Foundations for almost ring theory -- Release 7.5

TL;DR

This paper advances the foundations of almost ring theory by extending Faltings's methods, proving the almost purity theorem using Scholze's perfectoid spaces, and applying these results to generalize key conjectures.

Contribution

It provides a comprehensive foundation for almost ring theory, generalizes Scholze's perfectoid spaces, and extends the direct summand conjecture using new almost purity results.

Findings

Proved the almost purity theorem using Scholze's perfectoid spaces.

Generalized Y.André's perfectoid Abhyankar's lemma.

Extended the proof of the direct summand conjecture.

Abstract

This is release 7.5 of our project, aiming to provide a complete treatment of the foundations of almost ring theory, following and extending Faltings's method of "almost etale extensions". The central result is the "almost purity theorem", for whose proof we adapt Scholze's method, based on his perfectoid spaces. This release provides the foundations for our generalization of Scholze's perfectoid spaces, and reduces the proof of the almost purity theorem to a general assertion concerning the \'etale topology of adic spaces, whose proof uses previous work by the first author. As usual, this new release is a mix of corrections and various improvements, with a final chapter dedicated to applications; notably, we include a generalization of Y.Andr\'e's "perfectoid Abhyankar's lemma" which we use to give a proof of a generalization of the "direct summand conjecture", extending Andr\'e's recent work.