étale

alternative etale

1. not comparablemathematicsSuch that the natural homomorphism is an isomorphism.

   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

   3 more examples

   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

   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.

   2016, Thierry Vialar, Handbook of Mathematics, page 825