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