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

TL;DR

This paper proves that dagger and Berkovich Tate algebras in prime characteristic have intersection flat Frobenius, leading to new insights in tight closure theory and flatness properties of related rings.

Contribution

It establishes intersection flat Frobenius for dagger and Berkovich Tate algebras in prime characteristic, a novel result in non-Archimedean analytic geometry.

Findings

Dagger and Tate algebras have intersection flat Frobenius in characteristic p>0.

The Frobenius map makes the ring's extension flat and Mittag-Leffler.

Ideal-adic completions of certain rings have big test elements from tight closure theory.

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.