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Herzog ideals and $F$-singularities

Alessandro De Stefani, Linquan Ma, Matteo Varbaro

TL;DR

The paper investigates the interface between Herzog ideals and $F$-singularities. It proves that homogeneous Herzog ideals yield $F$-anti-nilpotent localizations and analyzes how being Herzog after a coordinate change relates to $F$-purity of reductions mod $p$ in various settings, including curves and low-degree hypersurfaces. It extends the study to annihilators of $F$-stable submodules of local cohomology, establishing uniform compatibility in many cases, and provides a counterexample to a broad generalization. Together, these results illuminate how combinatorial properties of initial ideals influence Frobenius-based singularities and potential characteristic $0$–to–characteristic $p$ transfer phenomena.

Abstract

In this paper we study the connection between Herzog ideals (i.e., ideals with a squarefree Gröbner degeneration) and $F$-singularities. More precisely, we show that, in positive characteristic, homogeneous Herzog ideals define $F$-anti-nilpotent rings, and we inquire, in characteristic 0, on a surprising relationship between being Herzog ideals after a change of coordinates and defining rings of dense open $F$-pure type.

Herzog ideals and $F$-singularities

TL;DR

The paper investigates the interface between Herzog ideals and -singularities. It proves that homogeneous Herzog ideals yield -anti-nilpotent localizations and analyzes how being Herzog after a coordinate change relates to -purity of reductions mod in various settings, including curves and low-degree hypersurfaces. It extends the study to annihilators of -stable submodules of local cohomology, establishing uniform compatibility in many cases, and provides a counterexample to a broad generalization. Together, these results illuminate how combinatorial properties of initial ideals influence Frobenius-based singularities and potential characteristic –to–characteristic transfer phenomena.

Abstract

In this paper we study the connection between Herzog ideals (i.e., ideals with a squarefree Gröbner degeneration) and -singularities. More precisely, we show that, in positive characteristic, homogeneous Herzog ideals define -anti-nilpotent rings, and we inquire, in characteristic 0, on a surprising relationship between being Herzog ideals after a change of coordinates and defining rings of dense open -pure type.
Paper Structure (7 sections, 13 theorems, 20 equations)

This paper contains 7 sections, 13 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Positive characteristic preliminaries
  3. Modules with a Frobenius action
  4. Local cohomology modules and $F$-singularities
  5. Herzog ideals and $F$-anti-nilpotency
  6. Herzog ideals versus $F$-purity for all $p\gg 0$
  7. Annihilators of $F$-stable submodules and uniformly compatible ideals

Key Result

Theorem 1.1

Let $R$ be an $\mathbb{N}$-graded algebra over an $F$-finite field $k$ of prime characteristic $p$ with homogeneous maximal ideal $\mathfrak{m}$. Write $R=S/I$ where $S$ is an $\mathbb{N}$-graded polynomial ring over $k$. If $I$ is a Herzog ideal, then $R_{\mathfrak{m}}$ is $F$-anti-nilpotent.

Theorems & Definitions (33)

  • Theorem 1.1: Theorem \ref{['t:HF-anti']}
  • Theorem 1.3: Proposition \ref{['p:curves']}
  • Theorem 1.4: Theorem \ref{['t:genova']}
  • Theorem 1.5: Proposition \ref{['p:radcomp']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • ...and 23 more