Another proof of the almost purity theorem for perfectoid valuation rings

TL;DR

This paper presents a new proof of the almost purity theorem for perfectoid valuation rings, utilizing the behavior of Faltings' normalized length under Frobenius, which is fundamental in perfectoid space geometry.

Contribution

It introduces a novel proof method for the almost purity theorem based on Faltings' normalized length and Frobenius, expanding the toolkit for perfectoid space theory.

Findings

New proof of the almost purity theorem for perfectoid valuation rings

Utilizes Faltings' normalized length and Frobenius map behavior

Enhances understanding of perfectoid space structures

Abstract

The almost purity theorem is central to the geometry of perfectoid spaces and has numerous applications in algebra and geometry. This result is known to have several different proofs in the case that the base ring is a perfectoid valuation ring. We give a new proof by exploiting the behavior of Faltings' normalized length under the Frobenius map.