Exactness of the reduction on \'etale modules

TL;DR

This paper proves the exactness of a reduction map between certain tale $(\u03b6,\u03b3)$-modules over localized group rings and Fontaine's ring, supporting a p-adic Langlands functor for reductive groups.

Contribution

It establishes the exactness of the reduction map and shows higher r functors vanish, providing evidence for the functor's role in p-adic Langlands correspondence.

Findings

Reduction map is exact for tale $(\u03b6,3)$-modules.

Higher r functors vanish in this setting.

Steinberg representation is acyclic for the functor in +1 3(5) groups.

Abstract

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 $d\geq 1$.