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

Alessandro De Stefani, Thomas Polstra, Austyn Simpson

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.

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

Abstract

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

Paper Structure

This paper contains 9 sections, 10 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graded ring considerations
  4. Hypersurface considerations
  5. F-singularities
  6. The Examples
  7. A characteristic 2 hypersurface
  8. Hypersurfaces of characteristic p>2
  9. Proofs of Main Theorems

Key Result

Theorem A

(= thm:f-inj-not-f-anti) For each prime integer $p>0$, there exists a local domain $(R,\mathfrak{m},k)$ of characteristic $p$, of dimension $2$, and essentially of finite type over $k$ such that $R$ is geometrically normal over $k$ (hence Cohen--Macaulay) and $F$-injective, but not $F$-anti-nilpoten

Figures (1)

  • Figure 1: Relationships between $F$-singularities for local rings $(R,\mathfrak{m},k)$ which are essentially of finite type over an $F$-finite field $k$.

Theorems & Definitions (17)

  • Theorem A
  • Theorem B
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.4
  • Example 3.1
  • Proposition 3.2
  • ...and 7 more