2)$, then $\mathcal{G} (R)$ is connected. In this note, we provide lower and upper bounds for the maximum diameter of Hochster-Huneke graphs of Stanley-Reisner rings satisfying $(S_2)$. These bounds depend on the number of variables and the dimension. Hochster-Huneke graphs of $(S_2)$ Stanley-Reisner rings are a natural abstraction of the $1$-skeletons of polyhedra. We discuss how our bounds imply new Hirsch-type bounds on $1$-skeletons of polyhedra.