A motivic version of the theorem of Fontaine and Wintenberger

TL;DR

This paper establishes an equivalence between categories of motives over perfectoid fields of mixed and equal characteristic, generalizing Fontaine and Wintenberger's theorem using Scholze's perfectoid space theory.

Contribution

It provides a motivic generalization of Fontaine and Wintenberger's theorem, connecting motives over different perfectoid fields.

Findings

Categories of motives over K and K^flat are equivalent.

The equivalence extends Fontaine and Wintenberger's Galois group isomorphism.

Uses Scholze's theory of perfectoid spaces as a key tool.

Abstract

We prove the equivalence between the categories of motives of rigid analytic varieties over a perfectoid field $K$ of mixed characteristic and over the associated (tilted) perfectoid field $K^{\flat}$ of equal characteristic. This can be considered as a motivic generalization of a theorem of Fontaine and Wintenberger, claiming that the Galois groups of $K$ and $K^\flat$ are isomorphic. A main tool for constructing the equivalence is Scholze's theory of perfectoid spaces.