Meaning of étale | Babel Free

Definitions

Such that the natural homomorphism is an isomorphism.

not-comparable

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.”

“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.”

“We will be interested in étale maps between stacks and étale covers of stacks.”

“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.”

“We discuss also on a more natural realization of the Galois representations in the etale cohomology of Artin-Schreier varieties..”

CEFR level