Word.ws English Dictionary

étale

étale adjective: such that the natural homomorphism is an isomorphism.

Adjective

étale (not comparable)

  1. Such that the natural homomorphism is an isomorphism.
    • 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) → Ôᵤ(U) is an isomorphism.1994, Aleksei Parshin, Igor Shafarevich, Algebraic Geometry IV: Linear Algebraic Groups Invariant Theory, page 278
    • 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] 12.2.1.2009, Rob de Jeu, James Dominic Lewis, Motives and Algebraic Cycles: A Celebration in Honour of Spencer J. Bloch, page 96

Examples of étale in a sentence

  • We will be interested in étale maps between stacks and étale covers of stacks.2014, Christopher Douglas, John Francis, André Henriques, Michael Hill, Topological Modular Forms, page 53:
  • For a scheme X, let Ét(X) be the category of all étale morphisms from a scheme to X.2016, Thierry Vialar, Handbook of Mathematics, page 825:

Forms of étale

alternative formetale

Origin of étale

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."

Questions about étale

What does étale mean?
Étale means such that the natural homomorphism is an isomorphism.
How do you use étale in a sentence?
We will be interested in étale maps between stacks and étale covers of stacks.
What part of speech is étale?
Étale is an adjective.

Words near étale in the dictionary

  1. Érin
  2. Érone
  3. étage
  4. étagère
  5. étagères
  6. étale
  7. Étampes
  8. étape
  9. étapes
  10. Étaples
  11. Étard reaction

Source and license

Adapted from Wiktionary: étale (page history and authors). Text available under the Creative Commons Attribution-ShareAlike 4.0 license. Word.ws reformatted the entry and added links. Wiktionary data of 2026-09-02.

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