Asymptotic Properties of Filtrations of Ideals

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.