An introduction to p-adic period rings

TL;DR

This paper provides an introduction to p-adic period rings, specifically B crys and B dR, and discusses comparison theorems linking étale, crystalline, and de Rham cohomologies for p-adic varieties.

Contribution

It offers an accessible introduction to p-adic period rings and their role in comparison theorems, bridging different cohomological theories in p-adic geometry.

Findings

Introduction of B crys and B dR rings

Statements of key comparison theorems

Connections between étale, crystalline, and de Rham cohomologies

Abstract

This paper is the augmented notes of a course I gave jointly with Laurent Berger in Rennes in 2014. Its aim was to introduce the periods rings B crys and B dR and state several comparison theorems between{\'e}tale and crystalline or de Rham cohomologies for p-adic varieties.