Cohomology of $(\varphi,\Gamma)$-modules over pseudorigid spaces

TL;DR

This paper investigates the cohomology of $(, )$-modules over pseudorigid spaces, establishing finiteness, duality, and classification results, and applies these to eigenvarieties and Galois representations.

Contribution

It proves finiteness of cohomology, classifies rank-1 modules, and extends triangulation interpolation to pseudorigid families, advancing the understanding of eigenvarieties and Galois representations.

Findings

Cohomology of $(, )$-modules is finite.

Eigenvarieties are proper at the boundary of weight space.

Certain Galois representations are trianguline at characteristic $p$ points.

Abstract

We study the cohomology of families of $(\varphi,\Gamma)$-modules with coefficients in pseudoaffinoid algebras. We prove that they have finite cohomology, and we deduce an Euler characteristic formula and Tate local duality. We classify rank-$1$ $(\varphi, \Gamma)$-modules and deduce that triangulations of pseudorigid families of $(\varphi,\Gamma)$-modules can be interpolated, extending a result of [KPX14]. We then apply this to study extended eigenvarieties at the boundary of weight space, proving in particular that the eigencurve is proper at the boundary and that Galois representations attached to certain characteristic $p$ points are trianguline.