The goal of this course is to give an introduction to (pro-)étale cohomology for schemes and explain how this theory defines a good notion of $\ell$-adic cohomology. The course will be in two parts: a first one about étale cohomology (with an introduction to sheaf theory, including sheaf cohomology and a few facts about derived functors, study of the étale site of a scheme, some properties of étale sheaves). In the second part, I will (partially) explain the paper "The pro-étale topology for schemes" of Bhatt and Scholze (notion of locally weakly contractible topoi, replete topoi, weakly étale morphisms, comparison between étale and pro-étale and if time permits, constructible sheaves and 6-functors formalism in this setting).

The lectures will take place on Tuesdays at 12:00pm in room WSC-S-U-3.01 and the exercise sessions on the same day at 2:30pm in room WSC-S-U-3.02.
The first lecture will be on the 8th of October.  


Main references:

- Étale cohomology:
Introduction to Étale Cohomology by G. Tamme.
Étale Cohomology by J. Milne, or his lecture notes.
There are other lecture notes available online. See for example these notes of J. de Jong or the corresponding chapter on the Stack Project.

 
- Pro-étale cohomology:  
The main paper of B. Bhatt and P. Scholze.
Pro-étale cohomology on the Stack Project.

Lecture notes: notes-29/10/2024

Exercise sheets:
- sheet1
- sheet2
- sheet3