Exactness of the reduction on étale modules (Q716472)

This paper belongs to the emerging theory of \(p\)-adic local Langlands correspondences, which relate irreducible \(p\)-adic Banach space representations of \(p\)-adic reductive groups to representations of \(p\)-adic Galois groups. In the work of \textit{P. Colmez} [Astérisque 330, 281--509 (2010; Zbl 1218.11107)] on the \(p\)-adic local Langlands correspondence for \(\text{GL}(2,\mathbb Q_p)\), an important role is played by the observation that Fontaine's étale \((\varphi, \Gamma)\)-modules, semilinear objects introduced to classify representations of \(\text{Gal}(\overline{\mathbb Q}_p / \mathbb Q_p)\), can be regarded naturally as representations of a monoid contained in the Borel subgroup of \(\mathbb{Q}_p\). More recently, Schneider and Vignéras (unpublished) have proposed a notion of an étale \((\varphi, \Gamma)\)-module for a parabolic subgroup in a split reductive group over \(\mathbb Q_p\), which recovers Fontaine's modules in the case of the Borel subgroup of \(\text{GL}(2,\mathbb Q_p)\). There is a natural reduction map from \((\varphi, \Gamma)\)-modules for a general \(P\) to Fontaine \((\varphi, \Gamma)\)-modules, which forms part of a \(\delta\)-functor \((D^i)_{i \geq 0}\) constructed by Schneider and Vignéras. This is intended to generalize part of Colmez's construction (the so-called ``Montreal functor'') to this setting. The key open problem in this theory is to determine conditions on a \(P\)-representation \(V\) under which the modules \(D^i(V)\) are finitely generated. In the present paper, the author proves that the reduction map is exact on étale \((\varphi, \Gamma)\)-modules for \(P\). He uses these results to calculate the Schneider-Vignéras modules \(D^i(V)\) where \(V\) is the Steinberg representation of \(\text{GL}_{n}\) for \(n \geq 2\), showing in particular that they are finitely generated in this case.