Exactness of the reduction on étale modules

Abstract

We prove the exactness of the reduction map from étale $(\varphi ,\Gamma )$-modules over completed localized group rings of compact open subgroups of unipotent $p$-adic algebraic groups to usual étale $(\varphi ,\Gamma )$-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 $(\varphi ,\Gamma )$-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 $(\varphi ,\Gamma )$-modules whenever our reductive group is $\GL_{d+1}(\mathbb{Q}_p)$ for some $d\ge 1$.

18 pages; some typos corrected and proof of Lemma 1 rewritten, to appear in Journal of Algebra