First Nonlinear Syzygies of Ideals Associated to Graphs

TL;DR

This paper investigates the emergence of nonlinear syzygies in ideals generated by degree-two monomials associated with graphs, precisely identifying the step and degree where they first appear, along with their Betti numbers.

Contribution

It determines the exact step and degree at which nonlinear syzygies first occur in such ideals, providing explicit Betti number calculations and multidegree characterizations.

Findings

Nonlinear syzygies appear at a specific step s in the resolution.

At this step, nonlinear syzygies are concentrated in degree s+3.

All relevant multigraded Betti numbers are equal to 1.

Abstract

Consider an ideal $I\subset K[x_1,..., x_n]$, with $K$ an arbitrary field, generated by monomials of degree two. Assuming that $I$ does not have a linear resolution, we determine the step $s$ of the minimal graded free resolution of $I$ where nonlinear syzygies first appear, we show that at this step of the resolution nonlinear syzygies are concentrated in degree $s+3$, and we compute the corresponding graded Betti number $\beta_{s,s+3}$. The multidegrees of these nonlinear syzygies are also determined and the corresponding multigraded Betti numbers are shown to be all equal to 1.