Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Strong $F$-regularity and the Uniform Symbolic Topology Property

Thomas Polstra

TL;DR

The paper addresses the problem of uniformly bounding symbolic powers by ordinary powers in prime characteristic, establishing the Uniform Symbolic Topology Property (USTP) for broad classes of singular rings. It introduces and leverages splitting ideals to translate Frobenius actions into uniform containments, and couples this with uniform Artin–Rees/Briançon–Skoda bounds and Chevalley-type arguments to derive global uniformity. A central outcome is that $F$-finite strongly $F$-regular domains satisfy USTP, and the authors develop a descent framework showing that USTP descends along finite extensions under suitable hypotheses, including isolated non-$F$-regular loci. These results connect to test-ideal theory, BCM notions, and Rees valuations, yielding robust, characteristic-$p$ tools for controlling symbolic powers in a wide range of singularities with potential implications for birational geometry and singularity theory in positive characteristic and reduction to characteristic $0$.

Abstract

We investigate the containment problem of symbolic and ordinary powers of ideals in a commutative Noetherian domain $R$. Let $R$ be a normal domain of prime characteristic $p>0$ that is $F$-finite or essentially of finite type over an excellent local ring. Assume there exists a finite extension $R\to S$ so that the non-strongly $F$-regular locus of $\mathrm{Spec}(S)$ consists only of isolated points, then there exists a constant $C$ such that for all ideals $I \subseteq R$ and $n \in \mathbb{N}$, the symbolic power $I^{(Cn)}$ is contained in the ordinary power $I^n$. In other words, $R$ enjoys the Uniform Symbolic Topology Property. Moreover, if $R$ is $F$-finite and strongly $F$-regular, then $R$ enjoys a property that is proven to be stronger: there exists a constant $e_0 \in \mathbb{N}$ such that for any ideal $I \subseteq R$ and all $e \in \mathbb{N}$, if $x \in R \setminus I^{[p^e]}$, then there exists an $R$-linear map $\varphi: F^{e+e_0}_*R \to R$ such that $\varphi(F^{e+e_0}_*x) \notin I$.

Strong $F$-regularity and the Uniform Symbolic Topology Property

TL;DR

The paper addresses the problem of uniformly bounding symbolic powers by ordinary powers in prime characteristic, establishing the Uniform Symbolic Topology Property (USTP) for broad classes of singular rings. It introduces and leverages splitting ideals to translate Frobenius actions into uniform containments, and couples this with uniform Artin–Rees/Briançon–Skoda bounds and Chevalley-type arguments to derive global uniformity. A central outcome is that -finite strongly -regular domains satisfy USTP, and the authors develop a descent framework showing that USTP descends along finite extensions under suitable hypotheses, including isolated non--regular loci. These results connect to test-ideal theory, BCM notions, and Rees valuations, yielding robust, characteristic- tools for controlling symbolic powers in a wide range of singularities with potential implications for birational geometry and singularity theory in positive characteristic and reduction to characteristic .

Abstract

We investigate the containment problem of symbolic and ordinary powers of ideals in a commutative Noetherian domain . Let be a normal domain of prime characteristic that is -finite or essentially of finite type over an excellent local ring. Assume there exists a finite extension so that the non-strongly -regular locus of consists only of isolated points, then there exists a constant such that for all ideals and , the symbolic power is contained in the ordinary power . In other words, enjoys the Uniform Symbolic Topology Property. Moreover, if is -finite and strongly -regular, then enjoys a property that is proven to be stronger: there exists a constant such that for any ideal and all , if , then there exists an -linear map such that .

Paper Structure

This paper contains 14 sections, 25 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. Rings of Prime Characterisitic
  4. Linear Equivalence of Ideal Topologies
  5. Splitting Ideals
  6. Connections with Test Ideals
  7. Uniform Properties of Noetherian Rings
  8. Symbolic Multipliers
  9. On the Powers of Ideals, Frobenius Powers of Ideals, and Splitting Ideals
  10. Techniques in proofs of main theorems
  11. Uniform Linear Equivalence of Topologies
  12. The Uniform Symbolic Topology Property and Finite Extensions
  13. On the Descent of the Uniform Symbolic Topology Property
  14. Proof of Main Theorem \ref{['Main theorem USTP in SFR rings']}

Key Result

Theorem 1.1

Let $R$ be a Noetherian non-singular ring and $I\subseteq R$ an ideal. If $h$ is the largest height of an associated prime of $I$, then for all $n \in \mathbb{N}$, Outline of theorem development:

Theorems & Definitions (63)

  • Theorem 1.1: Uniform Symbolic Topology in Non-Singular Rings
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1
  • Remark 4
  • Theorem 2.1: Properties of $F$-finite rings
  • Definition 2
  • Proposition 1: Generalization of HKVErratum
  • proof
  • ...and 53 more