Stabilization of Betti Tables

Abstract

Let $I\subseteq R=\kk[x_1,...,x_n]$ be a homogeneous equigenerated ideal of degree $r$. We show here that the shapes of the Betti tables of the ideals $I^d$ stabilize, in the sense that there exists some $D$ such that for all $d\geq D$, $\betti{i}{j+rd}(I^d)\neq 0\Leftrightarrow \betti{i}{j+rD}(I^D)\neq 0$. We also produce upper bounds for the stabilization index $\Stab(I)$. This strengthens the result of Cutkosky, Herzog, and Trung that the Castelnuovo-Mumford regularity $\reg(I^d)$ is eventually a linear function in $d$.