$F$-intersection flatness of dagger and Berkovich affinoid algebras

TL;DR

This work extends tight closure methods to non-Archimedean analytic settings by proving that dagger and Berkovich Tate algebras in prime characteristic have intersection flat Frobenius, i.e. $S^{1/p}$ is flat and Mittag-Leffler. Through a blend of relative Frobenius analysis, Ohm-Rush trace, and descent arguments, the authors establish $F$-intersection flatness for dagger algebras and for rings of convergent power series on polydiscs. Consequently, ideal-adic completions of reduced rings essentially of finite type over these algebras possess big test elements, broadening the scope of tight closure theory to Berkovich and dagger analytic contexts. The results hinge on the excellence and regularity of the analytic algebras and enable new uniform testing mechanisms in completions arising from non-Archimedean geometry, with potential applications to rigid analytic geometry and related cohomological methods.

Abstract

We show, using the techniques developed in arXiv:2504.06444 and arXiv:2305.11139, that dagger algebras and Tate algebras in the sense of Berkovich in prime characteristic $p > 0$ have intersection flat Frobenius. Equivalently, if $S$ is such a ring, then $S^{1/p}$ is a flat and Mittag-Leffler $S$-module. As a consequence, we deduce that any ideal-adic completion of a reduced ring that is essentially of finite type over a dagger algebra or a Berkovich Tate algebra in prime characteristic has big test elements from tight closure theory.

Theorems & Definitions (66)

• Theorem 1.1
• Definition 1
• Theorem 2.1
• Definition 2
• Remark 1
• Theorem 2.2
• Corollary 1
• proof
• Proposition 1
• proof