I’d like to take a quick (one-post) break from simplicial methods. This summer, I will be studying étale cohomology and the proofs of the Weil conjectures through the HCRP program. I have currently been going through the basic computations in étale cohomology, and, to help myself understand one point better, would like to mention a very pretty and elementary argument I recently learned from Johan de Jong’s course notes on the subject (which are a chapter in the stacks project).

1. Motivation via étale cohomology

When doing the basic computations of the étale cohomology of curves, one of the important steps is the computation of the sheaf ${\mathcal{O}_X^*}$ (that is, the multiplicative group of units), and in doing this one needs to know the cohomology of the generic point. That is, one needs to compute

$\displaystyle H^*(X_{et}, \mathcal{O}_X^*)$

where ${X = \mathrm{Spec} K}$ for ${K}$ a field of transcendence degree one over the algebraically closed ground field, and the “et” subscript means étale cohomology. Now, ultimately, whenever you have the étale cohomology of a field, it turns out to be the same as Galois cohomology. In other words, if ${X = \mathrm{Spec} K}$, then the small étale site of ${X}$ is equivalent to the site of continuous ${G = \mathrm{Gal}(K^{sep}/K)}$-sets, and consequently the category of abelian sheaves on this site turns out to be equivalent to the category of continuous ${G}$-modules. Taking the étale cohomology of this sheaf then turns out to be the same as taking the group cohomology of the associated ${G}$-module. So, if you’re interested in étale cohomology, then you’re interested in Galois cohomology. In particular, you are interested in things like group cohomologies of the form

$\displaystyle H^2(\mathrm{Gal}(K^{sep}/K), (K^{sep})^*).$ (more…)