Cohomology and Duality for (phi,Gamma)-modules over the Robba ring

TL;DR

This paper demonstrates how Galois cohomology of p-adic representations can be computed via associated (phi,Gamma)-modules over the Robba ring, extending duality results and Euler-Poincare formulas to a broader class of modules.

Contribution

It generalizes Herr's results by establishing cohomology computation and duality principles for not necessarily etale (phi,Gamma)-modules over the Robba ring.

Findings

Galois cohomology can be computed using (phi,Gamma)-modules over the Robba ring.

Analogues of Euler-Poincare characteristic formula are established for these modules.

Tate local duality is extended to non-etale (phi,Gamma)-modules.

Abstract

Given a p-adic representation of the Galois group of a local field, we show that its Galois cohomology can be computed using the associated etale (phi,Gamma)-module over the Robba ring; this is a variant of a result of Herr. We then establish analogues, for not necessarily etale (phi,Gamma)-modules over the Robba ring, of the Euler-Poincare characteristic formula and Tate local duality for p-adic representations. These results are expected to intervene in the duality theory for Selmer groups associated to de Rham representations.