Le lemme d'Abhyankar perfectoide

TL;DR

This paper generalizes the almost purity theorem for perfectoid algebras to include ramified extensions without restrictions, using a perfectoid version of Riemann's extension theorem and revisiting categorical aspects.

Contribution

It extends Faltings's almost purity theorem to ramified cases in perfectoid algebras, removing previous restrictions on the discriminant.

Findings

Extended almost purity theorem to ramified perfectoid extensions

Developed a perfectoid Riemann extension theorem

Revisited categorical properties of Banach and perfectoid algebras

Abstract

We extend Faltings's "almost purity theorem" on finite etale extensions of perfectoid algebras (as generalized by Scholze and Kedlaya-Liu) to the ramified case, without restriction on the discriminant. The key point is a perfectoid version of Riemann's extension theorem. Categorical aspects of uniform Banach algebras and perfectoid algebras are revisited beforehand.