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Extremal F-thresholds in regular local rings

Benjamin Baily

TL;DR

The paper resolves when the extremal lower bound $\mathrm{ft}^{\mathfrak m}(\mathfrak a)=\frac{1}{\mathrm{ord}_{\mathfrak m}(\mathfrak a)}$ is achieved for ideals in regular local rings of characteristic $p>0$, identifying a precise criterion in terms of a power-root structure after Frobenius-splitting scaling. By combining $F$-threshold theory, test ideals, and the critical-point framework, the author reduces to the principal case via Bertini-type reductions and utilizes Weierstrass preparation to obtain a root factorization, which then descends under suitable formal-fiber hypotheses. The main result states that if $d=qs$ with $q=p^{e}$ and $(p,s)=1$, and $\mathrm{ft}^{\mathfrak m}(\mathfrak a)=\frac{1}{d}$, then $\mathfrak a\widehat{R}=g^{s}\widehat{R}$ (and $\mathfrak a=h^{s}R$ under reduced formal fibers), yielding a sharp classification of extremal $F$-thresholds. These findings mirror and contrast with the characteristic-zero lct theory and pave the way for a broader understanding of thresholds in higher-height cases via future work.

Abstract

Let $(R, \mathfrak{m})$ be a regular local ring of characteristic $p > 0$. Among all proper ideals $\mathfrak{a}\subseteq R$ with a fixed order of vanishing $\text{ord}_{\mathfrak{m}}(\mathfrak{a})$, we classify the ideals for which the $F$-threshold $\text{ft}^{\mathfrak{m}}(\mathfrak{a})$ is minimal.

Extremal F-thresholds in regular local rings

TL;DR

The paper resolves when the extremal lower bound is achieved for ideals in regular local rings of characteristic , identifying a precise criterion in terms of a power-root structure after Frobenius-splitting scaling. By combining -threshold theory, test ideals, and the critical-point framework, the author reduces to the principal case via Bertini-type reductions and utilizes Weierstrass preparation to obtain a root factorization, which then descends under suitable formal-fiber hypotheses. The main result states that if with and , and , then (and under reduced formal fibers), yielding a sharp classification of extremal -thresholds. These findings mirror and contrast with the characteristic-zero lct theory and pave the way for a broader understanding of thresholds in higher-height cases via future work.

Abstract

Let be a regular local ring of characteristic . Among all proper ideals with a fixed order of vanishing , we classify the ideals for which the -threshold is minimal.
Paper Structure (10 sections, 25 theorems, 34 equations)

This paper contains 10 sections, 25 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Background on F-Thresholds and Test Ideals
  4. Critical Points, after Hernández and Teixeira
  5. Proof of Theorem 1.1
  6. Step (1)
  7. Step (2)
  8. Step (3)
  9. Step (4)
  10. Step (5)

Key Result

Proposition 2.7

Let $(R, \mathfrak m)$ be a regular local ring of dimension $n$ and characteristic $p > 0$. Let $\mathfrak a$ be an ideal of $R$ and let $\mathfrak b\subseteq R$ such that $\mathfrak a\subseteq \sqrt{\mathfrak b}$. Then the following hold.

Theorems & Definitions (61)

  • Definition 2.2
  • Definition 2.4: schwede_singularities_2024, § 1.1
  • Definition 2.5: schwede_centers_2010takagi_f-pure_2004
  • Definition 2.6: blickle_discreteness_2008hara_generalization_2003
  • Proposition 2.7
  • proof
  • Definition 2.11
  • Lemma 2.12
  • proof
  • Lemma 2.13
  • ...and 51 more