Herzog ideals and $F$-singularities

TL;DR

This paper explores the relationship between Herzog ideals, characterized by squarefree Gr"obner degenerations, and $F$-singularities, revealing new connections in positive characteristic and characteristic zero.

Contribution

It establishes that homogeneous Herzog ideals define $F$-anti-nilpotent rings in positive characteristic and links Herzog ideals to $F$-pure type in characteristic zero after coordinate changes.

Findings

Herzog ideals in positive characteristic define $F$-anti-nilpotent rings.

In characteristic zero, Herzog ideals relate to dense open $F$-pure type.

New connections between algebraic ideals and $F$-singularities are identified.

Abstract

In this paper we study the connection between Herzog ideals (i.e., ideals with a squarefree Gr\"obner 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.