Stabilization of Betti Tables

TL;DR

This paper proves that the Betti tables of powers of a homogeneous ideal stabilize in shape after a certain point, providing bounds for the stabilization index and extending known results on regularity.

Contribution

It establishes the stabilization of Betti table shapes for ideal powers and offers upper bounds for the stabilization index, advancing understanding of asymptotic algebraic invariants.

Findings

Betti table shapes stabilize after a finite index D

Provided upper bounds for the stabilization index

Extended previous results on regularity linearity

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 $d$.