Powers of Principal $Q$-Borel ideals

TL;DR

This paper investigates powers of principal $Q$-Borel ideals, revealing they coincide with symbolic powers, satisfy persistence of associated primes, and their analytic spread can be explicitly computed from the poset structure.

Contribution

It introduces the study of powers of principal $Q$-Borel ideals, establishing their equality with symbolic powers, persistence of associated primes, and providing a formula for their analytic spread.

Findings

All powers of $Q(m)$ equal their symbolic powers.

$Q(m)$ satisfies the persistence property for associated primes.

The analytic spread of $Q(m)$ is explicitly computed from the poset $Q$.

Abstract

Fix a poset $Q$ on $\{x_1,\ldots,x_n\}$. A $Q$-Borel monomial ideal $I \subseteq \mathbb{K}[x_1,\ldots,x_n]$ is a monomial ideal whose monomials are closed under the Borel-like moves induced by $Q$. A monomial ideal $I$ is a principal $Q$-Borel ideal, denoted $I=Q(m)$, if there is a monomial $m$ such that all the minimal generators of $I$ can be obtained via $Q$-Borel moves from $m$. In this paper we study powers of principal $Q$-Borel ideals. Among our results, we show that all powers of $Q(m)$ agree with their symbolic powers, and that the ideal $Q(m)$ satisfies the persistence property for associated primes. We also compute the analytic spread of $Q(m)$ in terms of the poset $Q$.