Almost purity and overconvergent Witt vectors

In a previous paper, we stated a general almost purity theorem in the style of Faltings: if R is a ring for which the Frobenius maps on finite p-typical Witt vectors over R are surjective, then the integral closure of R   in a finite étale extension of R[p−1]R[p−1] is “almost” finite étale over R  . Here, we use almost purity to lift the finite étale extension of R[p−1]R[p−1] to a finite étale extension of rings of overconvergent Witt vectors. The point is that no hypothesis of p-adic completeness is needed; this result thus points towards potential global analogues of p  -adic Hodge theory. As an illustration, we construct (φ,Γ)(φ,Γ)-modules associated with Artin Motives over QQ. The (φ,Γ)(φ,Γ)-modules we construct are defined over a base ring which seems well-suited to generalization to a more global setting; we plan to pursue such generalizations in later work.