Exactness of the reduction on étale modules

Abstract

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 GL d + 1 ( Q p ) for some d ⩾ 1 .