étale

See also: etale, étalé, and Etåle

English

Alternative forms

• etale

Etymology

First 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

étale (not comparable)

1. (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_u(U) is an isomorphism.

   • 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_U together with an isomorphism between the two pull-backs of F_U to 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

Pronunciation

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Verb

étale

1. inflection of étaler:
   1. first/third-person singular present indicative/subjunctive
   2. second-person singular imperative