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$F$-nilpotent rings and permanence properties

Jennifer Kenkel, Kyle Maddox, Thomas Polstra, Austyn Simpson

Abstract

We explore the singularity classes $F$-nilpotent, weakly $F$-nilpotent, and generalized weakly $F$-nilpotent under faithfully flat local ring maps. As an application, we show that the loci of primes in a Noetherian ring of prime characteristic which define either weakly $F$-nilpotent or $F$-nilpotent local rings are open with respect to the Zariski topology whenever $R$ is $F$-finite or essentially of finite type over an excellent local ring.

$F$-nilpotent rings and permanence properties

Abstract

We explore the singularity classes -nilpotent, weakly -nilpotent, and generalized weakly -nilpotent under faithfully flat local ring maps. As an application, we show that the loci of primes in a Noetherian ring of prime characteristic which define either weakly -nilpotent or -nilpotent local rings are open with respect to the Zariski topology whenever is -finite or essentially of finite type over an excellent local ring.

Paper Structure

This paper contains 9 sections, 20 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Local cohomology and the Nagel-Schenzel isomorphism, and Frobenius actions
  4. Frobenius actions on local cohomology, Frobenius closure, and tight closure
  5. Relative Frobenius Actions
  6. Dualizing complexes and Frobenius actions
  7. Permanence properties
  8. Open loci results
  9. Acknowledgments

Key Result

Theorem A

Let $R$ be a ring of prime characteristic $p>0$ which is either $F$-finite or essentially of finite type over an excellent local ring. Then the following subsets of $\operatorname{Spec}(R)$ are open with respect to the Zariski topology:

Theorems & Definitions (43)

  • Theorem A
  • Theorem B
  • Theorem \oldthetheorem: NagelSchenzel
  • Lemma \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem: RaduRelative, AndreRelative
  • Theorem \oldthetheorem: PolstraQuy
  • Proposition \oldthetheorem
  • ...and 33 more