On the subadditivity condition of edge ideal

TL;DR

This paper investigates the subadditivity condition of edge ideals in graded free resolutions, establishing new bounds and conditions under which the subadditivity property holds, especially for ideals with low projective dimension.

Contribution

It proves that if the maximal shift satisfies a certain growth condition, then the subadditivity condition holds for all degrees, extending known cases to broader classes of edge ideals.

Findings

Subadditivity holds if $t_b(S/I) \

Subadditivity is guaranteed for ideals with projective dimension at most 9.

Established bounds for the maximal shifts ensuring subadditivity.

Abstract

Let $S=K[x_1,\ldots,x_n]$, where $K$ is a field, and $t_i(S/I)$ denotes the maximal shift in the minimal graded free $S$-resolution of the graded algebra $S/I$ at degree $i$, where $I$ is an edge ideal. In this paper, we prove that if $t_b(S/I)\geq \lceil \frac{3b}{2} \rceil$ for some $b\geq 0$, then the subadditivity condition $t_{a+b}(S/I)\leq t_a(S/I)+t_b(S/I)$ holds for all $a\geq 0$. In addition, we prove that $t_{a+4}(S/I)\leq t_a(S/I)+t_4(S/I)$ for all $a\geq 0$ (the case $b=0,1,2,3$ is known). We conclude that if the projective dimension of $S/I$ is at most $9$, then $I$ satisfies the subadditivity condition.