Exactness of the reduction on étale modules

We prove the exactness of the reduction map from étale (φ,Γ)-modules over completed localized group rings of compact open subgroups of unipotent p-adic algebraic groups to usual étale (φ,Γ)-modules over Fontaine's ring. This reduction map is a component of a functor from smooth p-power torsion representations of p-adic reductive groups (or more generally of Borel subgroups of these) to (φ,Γ)-modules. Therefore this gives evidence for this functor—which is intended as some kind of p-adic Langlands correspondence for reductive groups—to be exact. We also show that the corresponding higher Tor-functors vanish. Moreover, we give the example of the Steinberg representation as an illustration and show that it is acyclic for this functor to (φ,Γ)-modules whenever our reductive group is GLd+1(Qp) for some d⩾1.