---
tag: 7YX76G0838
kind: concept
name: Étale cohomology
msc: [14F20]
sources:
  - url: https://en.wikipedia.org/wiki/%C3%89tale_cohomology
same_as:
  wikidata: Q8078429
  wikipedia: Étale cohomology
---

## Summary

Étale cohomology is a cohomology theory for schemes, built from sheaves on the étale site instead of the Zariski topology.
Grothendieck and Artin introduced it to obtain, for varieties over any field, groups that behave like the singular cohomology of a complex variety.
It contains [[Galois cohomology|9MPCZ3EAJM]] as the case of a point.
It is the setting of [[ℓ-adic cohomology|E1DVHN06CR]] and of Deligne's proof of the [[Weil conjectures|81TXP32VDM]].

## Explanations

### L1 human

The Zariski topology of a variety has too few open sets: on an irreducible variety, any two nonempty open sets meet.
So sheaf cohomology with constant coefficients in that topology sees almost nothing.
Étale cohomology allows "open sets" that are étale maps $U \to X$, which look like local isomorphisms, such as finite unramified covers.
These play the role of small discs in topology.
For a smooth projective complex curve of genus $g$, the first étale cohomology with coefficients $\Z/n\Z$ is $(\Z/n\Z)^{2g}$, as in topology.