---
date: 2022-01-23 18:18
modification date: Sunday 23rd January 2022 18:18:55
title: etale
aliases: ["etale", "etale cover", "etale cohomology", "etale algebra", "etale homotopy", "etale sheaf"]

---


---

- Tags 
	- #arithmetic-geometry 
- Refs: #resources 
	- <https://www.math.ru.nl/personal/bmoonen/Seminars/EtCohConrad.pdf/>
- Links:
	- [site](Unsorted/site.md)
	- [topos](Unsorted/topos.md)
	- [Luna etale slice theorem](Luna%20etale%20slice%20theorem.md)
	- [Unsorted/profinite integers](Unsorted/profinite%20integers.md)
	- [etale morphism](etale%20morphism)
	- [etale fundamental group](etale%20fundamental%20group.md)
	- [etale descent](etale%20descent.md)
	- [examples of computations of etale cohomology](Unsorted/examples%20of%20computations%20of%20etale%20cohomology.md)

---

# etale
New developments: [Berkovich integral etale cohomology](Berkovich%20integral%20etale%20cohomology) and [rigid analytic motivic cohomology](rigid%20analytic%20motivic%20cohomology).

# In algebra

![](attachments/Pasted%20image%2020221011190430.png)
![](attachments/Pasted%20image%2020221011190458.png)

# In AG
![](attachments/Pasted%20image%2020220407233147.png)

![](attachments/Pasted%20image%2020220505161603.png)

Note that being etale or [unramified](Unsorted/unramified.md) is [local on the base](local%20on%20the%20base) and [local on the target](local%20on%20the%20base).

![](attachments/Pasted%20image%2020220420101053.png)
![](attachments/Pasted%20image%2020220424124855.png)
![](attachments/Pasted%20image%2020221120161259.png)

## Explicit examples

- Any [open immersion](Unsorted/open%20immersion%20of%20schemes.md) is etale.
- A morphism $X\to \spec k$ is etale iff $X\cong \disjoint \spec K_i$ where each $K_i/k$ is a finite [separable field extension](separable%20field%20extension).
- Any finite [separable field extension](separable%20field%20extension) $k \injects K$ induces an etale cover $\spec K\to \spec k$.
- The localization cover $\spec R\localize{S} \to \spec R$ is etale.
- $\spec \\CC[t] \to \spec \RR[t]$ is etale:
	  ![](attachments/Pasted%20image%2020220407234842.png)

# Motivation

![](attachments/Pasted%20image%2020220404005403.png)
![](attachments/Pasted%20image%2020220404005418.png)

![](attachments/Pasted%20image%2020220220032838.png)
![](attachments/Pasted%20image%2020220220032912.png)
![](attachments/Pasted%20image%2020220220032922.png)

## Relation to Galois covers
![](attachments/Pasted%20image%2020220220033148.png)

# Etale Morphisms



The idea: like local diffeomorphisms of manifolds, so inducing isomorphisms on tangent spaces at every point:
$$
f:X\to Y \text{ etale} \leadsto df: \T_x X \iso \T_{f(x)} Y\quad \forall x\in X
$$
Thus some version of the implicit function theorem holds in the analytic setting.
The analog of a [covering space](covering%20space.md) is a finite [etale morphism](etale%20morphism.md).

![](attachments/Pasted%20image%2020220209190402.png)

![](attachments/Pasted%20image%2020220404011124.png)

## Standard etale morphisms and structure theorem


![](attachments/Pasted%20image%2020220407234729.png)
![](attachments/Pasted%20image%2020220407234904.png)

# Etale cover

An **etale cover** is a family of morphisms $\ts{U_i \to X}$ which are etale and  [locally of finite type](locally%20of%20finite%20type) such that $X \subseteq \Union U_i$.



# Etale algebra

Idea: a finite direct product of separable field extensions.

![](attachments/Pasted%20image%2020220123205031.png)
![](attachments/Pasted%20image%2020220123205051.png)

Every etale algebra is a [semisimple algebra](semisimple%20algebra.md).

Characterization of when an etale algebra is monogenic:
![](attachments/Pasted%20image%2020220123205831.png)


# Etale cohomology
![](attachments/Pasted%20image%2020220207120901.png)
![](attachments/Pasted%20image%2020220207120912.png)
# Etale homotopy

#todo 

# Relation to Galois cohomology

![](attachments/Pasted%20image%2020220424125128.png)
![](attachments/Pasted%20image%2020220424125135.png)