Betti table Stabilization of Homogeneous Monomial Ideals

TL;DR

This paper investigates how the Betti table shapes of powers of homogeneous monomial ideals stabilize as the exponent increases, introducing new concepts like the stabilization sequence to analyze these patterns.

Contribution

It introduces the stabilization sequence of a monomial ideal and explores its behavior through examples and specific classes of ideals, extending previous notions of stabilization index.

Findings

Betti table shapes stabilize after certain powers

Defined and analyzed the stabilization sequence for monomial ideals

Provided explicit stabilization indices for specific ideal families

Abstract

Given an homogeneous monomial ideal $I$, we provide a question- and example-based investigation of the stabilization patterns of the Betti tables shapes of $I^d$ as we vary $d$. We build off Whieldon's definition of the stabilization index of $I$, Stab$(I)$, to define the stabilization sequence of $I$, StabSeq$(I)$, and use it to explore changes in the shapes of the Betti tables of $I^d$ as we vary $d$. We also present the stabilization indices and sequences of the collection of ideals $\{I_{n}\}$ where $I_{n}=(a^{2n}b^{2n}c^{2n},b^{4n}c^{2n},a^{3n}c^{3n},a^{6n-1}b)\subseteq \Bbbk[a,b,c]$.