This is a small seminar we ran on Étale cohomology over the summer organized by Vidhu Adhihetty and myself. We had scheduling issues since most people were traveling so we couldn't finish.

Location: Room 622
Meeting Time: Thursdays 11:00am EST
Syllabus: Found here

1. Introduction

Vidhu Adhihetty 06/19/25

We introduce the Weil conjectures as motivation for Étale cohomology and go over étale morphisms. Notes available here.

2. Sites and Sheaves

Wenqi Li 6/26/25

We introduce the notion of a site and a Grothendieck topology. We then prove faithfully flat descent. Notes available here.

3. More on sheaves

Nicolás Vilches 7/3/25

We continue the discussion from last week, specializing to sheaves on the étale sites. Notes available here.

4. Sheaf Cohomology

Carlos Andrés Alvarado Álvarez 7/10/25

We prove that the category of sheaves is abelian and has enough injectives to define étale cohomology.

5. Cech Cohomology

Carlos Andrés Alvarado Álvarez 7/18/25

We define Cech covers and Cech cohomology and relate it to étale cohomology.

6. Torsors and H^1

Vidhu Adhihetty 7/23/25

We will explain étale G-torsors over X and explain how these torsors are in bijection with elements of H^1(X,G).