# étale

## English

edit### Alternative forms

edit### Etymology

editFirst applied in a mathematical context (in French) by Alexander Grothendieck to étale morphisms, apparently with reference to the phrase *mer étale* ("the sea at high or low tide"), the connection being that étale morphisms are, in an intuitive sense, calmly behaved or "spread out."

### Adjective

edit**étale** (*not comparable*)

- (mathematics) Such that the natural homomorphism is an isomorphism.
**1994**, Aleksei Parshin, Igor Shafarevich, Algebraic Geometry IV: Linear Algebraic Groups Invariant Theory, page 278:- Recall (Hartshorne [1977]) that a morphism
*ψ: U → V*is called**étale**if it is**étale**at each point*u*∈*U*, where being**étale**at*u*means that the natural homomorphism of local ring completions*ψ**:*Ô*→_{ψ(u)}(V)*Ô*is an isomorphism._{u}(U)

**2009**, Rob de Jeu, James Dominic Lewis, Motives and Algebraic Cycles: A Celebration in Honour of Spencer J. Bloch, page 96:- Because every Deligne-Mumford stack admits an
**étale**cover*π : U → 𝔛*by a scheme, to give a sheaf of sets*F*on the**étale**site of*𝔛*is equivalent to giving a sheaf*F*together with an isomorphism between the two pull-backs of_{U}*F*to_{U}*U*×_{𝔛}*U*satisfying the cocycle condition of [26] 12.2.1.

**2014**, Christopher Douglas, John Francis, André Henriques, Michael Hill, Topological Modular Forms, page 53:- We will be interested in
**étale**maps between stacks and**étale**covers of stacks.

**2016**, Thierry Vialar, Handbook of Mathematics, page 825:- For a scheme
*X*, let*Ét(X)*be the category of all**étale**morphisms from a scheme to*X*. An**étale**presheaf on*X*is a contravariant functor from*Ét(X)*to the category of sets.

## French

edit### Pronunciation

editAudio: (file)

### Verb

edit**étale**

- inflection of
*étaler*: