$F$-injectivity does not imply $F$-fullness in normal domains

TL;DR

This paper constructs examples of normal domains in prime characteristic that are $F$-injective but not $F$-full, revealing nuanced distinctions in Frobenius properties in algebraic geometry.

Contribution

It provides the first known examples demonstrating that $F$-injectivity does not imply $F$-fullness in normal domains, especially in three-dimensional cases.

Findings

Examples of 3D normal domains that are $F$-injective but not $F$-full

Examples of 2D normal domains that are $F$-injective but not $F$-anti-nilpotent

Analysis of $F$-injectivity behavior under purely inseparable base change

Abstract

We construct examples of noetherian three-dimensional local geometrically normal domains of prime characteristic which are $F$-injective but not $F$-full. Along the way, we find examples of two-dimensional local geometrically normal domains which are $F$-injective but not $F$-anti-nilpotent. A crucial theme of our constructions is the behavior of $F$-injectivity along a purely inseparable finite base change.