# B-pairs and (phi,Gamma)-modules

## TL;DR

This paper explores the category of B-pairs, establishing their equivalence to (phi,Gamma)-modules over the Robba ring, and discusses modifications to key theorems in p-adic Hodge theory.

## Contribution

It introduces and analyzes B-pairs, showing their equivalence to (phi,Gamma)-modules, with updates to existing theorems in the field.

## Key findings

- B-pairs form a category containing p-adic representations.
- B-pairs are equivalent to (phi,Gamma)-modules over the Robba ring.
- Theorem C has been modified in the latest version.

## Abstract

Main change from v1 : theorem C has been modified, see remark 3.1.7 (2). We study the category of B-pairs (W_e,W_dR^+) where W_e is a free B_cris^{phi=1}-module with a semilinear and continuous action of G_K and where W_dR^+ is a G_K-stable B_dR^+ -lattice in B_dR \otimes W_e. This category contains the category of p-adic representations and is naturally equivalent to the category of all (phi,Gamma)-modules over the Robba ring.