Perfectoid spaces

TL;DR

This paper introduces perfectoid spaces and rings, providing a framework that connects characteristic 0 and p, and applies this to prove cases of the weight-monodromy conjecture.

Contribution

It develops the theory of perfectoid spaces and rings, establishing a tilting operation and applying it to advance understanding of the weight-monodromy conjecture.

Findings

Established a framework for perfectoid spaces and rings.

Demonstrated a tilting operation exchanging characteristic 0 and p.

Proved cases of the weight-monodromy conjecture via reduction to equal characteristic.

Abstract

We introduce a certain class of so-called perfectoid rings and spaces, which give a natural framework for Faltings' almost purity theorem, and for which there is a natural tilting operation which exchanges characteristic 0 and characteristic p. We deduce the weight-monodromy conjecture in certain cases by reduction to equal characteristic.