étale

adj

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."

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.”

   “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.”