Some Computations for Binomial Edge Ideals and Koszul Duality

TL;DR

This paper investigates the algebraic properties of binomial edge ideals, focusing on Koszulness, graph constructions, and duality, providing new characterizations and computations related to their algebraic structure.

Contribution

It introduces new characterizations of Koszulness for binomial edge ideals, explores graph constructions, and computes dual algebras and syzygies, advancing understanding of their algebraic properties.

Findings

Koszulness characterized for cone graphs

First two syzygies are always linear

Dual algebra computed for arbitrary binomial edge ideals

Abstract

We make some observations on binomial edge ideals, with the characterization of their Koszulness as motivation. Inspired by results of Ene, Herzog and Hibi, we discuss building Koszul graphs from smaller pieces in a controlled manner. We characterize the Koszul property of cone graphs and compute the dual algebra of the quadratic algebra coming from an arbitrary binomial edge ideal. We compute the first two syzygies of the infinite minimal free resolution of the residue field over the algebra defined by a binomial edge ideal, and observe that they are always linear.