On the set of associated radicals of powers of monomial ideals

TL;DR

This paper investigates the long-term behavior of associated radical ideals of powers of monomial ideals, revealing conditions for stability and monotonicity, especially for square-free and hypergraph cover ideals.

Contribution

It establishes that associated radicals of powers do not always stabilize, but do stabilize for square-free monomial ideals and increase monotonically for cover ideals of balanced hypergraphs.

Findings

Associated radicals of powers may not stabilize for large s.

For square-free monomial ideals, associated radicals become constant eventually.

In cover ideals of balanced hypergraphs, associated radicals increase monotonically with s.

Abstract

Let $I$ be a monomial ideal in a polynomial ring. In this paper, we study the asymptotic behavior of the set of associated radical ideals of the (symbolic) powers of $I$. We show that both $\asr(I^s)$ and $\asr(I^{(s)})$ need not stabilize for large value of $s$. In the case $I$ is a square-free monomial ideal, we prove that $\asr(I^{(s)})$ is constant for $s$ large enough. Finally, if $I$ is the cover ideal of a balanced hypergraph, then $\asr(I^s)$ monotonically increases in $s$.