Asymptotic Properties of Filtrations of Ideals

TL;DR

This paper develops a unified framework for analyzing the long-term behavior of ideal filtrations in commutative algebra, introducing new persistence concepts and establishing their relationships.

Contribution

It introduces the notions of $$-persistence and $$-strong persistence for ideal filtrations, extending classical concepts and proving their implications.

Findings

Strong persistence of a filtration implies the strong persistence of its symbolic filtration.

Strong persistence of a filtration implies its persistence.

The framework unifies and extends classical persistence notions in algebraic filtrations.

Abstract

We introduce a unified framework for studying persistence phenomena in commutative algebra via filtrations of ideals. For a filtration $\mathcal{F} = \{I_i\}_{i \in \mathbb{N}}$, we define $\mathcal{F}$-persistence and $\mathcal{F}$-strong persistence, extending the classical notions for ordinary and symbolic powers of ideals. We show that if $\mathcal{F}$ is strongly persistent, then $\mathcal{F}_{\mathrm{sym}}$ is strongly persistent, where $\mathcal{F}_{\mathrm{sym}}$ denotes the symbolic filtration associated with the filtration $\mathcal{F}$. In addition, we prove that if $\mathcal{F}$ is strongly persistent, then $\mathcal{F}$ is persistent.