Stabilization of Boij-S\"oderberg Decompositions of Ideal Powers

TL;DR

This paper proves that the Boij-S"oderberg decompositions of powers of a homogeneous ideal with generators in one degree stabilize as the power increases, with consistent structure and polynomial coefficient behavior.

Contribution

It establishes the stabilization of Boij-S"oderberg decompositions for ideal powers, confirming conjectures and extending results to arbitrary coefficient chains.

Findings

Number of terms with positive coefficients stabilizes for large k

Pure diagrams in the decomposition have fixed shape for large k

Coefficients of diagrams are polynomial functions in k

Abstract

Given an ideal $I$ we investigate the decompositions of Betti diagrams of the graded family of ideals $\{I^k \}_k$ formed by taking powers of $I$. We prove conjectures of Engstr\"om and show that there is a stabilization in the Boij-S\"oderberg decompositions of $I^k$ for $k>>0$ when $I$ is a homogeneous ideal with generators in a single degree. In particular, the number of terms in the decompositions with positive coefficients remains constant for $k>>0$, the pure diagrams appearing in each decomposition have the same shape, and the coefficients of these diagrams are given by polynomials in $k$. We also show that a similar result holds for decompositions with arbitrary coefficients arising from other chains of pure diagrams.