A motivic version of the theorem of Fontaine and Wintenberger

We establish a tilting equivalence for rational, homotopy-invariant cohomology theories de fi ned over non-archimedean analytic varieties. More precisely, we prove an equivalence between the categories of motives of rigid analytic varieties over a perfectoid fi eld K of mixed characteristic and over the associated (tilted) perfectoid fi eld K-b 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-b are isomorphic.