Asymptotic Properties of Filtrations of Ideals
Mehrdad Nasernejad, Jonathan Toledo
TL;DR
The paper develops a unified framework for persistence phenomena in commutative algebra via filtrations of ideals, introducing F-persistence and F-strong persistence to generalize classical notions for ordinary and symbolic powers. It defines the symbolic filtration associated with a given filtration and proves foundational stability results: strong persistence of F implies strong persistence of its symbolic filtration and, in turn, persistence of F. A key technical link is established between colon-ideals and base-step conditions, namely (I_i:I_j)=I_{i-j} for all i≥j if and only if (I_{k+1}:I_1)=I_k for all k, with localization arguments clarifying associated primes. The work also analyzes how filtrations behave under direct-sum operations, showing that F⊕G is strongly persistent exactly when both F and G are, thereby providing a cohesive, extensible framework that unifies ordinary, symbolic, and filtration-based persistence and their implications for asymptotic prime behavior.
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.