A MOTIVIC VERSION OF THE THEOREM OF FONTAINE AND WINTENBERGER

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♭ 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♭ are isomorphic. A main tool for constructing the equivalence is Scholze's theory of perfectoid spaces.