• Tags
  • #arithmetic-geometry
• Refs: #resources
  • https://www.math.ru.nl/personal/bmoonen/Seminars/EtCohConrad.pdf/
• Links:
  • site
  • topos
  • Luna etale slice theorem
  • Unsorted/profinite integers
  • etale morphism
  • etale fundamental group
  • etale descent
  • examples of computations of etale cohomology

etale

New developments: Berkovich integral etale cohomology and rigid analytic motivic cohomology.

In algebra

In AG

Note that being etale or unramified is local on the base and local on the target.

Explicit examples

• Any open immersion is etale.
• A morphism \(X\to \operatorname{Spec}k\) is etale iff \(X\cong {\textstyle\coprod}\operatorname{Spec}K_i\) where each \(K_i/k\) is a finite separable field extension.
• Any finite separable field extension \(k \hookrightarrow K\) induces an etale cover \(\operatorname{Spec}K\to \operatorname{Spec}k\).
• The localization cover \(\operatorname{Spec}R \left[ { \scriptstyle { {S}^{-1}} } \right] \to \operatorname{Spec}R\) is etale.
• \(\operatorname{Spec}\\CC[t] \to \operatorname{Spec}{\mathbf{R}}[t]\) is etale:

Motivation

Relation to Galois covers

Etale Morphisms

The idea: like local diffeomorphisms of manifolds, so inducing isomorphisms on tangent spaces at every point: \begin{align*} f:X\to Y \text{ etale} \leadsto df: {\mathbf{T}}_x X { \, \xrightarrow{\sim}\, }{\mathbf{T}}_{f(x)} Y\quad \forall x\in X \end{align*} Thus some version of the implicit function theorem holds in the analytic setting. The analog of a covering space is a finite etale morphism.

Standard etale morphisms and structure theorem

Etale cover

An etale cover is a family of morphisms \(\left\{{U_i \to X}\right\}\) which are etale and locally of finite type such that \(X \subseteq \bigcup U_i\).

Etale algebra

Idea: a finite direct product of separable field extensions.

Every etale algebra is a semisimple algebra.

Characterization of when an etale algebra is monogenic:

Etale cohomology

# Etale homotopy

#todo

Relation to Galois cohomology