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Variants on Frobenius Intersection Flatness and Applications to Tate Algebras

Rankeya Datta, Neil Epstein, Karl Schwede, Kevin Tucker

TL;DR

This work broadens Frobenius-based singularity theory beyond $F$-finite settings by introducing FOR, FIF, and FORT, and applies these notions to nonlocal regular rings such as Tate algebras. It proves Tate algebras are Frobenius intersection flat, yielding test elements for reduced affinoid algebras, and develops descent and openness results for $F$-purity and related regularity notions. The paper also establishes that Tate algebras over non-Archimedean fields are FIF or even FORT under natural hypotheses, and demonstrates robust base-change and completion behavior, enabling a framework to study $F$-singularities without resorting to the $ extGamma$-construction. Together, these results provide new tools for understanding singularities in rigid-analytic and non-Archimedean settings and pave the way for a broader theory of test ideals and $F$-compatible structures in affinoid and related algebras.

Abstract

The theory of singularities defined by Frobenius has been extensively developed for $F$-finite rings and for rings that are essentially of finite type over excellent local rings. However, important classes of non-local excellent rings, such as Tate algebras and their quotients (affinoid algebras) do not fit into either setting. We investigate here a framework for moving beyond the $F$-finite setting, developing the theory of three related classes of regular rings defined by properties of Frobenius. In increasing order of strength, these are Frobenius Ohm-Rush (FOR), Frobenius intersection flat, and Frobenius Ohm-Rush trace (FORT). We show that Tate algebras are Frobenius intersection flat, from which it follows that reduced affinoid algebras have test elements using a result of Sharp. We also deduce new cases of the openness of the $F$-pure locus.

Variants on Frobenius Intersection Flatness and Applications to Tate Algebras

TL;DR

This work broadens Frobenius-based singularity theory beyond -finite settings by introducing FOR, FIF, and FORT, and applies these notions to nonlocal regular rings such as Tate algebras. It proves Tate algebras are Frobenius intersection flat, yielding test elements for reduced affinoid algebras, and develops descent and openness results for -purity and related regularity notions. The paper also establishes that Tate algebras over non-Archimedean fields are FIF or even FORT under natural hypotheses, and demonstrates robust base-change and completion behavior, enabling a framework to study -singularities without resorting to the -construction. Together, these results provide new tools for understanding singularities in rigid-analytic and non-Archimedean settings and pave the way for a broader theory of test ideals and -compatible structures in affinoid and related algebras.

Abstract

The theory of singularities defined by Frobenius has been extensively developed for -finite rings and for rings that are essentially of finite type over excellent local rings. However, important classes of non-local excellent rings, such as Tate algebras and their quotients (affinoid algebras) do not fit into either setting. We investigate here a framework for moving beyond the -finite setting, developing the theory of three related classes of regular rings defined by properties of Frobenius. In increasing order of strength, these are Frobenius Ohm-Rush (FOR), Frobenius intersection flat, and Frobenius Ohm-Rush trace (FORT). We show that Tate algebras are Frobenius intersection flat, from which it follows that reduced affinoid algebras have test elements using a result of Sharp. We also deduce new cases of the openness of the -pure locus.

Paper Structure

This paper contains 26 sections, 52 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conventions and abbreviations
  4. Ohm-Rush, Ohm-Rush trace, and intersection flatness
  5. Pure maps
  6. Weak versions of the excellence property
  7. The relative Frobenius
  8. Frobenius Ohm-Rush (trace) and F-intersection flatness
  9. Definitions and first properties
  10. FORT rings
  11. Quotients of FORT rings
  12. Descent of F-intersection flatness and Frobenius Ohm-Rush
  13. On the openness of pure loci of --linear maps
  14. Openness of F-pure locus
  15. The variants of strong F-regularity
  16. ...and 11 more sections

Key Result

Proposition 1

Let $R$ be a Noetherian ring. Then the following are equivalent: Moreover, the equivalent conditions imply that $R$ is J-1.

Theorems & Definitions (132)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Definition 4
  • Proposition 1
  • proof
  • Theorem 2.5.1
  • proof : Indication of proof of \ref{['thm:Radu-Andre-Dumitrescu']}
  • Proposition 2
  • ...and 122 more