Exactness of the reduction on \'etale modules

We prove the exactness of the reduction map from 'etale $(\varphi ,\mathrm{\Gamma })$-modules over completed localized group rings of compact open subgroups of unipotent $p$-adic algebraic groups to usual 'etale $(\varphi ,\mathrm{\Gamma })$-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 $(\varphi ,\mathrm{\Gamma })$-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 $\text{Tor}$-functors vanish. Moreover, we give the example of the Steinberg representation as an illustration and show that it is acyclic for this functor to $(\varphi ,\mathrm{\Gamma })$-modules whenever our reductive group is ${\text{GL}}_{d+1}({\mathbb{Q}}_p)$ for some $d\ge 1$.